# Dynamic system of strategic games

Madjid Eshaghi Gordji11affiliationmark:  and Gholamreza
Askari11affiliationmark:  11affiliationmark:  Department of Mathematics,
Semnan University P.O. Box 35195-363, Semnan, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Maybe an event can’t be modeled completely through one game but there is more
chance with several games. With emphasis on players’ rationality, we present
new properties of strategic games, which result in production of other games.
Here, a new attitude to modeling will be presented in game theory as dynamic
system of strategic games and its some applications such as analysis of the
clash between the United States and Iran in Iraq will be provided. In this
system with emphasis on players’ rationality, the relationship between
strategic games and explicitly the dynamics present in interactions among
players will be examined. In addition, we introduce a new game called trickery
game. This game shows a good reason for the cunning of some people in everyday
life. Cooperation is a hallmark of human society. In many cases, our study
provides a mechanism to move towards cooperation between players.

###### keywords:

Dynamic system, Game theory, Second Persian Gulf War, Cooperation,
Rationality.

##  1 Introduction

In recent years, number of international conflicts have been increased and
this issue can lead to dangerous political games. Countries prefer to avoid
military confrontation, this problems settled through peaceful negotiation.
Cooperation is only profitable to each government if the other government
reciprocates keohane1986reciprocity ; kydd2000trust . For this purpose, from
game models are used to describe strategic interaction between countries
lejano2012modeling ; micozzi2016empirical . There are aspects to a conflict
which cannot be properly modelled using static analysis. It is necessary to
use a dynamic model for games where there are constraints on players’ actions
as time passes fraser1982dynamic .

In game theory, players are two groups, first class is rational players and
the second class is irrational players. The environment in which rational
players interact is called strategic environment. The essential assumption of
a player’s rationality is that he/she considering his decision probable impact
on other players, makes a decision along with his own benefits
belot2013players ; brams1988game ; van1984relation . Each player thinks about
the game continuation according to his rationality. The environment in which
irrational players interact is called evolutionary environment nowak2006five ;
smith1973lhe .

Study of games can be classified into two groups. First class is the studies
that speak about simple games with a few players and a few possible actions
and way of modeling by one game nash1951non ; nash1950equilibrium ;
selten1975reexamination ; von2007theory . Second class examines relationship
between games. Most of the researches conducted so far are of the first type
and second type research is very rare. Among the most important research of
second type can be Meta games and topology of the $2\times 2$ games. In Meta
games initially a game is selected and using Meta strategies of $n$ type, Meta
games are developed and this structure is used to analyze game
howard1971paradoxes . In topology of the $2\times 2$ games, the games are
classified and using reflections, rotations and moves from one game reach to
another and then the model result is applied in applied problems
robinson2005topology . At the Theory of Moves, Steven Brams write: TOM is by
no means the be-all and end-all of applied game-theoretic modeling. The
dynamic analysis of ordinal games still has gaps that need to be filled and
details that need to be worked out brams1994theory .

At the Theory of Games and Economic Behavior, von Neumann and Morgenstern
write: We repeat most emphatically that our theory is thoroughly static. A
dynamic theory would unquestionably be more complete and therefore preferable
von2007theory . So, here we examine relationship between strategic $2\times 2$
games by presenting a new system called dynamic system of strategic games. The
dynamic system of strategic games is a dynamic model of $2\times 2$ games. The
most important significance of this modeling is consideration of the impact of
these games on each other. In this modeling, games, strategies, and the pair
of rational actions that are created in the heart of a game will be raised and
examined.

As an application of dynamic system of strategic games, we seek modeling of
conflict between United States of America and Islamic Republic of Iran in Iraq
before attack to Iraq until complete withdrawal of U.S. troops in 2011. During
Cold War, Iraq was one of the few allies of Soviet at the critical region of
Middle-East. During 1980s, Baghdad-Washington relations s improved and reached
a very good position. Yet, Iraq invasion of Kuwait in 1991 put it in a full
hostility with America and finally after September 11 attacks, America decided
to overthrow Saddam regime. The US invasion of Iraq and the policy guidelines
that led to it thus reflect a world view whereby the United States is thought
to be both so powerful and so benevolent that it has the ability to spread
democracy throughout the world, which can be achieved by military force if
necessary heinze2008new . The US invasion of Iraq in 2003 is likely to become
one of the most consequential American foreign policy decisions of our time
boyle2004utopianism ; heinze2008new . After the war termination by collapse of
Iraq Baathist regime, America entered into Iraq and took its control. New Iraq
has found a position different from Iraq of Saddam Hossein period in foreign
policy of Iran. Iran considered the circumstances is appropriate to enter in
Iraq in view of being adjacent with this country, collapse of Baathist regime,
help to people suffered from war and establishment of government in Iraq and
entered in Iraq. Both countries also belong to different models of
international relation and political systems. In the following, we introduce
Trickery game. This game is an asymmetric game $2\times 2$ which one of
players can with cunning change his action that reduces the payoff other
player. Game theorists have introduced a variety of games to express the
circumstances of the event.

