###### Abstract

This paper presents a succinct review of attempts in the literature to use
game theory to model decision making scenarios relevant to defence
applications. Game theory has been proven as a very effective tool in
modelling decision making processes of intelligent agents, entities, and
players. It has been used to model scenarios from diverse fields such as
economics, evolutionary biology, and computer science. In defence
applications, there is often a need to model and predict actions of hostile
actors, and players who try to evade or out-smart each other. Modelling how
the actions of competitive players shape the decision making of each other is
the forte of game theory. In past decades, there have been several studies
which applied different branches of game theory to model a range of defence-
related scenarios. This paper provides a structured review of such attempts,
and classifies existing literature in terms of the kind of warfare modelled,
the types of game used, and the players involved. in terms of the warfares
modelled, we recognise that most papers which apply game theory in defence
settings are concerned with command and control warfare, and can be further
classified into papers dealing with (i) Resource Allocation Warfare (ii)
Information Warfare (iii) Weapons Control Warfare (iv) Adversary Monitoring
Warfare. We also observe that most of the reviewed papers are concerned with
sensing, tracking, and large sensor networks, and the studied problems have
parallels in sensor network analysis in the civilian domain. In terms of the
games used, we classify the reviewed papers into papers that use non-
cooperative or cooperative games, simultaneous or sequential games, discrete
or continuous games, and non-zero sum or zero-sum games. Similarly, papers are
also classified into two-player, three-player or multi-player game based
papers. We also explore the nature of players and the construction of payoff
functions in each scenario. Finally, we also identify gaps in literature where
game theory could be fruitfully applied in scenarios hitherto unexplored using
game theory. The presented analysis provides a concise summary about the
state-of-the-art with regards to the use of game theory in defence
applications, and highlights the benefits and limitations of game theory in
the considered scenarios.

###### keywords:

Decision making; Game theory; Defence Science; Ground warfare; Maritime
warfare; Tracking

xx 1 5 Received: date; Accepted: date; Published: date Game Theory in defence
applications: a review Edwin Ho1, Arvind Rajagopalan2, Alex Skvortsov2,
Sanjeev Arulampalam2, and
Mahendra Piraveenan1,∗ Mahendra Piraveenan Correspondence:
<EMAIL_ADDRESS>
##  1 Introduction

Game Theory has become one of the conventional theoretical frameworks to model
important decision making processes in many aspects of our life. The well-
known examples include economy, social sciences, finance, population dynamics
and epidemics (see (Piraveenan, 2019; Peldschus, 2008; Schotter, 2008; Morrow,
1994; Hammerstein and Selten, 1994) and references therein). Since the seminal
work of John Von Neumann, John Nash, and others (Von Neumann and Morgenstern,
1944; Nash et al., 1950; Nash, 1951), it has been well recognised that there
is an optimal strategy in the context of complex interactions (games) between
two or more parties (players) that can lead to a predictable outcome (payoff).
In practical situations this outcome can often be quantitative and amenable to
arithmetic operations (cost, number of infected people, number of vaccinated
people etc), but often it can be qualitative in nature (such as risk,
readiness level, health state etc).

Application of game theory in defence has a long and diversified history
ranging from the design of real time military systems (e.g, applied in missile
interception) to the support of strategic decisions on large defence
investments and acquisitions. There is extensive literature on specific
theoretical methods and tools and their defence applications (Martin, 1978;
Machiavelli, 1984; Lattimore, 1962; Tzu, 2014). We believe that the review of
this literature is of interest to the community dealing with operational
analysis and data-driven decision support. This is the main motivation for the
presented study.

Game theory (Myerson, 2013; Osborne et al., 2004) elevates military strategies
and decisions with a more holistic and value-laden analysis of the situation
(Martin, 1978). For the military, the potential scenarios amenable to game
theoretic analysis include the rapidly growing applications of autonomous
intelligent systems, and game theory provides a comprehensive mathematical
framework that greatly enhances the capabilities of decision making of the
people who use these systems. Because of its potential, research into game
theory is burgeoning, with a few papers beginning to emerge in the literature
for this military research niche. The aim of this review is to assist
researchers in utilising the body of knowledge in game theory to develop
smarter and safer software systems for defence practitioners. Given that the
state of such research is still in an incipient phase, we do this by drawing
connections between existing military knowledge with the nascent possibilities
that game theory offers so that it can become a more widely understood and
considered framework in military software systems.

Although it is not overly extensive, the body of literature around game theory
in the military has covered a notable portion of the different forms of
engagement and combat. These papers cover past, present and future scenarios:
from predictive strategies in potential hostile situations to analytical
assessments in hindsight of military standoffs thousands of years ago. Game
theory has demonstrated the capacity to be useful in any such military
scenario. However, rapid technological progression has led to consistently new
frontiers of military engagement, each of these possessing its own complex
systems. The overarching areas that have been addressed are Tracking systems
(across all domains), Aerial combat, Ground combat, Homefront warfare, Cyber
warfare and Space systems. Notably, applications of game theory in Naval
warfare have been few, and an exploration into the future research into areas
like this will be discussed later in the review. Within each of these areas
there are a myriad of possibilities for new and innovative systems: different
agents, different weapons, different control structures - and each of these
could be enriched with game theoretic analysis. While Haywood’s and Thunholm’s
treatises on game theory used in military decision making provide coverage of
several different game types (Haywood Jr, 1954; Thunholm, 2005), there does
not seem to be a paper which addresses the use of game theory in the military
across each of the respective fields in the new context of military systems
built on complex and high performance processing and algorithms. Our aim is to
present the literature in such a way that it addresses all of the functions of
game theory in military software systems in each key domain.

This review highlights the scope and utility of each analysed paper by
presenting it in terms of the essential game components: players, game types,
strategies and the key parameters of their payoff functions. It will act as
both an annotated bibliography as well as a framework to understand and plan
further research into the area. It will also layout the fundamental tenets
that are considered by players in every military decision making scenario, as
well as how they impact the decisions that are made by military personnel and
systems, either while competing with hostile players or while cooperating with
friendly players. This will make it possible for most military scenarios to be
viewed as games, and can provide, at the very least, an interesting new
perspective on familiar military situations.

The rest of the paper is structured as follows. Section 2 will discuss the
basic defence principles which are elaborated by the papers that we review, as
well as introduce basic concepts of game theory. Section 3 investigates and
analyses the literature and summarises the findings and associations in each
of the papers. Section 4 elaborates our multi-dimensional classification of
the literature based on the observations made in the previous section. Section
5 identifies the gaps in the literature, and based on this, highlights
opportunities for future research in niche, particularly areas of defence
research which could benefit from the application of game theory where game
theory has not been often applied so far. Finally, section 6 summarises our
findings and classifications, and provides broad conclusions.

##  2 Background

Ideologies, beliefs and knowledge about war have been shaping human knowledge
and philosophy for centuries. The great works of Sun Tzu, Homer and
Machiavelli (Machiavelli, 1984; Lattimore, 1962; Tzu, 2014) have not only
established a foundation for knowledge etched into the essence of military
decision making, but also provided insight into sociology and social
psychology (Andrzejewski, 2013). The military forms a core power bloc for many
civilisations, and is instrumental to both the growth of influence for
existing nations, and the birth of new nations (Janowitz, 1964). The military
deals with conflicts in real time, plans for the future, as well as reviews
past engagements - and every single one of these activities has an impact on
society (DeGrasse, 2016). This review therefore by necessity addresses many
facets of military conflict across multiple physical domains, and the major
decisions which define each domain will be further clarified below. With
respect to game theory for all of these domains, the value of targets, the
value of resources, and the priority of objectives are usually the key
parameters that shape the payoff functions and strategies which in turn define
the games that we use in modelling.

In this section, we discuss the concepts in defence science and technology, as
well as game theory, which are necessary to understand and analyse the
literature in the presented niche. First of all, let as consider the broad
domains of defence and national security which are considered in this review.
They can be summarised, as shown in Table LABEL:classstructure.

Table 1: Classification System used in this review Military’s Main Method | Command and Control Warfare
---|---
Period | Traditional (T) | Modern (M)
Military Field | Land (L) | Air (A) | Sea (S) | Cyber (C) | Space (S)
Major Military Field |  | Resource Allocation Warfare (RAW)
---
or
Information Warfare (IW)
or
Weapon Control Warfare (WCW)
or
Adversary Monitoring Warfare (AMW)
Game Theory Category 1 |  | Non-Cooperative (NCo)
---
or
Cooperative (Co)
Game Theory Category 2 |  | Sequential (Seq)
---
or
Simultaneous (Sim)
Game Theory Category 3 |  | Discrete (D)
---
or
Continuous (C)
Game Theory Category 4 |  | Zero Sum (ZS)
---
or
Non-Zero Sum (NZS)
Number of Players |  | 2 player (2P)
---
or
3-player (3P)
or
N-player (NP)

As shown in the table LABEL:classstructure, in this review, the focus is
primarily on ‘command and control’ warfare, where decision making is critical.
However, command and control warfare has applicability in traditional modes of
warfare, such as land, sea, and air warfare, as well as modern modes of
warfare, such as space and cyber warfare. At an orthogonal level, command and
control warfare could also be sub-divided into Resource Allocation Warfare
(RAW), Information Warfare (IW), Weapons Control Warfare (WCW), as well as
Adversary Monitoring Warfare (AMW). Since these concepts are extensively used
in our classification of literature, let us briefly introduce them first.

###  2.1 Warfare types

####  2.1.1 Land warfare

Technology is a dictating force of warfare, and technology is not as
imperative to land warfare as it is to other domains(Roland, 1991). The
technology that has impacted land warfare has been static, and avoids the
exposure of human resources if possible (Paxson et al., 1979; Edmonds, 1992).
Interpersonal combat at a physical level is much less prevalent nowadays,
making way for a greater focus on positioning strategy. The literature which
applied game theory to ground warfare includes a strong repository of Weapon-
Target Allocation papers (which touch upon Weapon Control Warfare and Resource
Allocation Warfare in the modern context), as well as papers that address
ancient ground engagements and guerilla warfare. Where human lives are
vulnerable, their protection is the most important element of these games, and
the next priority is the protection of ground based assets.

####  2.1.2 Sea (Naval) warfare

Given the importance of navies for the projection of power globally, there is
a surprising paucity of publicly available literature on naval warfare - with
or without the application of game theory. There is often mention of naval
warfare in papers dealing with target tracking, but a discussion on military
naval strategy is limited to outdated literature or discussion of bare
essentials (Harding, 2002; Speller, 2018). We will review the available papers
in this regard, and highlight this an area where there is a sizeable gap in
the literature.

####  2.1.3 Aerial warfare

It was not long after the Wright Brothers’ invention of the aeroplane that
aerial warfare became a critical factor in combat and military campaigns
(Paris, 1992). In a combat medium rarely impeded by obstacles or dimension,
the nature of aerial combat is fast paced, intuitive and incredibly
treacherous, with unpredictable ‘rules’ for engagement (Colby, 1925; Garner,
1924). In the present day, the factors to consider are vastly more
complicated, and there is no shortage of resources available to military
forces to conduct aerial combat- both human and machine (Gingras and Ruby,
2000; Dunn, 2013; Stoyen, 2008). The literature shows that as a result of this
abundance of arsenal, the intrinsic and potential value of both the targets
and the resources used to engage is of particular importance in aerial warfare
scenarios. Decisions about those values for both sides of the conflict need to
be made when evaluating strategies for combat. As such, there are several
papers which deal with the use of game theory in aerial warfare.

####  2.1.4 Space warfare

While the notion of warfare in space has existed for almost a century, neither
physical execution nor established strategic theory of space warfare has been
established (Klein, 2012). Nevertheless, this has not stopped military forces
chasing the stars (literally and figuratively) (Wortzel, 2008; Johnson-Freese,
2016), and has inevitably lead to concepts from game theory being used in
space warfare strategic thinking. This is currently mostly limited to
satellite networks, where the key parameters of the game are optimised power
use and signal strength across the network. The field is still quite young,
and further military development in space seems to be inevitable, with which
the corresponding literature dealing with applications of game theory in space
warfare will also grow.

####  2.1.5 Cyber warfare

Cybersecurity is the protection of IT systems and networks from being
damaged/disrupted/subjected to information theft. Cyber warfare deals with the
concept of Information and communication Systems being deliberately attacked
to obtain a military advantage. While Cybersecurity has been an important
field in computer science for many decades now, literature about cyber warfare
as such is more scarce, and in any case heavily overlaps with applications of
game theory in computer science in areas related to cybersecurity. This review
presents and analyses some papers which are specifically concerned with cyber
warfare.