##  2 Results

### Concepts and terms. Game-maker games

Here we examine the games dynamic system with rational players. Before start
of modeling, we present required concepts and terms. If a game produces other
games, it is called a game-maker game. In general, if the games
$g_{1},g_{2},...,g_{n}$ generate games
$g^{{}^{\prime}}_{1},g^{{}^{\prime}}_{2},...,g^{{}^{\prime}}_{m}$, then
$g_{i}$\- and $g^{{}^{\prime}}_{i}$-s are called producer and produced,
respectively. We call the form of displaying game-maker games as dynamic
system of strategic games.

### Strategy-maker game

We consider strategic $2\times 2$ games with perfect information. If a game
create one or more strategies is will be called strategy-maker game. Each
strategy has at least two pairs of actions. Each pair of actions includes
players’ payoffs. The produced strategies can be dominant strategy, dominated
strategy, weakly dominant strategy and weakly dominated strategy. Therefore,
each dominant action of a player can be called dominant strategy of a player.
If a game doesn’t generate any strategy, the game isn’t strategy maker.

In a strategic game with ordinal preferences, player $i$s action
$a^{{}^{\prime\prime}}_{i}$ strictly dominates her action
$a^{{}^{\prime}}_{i}$ if

$u_{i}(a^{{}^{\prime\prime}}_{i},a_{-i})>u_{i}(a^{{}^{\prime}}_{i},a_{-i})\;\
for\;\ every\;\ a_{-i}\in A_{-i},$

where $u_{i}$ is a payoff function that represents player i’s preferences
osborne2004introduction . If for player $i$ the action
$a^{{}^{\prime\prime}}_{i}$ is preferred to action $a^{{}^{\prime}}_{i}$ per
every choice of action of other players, it is called dominant strategy and is
shown by $S^{j}_{i}$ where $S^{j}_{i}$ shows $j$-th strategy of $i$-th player.

In a strategic game with ordinal preferences, player $i$’s action
$a^{{}^{\prime\prime}}_{i}$ weakly dominates her action $a^{{}^{\prime}}_{i}$
if

$u_{i}(a^{{}^{\prime\prime}}_{i},a_{-i})\geq
u_{i}(a^{{}^{\prime}}_{i},a_{-i})\;\ for\;every\;\ a_{-i}\in A_{-i}$

and

$u_{i}(a^{{}^{\prime\prime}}_{i},a_{-i})>u_{i}(a^{{}^{\prime}}_{i},a_{-i})\;\
for\;\ some\;\ a_{-i}\in A_{-i},$

where $u_{i}$ is a payoff function that represents player i’s preferences
osborne2004introduction . If for player $i$ the action
$a^{{}^{\prime\prime}}_{i}$ is preferred over action $a^{{}^{\prime}}_{i}$ for
each action choice of other players, it is called weakly dominant strategy and
will be represented by $S^{j}_{i}$.

$g_{1}$ |  |
---|---|---
| C | D
C | 3,2 | 2,4
D | 4,1 | 1,3
$g_{2}$ |  |
---|---|---
| C | D
C | 1,-1 | -1,1
D | -1,1 | 1,-1
Figure 1: Strategy maker games. The Bully game $g_{1}$ is a strategy maker
game of order $(2,1)$. Matching Pennies $g_{2}$ is a strategy maker game of
order $(2,0)$.

If a game with $n$ players is strategy maker for $k$ players ($1\leq k\leq n$)
it is called strategy maker game of order $(n,k)$. If a game with $n$ players
isn’t strategy maker it is called strategy maker game of order $(n,0)$. In
other word, we can consider a strategy maker game of order $(n,0)$ as a game
which is not strategy maker. In Fig. 1, we consider row player as player $1$
and column player as player $2$. The Bully game $g_{1}$ is a strategy maker
game of order $(2,1)$. This game generates dominant strategy of defect $D$ and
dominated strategy of cooperation $C$ for player $2$, but this game isn’t a
strategy maker for player $1$. Matching Pennies $g_{2}$ isn’t a strategy maker
game. The game $g_{2}$ is a strategy maker game of order $(2,0)$.

The following example shows how a strategy maker game can produce other games.
Consider two players that play the Prisoner’s Dilemma $g_{1}$ in Fig. 2. Each
player has two actions. Players can choose cooperat action ${{}_{1}C}$ or
defect action ${{}_{1}D}$, where ${{}_{k}C}$ and ${{}_{k}D}$ shows actions of
players from $k$-th game. Players choosing each action obtain a payoff. In the
game $g_{1}$ for player $1$, the dominant strategy ${{}_{1}S^{1}_{1}}$ is
defect and dominated strategy ${{}_{1}S^{2}_{1}}$ is cooperation. In this
game, for player $2$ the dominant strategy ${{}_{1}S^{1}_{2}}$ is defect and
dominated strategy ${{}_{1}S^{2}_{2}}$ is cooperation. In other words, the
Prisoner’s Dilemma is a strategy maker game of order $(2,2)$. Player $1$ can
do the game continuation process by strategy ${{}_{1}S^{1}_{1}}$ or strategy
${{}_{1}S^{2}_{1}}$. Player $2$ can do the game continuation process by
strategy ${{}_{1}S^{1}_{2}}$ or strategy ${{}_{1}S^{2}_{2}}$. Based on the
assumption of players rationality, player $1$ selects strategy
${{}_{1}S^{1}_{1}}$ and player $2$ selects strategy ${{}_{1}S^{1}_{2}}$ to
continue the game. The strategy ${{}_{1}S^{1}_{1}}$ ends to deadlock $g_{2}$,
i.e player $1$ has designed game $g_{2}$ for game continuation. The strategy
${{}_{1}S^{1}_{2}}$ ends to chicken, i.e. player $2$ has designed game $g_{3}$
to continue the game.

$g_{1}$ |  |
---|---|---
| ${{}_{1}C}$ | ${{}_{1}D}$
${{}_{1}C}$ | 3,3 | 1,4
${{}_{1}D}$ | 4,1 | 2,2
${{}_{1}S^{1}_{1}}$ $g_{2}$ |  |
---|---|---
| ${{}_{2}C}$ | ${{}_{2}D}$
${{}_{2}C}$ | 2,2 | 1,4
${{}_{2}D}$ | 4,1 | 3,3
${{}_{1}S^{1}_{2}}$ $g_{3}$ |  |
---|---|---
| ${{}_{3}C}$ | ${{}_{3}D}$
${{}_{3}C}$ | 3,3 | 2,4
${{}_{3}D}$ | 4,2 | 1,1
$({{}_{1}C},{{}_{1}C})_{1,2}$ $g_{4}$ |  |
---|---|---
| ${{}_{4}C}$ | ${{}_{4}D}$
${{}_{4}C}$ | 4,4 | 1,3
${{}_{4}D}$ | 3,2 | 2,1
Figure 2: Game-maker game. Each player has two action ${{}_{k}C}$ or action
${{}_{k}D}$, where $({{}_{k}C},{{}_{k}D})_{i,j}$ shows rational actions pair
of players $i$ and $j$ from $k$-th game and ${{}_{k}S^{j}_{i}}$ shows $j$-th
strategy of player $i$ from $k$-th game.

### Pair of rational actions

Here we introduce pair of rational actions. Players’ preferences on pairs of
rational actions are based on payoffs that they obtain.

######  Definition 1.

(Pair of rational actions) A pair of actions is called rational if at least
hold true in one of the following conditions:

  * •

would be Nash equilibrium;

  * •

pair of actions, would be Pareto dominant for both players over other pairs of
actions;

  * •

for each game that is strategy maker of order $(2,1)$, pairs of rational
actions for one player is responses to dominant strategy or weakly dominant
strategy produced for other player.

In a strategy maker game of order $(2,2)$ where both players have dominant
strategy and the game hasn’t Pareto action pairs over Nash equilibrium, the
Nash equilibrium of game is the only rational actions pair. For example, in
game $g_{2}$ in Fig. 2, the only rational actions pair is
$({{}_{2}D},{{}_{2}D})$.

In Fig. 2, chicken $g_{3}$ is a strategy maker of order $(2,0)$. Pairs of
actions $({{}_{3}C},{{}_{3}C})$, $({{}_{3}C},{{}_{3}D})$ and
$({{}_{3}D},{{}_{3}C})$ are rational for players. In Fig. 2, the Low Conflict
game $g_{4}$ is a strategy maker of order $(2,1)$. Dominant strategy of player
$2$ is cooperation ${{}_{4}C}$. Player $1$ response to this strategy makes
pairs of his rational actions. Therefore, pairs of actions
$({{}_{4}C},{{}_{4}C})$ and $({{}_{4}D},{{}_{4}C})$ are rational for player
$1$. So player $1$ can select one of the pairs of rational actions according
to his rationality to continue the game. In Fig. 1, Matching Pennies $g_{2}$,
the strategy maker is of order $(2,0)$ and also there isn’t Pareto pair of
actions property, on the other hand there isn’t Nash equilibrium. Therefore
pair of the actions haven’t rationality.

Classic dynamic games are ones in which players make decision consecutively,
that is, each player must make his choice after previous player’s choice. The
extensive form is applied to show a classic dynamic game
cressman2014replicator ; harsanyi1986rational ; huang2012emergence . As
mentioned above, game-maker games display form is called dynamic system of
games. Every dynamic system of games includes players set, strategies set, set
of rational actions pairs, system history, node preference and systemic
preference of players.

A tools that can display dynamic system of strategic games is games graph.
Within each node of a graph, there is a strategic game in which players can
make decisions. Each node of this graph can be generator of the next game
through the two following methods and be connected to it:

  1. 1.

strategies,

  2. 2.

pair of rational actions.