####  2.1.6 Mixed/ other warfare

There are several papers which address specific niches of defence applications
of game theory, and yet cannot be classified as papers analysing a certain
type of warfare. In some of these papers, the focus is more on the technology
that is used: for example, target tracking. In others, the nature of the
hostile actors against whom the defence needs to be conducted changes: for
example, national security operations which target domestic terrorism threats.
There are several papers which deal with the use of game theory in such
scenarios.

Target tracking systems: Target tracking in the military is the observation of
a moving target and the surveillance of its position and manoeuvres (Blackman,
1986; Lipton et al., 1998; Li and Jilkov, 2003). Success in this domain relies
on accuracy in the observed metrics and data, as well as efficient
distribution and processing of all collected information (Kasthurirathna et
al., 2014). With the advent of intelligent targets, the military must also
incorporate predictive methods to maintain ideal tracking performance. The
literature reviewed in tis regard covers topics from tracking strike missiles
to theatre ballistic missiles, and tracking unknown intelligent agents to
enemy aircraft. Key considerations in this area which shape the games played
involve whether or not the target is ‘intelligent’ / can take evasive action,
weather or not target will have an optimal trajectory, and whether or not the
target will have defenders (Bar-Shalom and Li, 1996). Target tracking
applications of game theory mostly occur in aerial and naval warfare,
including underwater surveillance.

National Security applications : Game theory often finds application in
national security and anti-terrorism related applications. This includes
predicting and preparing for terrorist attacks, as well as resource allocation
scenarios for protection of key personnel and landmarks / other potential
targets for terrorist activity. While the value of potential targets and the
likelihood of attacks are obviously key parameters governing the payoff
functions of games in this niche, the subsequent social, economic and
political ramifications are equally instrumental in modelling games in this
area (Gandhi et al., 2011; Harris, 1992). Few military conflicts have as much
exposure as those on the home front (Maltby and Keeble, 2007), and the fallout
from terrorist attacks and their effect on public mood and confidence in the
security apparatus are often taken into account in modelling payoff functions
in this area.

###  2.2 Game Theory

Game theory, which is the study of strategic decision making, was first
developed as a branch of micro-economics (Von Neumann and Morgenstern, 1944;
Barough et al., 2012; Kasthurirathna and Piraveenan, 2014; Myerson, 1991; Von
Neumann, 1928; Osborne et al., 2004; Von Neumann and Morgenstern, 1944).
However, later it has been adopted in diverse fields of study, such as
evolutionary biology, sociology, psychology, political science, project
management, financial management and computer science (Rasmusen and Blackwell,
1994; Kasthurirathna and Piraveenan, 2015; Kasthurirathna et al., 2013;
Thedchanamoorthy et al., 2014; Martin, 1978; Piraveenan, 2019; Peldschus,
2008; Schotter, 2008; Morrow, 1994; Hammerstein and Selten, 1994; Thakor,
1991; Myerson, 2013). Game theory has gained such wide applicability due to
the prevalence of strategic decision making scenarios across different
disciplines. Game Theory provides insight into peculiar behavioural
interactions like the cooperative interactions within groups of animals
(Bshary and Oliveira, 2015; Hadjichrysanthou and Broom, 2012), the bargaining
and exchange in a marriage (Lundberg and Pollak, 1994; Manser and Brown, 1980;
Grossbard-Shechtman, 1995) or the incentivisation of Scottish salmon farmers
(Murray, 2014). A typical game defined in game theory has two or more players,
a set of strategies available to these players, and a corresponding set of
payoff values (sometimes called utility values) for each player (which are, in
the case of two-player games, often presented as a payoff-matrix)(Rasmusen and
Blackwell, 1994; Gibbons, 1992; Shubik, 1970).

####  2.2.1 Pure vs Mixed Strategies

A pure strategy in a game provides a complete definition of how a player will
play a game. A player’s strategy set is the set of pure strategies available
to that player(Polak, 2007).

A mixed strategy is a combination of pure strategies where a particular
probability $p$ (where $0\leq p\leq 1$ ) is associated with each of these pure
strategies. Since probabilities are continuous, there are infinitely many
mixed strategies available to a player. A totally mixed strategy is a mixed
strategy in which the player assigns a strictly positive probability to every
pure strategy. Therefore, any pure strategy is actually a degenerate case of a
mixed strategy, in which that particular strategy is selected with probability
$1$ and every other strategy is selected with probability $0$. A totally mixed
strategy is a mixed strategy in which every pure strategy has a strictly
positive (non-zero) value.

####  2.2.2 Nash Equilibrium

Nash equilibrium is one of the core concepts in game theory. It is a state (a
set of strategies) in a strategic game from which no player has an incentive
to deviate unilaterally, in terms of payoffs. Both pure strategy and mixed
strategy Nash equilibria can be defined. A strategic game can have more than
one Nash equilibrium (Nash et al., 1950; Kasthurirathna et al., 2014). It is
proven that every game with a finite number of players in which each player
can choose from finitely many pure strategies has at least one Nash
equilibrium (Nash et al., 1950).

The formal definition of Nash equilibrium is as follows. Let $(S,f)$ be a game
with $n$ players, where $S_{i}$ is the strategy set of a given player i. Thus,
the strategy profile $S$ consisting of the strategy sets of all players would
be, $S=S_{1}$ x $S_{2}$ x $S_{3}....$ x $S_{n}$. Let
$f(x)=(f_{1}(x),.....,f_{n}(x))$ be the payoff function for strategy set $x\in
S$. Suppose $x_{i}$ is the strategy of player i and $x_{-i}$ is the strategy
set of all players except player i. Thus, when each player $i\in 1,.....,n$
chooses strategy $x_{i}$ that would result in the strategy set
$x=(x_{1},....,x_{n})$, giving a payoff of $f_{i}(x)$ to that particular
player, which depends on both the strategy chosen by that player ($x_{i}$) and
the strategies chosen by other players ($x_{-i})$. A strategy set $x^{*}\in S$
is in Nash equilibrium if no unilateral deviation in strategy by any single
player would return a higher utility for that particular player
(Kasthurirathna et al., 2015). Formally put, $x^{*}$ is in Nash equilibrium if
and only if:

$\forall i,x_{i}\in S_{i}:f_{i}(x_{i}^{*},x_{-i}^{*})\geq
f_{i}(x_{i},x_{-i}^{*})$ (1)

####  2.2.3 Normal-form games and extensive-form games

In a normal-form game, only a single round of decision making takes place,
where all players make decisions simultaneously. An extensive-form game is, on
the other hand, an iterative game where there are several rounds of decision
making (Binmore, 2007; Hart, 1992). On each round, players could make
decisions simultaneously or in some pre-defined order. An extensive-form game
is often represented by a game tree, where each node (except terminal nodes)
is a decision point, and each link corresponds to a decision or a set of
decisions that could be made by the relevant player / players at that point.
The terminal nodes represent an end to the extensive-form game, with
corresponding payoffs for each player involved.

####  2.2.4 Non-cooperative games and cooperative games

Typically, games are taken to be played for the self-interest of the players,
and even when the players cooperate, that is because cooperation is the best
strategy under the circumstances to maximise the individual payoffs of the
players. In such games, the cooperative behaviour, if emerges, is driven by
selfish goals and is transient. These games can be termed ’non-cooperative
games’. These are sometimes referred to, rather inaccurately, as ‘competitive
games’. Non-cooperative game theory is the branch of game theory that analyses
such games. On the other hand, in a cooperative game, sometimes also called a
coalitional game, players form coalitions, or groups, often due to external
enforcement of cooperative behaviour, and competition is between these
coalitions (Myerson, 2013; Nash et al., 1950; Nash, 1951; Ritzberger et al.,
2002; Kasthurirathna et al., 2014). Cooperative games are analysed using
cooperative game theory, which predicts which coalitions will form and the
payoffs of these coalitions. Cooperative game theory focusses on surplus or
profit sharing among the coalition (Peleg and Sudhölter, 2007), where the
coalition is guaranteed a certain amount of payoff by virtue of the coalition
being formed. Often, the outcome of a cooperative game played in a system is
equivalent to the result of a constrained optimisation process (Bell et al.,
2017), and as such, some papers that we review use a linear programming
framework to solve the cooperative games they model.

####  2.2.5 Zero-sum games

Zero-sum games are a class of competitive games where the total of the payoffs
of all players is zero. In two player games, this implies that one player’s
loss in payoff is equal to another player’s gain in payoff. A two player zero-
sum game can therefore be represented by a payoff matrix which shows only the
payoffs of one player. Zero-sum games can be solved with the mini-max theorem
(Kuhn et al., 1953), which states that in a zero-sum game there is a set of
strategies which minimises the maximum losses (or maximises the minimum
payoff) of each player. This solution is sometimes referred to as a ‘pure
saddle point’. It can be argued that the stock market is a zero-sum game. In
contrast, most valid economic transactions are non zero-sum since each party
considers that, what it receives is more valuable (to itself) than what it
parts with.

####  2.2.6 Perfect vs Imperfect Information Games

In a perfect information game, each player is aware of the full history of the
previous actions of all other players, as well as the initial state of the
game. In imperfect information games, some or all players do not have access
to the entirety of information about other players’ previous actions.

####  2.2.7 Simultaneous games and sequential games

A simultaneous game is either a normal-form game or an extensive-form game
where on each iteration, all players make decisions simultaneously. Therefore,
each player is forced to make the decision without knowing about the decisions
made by other players (on that iteration). On the contrary, a sequential game
is a type of extensive-form game where players make their decisions (or choose
their strategies) in some predefined order (Binmore, 2007; Hart, 1992). For
example, a negotiation process can be modelled as a sequential game if one
party always has the privilege of making the first offer, and the other
parties make their offers or counteroffers after that. In a sequential game,
at least some players can observe at least some of the actions of other
players before making their own decisions (otherwise, the game becomes a
simultaneous game, even if the moves of players do not happen simultaneously
in time). However, it is not a must that every move of every previous player
should be observable to a given player. If a player can observe every move of
every previous player, such a sequential game is known to have ‘perfect
information’. Otherwise the game is known to have ‘imperfect information’.
Sequential games are often used by papers that we are reviewing here, to model
bargaining or negotiation processes.

####  2.2.8 Differential Games

Differential Games are often extensive form games, but instead of having
discrete decision points, they are modeled over a continuous time frame
(Kamien and Schwartz, 2012). In such games, the each state variable evolves
continuously over time according to a differential equation. Such games are
ideal for modelling rapidly evolving defense scenarios where each player
engages in selfish optimisation of some parameter. For example, in missile
tracking problems, the pursuer and the target both try to control the distance
between them, whereas the pursuer constantly tries to minimise this distance
and the target constantly tries to increase it. In such a scenario, iterative
rounds of decision making are much too discrete to model the continuous
movements and computations of each player. Differential games are ideal to
model such scenarios.

####  2.2.9 Stackelberg games

A Stackelberg game is a particular type of two player sequential game commonly
used in economics (Von Stackelberg, 2010). In a Stackelberg game, there is a
leader and a follower, which are typically companies operating in a market.
The leader firm has some sort of market advantage that enables it to move
first and make the first decision, and the follower’s optimal decision depends
on the leader’s decision. If a follower chose a non-optimal action given the
action of the leader, it will adversely affect not only the payoff of the
follower, but the payoff of the leader as well. Therefore, the optimal
decision of the leader is made on the assumption that the follower is able to
see the leader’s action and will act to maximise its own payoff given the
leader’s action. Several papers we review here have used Stackelberg games to
model project management scenarios. Stackleberg games might be relevant in
defence applications where there are leaders and followers, such as naval,
airplane fighter or tank formations.

####  2.2.10 Common interest games

Common interest games are another class of non-cooperative games in which
there is an action profile that all players strictly prefer over all other
profiles (Calcagno et al., 2014). In other words, in common interest games,
the interests of players are perfectly aligned. It can be argued that common
interest games are the antithesis of zero-sum games, in which the interests of
the players are perfectly opposed so that any increase in fortune for one
player must necessarily result in the collective decrease in fortune for
others. Common interest games were first studied in the context of cold war
politics, to understand and prescribe strategies for handling international
relations (Schelling, 1958, 1980; Lewis, 2008).

It makes sense to classify non-cooperative games into common interest games
and non-common interest games, just as much as it makes sense to classify them
into zero-sum games and non-zero sum games, since these two concepts (zero-sum
games and common interest games) represent extreme cases of non-cooperative
games. However, the papers that we review do not use common interest games to
model project management scenarios, and indeed it would be rare to find
scenarios in project management where the interests of players are perfectly
aligned. Therefore, we do not use the common interest games-based
classification in our classification process, as it would add another
dimension and increase the complexity of classification needlessly.