Players to move from one node to another nod proceed by selecting strategy or
pair of rational actions. Moving to the next node by the made strategy is
always preference of one of players but continuing game with rational action
pair may be preference of one player or both of them. In Fig. 2, game $g_{1}$
through strategies is connected to games $g_{2}$ and $g_{3}$ and through pair
of rational actions is connected to games $g_{4}$. Strategy
${{}_{1}S^{1}_{1}}$ is preference of player $1$ and strategy
${{}_{1}S^{1}_{2}}$ is preference of player $2$. Pair of rational actions
$({{}_{1}C},{{}_{1}C})$ is preference of two players. In fact, methods $(1)$
and $(2)$ are edges of the concerned graph and each edge is created by
decision and preference of one or two players.

In each node, player can decide whether to move to the next node or not. Nodes
that a player has built through strategy or pair of rational actions, desire
to continue the game are called move node. A number of edges originate from
each move node. Each edge may end to a move node or a final node. Final node
is a node that players have no appetence to continue. If an edge ended to a
final node, play (system) finishes in that edge. If an edge isn’t end to a
final node, games system continues yet. Also it is possible that some edges
end to one node. In the case that all edges end to final nodes, games system
finishes.

Now, we introduce a model of dynamic system of strategic games. Also
definitions and components of system is stated formally.

##  3 Description of the model

Dynamic system of strategic games is a model to examine interaction between
decision makers more exactly. Every decision maker is a player in this model.
To describe this system, we use the graph defined in above. In each node of
graph there is a strategic game with perfect information. A strategic game
includes players set, actions set and preferences on pairs of actions.

Now using graphs, we introduce mathematical model as follows. Graph
$\mathcal{G}$ is binary of $(G,M)$ that first coordinate
$G=\\{g_{1},g_{2},...,g_{n}\\}$ is a finite set of nodes that each node of
this graph is a strategic game. Second coordinate is a finite set named edges
that edges of this graph are produced strategies or pair of rational actions.

Set of all strategies produced by $k$-th game is represented by
${{}_{k}\mathcal{S}}={{}_{k}\mathcal{S}_{1}}\cup{{}_{k}\mathcal{S}_{2}}\cup\emptyset$.
Set of all pairs of actions Players’ in $k$-th game is shown with
${{}_{k}\mathcal{A}}={{}_{k}A_{1}}\times{{}_{k}A_{2}}$. Set of all pairs of
rational actions for player $i$ is shown with
${{}_{k}\mathcal{A}^{{}^{\prime}}_{i}}$ that is a subset of
${{}_{k}\mathcal{A}}$, for all $k\in\\{1,2,...,n\\}$.

Let
$\mathcal{A}={{}_{1}\mathcal{A}}\cup{{}_{2}\mathcal{A}}\cup...\cup{{}_{n}\mathcal{A}}\cup\emptyset$
and
$\mathcal{S}={{}_{1}\mathcal{S}}\cup{{}_{2}\mathcal{S}}\cup...\cup{{}_{n}\mathcal{S}}$
be two sets. The set valued functions, rational actions pair
$\phi^{{}^{\prime}}_{i}:G\to\mathcal{A}$ and strategy maker
$\phi_{i}:G\to\mathcal{S}$ for players $i$’s are defined as follows:

$\phi^{{}^{\prime}}_{i}(g_{k})={{}_{k}\mathcal{A}^{{}^{\prime}}_{i}}=\left\\{\begin{array}[]{rl}\\{({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i}|({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i}\in{{}_{k}\mathcal{A}}\\}\
\ \ \ if~{}g_{k}~{}has\\\ pair~{}of~{}rational~{}actions\\\ \emptyset\ \ \ \
~{}if~{}g_{k}~{}has^{\prime}nt~{}pair~{}of~{}rational~{}actions,\end{array}\right.$

$\phi_{i}(g_{k})={{}_{k}\mathcal{S}_{i}}=\left\\{\begin{array}[]{rl}\\{{{}_{k}S^{j}_{i}}|{{}_{k}S^{j}_{i}}\in{{}_{k}\mathcal{S}}\\}\
\ \ \ \ \ if~{}g_{k}~{}is~{}strategy~{}maker\\\ for~{}player~{}i\\\ \emptyset\
\
~{}if~{}g_{k}~{}is^{\prime}nt~{}strategy~{}maker~{}for~{}player~{}i,\end{array}\right.$

for all $i\in N$ and $j,k\in I=\\{1,2,...,n\\}$, where $g_{k}$ shows $k$-th
game, $({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i}$ shows rational actions pair of
$i$-th player from $k$-th game and ${{}_{k}S^{j}_{i}}$ shows $j$-th strategy
of $i$-th player from $k$-th game.