####  2.2.11 Nash bargaining

In a Nash bargaining game (Nash Jr, 1950; Myerson, 1999), sometimes referred
to as a bargaining problem or bargaining game, two players could choose from
an identical set of alternatives, however each alternative has varying payoffs
for the players. Typically, some alternatives have better payoff for one
player, while other alternatives have better payoff for the other player. If
both players choose the same alternative, then each get the payoff
corresponding to that alternative. If they choose differing alternatives, then
there is no agreement, and they each get a fixed payoff which corresponds to
the cost of non-agreement, and therefore typically negative. Thus, there is
incentive to choose an alternative which may not necessarily be the best for a
player. If there is perfect information, that is, the full set of alternatives
and payoffs is known to both players, then there is an equilibrium solution to
the Nash bargaining game (Rubinstein, 1982).

####  2.2.12 Subgames

A subgame is a subset of a sequential game such that at its (the subgame’s)
beginning, every player knows all the actions of the players that preceded it
(Osborne et al., 2004). That is, a subgame is a section of the game tree of a
sequential game where the first decision node that belongs to this section has
perfect information.

####  2.2.13 Subgame perfect Nash equilibrium

In a sequential game, a subgame perfect Nash equilibrium is a set of
strategies representing each player such that they constitute a Nash
equilibrium for every subgame of that sequential game (Osborne et al., 2004).
Thus, when a game tree of the sequential game is considered, if a set of
strategies could be identified so that they represent a Nash equilibrium for
every branch of game tree originating from a node which represents a point
where every player knows all preceding actions of all players, such a set of
strategies represent a subgame perfect Nash equilibrium for that sequential
game. For example, when two players are bargaining, they are in a subgame
perfect Nash equilibrium if they are presently employing a set of strategies,
which will represent a Nash equilibrium between them at any future stage of
the bargaining process given that they are aware of the full history of the
bargaining process up to that point.

####  2.2.14 Behavioural Game Theory

Behavioural Game Theory combines classical game theory with experimental
economics and experimental psychiology, and in doing so, relaxes many
simplifying assumptions made in classical game theory which are unrealistic.
It deviations from several simplifying assumptions made in classical game
theory such as perfect rationality(Camerer, 2011; Kasthurirathna and
Piraveenan, 2015), the independence axiom, and the non-consideration of
altruism or fairness as motivators of human decision making(Aumann and Hart,
1992; Aumann, 2008). We will show in this review that the approaches related
to behavioral game theory are crucial in modelling military scenarios, such as
in signaling games.

####  2.2.15 Evolutionary game theory

Evolutionary game theory is an outcome of the adoptation of game theory into
the field of evolutionary biology (Smith, 1993). Some of the critical
questions asked in evolutionary game theory include: which
populations/strategies are stable? which strategies can ‘invade’ (become
popular) in populations where other strategies are prevalent? How do players
respond to other players receiving or perceived to be receiving better payoffs
in an iterated game setting? etc. Evolutionary games are often modelled as
iterative games where a population of players play the same game iteratively
in a well-mixed or a spatially distributed environment (Le and Boyd, 2007;
Kasthurirathna et al., 2015).

A strategy can be identified as an evolutionary stable strategy (ESS) if, when
prevalent, it has the potential to prevent any mutant strategy from
percolating its environment (Kasthurirathna et al., 2015). Alternatively, an
ESS is the strategy which, if adopted by a population in a given environment,
cannot be invaded by any alternative strategy. Hence, there is no benefit for
a player to switch from an ESS to another strategy. Therefore, essentially, an
ESS ensures an extended Nash equilibrium. For a strategy $S_{1}$ to be ESS
against another ‘invading’ strategy $S_{2}$, one of the two conditions
mentioned below needs to be met, in terms of expected payoff $E$.

  1. 1.

$E(S_{1},S_{1})>E(S_{2},S_{1})$ : By unilaterally changing strategy to
$S_{2}$, the player will lose out against another player who sticks with the
ESS $S_{1}$.

  2. 2.

$E(S_{1},S_{1})=E(S_{2},S_{1})$ & $E(S_{1},S_{2})>E(S_{2},S_{2})$ : a player,
by converting to $S_{2}$, neither gains nor loses against another player who
sticks with the ESS $S_{1}$, but playing against a player who has already
‘converted’ to $S_{2}$, a player is better off playing the ESS $S_{1}$.

If either of these conditions are met, the new strategy $S_{2}$ is incapable
of invading the existing strategy $S_{1}$, and thus, $S_{1}$ is an ESS against
$S_{2}$. Evolutionary games are typically modelled as iterative games, whereby
players in a population play the same game iteratively.

####  2.2.16 Signaling Games

A signaling game (Cho and Kreps, 1987) is an incomplete information game where
one player has perfect information and another does not. The player with
perfect information (the Sender $S$) relays messages to the other player (the
Receiver $R$) through signals, and the other player will act on those signals
after inferring the information hidden in the messages. The Sender $S$ has
several potential types, of which the exact type $t$ in the game is unknown to
the Receiver $R$. $t$ determines the payoff for $S$. $R$ has only one type,
and their payoff is known to both players.

The game is divided into the sending stage and the acting stage. $S$ will send
one of $M=\\{m_{1},m_{2},m_{3},...,m_{j}\\}$ messages. $R$ will receive that
message and respond with an action from a set
$A=\\{a_{1},a_{2},a_{3},...,a_{k},\\}$. The payoff that each player receives
is determined by the combination of the Sender’s type and message, as well as
the action that the Receiver responds with. An example of the signaling game
is the Beer-Quiche Game (Cho and Kreps, 1987), in which Player B, the
receiver, makes a choice of whether or not to duel Player A. Player A is
either surly or wimpy, and Player B would only like to duel the latter. Player
A chooses to have either a Beer or a Quiche for breakfast. While they prefer a
quiche, a quiche signals information from a stereotype that quiche eaters are
wimpy. Player B must analyse how each decision, duel or not duel, will give
them payoff depending on which breakfast Player A chooses.

With this background, we now review the available literature which deals with
the applications of game theory in defence science and technology.

##  3 Use of game theory in defence science and technology applications

As mentioned earlier, the primary parameters that influence the payoff matrix
in games modelling defence scenarios are the value of targets, the value of
resources and the priority of objectives. Other than this, the games used in
defence applications can vary greatly, as we will see below. For this reason,
this section is structured based on the domain (type of warfare) each paper
covers Where a paper covers more than one domain, it is included in the most
relevant subsection / domain. We however analyse in detail the type of games
used, the way the payoff functions were structured, the available strategies
and equilibria etc for each paper.

###  3.1 Papers dealing with Land Warfare

In Land warfare related applications of game theory, most studies focus on
defensive warfare, whereby the military makes decisions on how to best
allocate their ground defences to multiple threats. Some studies also focus on
historical land-based conflicts and provide game theoretical analysis in
hindsight, revealing how some decisions made from intuition in historical
conflicts had a rational and mathematical justification. Land warfare can
result in very heavy casualties, so understanding how to best minimise human
losses is a key component (though not the only objective) of land warfare.
Quite often, prioritising military resources is also fundamental to success
and often features prominently in strategic decisions. Furthermore, often in
scenarios involving ground warfare, it is important to assess the knowledge
about opponents , their possible tactics, or terrain: it may become necessary
to combat airborne forces being inserted at certain places, or it may be
needed to traverse uncertain territory. In each of these situations,
understanding where a force has imperfect information will help that force to
make rational decisions.

Bier et al. (Bier et al., 2007-08) design a game to best assign defensive
resources to a set of locations / resources that need to be protected. The
attacker must then decide how they choose to split their force to attack the
different targets. The game is modelled as a two player game of normal form.
Payoff in this game is absolute, and an attack on a location $i$ is either a
success or a failure, where the attacker gains $a_{i}$ and the defender loses
$d_{i}$. Since orders for an attack are confirmed ahead of an attack,
attackers must use a set of pure strategies. The game can be played both
simultaneously or sequentially. That is, the game can be played depending on
whether or not the attacker knows how the defender has assigned their
resources before making their decision. This leads to the ideal strategy being
to leave some targets undefended, centralising defenses in key areas by
leaving some areas vulnerable.

There are several studies which analyse historical conflicts, which occurred
predominantly on land, using a game theoretic prism. For example, Cotton and
Liu’s (Cotton and Liu, 2011) describe two ancient Chinese military legends and
model them as signaling games. In both games, legendary military leaders are
faced with formidable opposing armies with much greater numbers and strength
than their own, but instead of retreating, they prepare to engage, acting as
if they are setting up for an ambush. Their opponents with imperfect
information are left only with the messages they can infer from their
opponents’ actions; spooked by the perceived confidence and the reputation
that these generals carried, the opposing armies, though in actuality are of
superior strength, choose not to engage. Through a brave and ingenious bluff,
both generals achieve an equilibrium solution in their favour by standing
their ground. They do this by creating deception without direct communication,
which follows the template of the aforementioned Beer-Quiche signaling game.

The first game described by Cotton and Liu is the "100 Horsemen" game. They
describe a piece of history where a hundred Han horsemen travelling alone
encounter a large Xiongnu force numbering in the thousands. Their available
strategies are to retreat or engage. If they retreat, and the enemy engage,
they will very likely be run down and defeated; and if they engage and the
enemy also engage, they will be eliminated in battle. The best outcome for
them is to somehow force an enemy retreat. The enemy is uncertain if the
horsemen are travelling with a greater army. They see the horsemen move to
engage, and decide not to take the risk, and retreat. The situation is
translated into a two player game, with two strategies. It is represented in
Figure 1 below:

Figure 1: The 100 Horsemen signaling game (Cotton and Liu, 2011)

In Figure 1 :

LG represents the decision point for the General Li Guang, of the Han forces.

GenX represents the decision point for the opposing Xiongnu force.

Payoffs are listed as (LG, GenX)

$\lambda\in(0,1)$ represents the General’s ability, as either strong or weak

$\alpha$ and $\beta$ represents the proportion of Han horsemen killed in a
retreat

$w$ is a positive parameter

The second game is very similar to the first. In this game, a small city is
guarded by the formidable General Zhuge Liang. He learns that a great hostile
army is approaching the city. He is faced with two options. He could run,
after which he would secede the city and likely be chased down by the
approaching army, or he could stay and defend the city. If he chose the
latter, and the army were to engage, he would likely lose his life, his army
and the city. Faced with this dilemma, he orders his men to hide out of sight,
so that the city appears empty from the outside. He climbs to the top of the
foremost tower of the city and plays music. The opposing general, aware of
General Liang’s experience and prowess, suspects that the General has taken
this unassuming position in the tower in the empty city to ambush his army,
and they move away from the city to avoid being ambushed. General Liang sent
effectively two signals here. The first was his reputation, a signal
encompassing his strategic and military strength. The second was his choice to
stay and defend the city. With these two pieces of information, and nothing
else about the whereabouts or magnitude of General Liang’s army, the opposing
army chooses the safe option of zero loss and leaves. This piece of history is
modelled as another two player signaling game, shown in Figure 2 below:

Figure 2: The Empty City signaling game (Cotton and Liu, 2011)

In Figure 2 :

ZL represents the decision point for the General Zhuge Liang

Payoffs are listed as (ZL, Opposing Army)

$\lambda\in(0,1)$ represents the General’s ability, as either strong or weak

$c$ represents the value of the city

$w$ represents the gains if ZL’s army matched the opposing army

$y$ represents the losses if ZL’s army is weaker than the opposing army, and
$y>c$ since it encompasses losing the city

Both pieces of history represent distinguished military decision making in the
face of near certain defeat, and are in fact examples of Generals with a
strong understanding of the nuances of signals and rational decision making in
strategic interactions forcing a favourable outcome to themselves.

The next paper we review is Gries et al (Gries et al., 2016) which is a
comprehensive investigation into the utility of game theory principles in
guerilla/destabilisation warfare. The significant factors they model are:
destabilisation insurgents often attack randomly, creating a continuous threat
that must have a continuous mitigation and detection strategy; duration of a
war is important to consider, and will change the value that is assigned to
targets and assets; time preferences play a critical role in setting
priorities, as always judgements of value determine strategic decisions which
in turn determine success or failure. The game model they propose involves
both a sequential non cooperative game and a simultaneous non cooperative
game, in each of which the two players are the guerilla force and the
government. For these conflicts, the economic and social impacts are much more
significant than military losses and gains, and therefore play a much more
significant role in calculating the value of outcomes.

The game specifically looks at moments when each side looks to try and find
peace or conflict with the other. At each of these moments, the government
force must consider the financial cost of each option, while the rebels must
on the order of priority of the engagements, and what portion of their
fighting force they will make available for each engagement. Figure 3
demonstrates an example of the decision tree to emerge from these moments in
destabilisation warfare, where G represents the Government decisions and R
represents the Rebel decisions.