Every move of system as a member of set $M$ is as follows:

$\displaystyle M:=\Big{\\{}m^{j}_{k}|\;\ m^{j}_{k}={{}_{k}S^{j}_{i}}\;\ or\;\
m^{j}_{k}=({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i}\ \ or$ $\displaystyle\ \ \ \ \ \
\ \ \ \ \ \ \ \ \ m^{j}_{k}=({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i,j}\ \ \forall\;\
{{}_{k}S^{j}_{i}}\in{{}_{k}\mathcal{S}_{i}},$ $\displaystyle\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ ({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i}\in{{}_{k}\mathcal{A}_{i}},\ \
({{}_{k}a_{i}},{{}_{k}a_{-i}})_{j}\in{{}_{k}\mathcal{A}_{j}}\Big{\\}},$

where $m^{j}_{k}$ shows $j$-th move of $k$-th game and
$({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i,j}$ shows the pair of rational action
selected by players $i$ and $j$ of $k$-th game. Players’ move function
$\varphi_{i}:M\to G^{2}$ and $\varphi_{i,j}:M\to G^{2}\cup\emptyset$ with
$\varphi_{i,j}({{}_{k}S^{j}_{i}})=\emptyset$ is defined as following:

$\varphi_{i}(m^{j}_{k})=\left\\{\begin{array}[]{rl}(g_{k},g_{p})=g_{k}g_{p}&~{}~{}if~{}~{}m^{j}_{k}={{}_{k}S^{j}_{i}}\\\
(g_{k},g_{q})=g_{k}g_{q}&~{}~{}if~{}~{}m^{j}_{k}=({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i},\end{array}\right.$
$\varphi_{i,j}(m^{j}_{k})=\left\\{\begin{array}[]{rl}\emptyset&~{}~{}if~{}~{}~{}~{}m^{j}_{k}={{}_{k}S^{j}_{i}}\\\
(g_{k},g_{s})=g_{k}g_{s}&~{}~{}if~{}~{}~{}~{}m^{j}_{k}=({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i,j}.\end{array}\right.$

The above function shows by what move two play nodes have been connected to
each other by one or both players. Consequently, it can be said that in move
$m^{j}_{k}={{}_{k}S^{j}_{i}}$, nodes $g_{k}$ and $g_{p}$ have been connected
through the strategy selected by player $i$ to each other. In move
$m^{j}_{k}=({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i}$ the nodes $g_{k}$ and $g_{q}$
have been connected by pair of rational action selected by player $i$ to each
other. In move $m^{j}_{k}=({{}_{k}a_{i}},{{}_{k}a_{-i}})_{i,j}$ the nodes
$g_{k}$ and $g_{s}$ have been connected through a pair of rational actions
selected by players $i$ and $j$ to each other.

Consider that $H$ is a set including all series (finite and infinite) that
hold true in the following conditions:

  1. 1.

$\emptyset$ is member of $H$.

  2. 2.

Sequence $\big{\\{}m^{j}_{i},\\{g_{k},m^{j}_{k}\\}\big{\\}}_{i,j,k\in I}$ for
all $i,j,k\in\\{1,2,...,n\\}$, is a member of $H$. Each member of $H$ is
called a history and is represented by $h$.

  3. 3.

History $h=\big{\\{}m^{j}_{i},\\{g_{k},m^{j}_{k}\\}\big{\\}}_{i,j,k\in I}$ is
called final history if it is infinite or there isn’t $g_{k+1}$ that is a
member of h.

The set $H$ is called system history. In Fig. 2, the system history is as
follows:

$\displaystyle
H=\Big{\\{}\emptyset,\big{\\{}g_{1},{{}_{1}S^{1}_{1}},{{}_{1}S^{1}_{2}},({{}_{1}C},{{}_{1}C})_{1,2}\big{\\}},\big{\\{}{{}_{1}S^{1}_{1}},\\{g_{2}\\}\big{\\}},$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
\big{\\{}{{}_{1}S^{1}_{2}},\\{g_{3}\\}\big{\\}},\big{\\{}({{}_{1}C},{{}_{1}C})_{1,2},\\{g_{4}\\}\big{\\}}\Big{\\}}.$

Preferences of each node of a games system that are exactly the same
preferences on the pairs of a strategic game actions are called node
preferences or tactical preferences. Preferences on strategies set or set of
rational actions pair of a game is called systemic preferences or strategic
preferences.

######  Definition 2.

(Dynamic system of strategic games) A dynamic system of strategic games with
perfect information including:

  * •

a set of players

  * •

for each player, a set of strategies

  * •

for each player, a set of rational actions pair

  * •

system history

  * •

node preferences (tactical preferences) on set of all actions pairs

  * •

systemic preferences (strategic preferences) on strategies or pairs of
rational actions.

In dynamic system of strategic games, players using conditions of producer
game and generated strategies and pairs of rational actions decide what move
the do along with their benefits and what game they design and where they
stand. Also this system allows players to select among strategies and rational
actions pair which result in his most benefits based on their abilities and
future conditions using available information, according to their rationality
and strategic preferences. Hence, players can agree with each other on the
next move and choose a move that favors all or choose a move according to
personal benefits. Players can choose several moves at the same time that may
one has a personal benefit and other has a collective benefit.

In Fig. 2, by starting the first round of negotiations and choosing tactical
preferences in the first node, game $g_{1}$ or first round of negotiations
ends. Game $g_{1}$ is strategy maker of order $(2,2)$, that is, producer of
dominant strategies ${{}_{1}S^{1}_{i}}$ and dominated strategies
${{}_{1}S^{2}_{i}}$ for playe$i$ and has pair of rational actions
$({{}_{1}D},{{}_{1}D})_{1,2}$ that is game Nash equilibrium and pair of
actions $({{}_{1}C},{{}_{1}C})_{1,2}$ that is dominant Pareto compared to game
Nash equilibrium for both players. In other words, game $g_{1}$ provide
players with the above information. Based on players being rational, the
player $i$ to continue game can choose his dominant strategy or pair of
rational actions, Pareto dominant. Players using analysis of game $g_{1}$ in
the first round of negotiations and information obtained from this stage
determine their systemic preferences and predict their motion path. Player $1$
according to dominant strategy ${{}_{1}S^{1}_{1}}$ design deadlock game.
Player 2 according to dominant strategy ${{}_{1}S^{1}_{2}}$ design chicken
game. Players agreeing to choose a pair of rational action
$({{}_{1}C},{{}_{1}C})_{1,2}$ enter in win-win game and wish this process
occur in the next round. Therefore, each player has designed two games to
continue negotiation process in the next stage. Player $i$ uses strategy
${{}_{1}S^{1}_{i}}$ as a believable or unbelievable threat to continuation of
the negotiation process and in the case of not reaching result and leaving
negotiations, they will choose the strategy. Choosing systemic preference of
game $g_{1}$ and moving from this node, players enter in the next nodes and
determine their tactical preferences in the new node. Players along with their
benefits in nodes $g_{2}$, $g_{3}$ and $g_{4}$ prefer an action that has more
income and as much as possible would be along strategic benefits and
preferences. This process of choices is performed in the next stages nodes as
well.

In general, most of the games existing in nature or among mans and human
communities can be modeled by the dynamic system of games. For example, games
dynamic system within uterus that includes games inside uterus before a baby
birth or period of a person’s lifetime with all events can be considered as
games dynamic system. As another example, the diplomatic relationships between
two countries during a certain period of time can be modeled by games dynamic
system. Inside a dynamic system by specifying times and subjects, there is
statics as well. Therefore, the existence world is a combination of both of
them.

In the following example, there are conditions which show how players can with
help of dynamic system of strategic games reach the satisfactory conditions
for cooperation by bargaining from the conditions where there is unwanted
unfair situation and dissatisfaction. For more accurate expression, the row
player is called player $1$ and column player is called player $2$. In the
conditions of example, players can be two countries, wife and husband, two
companies, … and set of players’ actions include cooperation action
${{}_{i}C}$ and non-cooperation action ${{}_{i}D}$.

Consider two players that play Unfair game $g_{1}$ in Fig. 3. Each player has
two actions. Players can choose cooperation action ${{}_{1}C}$ or defect
${{}_{1}D}$. Players by choosing each action obtain a payoff. In game $g_{1}$
dominant strategy ${}_{1}S^{1}_{2}$ for player $2$ is defect and dominated
strategy ${}_{1}S^{2}_{2}$ is cooperation. Game $g_{1}$ isn’t strategy maker
for player $1$. In e other words, game $g_{1}$ is strategy maker of order
$(2,1)$. Player $2$ can continue the process of game by strategy
${}_{1}S^{1}_{2}$ or ${}_{1}S^{2}_{2}$. Nash equilibrium of the game $g_{1}$
is $({{}_{1}C},{{}_{1}D})$. Pair of rational action for player $2$ is
$({{}_{1}C},{{}_{1}D})_{2}$. Given the dominant strategy ${}_{1}S^{1}_{2}$ for
player $2$ produces responses of player $1$ to this strategy pairs of rational
actions. So pairs of rational actions for player $1$ is
$({{}_{1}D},{{}_{1}D})_{1}$ and $({{}_{1}C},{{}_{1}D})_{1}$. Based on players
being rational, player $2$ chooses dominant strategy ${}_{1}S^{1}_{2}$ and
player $1$ chooses pair of rational actions $({{}_{1}C},{{}_{1}D})_{1}$ to
continue the game. As player $2$ wish to maintain his superiority, strategy
${}_{1}S^{1}_{2}$ end to Bluff game and pair of rational actions
$({{}_{1}C},{{}_{1}D})_{1}$ ends to Mixed Harmony game. As a result, game
$g_{1}$ is producer of Mixed Harmony game $g_{2}$ and Bluff game $g_{3}$.