Figure 3: Destabilisation Warfare game (Gries et al., 2016)

Krisnamurthy et al. (Krishnamurthy et al., 2007) investigate a game-theoretic
control of dynamic behaviour of an unattended ground sensor network (UGSN) in
order to acquire information about intruders. Each sensor in this network is
capable of receiving measurements, with specified accuracy, of the range and
bearing of nearby targets which they then transmit to a local hub for data
fusion. In this framework, while more sensor measurements and larger volumes
of transmission of measurements may lead to better target awareness, this also
results in the undesirable effect—greater consumption of limited battery
power. Hence, the goal to which game-theory is applied is to optimally trade-
off target awareness, data transmission and energy consumption using a two-
time scale, hierarchical approach.

The authors demonstrate that the sensor activation and transmission scheduling
problem can be decomposed into two coupled decentralized algorithms. In
particular, the sensors are viewed as players in a non-cooperative game and an
adaptive learning strategy is proposed to activate the sensors according to
their proximity to targets of interest. This turns out to be a correlated
equilibrium solution of this non-cooperative game. Next, the transmission
scheduling problem, in which each sensor has to decide at each time instant,
whether to transmit data and waste battery power, or to wait and increase
delay, is formulated as a Markov Decision process with a penalty terminal
cost. The main result of this formulation is to show that the optimal
transmission policy has a threshold structure which is then proved using the
concept of supermodularity.

###  3.2 Papers dealing with Naval Warfare

Naval warfare predates aerial warfare and has been prevalent for a longer
duration in human history. In the 18th and 19th centuries, the powerful
nations of the time built warships with cannons positioned along their sides.
It meant that ships could attack typically only to their sides. When sailing
as an armada, the standard approach was to form a ‘line of battle’ i.e. a
column of allied naval ships sailing in a direction such that their sides
would face the enemy, also positioned in a line. The two parallel opposing
fleets could blast one another with large number of cannons. The ‘line of
battle’ strategy is considered to be a Nash-equilibrium because neither fleet
would gain from performing raking (a tactic of the era, whereby an attacking
ship would attempt to sail across its adversary’s stern, concentrating cannon
fire there while the enemy could only respond minimally due to having less
cannon defences in the stern. The attacking ship would damage both the stern
and some of the broadsides of its adversary). According to (Levine, 1995)
raking was not preferred in a fleet as this would mean having to first sail
ahead of its enemy and then turn towards it—a challenging task when the ships’
speeds were roughly equal and maneuvering was difficult. As neither fleet
would gain from turning towards the enemy and neither would get ahead, Levine
concludes that this strategy—forming a line of battle and sailing parallel to
the other fleet—was each fleet’s best response, and thus represented a Nash
equilibrium.

Levine goes on to mention battles in which English fleets deviated from the
above strategy and sailed orthogonally towards a French and Franco-Spanish
fleet. In the first battle Levine mentions, it was likely unplanned. In the
second—the 1805 Battle of Trafalgar—it was by careful design: the English
fleet divided itself into two columns, each of which sailed orthogonally to
the Franco-Spanish line, raking fire for about 45 minutes before crashing
through it and beginning a general melee. The English would go on to isolate
the middle of the Franco-Spanish fleet to score a decisive victory. Levine
considers both battles to be counterexamples to his thesis. However, in the
Battle of Trafalgar, its is possible that the English strategy was a best
response to the likely Franco-Spanish strategy of forming an orthodox line of
battle. The English admiral, Lord Nelson, desired to keep the Franco-Spanish
fleet from escaping—which they could if both fleets formed parallel lines of
battle—thus reducing the reward he would get for forming his own fleet into a
line of battle. Moreover, he may have estimated that the poor gunnery of the
French and Spanish ships would lessen the effect of the raking fire, thus
reducing the negative reward he would get for directly charging the Franco-
Spanish fleet. In his eyes, this may have made the unorthodox option a better
response to the likely Franco-Spanish strategy than the orthodox line of
battle. While Levine did not explicitly attribute these strategies in naval
battles of the era to game theory, it is quite possible that the adopted
strategies had a game-theoretic basis.

Maskery et al. (Maskery and Krishnamurthy, 2007) study the problem of
deploying counter-measures against anti-ship missiles using a network enabled
operations (NEOPS) framework, where multiple ships communicate and coordinate
to defend against a missile threat. Here, the missile threats are modelled as
a discrete Markov process and they appear at random positions within a fixed
physical space and move towards the ships obeying some known target dynamics
and guidance laws. The ships which are equipped with counter-measures (CM)
such as decoys and electromagnetic jamming signals, are modelled as the
players of a transient stochastic game, where the actions of the individual
players include the use of CM to maximize their own safety while cooperating
with other players which are essentially aiming to achieve the same objective.
The optimal strategy of this game-theoretic problem is a correlated
equilibrium strategy, and is shown to be achieved via an optimization problem
with bilinear constraints. This is in contrast to the Nash equilibrium
solution proposed in (Maskery et al., 2007) to a related problem but one
without player coordination. A noteworthy contribution of this paper is that
it also quantifies the amount of communication necessary in order to implement
the NEOPS equilibrium strategy. This paper highlights the utility of game-
theoretic methods in analysing optimal strategies in network enabled systems
which are critical in modern warfare.

In (Maskery et al., 2007) Maskery et al. consider the problem of network
centric force protection of a task group against anti-ship missiles. The
decision makers in this model are the ships equipped with hard-kill/soft-kill
weapons (counter-measures) and these ships are also considered the players in
the formulation of this problem as a game-theoretic setting. The platforms
must make critical decisions independently on the optimal deployment of the
counter-measures while they simultaneously work towards a common goal of
protecting the members of the task group. Essentially this is a decentralised
missile deflection problem in a naval setting which is formulated as a
transient stochastic game for which the ships may compute a joint counter-
measure policy that is in Nash equilibrium. Here the ships play a game with
each other instead of with a missile. This approach naturally lends itself to
decentralised solutions which may be implemented when full communication is
not feasible. Moreover, this formulation leads to an interpretation of the
problem as a stochastic shortest past game for which Nash Equilibrium
solutions are known to exist.

Bachmann et al. (Bachmann et al., 2011) analyse the interaction between a
radar and jammer using a noncooperative two-player, zero-sum game. In their
approach, the radar and jammer are considered “players” with opposing goals:
the radar tries to maximize the probability of detection of the target while
the jammer attempts to minimize its detection by the radar by jamming it.
Bachmann et al. (Bachmann et al., 2011) assume a Swerling Type II target in
the presence of Rayleigh distributed clutter, for which certain utility
functions are described for cell-averaging (CA) and order-statistic (OS) CFAR
processors in different cases of jamming. This game-theoretic formulation is
solved by optimizing these utility functions subject to constraints in the
control variables (strategies), which for the jammer are jammer power and the
spatial extent of jamming while for the radar the available strategies include
the threshold parameter and reference window size. The resulting matrix-form
games are solved for optimal strategies of both radar and jammer from which
they identify conditions under which the radar and jammer are effective in
achieving their individual goals.

###  3.3 Papers dealing with Aerial Warfare

Aerial combat is often a normal-form game where decisions about utilised
resources are made before the engagement, based on assumptions and knowledge
about the strength of different elements of the arsenal. For example,
Suppression of Enemy Air Defense vehicles (SEADs) are effective against
ground-to-air defenses and Surface to Air Missiles (SAMs), but will not be
useful against fighter aircraft. Therefore, when military personnel decide
which resources to use in an engagement, they need to weigh how valuable each
of their resources is, as well as how important the objective is to both sides
of the conflict. If the attacking force values a target much more than it is
actually worth, then their increased resource expenditure may be detrimental
to their military campaign as a whole. With humans operating the aerial
weaponry usually, their respective abilities and skill sets need to be
considered, and the likelihood of them executing their mission.

There is limited literature on aerial combat modelled with game theory.
Hamilton (Hamilton and Mesic, 2004) writes a comprehensive guide to
applications of game theory on multiple Aerial warfare situations. Hamilton
suggests using game theory to devise strategies not only based on one’s own
military options, but also expectations around enemy actions as well. Game
theory accounts for different interactions with the enemy, rather than simply
considering which side had superior maximum-effort power. Nowadays, many
military forces are able to adapt to instantly changing situations and adjust
their actions based on those new circumstances. As such, Hamilton suggests
first determining all of the tactical options available to each side. As
stated earlier, one of the most fundamental elements of using game theory for
the military is understanding exactly how much value each asset holds - and
detailing the inventory and strategic possibilities of both sides will best
clarify all strategic options. For each option, Hamilton suggests assigning a
numerical value - a Measure of Effectiveness (MoE). Decisions about MoEs are
important because being accurate with MoEs will underpin the choices that are
made strategically. Incorrect MoEs can lead to incorrect strategic decisions,
and perhaps also resulting in poor understanding of why the decision was
wrong. An example of this (although not in the aerial warfare context) was the
Vietnam War, where the early US strategy was maximising the neutralisation of
Viet Cong soldiers. Since the Northern Vietnamese leadership did not place
great emphasis on their infantry, the US strategy ultimately led to a loss in
the war. Next, Hamilton suggests calculating the combined value for all
possible interactions between strategies of both sides of the conflict. This
will generate a matrix of payoffs, from which it is possible to derive the
optimum or dominant strategy for each player, and then an equilibrium
solution. Thus, ahead of any engagement a military leader may partake in, they
have a well-formed idea of the expected result of the game. A caveat that
Hamilton adds to these guidelines is to consider the length of a military
campaign as a whole. The values that can be assigned to a resource for one
battle or strike attack may be small if they are low cost or numerous asset,
but depending on how many of these skirmishes happen over the course of a
campaign, those resources may be come pivotal over the course of a war.

To illustrate these points, Hamilton applies them to a standard aerial warfare
game of SEADs and time critical targets. In this combat, the ’Blue’ side is
trying to eliminate some ground based targets. To do this, they use SEADs. In
response, the ’Red’ side will fire SAMs, which SEADs struggle to avoid.
However, expecting this response, the Blue side also has Strike aircraft which
can defend the SEADs and counteract the SAMs, but are unable to attack the
targets. The questions for the Blue team are: what is the value of the target
and what ratio of SEADs and Strike aircraft should be deployed for the
targets? Likewise, for the Red team: how valuable is the target and how many,
if any SAMs should be fired? Hamilton contends that the optimal Red strategy
is to fire only for a fraction of engagement which is equal to:

$\frac{\text{Value of Target}}{\text{Value of Target}+\text{Value of SEAD
}\times Pk_{s}+\text{Value of SAM }\times Pk_{A}}$ (2)

and the optimal Blue Strategy is to assign a fraction of the planes as SEADs
which is equal to:

$\frac{\text{Value of SAM }\times Pk_{A}}{\text{Value of SAM }\times
Pk_{A}+\text{Value of SEAD }\times Pk_{s}+\text{Value of Target}}$ (3)

where

$Pk_{s}$ is the probability of the SAMS destroying the SEADs

$Pk_{A}$ is the probability of the Strike aircraft destroying the SAMs

This formulation gives a concise prediction of the likely outcome of an
engagement given every possible assignment of aircraft and missile launches.
It must be noted that it is incredibly difficult in practice to accurately
quantify the numerical value of different targets and resources.

Deligiannis et al. (Deligiannis et al., 2016) consider a competitive power
allocation problem in a MIMO radar network in the presence of multiple
jammers. The main objective of the radar network is to minimize the total
power emitted by the radar while achieving a specific detection criterion for
each of the targets. In this problem, the radars are confronted by intelligent
jammers that can observe the radar transmitted power and thereby decide its
jamming power to maximize interference to the radar. Here they treat this
power allocation problem as a non-cooperative game where the players are the
central radar controller and the jammers, and solve this using convex
optimization techniques. Moreover, they provide a proof for the existence and
uniqueness of the Nash equilibrium stable point, where no player can further
profit by changing its power allocation.

Garcia et al. (Garcia et al., 2019) investigate the problem of defending a
maritime coastline against two enemy aircraft whose main objective is to
invade the territory controlled by the defending aircraft. The defender, on
the other hand, attempts to prevent this by trying to intercept both enemy
aircraft in succession as far as possible from the border. This is a typical
pursuit-evasion scenario and is representative of many important problems in
robotics, control and defence. In this paper, Garcia et al. formulate this
problem as a zero-sum differential game, where the defender/pursuer tries to
successively capture the two attackers/evaders as far as possible from the
defended coastline while the attackers cooperate and minimize their combined
distance from the border before they are captured. They then find the optimal
strategies for the attackers and the defender in this one-defender two-
attacker pursuit-evasion game by solving a set of nonlinear equations. The
cooperative strategy discussed in this paper provides an important
coordination approach for less capable (perhaps slower) agents when they are
tasked to carry out a mission.