Game $g_{2}$ is a strategy maker of order $(2,2)$. In this game dominant
strategy ${}_{2}S^{1}_{i}$ for player $i$ is cooperation and dominated
strategy ${}_{2}S^{2}_{i}$ is defect. The game Nash equilibrium is
$({{}_{2}C},{{}_{2}C})$. Pareto dominant Nash equilibrium compared to pair of
actions is $({{}_{2}D},{{}_{2}D})$. As a result, the only pair of rational
actions for both players is $({{}_{2}C},{{}_{2}C})_{1,2}$. Based on players
being rational, the player $1$ chooses dominant strategy ${}_{2}S^{1}_{1}$ and
player $2$ chooses pair of rational actions $({{}_{2}C},{{}_{2}C})_{2}$ to
continue the game. As player $1$ desires cooperation, the dominant strategy
${}_{2}S^{1}_{1}$ ends to Pure Harmony game $g_{4}$. Pair of rational actions
$({{}_{2}C},{{}_{2}C})_{2}$ ends to Stag Hunt $g_{5}$. Consequently, game
$g_{2}$ is producer of games $g_{4}$ and $g_{5}$.

Game $g_{3}$ is strategy maker of order $(2,1)$. In this game dominant
strategy ${}_{3}S^{1}_{2}$ for player $2$ is defect and dominated strategy
${}_{3}S^{2}_{2}$ is cooperation. Game $g_{3}$ for player $1$ isn’t strategy
maker. Nash equilibrium of game $g_{3}$ is $({{}_{3}C},{{}_{3}D})$. Pair of
rational actions for player 2 is $({{}_{3}C},{{}_{3}D})_{2}$. Considering
dominant strategy ${}_{3}S^{1}_{2}$ player $2$, produces responses of player
$1$ to this strategy pair of rational actions. So, pairs of rational actions
for player $1$ is $(D_{3},{{}_{3}D})_{1}$ and $({{}_{3}C},{{}_{3}D})_{1}$.
Based on rationality of players, and also player $2$ desires to maintain his
superiority chooses dominant strategy ${}_{3}S^{1}_{2}$ to continue the game.
Dominant strategy ${}_{3}S^{1}_{2}$ ends to game $g_{5}$. As player $1$
desires cooperation in every stage doesn’t choose a motion from this game to
continuation.

Game $g_{4}$ is strategy maker of order $(2,2)$. In this game dominant
strategy ${}_{4}S^{1}_{i}$ for player $i$ is cooperation and his dominated
strategy ${}_{4}S^{2}_{i}$ is defect. Pair of action $({{}_{4}C},{{}_{4}C})$
is Pareto dominant compared to pair of actions $({{}_{4}D},{{}_{4}D})$. Nash
equilibrium of game and the only pair of rational action for both players is
$({{}_{4}C},{{}_{4}C})_{1,2}$.

Game $g_{5}$ produced by dominant strategy ${}_{3}S^{1}_{2}$ of game $g_{3}$
and pair of rational actions $({{}_{2}C},{{}_{2}C})_{2}$ of game $g_{2}$.
Therefore, it can be concluded that player $2$ for the game continuation has
involved in dichotomy between choosing dominant strategy and pair of rational
action that the dichotomy results in choosing of game $g_{5}$ for play
continuation. Game $g_{5}$ is strategy maker of order $(2,0)$. Pair of actions
$(C_{5},C_{5})$ is dominant Pareto compared to pair of actions
$({{}_{5}D},{{}_{5}D})$. Nash equilibrium of game and players’ pairs of
rational actions is $({{}_{5}C},{{}_{5}C})_{1,2}$ and
$({{}_{5}D},{{}_{5}D})_{1,2}$.

Based on players being rational, they conclude from game $g_{4}$ and $g_{5}$
to cooperate with each other. Hence, in game $g_{4}$ pair of rational action
$({{}_{4}C},{{}_{4}C})_{1,2}$ and in game $g_{5}$, they choose pair of
rational action of dominant Pareto $({{}_{5}C},{{}_{5}C})_{1,2}$. As a result,
these two games are producers of the No Conflict game $g_{6}$. In this step,
the players have no appetence to continue. In the following, we obtain
strategies and pairs of rational actions in Fig. 3, for game $g_{1}$, we have:

$\displaystyle\phi_{1}(g_{1})={{}_{1}\mathcal{S}_{1}}=\emptyset$
$\displaystyle\phi_{2}(g_{1})={{}_{1}\mathcal{S}_{2}}=\\{{{}_{1}S^{1}_{2}},{{}_{1}S^{2}_{2}}\\}$
$\displaystyle\phi^{{}^{\prime}}_{1}(g_{1})={{}_{1}\mathcal{A}^{{}^{\prime}}_{1}}=\\{({{}_{1}C},{{}_{1}D)_{1}},{(_{1}D},{{}_{1}D)_{1}}\\}=\\{(2,4),(1,3)\\}$
$\displaystyle\phi^{{}^{\prime}}_{2}(g_{1})={{}_{1}\mathcal{A}^{{}^{\prime}}_{2}}=\\{({{}_{1}C},{{}_{1}D})_{2}\\}=\\{(2,4)\\}$