He et al (He and Su, 2019) consider the radar counter-measure problem in a
multistatic radar network, where a game-theoretic formulation of joint power
allocation and beamforming is studied in in the presence of a smart jammer.
The goal of each radar in this network is to meet the expected detection
performance of the target while minimizing its total transmit power and
mitigate the potential interferences. On the other hand, the goal of the
jammer is to adjust its own transmit power to interfere the radar so as to
protect the target from detection. First, they study the power allocation game
with strategy sets of each player (radar and jammer) consisting of their
respective transmit powers. For this problem, they proceed to solve the
corresponding optimization problems to work out the best response function for
the radar and the jammer and show the existence and uniqueness of the Nash
equilibria. Next, they consider the joint power allocation and beamformer
design problem in the presence of jammers again as a non-cooperative game and
propose a power allocation and beamforming algorithm which is shown to
converge to its Nash equilibrium point.

McEneaney et al. (McEneaney et al., 2004) investigate the command and control
problem for unmanned air vehicles (UCAVs) against ground-based targets and
defensive units such as surface-to-air missile (SAM) systems. The motivation
for this work arises from the requirement for operations planning and real-
time scheduling in an unmanned air operations scenario. The problem is
modelled as a stochastic game between blue players (UCAVs) and red players
that comprise the SAMs and ground based targets. The game objective may vary,
an example of which for a blue player is to destroy a strategic target while
minimizing damage to itself. The red players, on the other hand, attempt to
inflict maximum damage on the UCAV while protecting themselves from attack by
the UCAVs.

The control strategies for the UCAVs consist of a set of discrete variables
that correspond to the specific target or SAM to attack while that for the
SAMs are to switch their radar “on” or “off”. Note that when the radar is
“on”, the probability of the SAM causing damage to the blue players increases
as does the probability that the blue players inflict damage on the SAM. The
solution to this stochastic game is obtained via dynamic programming and
illustrated on some numerical examples. A main contribution of this work is
the analysis of a risk-sensitive control based approach for stochastic games
under imperfect knowledge. In particular, this approach not only handles noisy
observations due to random noise, but also deals with cases that include an
adversarial component in the observations.

Wei et al. (Wei et al., 2009) have developed a mission decision making system
for multiple uninhabited combat aerial vehicles (UCAVs) working together. The
UCAVs weapons are air-to-air missiles. In the paper a red-UCAV team consisting
of an unmanned fighter-bomber flanked by two UCAVs attempts to strike a blue-
team ground target. The blue-team has its own set of UCAVs that are directed
to defeat the red-team. The success of a given missile against its chosen
threat is determined by the distance between the attacker and threats, their
relative speed, and relative angles. The blue team versus red-team scenario is
represented as a simultaneous normal form game with the strategies for the
team corresponding with allocations of blue team entities against red-team
entities and vice versa. In the paper the payoff for red or blue team is based
upon considering the effectiveness and ineffectiveness of a given allocation,
which is turn is dependant upon the relative geometry between the opposing
team allocation groupings. Dempster-Shafer (D-S) theory is applied where the
D-S combinatorial formula is harnessed to formulate the payoff. These payoffs,
calculated for each strategy, for each team is then placed into bi-matrices
i.e., one for each team and solved using a linear programming optimisation
approach. If an optimal Nash equilibrium point is not present, mixed strategy
approach is applied and solved for. The authors have developed some mission
scenarios with differing geometries and illustrated the use of their game-
theoretical allocation strategy. They use annotated diagrams of entity
geometry containing red and blue teams in proximity to one another to justify
that the allocation strategy determined by their payoff formulation is
satisfactory.

Ma et al. (Ma et al., 2019) have developed a game-theoretic approach to
generate a cooperative occupancy decision-making method for multiple unmanned
aerial vehicle (UAV) teams engaged against each other in a beyond-visual-range
(BVR) air combat confrontation. BVR combat is targeted because of developments
in missile technology enabling long-range engagements. In the paper, the team
on each side first decides the occupancy positions (cubes in Cartesian space)
of its UAV entities followed by selecting targets for each UAV team member to
engage. The goal is for each side to obtain the greatest predominance while
experiencing the smallest possible threat condition. A zero-sum simultaneous
bi-matrix game is applied to express the problem. For a given occupancy of a
UAV, height and distance predominance formulas that factor in range and weapon
minimum/maximum performance criteria are used to generate payoff values for
the utility functions. Since, the scale of the game leads to a explosion in
size (and thus strategy) as the number of occupancy cubes and UAVs for each
team is increased, the authors have chosen to augment the Double Oracle (DO)
algorithm, that was designed to solve large scale zero sum game problems in
earlier works by combining it with a neighbourhood search (DO-NS) algorithm.
Through simulations, the authors illustrate that the results show the DO-NS
algorithm outperforming the DO algorithm in terms of computational time and
solution quality.

Evers et al. (Evers et al., 2013) have developed the design for a Cooperative
Ballistic Missile Defence Game (CBMDG). This game has been designed to support
strategic negotiations between a threatened nation and a possible coalition of
nations that can be integrated together to form a layered BMD to protect the
threatened nation. Through the game, the assignment of ballistic missile
interceptors to the coalition nations is determined which minimizes the
expected number of interceptors required to achieve a desired defence level in
case of an attack from an enemy. The paper shows that the number of
interceptor required by a coalition is smaller compared to utilising
interceptors solely within the threatened nation to engage the BM threat,
hence motivating it to seek out a coalition based layered defence. For the
negotiation, the cost savings that threatened nation gains due to having to
spend less on interceptors by leveraging cooperation via a coalition
corresponds with the bargaining power that the nations in the coalition can
demand as compensation. Using the interceptor savings game formulation devised
in the paper, the savings introduced through each nation participating in the
coalition can be determined and the fair compensation can be allocated to each
aiding participating nation.

Garcia et al. (Garcia et al., 2017) consider an air combat scenario where a
target aircraft that is engaged by an attacking missiles utilises a defending
missiles as a countermeasure to defend itself as it attempts to escape the
attacker by maximising my the distance between itself and the attacker when
the defender reaches as close at it can to the attacking missile. The game is
referred to as an active target defense differential game (ATDDG). In the
paper, the authors extend previous works performed on this three party problem
to develop a closed-form analytical solution for the ATDDG where the Defender
missile is able to defeat the attacker if it enters within a with a capture
circle with specified radius rc > 0\. Additionally, the closed-form optimal
state feedback solution generated in the paper is supposed to work in spite of
the attacker employing an unknown guidance law rather than assuming it is
Propotional Navigation (PN) or pursuit (P). Finally, the authors provide the
set of initial conditions for the target aircraft where its survival is
guaranteed if the target-defender team plays optimally depsite the unknown
guidance law employed by the attacking missile.

Han et al. (Han et al., 2016) present an Integrated Air and Missile Defence
(IADS) problem where surface-to-air (SAM) batteries equipped with interceptor
missiles (IM) engage the attacker missiles (AM) targeted at damaging cities
being protected. The problem is cast as a simplified two-party zero sum
(equally valued targets) game with perfect information and has three stages.
By perfect information, it is implied that both players know what has happened
in the previous stages of the game The three stages correspond with defender
setting up their allocation of SAMs to cities, followed by attacker allocating
their missile salvo against cities and finally the defender in response
allocating interceptor missiles to counter attacker missile salvo. The
simplifying assumptions made in this problem are that there is only one SAM
allocated near a city and only one installed per site. Additionally, no more
than one interceptor is launched against each attacking missile. Additionally,
only one IM can be allocated to one DM, each each SAM has the same number and
type of IMs, AMs are identical and are fired in a single salvo. It is
attempted to be solve the tri-level game using an extensive form game tree,
$\alpha-\beta$ pruning and using the Double Oracle (DO) algorithm for a 6 city
network that needs to be protected. Only DO is a heuristic based approach and
is not guaranteed to find the SPNE. The efficiency with which sub-game Nash
equilibrium is reached by each choice of algorithm is studied. For the game-
tree approach, the conclusion made is that size of the strategy space is
determined to increase to an intractable size because of the combinatorial
nature of the problem. When applying $\alpha-\beta$ pruning, the author
determines that the when scaling to change number to increase SAM batteries,
AMs and IMs, it does not scale well in terms of computation time compared to
DO. However DO does fail to find the SPNE in a small number of instances. The
author preferences DO despite its lack of guarantee to reach the SPNE since it
is shown to not violate monotonicity (increase in payoff) and the solution
quality trends (non exponential increase in computational time) even when
increasing the size of the problem from 6 cities to 55 cities.

The work of Başpınar, Barış et al. (Başpınar and Koyuncu, 2020) focuses on
modeling of air-to-air combat between two unmanned aerial vehicles (UAVs)
using an optimization-based control and game-theoretic approach. In this work,
the ability to provide a controller despite the presence of complex non-linear
dynamics for a UAV is achieved through avoiding integration of the
differential equations involved. It is possible to do this because the
differential flatness theory can be successfully harnessed to describe the
movements of the UAV in the horizontal and vertical planes. In particular,
vehicle motion is expressed in terms of specific variables and their
derivatives through parameterized curves. Then, any trajectory planning, for
moving from one waypoint to another can be solved by determining the smooth
curves satisfying defined conditions in flat output space. Following
determination, all variables involved to describe the smooth curve can be
reverted to the original state/input space. The impact is a speed-up in the
solving of any trajectory optimization through reducing the number of
variables required. Game theory is then harnessed where the aerial combat
between the two UAVs is modelled as a zero-sum game using a minimax approach.
That is the each party tries to maximise its payoff when the opponent plays
its best strategy. Here, The objective is for each UAV is to get directly
behind the other party and within a range threshold in order to satisfy
onboard weapon effective range constraints. In the paper, the authors provide
cost functions associated with degree of being in tail-chase to the target
based on aspect and bearing angles as well as the cost functions associated
with generating a maximum score when the opponent is within some threshold of
the optimum shooting range. The cost functions are multiplied together to
create the total cost. The cost functions are put into a receding horizon
control scheme where the trajectory planning determined through selection of
the controls is performed for a given look-ahead time period where both
players are utilising opposite strategies. Each player considers its opponents
reachable sets within the horizon and uses this to select its choice of
controls to maximise its payoff. This process is repeated every few control
steps. For selecting the control actions, that authors mention the use the
full set of control inputs within the performance envelope rather than subset
(e.g. turn, maintain hading, roll left at particular angle, immelman, split S
or spiral dive) unlike most other works and thus point to generating a more
optimal solution for each player selecting their respective strategies. Two
simulation scenarios are provided with the first being the case where neither
UAV starts off in air-superiority position and then exercises the receding
horizon cost function optimisation to get into tail-chase with its opponent
within optimum firing range. Neither succeed. The authors show the speed, load
factor and bank-angle when applying the controls do not violate bounds during
the flights and that feasible trajectories are generated. For the second
simulation, the UAVs are initially in a tail-chase except not satisfying the
within shooting range criterion. The opponent being chased manevouers to
escape through applying the cost function while the chaser continues chasing.
At the end of the engagement, within shooting range criteria are met and the
target is directly in front but at the sub-optimal aspect, which leads to its
escape. These scenarios are used to demonstrate the validity of the control
strategy developed and thus provide the automatic selection of combat strategy
for two unmanned aerial vehicles engaged in combat against one another.

Casbeer et al. (Casbeer et al., 2018), consider a scenario where an attacker
missile pursuing an unmanned aerial vehicle target is engaged by two defending
missiles launched from entities allied to the target which cooperate with the
target. So, this scenario is an extension from a three party game where there
is only one single defending missile engaging the attacker that cooperates
with the target. The three party game is referred to as the Active Target
Defence Differential Game (ATDDG). Besides computing the optimal strategy for
the players in the extension to ATDDG, the paper is directed to determine the
degree of reduction in vulnerability of the target when it uses two defenders
rather than one. A constrained optimisation problem is formulated to setup the
extension. It is shown that the target through having the choice to cooperate
with either defender can more successfully escape the attacker. Additionally
the presence of two defenders enables the attacker to be more easily
intercepted. When the two defender missiles ae well positioned, both are able
to intercept the attacker.

###  3.4 Papers dealing with Cyber Warfare

Keith et al. (Keith and Ahner, 2021) consider a multi-domain (cyber combined
with air-defence) defence security game problem. There are two players
engaging each other in a zero sum extensive form game, a defender,
representing an integrated air-defence system combined with cyber security
protection and an attacker, capable of unleashing air-to-ground threats
(missiles, bombs) as well cyber-attacks (against IADS network). Here, the
payoff has been selected as the expected loss of life. The defender wants to
minimise this while attacker wants to maximise it. The cyber security game
problem to protect the IADS is nested within the physical security game
problem. The actions of the players correspond with allocations to activate
IADS/cyber security responses nodes corresponding with population centres for
the defender and allocations to attack IADS/associated cyber-security nodes by
the attacker. The realism of the game is increased through provisioning in
imperfect information, which is where the defender and attacker are not fully
aware of vulnerability of nodes (selected randomly by a nature model which
picks them in each game). Additionally, the defender is only able to sense
cyber attacks on nodes probabilistically which has the implication that its
allocation of cyber defence teams to particular IADS is only effective
probabilistically. For the attacker, it can also determine the effectiveness
of its cyber-attacks following physically attacking a node. This work is
directed to advances security game literature by introducing the integrated
domain and multiple periods for agent actions as well enabling continuous
mixed form strategies. Also, the author considers it the first work where
Monte Carlo (MC), discounted and robust counterfactural regret minimisation
based approaches have been compared in security game that are applied and
evaluated following increasing scale of the problem. Initially, for a small-
scale problem the Nash equilibrium (NE) in the form of a sequence-form linear
program is determined for the defender. Then, the problem is gradually scaled
to expand to number of population centres to be defended up to an upper limit.
Then the author, applies the approximate counterfactual regret minimisation
(CFR) algorithm to reduce computation time while preserving the optimality of
the strategy as much as possible. When the scale is further increased a
discounted CFR is introduce which further reduces computation time. The
parameter space of the problem and algorithm are explored to select best
choice of tuning parameters and extract best performance per algorithm. The
impact of limiting the rationality of the players, through introducing bounded
rationality where the players do not necessarily make optimal robust best
response moves. They can only manage approximate robust best response moves. A
robust best response for a player is defined as the compromise between the
completely conservative NE strategy and tje completely aggressive best
response strategy. It means that the strategy of the players has an
exploitability where the opponent can capitalise on non-best responses being
played by them. With respect to a player, the capability of their strategy to
capitalise over opponent is referred to as exploitation. Conversely, the
vulnerability of their strategy with respect to an opponent is referred to as
exploitability. When running all the different algorithms, the results show
the Nash equilibrium solution being the safest strategy since the best moves
are being played which are neither exploitable. The performance charts show
that the robust linear program generates the highest mean utility and highest
exploitation/exploitabiltiy ration while also introducing the maximum
computational time. The Data biased CFR is seen to offer the best trade-ff by
offering a high mean utility, a explotiation to exploitability ration in
favour of exploitation while producing the lowest computational time.

###  3.5 Papers dealing with Space Warfare

In the domain of space warfare, human resources and risks are much less
prevalent here, and instead the focus is more on network strength and
interaction between independent autonomous agents, connected or otherwise.
Ultimately, warfare in these aspects will operate at a pace and in dimensions
far beyond human cognitive capacity. Since the rapidity and complexity of
decisions within engagements will almost certainly outscale military
personnel’s understanding, game theory will take the place of decision makers
as part of the overall software system, and imbue future technology to take
human/ social factors into consideration when making calculations. With a
greater focus on connectivity and networking, the key to success in these
areas relies on effective communication channels and a unanimously shared goal
across the system.

Zhong et al (Zhong et al., 2018) set the ambitious goal of optimising
bandwidth allocation and transmission power across a satellite network. They
base their research on bargaining game theory, and have to achieve compromise
across: interference constraints, Quality of Service requirements, channel
conditions, and transmission and reception capabilities for satellites at
every point in the network. Interference constraints and bandwidth limitation
are the surplus that need to be negotiated in the bargaining game, with each
satellite using different strategies to improve their utility/ share of
resources. This quickly escalates in complexity, with the most important
takeaway from the model being the mapping of a problem to the cooperative
bargaining game framework.

Similarly, Qiao and Zhao (Qiao and Zhao, 2013-12) detail some key issues with
the finite energy availability for the nodes in satellite networks. Their
paper offers a solution through a game theoretical model of a routing
algorithm, and use it to find an equilibrium solution to the uneven network
flow. The model locates certain network hot spots which are reserving a lot of
energy, and takes measures to evenly distribute the resources. This is another
case of a bargaining/ cooperative game across multiple players in a network.

###  3.6 Papers dealing with Target Tracking

Gu et al. (Gu, 2010) study the problem of tracking a moving target using a
sensor network comprising of sensors capable of providing some position-
related target measurements. Each sensor node has a sensor to observe the
target and a processor to estimate its state. While there is some
communication available among sensors, this ability is limited in the sense
that each sensor node can only communicate with its neighbors. The problem is
further compounded by the fact that the target is an intelligent agent capable
of minimizing its detectability by the adversary and thereby has the potential
to increase the tracking error of the tracking agent. Gu et al. (Gu, 2010)
solve this problem within the framework of a zero-sum game, and by minimizing
the tracking agent’s estimation error, a robust minimax filter is developed.
Moreover, in order to handle the limited communication capability of the
sensor nodes, they propose a distributed version of this filter for which each
node only requires information in the form of current measurement and
estimated state from its immediate neighbors. They then demonstrate the
performance of their algorithm on a simulated scenario with an intelligent
target and show that while the standard Kalman filter errors diverge, the
minimax filter which takes into account the adversary’s noise, is able to
significantly outperform the Kalman filter.

Qilong et al.(Sun et al., 2018) similarly address the issue of tracking an
intelligent target, but they model a scenario where the tracking players are
also in pursuit, and the focus is on protecting the target. Additionally, the
target is able to fire a defensive missile at the attacker/ tracker. The
attacker has line of sight of both the target and the defensive missile. The
target’s plan is to allow the tracker to slowly close the distance between
itself and the target, all the while maneuvering to develop and understanding
of how the attacker reacts. When the attacker is close to collision, the
defensive missile is released. The target and the missile then communicate,
use the knowledge of the attacker’s movement patterns, and adhere to an
optimal linear guidance law to destroy the attacker. This was modelled as a
zero sum competitive game between the attacker, the target, and tjhe defensive
missile. However, the paper also focuses on the cooperative game played
between the target and the defensive missile, which is a non-zero sum game.
For them, the payoff is calculated by minimised miss distance (which ideally
equals zero - a collision with the attacker), as well as the control effort
required to guide the defensive missile.

Faruqi (Faruqi, 2017) discusses the general problem of applying differential
game theory to missile guidance. They state that missile trajectory follows
Proportional Navigation (PN), a set of navigational systems using specific
guidance laws for tracking targets. The performance of these systems is
measured by a Linear System Quadratic Performance Index (LQPI). With respect
to differential game theory, they model the missile guidance problem by
representing the missile navigation and trajectory with a set of differential
equations. The general form of this problem is:

$\underline{\dot{x}}_{ij}=F\underline{x}_{ij}(t)+G\underline{u}_{i}(t)-G\underline{u}_{j}(t)$
(4)

and

$J(...)=\frac{1}{2}\underline{x}_{ij}^{T}(t_{f})S\underline{x}_{ij}(t_{f})+\frac{1}{2}\int_{t_{0}}{t_{f}}[\underline{x}_{ij}^{T}QG\underline{u}_{i}+\underline{u}_{i}^{T}R_{i}\underline{u}_{i}+\underline{u}_{j}^{T}R_{j}\underline{u}_{j}]$
(5)

where

$\underline{x}_{ij}(t)=x_{i}(t)-x_{j}(t)$: is the relative state of player $i$
w.r.t player $j$

$\underline{u}_{i}(t)$: is the player $i$ input

$\underline{u}_{j}(t)$: is the player $j$ input

$F)$: is the state coefficient matrix

$G$: is the player input coefficient matrix

$Q)$: is the Performance Index (PI) weightings matrix for current relative
states

$S)$: is the PI weightings matrix for final relative states

$R_{i},R_{j}$: PI weightings matrices on inputs

Faruqi mainly focuses on two player and three player games, while the utility
functions are modelled based on the relative distance vectors between missiles
and targets. Faruqi shows that game theory can be effectively used in missile
guidance tasks involving PNs in modern missiles.

Evers (Evers et al., 2013) on the other hand analyses defence against Theater
Ballistic Missiles (TBMs) using game theory. The proliferation of ballistic
missiles and nuclear technology has important consequences for military
conflict, where the cost of failure can lead to the destruction of entire
cities. It can be hard to pinpoint their launch, since they have a big range
are very powerful, though their payload can vary considerably. In combating
this threat, the defending nation does have the advantage that there is
usually a long flight trajectory, typically split into three phases, during
which the TBM can be intercepted. The Boost phase marks launch and the
majority of the TBM ascent. The end of the boost phase is marked by the burn-
out after which the TBM enters its midcourse phase. This phase is the longest
phase of the flight, and affords the best opportunity for defenders to
intercept the TBM. After the midcourse phase, the TBM enters its terminal
phase from re-entry into the atmosphere. This is the last opportunity for
defenders to intercept the missile. The flight path is illustrated in Figure 4
below.

Figure 4: Flight path of Theater Ballistic Missiles (Evers et al., 2013)

The missile travels a long distance over a reasonably extended flight time.
However, from its physical geographical location, a defending military force
or nation can only apply its resources to defend against it in the termination
phase of flight, where the risk is much higher and the cost of failure is at
its greatest. For this reason, Evers proposes a cooperative strategy, where
the defending nation forms alliances with nations around itself so that they
too can attempt to intercept the TBM in its earlier phases as it travels to
the impact location. Therefore, the game is divided into two smaller games:
the first being a cooperative multiplayer game devising a set of strategies
for the coalition of nations to utilise throughout a TBMs flight path, and the
second is the bargaining and cooperation game between the defending nations
and potential allies.

The basis of the cooperative game to shoot down the TBM is a strategy called
’shoot-look-shoot.’ It relies on a set of $N$ nations attacking the target
using a set of strategies - their interceptor missiles - $M$ each of which has
its own Probability of Interception $P_{i}$. As the TBM flies, each nation $n$
in $N$ will fire its missile(s) $m_{n}$ to intercept the TBM, then look to see
if it has successfully neutralised the threat. If it has failed, the next
interceptor $m_{n+1}$ will be fired. The game’s problem is then reduced to
optimising the probability of interception of the whole set of strategies,
such that it has a feasible likelihood of stopping the TBM. Game theory is
useful here because the principles of cooperative game theory provide a strong
mathematical framework by which an equilibrium solution can be reached for the
set of cooperating nations as a whole.

The second game described is based on negotiating with other nations to form
an alliance. For these other nations, participation in this game is a risk
because it makes them another potential target for the attacking force. To
solve this game, the defending nation must accurately evaluate the interceptor
cost savings, that is, how much there is to gain by preventing the impact of
the TBM. With these savings becoming the surplus that cooperating nations can
share in, potential allies then negotiate over how those savings will be
shared, in proportion to what interception resources they have to offer.

Shinar and Shima (Shinar and Shima, 1996) continues the research of both
pursuit-evasion games and ballistic missile defence with a zero sum game of a
highly maneuverable ballistic missile avoiding an interceptor missile. More
specifically, it ties in an imperfect information element to the game, where
the ballistic missile knows it is being attacked by anti-missle missiles, but
knows little about their trajectory or launching location. In this game, the
two players are the ballistic missile and the interceptor. If the ballistic
missile uses a pure strategy, it will likely be hit because it either (a)
cannot react quickly enough to an opponent it has little information about or
(b) will move predictably and allow for a straightforward trajectory towards
collision. Therefore, the best solution to the game for the ballistic missile
is in mixed strategies.

The mixed strategy will incorporate stochasticity in its flight pattern,
assigning a probability distribution over a set of pure strategies. These pure
strategies will be based on essential navigational heuristics, which will
likely be known or easily discovered by the interceptor. By applying a small
number of rapid and random switches in strategy, the ballistic missile can
maximise its potential for avoiding interception, and force the complexity of
timing calculation back onto the interceptor.

Bogdanovic et al. (Bogdanović et al., 2018) investigate a target selection
problem for multi-target tracking using a game-theoretic perspective. This is
an important problem in a multi-function radar network as it needs to perform
multiple functions such as volume surveillance and fire control simultaneously
while effectively managing the available radar resources to achieve specified
objectives. Thus, in effect they tackle a radar resource management problem in
(Bogdanović et al., 2018) and use non-cooperative game-theoretic approaches to
find optimal solutions for this problem. They formulate the problem in a
framework where each radar is considered to be autonomous; there is no central
control engine informing the radars of their optimal strategies nor is there
any communication among the radars. First they consider a case where all
radars share common interests with respect to the targets and for this problem
they propose a distributed algorithm based on the best response dynamics to
find the Nash equilibria points. This problem is then extended to a more
realistic case of heterogeneous interests among radars and partial target
observability. For this case, they employ the solution concept of correlated
equilibria and propose an efficient distributed algorithm based on regret
matching which is shown to achieve comparable performance to the more
computationally intensive centralized approach.

Finally, Parras et al (Parras et al., 2017) examine a pursuit-evasion game,
involving anti-jamming strategies for Unmanned Aerial Vehicles (UAVs). The
game operates within a continuous time frame and is therefore dynamic, being
solved with the help of differential Game Theory. In somewhat of a culmination
of the aforementioned work, it combines elements of communication
optimisation, sensor evasion and navigation. Given that UAVs require strong
communication for control and the relaying of information, this dependency
makes UAVs incredibly susceptible to jamming attacks. There are multiple
strategies to both jam and anti-jam these communications, and this can be
considered a zero-sum game where the UAV must attempt to optimise its
communication capacity. There is usually uncertainty about the positioning and
movement of the jamming agent, so the game is an imperfect information
differential game. The most important payoff for the UAV is avoiding losing
communication to jamming, and it can do this by maneuvering to make
approximations for the distance of jamming agents, and thus avoiding them.

###  3.7 Papers dealing with Homeland Security

The key components of homeland security addressed by game theory are cyber
security, modelling terrorism threats and defence contracts. With many
applications in computer science (Halpern, 2008; Apt and Grädel, 2011; Shoham,
2008), game theory fits well into cyber security problems. Game theory
combines the strict mathematical rigour of computer science, with more
psychological and philosophical elements like attacker incentives and mindset,
as well as human vulnerabilities in cyber security. Terrorism modelling
similarly benefits the human considerations of game theory, since so much of
their impact is not easily quantifiable, including social, economic and other
spheres affected by terrorist threats, all of which are modled in a game-
theoretic setting. These variables are necessary to understand the potential
of terrorist threats and attacks, and to formulate strategies to best deal
with them. Finally, game theory is suitable for a topic such as contracting
and subcontracting because it captures interactions between selfish
individuals effectively(Piraveenan, 2019; Peldschus, 2008).

The paper by Tom Litti (Tom, 2015) provides a brief summary of how traditional
network security heuristics can be updated with more precision, and how game
theory can help network security engineers design strategies to properly
predict, mitigate, and handle threatened networks. He develops a qualitative
method for valuing the potential risks and costs of attacks on networks. While
a fairly short paper, it does provide some cyber security situational examples
of game theory in practice. For example, he models a two player zero sum game
to represent an attacker and a security system. The nodes have their own
interdependence, vulnerabilities and security assets, but cooperate to
minimise the attacker’s potential to compromise the system. This review is a
great introduction to understanding how cyber security scenarios can be
modeled in a game-theoretic framework.

Jhawar et al (Jhawar et al., 2016) cover a more specific approach of game
theory, namely, Attack-Defence Trees (ADTs) to model scenarios. Here ADTs are
used to map potential attack and defence scenarios on a system which is
equipped with automatic defence protocols. The system needs to be
comprehensive in addressing all possible vulnerabilities, as well as
generating responses which adapt to the aggressively evolving situation of
cyber security attacks. Currently, ADTs only provide upfront system analyses.
Having a reactive strategy to cyber security is important because Attackers
constantly change their attack strategies for offence, and so the split second
judgements that are made in response can make the difference between a
successful and a failed defence of a system.

In (Jhawar et al., 2016) they model a simple game of Attacker and Defender—a
hacker and a security network administrator. The hacker attempts to breach the
integrity of the system, and for each move they make, the administrator will
have devised a reactive strategy based on the attacker’s attempt. The greatest
utility of this method comes from the ability to convert long extensive form
games into a graphical layout for easier understanding and communication.

Gonzalez (Gonzalez, 2013) clearly outlines a standard two player competitive
game of attacker and defender, and then addresses two aspects—Instance Based
Learning Theory and Behavioural game theory. The former compiles cognitive
information into a representation known as an instance. Each instance has a
three part structure of situation, decision and utility—a standard game.
However, it is the interaction between instances which is critical to this
approach. Instance Based Learning Theory uses the learning from the outcome of
each instance to feedback into the situation of the next instance, hopefully
leading to better decisions in later iterations. This is notably similar to
the reinforcement learning technique in machine learning. On the other hand,
Behavioural game theory involves devising a strategy where we assess a variety
of factors, to make more precise long term evaluations of targets and
resources so that utility scores more closely reflect real life value. Once
again, game theory allows us to draw upon social information in cyber security
situations, and assess how that will affect behaviour for both players in the
game. Other such factors include motivational factors of the players,
completeness of information for each player, technological constraints and
inefficiencies between player and technology. Gonzalez stresses that the
importance of accommodating for these factors in any cyber security model will
help make more realistic and useful policies for cyber defence.

One common use of cyber security is in the prevention of terrorism. Hausken et
al (Hausken et al., 2009) cover both terrorism and natural disaster modelling
with some guiding game theory principles. Terrorism and Natural Disasters are
addressed by defending with anti-terrorism, anti-disaster, and anti-all-hazard
investments. Making projections about the likelihoods of each of these
occurrences, defenders must make strategic decisions towards the amount of
investment dedicated to each defence. Costs to be considered in the utility
function of each of these situations include: the intelligence of terrorists
or the randomness/ environmental control of natural disasters; the intensity
of an attack/ disaster; and differences in evaluation of target value between
a terrorist and defender. The game-theoretic approach used in this analysis
captures the defender’s effort in combating each threat. Depending on the
likelihood of each event, combined with the costs of each defensive system,
defenders can derive an optimal division of funds.

Kanturska et al. (Kanturska et al., 2009) write a rigorous inspection of how
transport network reliability can be assessed using game theory, when attack
probability on different locations is unknown. The approach favoured using a
minimax algorithm to distribute risk across multiple paths as long as the
travel costs are small relative to potential losses incurred by an attack.
This would be useful in negating the potential risk associated with safely
escorting a VIP through a city. Game theory is helpful in this situation
because it can analyse network reliability when the attack probabilities are
unknown.

Bier (Bier, 2006) presents useful game theory based suggestions for policy
insights and investment decisions, insurance policy premiums and more. Her
work discusses the weakest link model: a strategy focusing all resources on
preventing the worst utility scenario. This is generally weak in practice, and
she suggests instead to consider hedging those investments with a variety of
defense strategies for different potential targets. The paper looks into
standard terrorist/ defender games, and how security investments can change
the landscape of attacker-defender interactions for the whole community. This
is mainly done through its own scoping study, with one of the key takeaways
being that terrorism mitigation systems can benefit from game theory because
it adds an extra layer of consideration of the terrorists’ response to any
defence mechanisms. Hence, game theory, combined with the holistic approach of
risk and reliability analysis over all systems, can provide a more
comprehensive assessment of all the potential risks and vulnerabilities in
counter terrorism strategies.

Cioaca (Cioacă et al., 2013) investigates a similar question to Bier et al
(Bier et al., 2007-08) as mentioned earlier, but specifically towards aviation
security. For Cioaca, the problem is summarised by targeting both the cost of
security measures for airports and the cost of maintaining a stable and
resilient system of defence. Key strategies are: preventing the attack or
threat entirely (perhaps by removing all access to the target location or
restricting airlines permissions if they fail to adhere to guidelines to
helping); managing the temporal dimension of the attack (the length of the
attack and the subsequent time to recover from it); understanding all direct
and indirect losses (both casualties and related damages like contamination or
infection, compromised secondary security measures, or reputational/ signal
ramifications); and the costs of mitigation, response and recovery.

The model is built around several factors and parameters. The first and most
critical is human losses and material damages $C(H,D|b_{t})$. The most obvious
and direct casualties of an attack, these two losses are highly negative
payoffs in such an attack, and are often greater than any cost to prevent
them. Human losses $H$ are hard to numerically quantify, and therefore when
making proper assessments about resource division, understanding how to
minimise human losses across different groups of humans and in different
dimensions is one of the most difficult aspects of this problem. Material
damages $D$ can be monetarily quantified, but often the run-on effects of such
damages are where the significant losses are incurred. These losses can lead
to total infrastructure shutdowns, ceasing operation of the facility, loss of
jobs of workers and even potentially the slow decline and shutdown of the
facility altogether. The second major factor in this game is the budget
allocated to security systems $b_{t}$. Organisational and managing bodies will
only have a certain amount of resources allocated to a security system $T$.
The next factor is the number $s$ of security system components as this will
be how the budget is composed. Each of these components are partitioned into
one of $n$ separate system sub-components. These components are divided
amongst a number of targets $t_{ik}$, and each one of these targets is
assigned a probability of being attacked $p_{a_{j}}^{t_{k}}(b_{t_{k}})$ and a
value $w_{t_{ik}}$. This can be formally expressed as:

$\min
C(H,D|b_{t})=\sum_{j=1}^{l}\sum_{i=1}^{n}\sum_{k=1}^{s}p_{a_{j}}^{t_{ik}}\cdot
p_{v_{j}}^{t_{ik}}\cdot w_{t_{ik}}$ (6)

For any system of resource division, Ciaoca advocates establishing dimensions
of measuring system resilience. This is divided into static resilience, the
efficient allocation of resources; and dynamic resilience, recovery speed of
the system after the shock, including long term investment in-flows. These two
forms of resilience signify the strength of a system both before, during and
after an attack. With respect to game theory, Ciaoca’s study defines a game
clearly, and incorporates a myriad of complex and interconnected parameters to
outline an effective and calculable game model.

The final paper we discuss on home land security is by Gardener and Moffat
(Gardener and Moffat, 2008). A distinct and unique approach to game theory in
defence, this paper covers the notion of developing a strategy to assess
defence contractors and their potential performance/ ability to meet
contractual obligations. In game theory terms, this problem is a question of
cooperation and defection. It suggests quantitative methods through which
defence departments can more rigorously assess contracts and bidding
scenarios, and therefore wisely select contractors, and protect their budget.
Gardener and Moffat further the understanding of change requirements in
project management at different bidding stages of defence acquisition
projects. The factor they focus on is the conspiracy of optimism, whereby
projects spiral out of control—past budget limits and necessary deadlines—due
to irrational expectations of project progress. Often this ’conspiracy’ is a
drive towards making short term gains, and in fact leads to overall losses.
The bidding game that is played becomes less about the success of the project,
and more about profit capitalisation. The paper presents a guided approach to
understanding how ’cooperative’ games still lead to selfish motivations. Such
is the unreliability of contract outcomes that the game further degenerates to
a two player game against the industry of contractors as a whole.

###  3.8 Papers dealing with Other / Mixed warfare

Zhang and Meherjerdi (Zhang and Mehrjerdi, 2013) investigate how groups of
multiple unmanned vehicles can be used and controlled using game-theoretic
methods in different communication frameworks. Dividing a mission for a single
unmanned vehicle amongst multiple unmanned vehicles yields more effective task
allocation and performance. The separation of labour away from one powerful
single vehicle to several smaller vehicles provides flexibility, adaptability
and improved fault tolerance. The uses of such a network are surveillance,
exploration, satellite clustering, combining Unmanned Underwater Vehicles
(UUVs) and submarines, planes and UAVs, and cooperative robot reconnaissance.
As evident from this list, the strategy is incredibly powerful because if its
ability to combine resources across multiple domains. To be able to operate a
cohesive unit of resources across multiple unmanned systems for combat or
exploration brings an unprecedented level of information and control which
would accelerate military performance tremendously.

Search is a ‘hide and seek’ game with a long history in military applications
(Stone, 1981, 1980; Washburn et al., 2014; Washburn and Karataş, 2015; Jing et
al., 2021; Ristic and Skvortsov, 2020; Ristic et al., 2016, 2020; Hutchinson
et al., 2018; Park and Oh, 2020). The theory was pioneered by Koopman (Stone,
1981) primarily in the military context (search for an escaping target),
followed by developments by Stone at al (Stone, 1980). The applications
include submarine hunting, mine detection, rescue operations, risk for the
first responders, and backtracking of a hazardous source (Stone, 1981, 1980;
Washburn et al., 2014; Washburn and Karataş, 2015; Jing et al., 2021; Ristic
and Skvortsov, 2020; Ristic et al., 2016, 2020; Hutchinson et al., 2018; Park
and Oh, 2020). This framework provides the optimal a priori search plan for a
given detection model, target motion and the cost of the search. The cost of
the search may include time of the search, probability of escape (for a
target), exposure risk (for a searcher), information entropy, or situation
awareness (map of probability of target location). The searcher can be a
moving platforms (UAV, UV, patrolling boats, helicopters, robots, people) and
the targets can be static, movable, blind, silent, or emitting. In this
context the simultaneous localisation and mapping (SLAM) algorithms are often
used (Durrant-Whyte and Bailey, 2006).

The new direction of this research (inspired by some biological applications)
employs the ideas of infotaxis (Vergassola et al., 2007), or real-time control
of the searchers movement based on information (entropy) gain extracted from
the environment (sporadic measurements, forbidden areas, communication between
searchers). This method resembles analogy with navigation of some animals
(Vergassola et al., 2007; Bénichou et al., 2011).

##  4 Classification of papers

In the previous section, it would have been noticeable that many papers have
applicability in more than one domain, and use myriad types of games and model
a range of players. It is therefore imperative to classify the reviewed papers
in a principled manner. To do so, we use the classification scheme already
introduced in Table LABEL:classstructure in section 2.

In particular, the reviewed papers could be classified based on (1) the domain
or type of warfare (2) the type of game or games used in the paper (3) the
nature of players modelled in the paper. The domain can be broadly classified
into Traditional (T) or Modern (M), and more specifically into Land warfare,
Naval warfare, Aerial warfare, Cyber warfare and Space Warfare. The type of
games used can also have a complex classification, based on whether the games
were Non-cooperative or cooperative, sequential or simultaneous, discrete or
continuous, zero sum or non-zero sum. Finally, the games can be two-player,
three-player, or multi-player games. All of this is succinctly captured in
Table LABEL:classstructure.

In Table LABEL:table:MilGTPapersClassified, we provide a self-explanatory,
elaborate classification of all the reviewed papers based on the above-
mentioned classification scheme.

Table 2: Defence Game Theory Classifed using Table LABEL:classstructure Title | Authors |  | Classification Code using
---
Military Defence
Game Theory
Classification System
Game Theoretic analysis of adaptive radar jamming | Bachmann et al | T-N-IW-NCo-Sim-D-ZS-
| Target selection for tracking in multifunction radar
---
networks: Nash and Correlated equilibria
Bogdanovic et al | T-A-IW-NCo-Sim-D-ZS-
| Power allocation game between a radar network
---
and multiple jammers
Deligiannis et al | T-A-IW-NCo-Sim-D-ZS-
| Strategies for defending a coastline against multiple
---
attackers
Garcia et al | T-A-IW-NCo-Sim-D-ZS-
| A game theory approach to target tracking in
---
sensor networks
Gu et al | T-L-IW-NCo-Sim-D-ZS-
| Joint Power allocation and beamforming between
---
a multistaticradar and jammer based on game theory
He et al | T-A-IW-NCo-Sim-D-ZS-
| Game theoretic situation and transmission in
---
unattended ground sensor networks: a correlated
equilibrium approach
Krishnamurthy et al | T-L-(RAW & AMW)-NCo-Sim-D-ZS-
| Network enabled missile deflection: games and
---
correlated equilibrium
Maskery et al | T-N-(RAW & IW)-Co-Sim-D-NZS-
| Decentralised algorithms for netcentric Force
---
Protection against anti-ship missiles
Maskery et al | T–N-(RAW & IW)-NCo-Sim-D-ZS-
Stochastic game approach to air operation | McKeaney et al | T-A-RAW-NCo-Sim-D-ZS-
| Hybrid Game Theory and D-S Evidence Approach
---
to Multiple UCAVs Cooperative Air Combat
Decision
Xingxing Wie et al | T-A-RAW-NCo-Sim-D-ZS-
| Cooperative Occupancy Decision Making of
---
Multiple-UAV in Beyond-Visual Range Air-Combat:
A Game Theory Approach
Yingying Ma et al | T-A-RAW-NCo-Slt-D-ZS-
| A Simple Game-Theoretic Approach to Suppression
---
of Enemy Defenses and Other Time Critical Target
Analyses
Thomas Hamilton et al | T-A-RAW-NCo-Slt-D-ZS-
The Cooperative Ballistic Missile Defence Game | Lanah Evers et al | T-A-RAW-NCo-Slt-D-ZS-
| Cooperative Missile Guidance for Active Defense
---
of Air Vehicles
Eloy Garcia et al | T-A-RAW-NCo-Slt-D-ZS-
| Counterfactual regret minimization for integrated
---
cyber and air defense resource allocation
Andrew Keith et al | M-C-RAW-NCo-Slt-D-ZS-
| Game Theoretic Model for the Optimal Disposition
---
of Integrated Air Defense System Assets
Chan Y. Han et al | T-A-RAW-NCo-Seq-D-ZS-
| Differential game theory and applications
---
to Missile Guidance
Faruqi | T-A-WAW-NCo-Slt-C-ZS-(2P,3P,NP)
| Assessment of Aerial Combat Game via
---
Optimisation-Based Horizon Control
Baspınar, Barıs et al | T-A-RAW-NCo-Slt-D-ZS-
Target Differential Game with Two Defenders | David Casbeer | T-A-WAW-NCo-Slt-C-ZS-3P
The optimal search for weak targets | Koopman, Stone et al | T-N-AMW-NCo-Slt-D-ZS-
| Optimal Strategy for Target Protection with a
---
defender in the pursuit-evasion scenario
Qilong et al |
| Differential game theory with applications to
---
missiles and autonomous systems guidance
Faruqi | T-A-(AMW,WCW)-NCo-Slt-C-ZS-
| A game theoretical interceptor guidance law
---
for ballistic missile defence.
Shinar et al | T-A-(AMW,WCW)-NCo-Slt-C-ZS-
| Pursuit-Evasion games: a tractable framework for
---
anti-jamming in aerial attacks
Parras et al | T-A-AMW-Nco-Slt-C-ZS-
| A simple game theoretic approach to suppression of
---
enemy defences and other time-critical target
analyses
Hamilton et al | T-A-(RAW,WCW)-NCo-Slt-D-ZS-
| Choosing What to Protect: Strategic Defence
---
allocation against an unknown attacker
Bier et al | T-L-RAW-NCo-Slt-D-ZS-
| Considerations on Optimal Resource allocation
---
in avation security
Cioaca | T-L-RAW-NCo-Slt-D-ZS-
| Horsemen and the empty city: A game theoretic
---
examination of deception in Chinese military
legend
Cotten et al | T-L-IW-NCo-Slt-D-ZS-
An Economic Theory of Destabilisation War | Gries et al | T-L-IW-NCo-Seq-D-ZS-
| Game theoretic approach towards network security:
---
A review
Tom Litti | M-C-IW-NCo-Slt-D-ZS-
| Automating cyber defence responses using
---
attack-defence trees and game theory
Jhawar et al | M-C-WCW-NCo-Slt-D-ZS-
| From individual decisions from experience to
---
behavioural game theory
Gonzalez | M-C-IW-NCo-Slt-D-ZS-
| Game Theoretic Approaches to Attack Surface
---
Shifting. Moving Target Defense II
Manadhata |
| Improving reliability through Multi-Path routing
---
and Link Defence: An Application of Game Theory
to Transport
Kanturska et al | M-C-WCW-NCo-Slt-D-ZS-
| Game Theoretic and Reliability models in
---
counter-terrorism and security
Bier et al | M-C-(IW,AMW,WCW)-NCo-Slt-D-ZS-
| Changing behaviours in defence acqusition:
---
a game theory approach
Gardener et al |
| Joint Transmit Power and Bandwidth Allocation
---
for Cognitive Satellite Network based on Bargaining
Game Theory
Zhong et al | M-S-RAW-Co-Slt-D-NZS
| The Research and Simulation of a Satellite Routing
---
Algorithm based on Game Theory
Qiao et al | M-S-WCW-NCo-Slt-D-ZS
| A survey of multiple unmanned vehicles formation
---
control and coordiation. Normal and fault situations
Zhang et al | T-A-WCW-Co-Slt-D-NZS

##  5 Opportunities for Further Research

The reviewed papers haver shown that game theory can provide a unifying
framework to analyse decision making behaviours of agents in defence contexts.
In this section, we briefly discuss a range of potential defence scenarios
where game theory has hitherto not been applied, but would make a useful
contribution if applied in future.

A recent investigation by Defence Advanced Research Projects Agency (DARPA)
into ‘Mosaic warfare’ (DARPA, ) is an example of such a potential future
application of game theory. The idea is mentioned in Zhang (Zhang and
Mehrjerdi, 2013) in the context of operating multiple unmanned vehicles, and
proposes having a lot of smaller cost-efficient resources interconnected in a
’mosaic’ network, such that if several units are destroyed, the overall nature
of the network stands, much like how a mosaic retains its image even if a few
tiles are removed. The goal is that such a vast array of resources with
different capabilities will be able to overwhelm the enemy with its
completeness and complexity. It utilises principles of concurrency to address
the intricacy of the connections in systems of millions of sensors and
actuators. These systems in turn must handle inter-system communication. If
successfully implemented, such a system of systems can provide military
strategists with an overwhelmingly powerful network of weaponry and resources,
which can defeat opponents with the sheer scale and complexity of their
dynamics. This method of combining different parts of the arsenal maximises
the benefits of each, and reintroduces a focus on expendability of resources,
rather than focussing on a few pieces of high value weaponry. This in turn
builds resilience and adaptability into the strategy, a shift away from
heavyweight, single focus attack methods. Since there are a high number of
lower cost resources which need to cooperate for the best outcome, this
scenario could be modelled as a multi-player co-operative game at one level,
while the strife with the opponent(s) can be modelled as a muti-player non-
cooperative game. it cane be noted that the concept of ‘Mosaic warfare’ is
essentially similar to the more general concept of agent-based modelling,
which has already been used in several diverse contexts, ranging from Ageless
Aerospace Vehicle design (Prokopenko et al., 2005; Piraveenan et al., 2005) to
modelling of infectious disease dynamics (Cliff et al., 2018), and game theory
has already been used successfully in some of these contexts (Chang et al.,
2020; Piraveenan et al., 2021).

Another area, within the context of naval warfare, where game theory could be
fruitfully applied in naval susceptibility. in analysing naval susceptibility,
naval vessels factor in their environment, movement patterns and potential
adversary sensors to calculate their risk of detection when moving covertly
(Vaitekunas, 2010). Such an application has overlaps between commonly studied
tracking problems in defence science, as explained in Gu’s (Gu, 2010) which
describes tracking using sensor networks. Such a scenario, as elaborated
before, could be modelled as a two-player non-cooperative differential game,
with detection being the main payoff parameter for each player.

Indeed, ground based tracking problems could benefit from the application of
game theory as well, and papers in this area so far has been few. Ground based
tracking problems may appear in both ground based military applications
(classified here as ground warfare), as well as domestic security and anti-
terrorism applications (classified here as home-front warfare), where the the
ability of security agencies to track individuals’ movements throughout a
society - including their locations, their social network and their
motivations - is a crucial capability(Zalud, 2010). The later scenario could
be modelled as a two player game of pursuit and evasion, or perhaps just
pursuit and reconnaissance with the aim of not revealing the pursuit to the
target, while the target would try to identify pursuit. The amount of
predictive information gained from covert tracking would be the payoff in this
situation.

Modelling cyber warfare is another area where game theory could be applicable,
and again there have been few papers addressing this niche, other than papers
coming primarily from the computer science domain and focus primarily on cyber
security. Kim et al (Kim et al., 2019) describe cyber warfare scenarios which
are integral to all military operations, and highlight the critical role
played by new technological paradigms such as Internet of Things (IoT) and
Brain Computer Interfaces. Defence experts increasingly have a need to predict
and preempt cyber warfare strategies of hostile players. Modelling decision
making scenarios involving Cyber warfare scenarios with novel technological
interfaces is an area where game theory can play a vital role.

##  6 Conclusions

Game theory has proven itself as a versatile and powerful tool for obtaining
vital insights about the decision making processes of agents and players in a
number of fields. In this review paper, we elaborated several scenarios in
which game theory could be applied in defence science and technology, and
presented a succinct review of existing research in this direction. We
introduced an extensive classification for the reviewed papers, based on the
kind of warfare studied, type of games used, and the nature of players. Based
on the observations made, we identified gaps in the literature where game
theory has not been applied extensively so far but has great potential to be
applied fruitfully in future, and we discussed the potential directions in
which defence applications of game theory could expand in the future

The domain based classification was the primary mode of classification that
employed, and in this context we grouped the reviewed papers into papers about
tracking systems, aerial warfare, ground warfare, home front warfare and
space/ other warfare. For each paper considered, the number and roles of
players and game types were defined, and where possible, strategies and payoff
functions were discussed. The goal of this exercise was to identify the most
commonly analysed domains as well as frequently used game types, and use this
knowledge to identify gaps in literature and cross-pollinate ideas across
various domains and types of warfare within the defence context.

As the world deals with emerging challenges to peace and stability, the future
of humanity depends on our ability to solve problems peacefully. While this is
a lofty goal to achieve, the projection of power is decidedly better than an
actual armed conflict which will be very costly at many levels, and game
theory could indeed play a part in deciding some of the ‘soft conflicts’ which
could play out in the coming years and decades. As the focus on defence
strategies and capabilities are likely to increase in the coming years, game
theory can serve as an additional tool that defence scientists can use at many
levels of abstraction to solve deployment, sensing, tracking, and resource
allocation problems.

MP, SA, and AS conceptualised the research. All authors undertook the
literature survey and wrote the manuscript.

###### Acknowledgements.

The authors thank Dr. Farhan Faruqui from the Defence Science and Technology
Group, Australia for many fruitful discussions The authors declare no conflict
of interest. References yes

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