For game $g_{2}$, we have:

$\displaystyle\phi_{1}(g_{2})={{}_{2}\mathcal{S}_{1}}=\\{{{}_{2}S^{1}_{1}},{{}_{2}S^{2}_{1}}\\}$
$\displaystyle\phi_{2}(g_{2})={{}_{2}\mathcal{S}_{2}}=\\{{{}_{2}S^{1}_{2}},{{}_{2}S^{2}_{2}}\\}$
$\displaystyle\phi^{{}^{\prime}}_{1}(g_{2})={{}_{2}\mathcal{A}^{{}^{\prime}}_{1}}=\\{{(_{2}C},{{}_{2}C)_{1}}\\}=\\{(4,4)\\}$
$\displaystyle\phi^{{}^{\prime}}_{2}(g_{2})={{}_{2}\mathcal{A}^{{}^{\prime}}_{2}}=\\{(_{2}C,{{}_{2}C})_{2}\\}=\\{(4,4)\\}$

For game $g_{5}$ we have:

$\displaystyle\phi_{1}(g_{5})={{}_{5}\mathcal{S}_{1}}=\emptyset$
$\displaystyle\phi_{2}(g_{5})={{}_{5}\mathcal{S}_{2}}=\emptyset$
$\displaystyle\phi^{{}^{\prime}}_{1}(g_{5})={{}_{5}\mathcal{A}^{{}^{\prime}}_{1}}=\\{({{}_{5}C},{{}_{5}C})_{1},{(_{5}D},{{}_{5}D)_{1}}\\}=\\{(4,4),(2,2)\\}$
$\displaystyle\phi^{{}^{\prime}}_{2}(g_{5})={{}_{5}\mathcal{A}^{{}^{\prime}}_{2}}=\\{({{}_{5}C},{{}_{5}C})_{2},(_{5}D,{{}_{5}D})_{2}\\}=\\{(4,4),(2,2)\\}$

Functions of the game move are as follows:

$\displaystyle\varphi_{2}(m^{1}_{1})=\varphi_{2}({{}_{1}S^{1}_{2}})=g_{1}g_{3}$
$\displaystyle\varphi_{1}(m^{1}_{1})=\varphi_{1}({{}_{2}S^{1}_{1}})=g_{2}g_{4}$
$\displaystyle\varphi_{1,2}(m^{2}_{1})=\varphi_{1,2}(({{}_{5}C},{{}_{5}C})_{1,2})=g_{5}g_{6}$

The system history is as follows:

$\displaystyle
H=\Big{\\{}\emptyset,\big{\\{}g_{1},{{}_{1}S^{1}_{2}},({{}_{1}C},{{}_{1}D})_{1}\big{\\}},\big{\\{}({{}_{1}C},{{}_{1}D})_{1},\\{g_{2},{{}_{2}S^{1}_{1}},({{}_{2}C},{{}_{2}D})_{1,2}\\}\big{\\}},\big{\\{}{{}_{1}S^{1}_{2}},\\{g_{3},{{}_{3}S^{1}_{2}}\\}\big{\\}},$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\
\big{\\{}{{}_{2}S^{1}_{1}},\\{g_{4},({{}_{4}C},{{}_{4}C})_{1,2}\\}\big{\\}},\big{\\{}{{}_{3}S^{1}_{2}},({{}_{2}C},{{}_{2}C})_{2},\\{g_{5},({{}_{5}C},{{}_{5}C})_{1,2}\\}\big{\\}},$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\
\big{\\{}({{}_{4}C},{{}_{4}C})_{1,2},({{}_{5}C},{{}_{5}C})_{1,2},\\{g_{6}\\}\big{\\}}\Big{\\}}$

$g_{1}$ |  |
---|---|---
| ${{}_{1}C}$ | ${{}_{1}D}$
${{}_{1}C}$ | 3,1 | 2,4
${{}_{1}D}$ | 4,2 | 1,3
(${{}_{1}C},{{}_{1}D})_{1}$ $g_{2}$ |  |
---|---|---
| ${{}_{2}C}$ | ${{}_{2}D}$
${{}_{2}C}$ | 4,4 | 3,1
${{}_{2}D}$ | 1,3 | 2,2
${}_{1}S^{1}_{2}$ $g_{3}$ |  |
---|---|---
| ${{}_{3}C}$ | ${{}_{3}D}$
${{}_{3}C}$ | 3,3 | 2,4
${{}_{3}D}$ | 4,1 | 1,2
${}_{2}S^{1}_{1}$ $g_{4}$ |  |
---|---|---
| ${{}_{4}C}$ | ${{}_{4}D}$
${{}_{4}C}$ | 4,4 | 3,2
${{}_{4}D}$ | 3,2 | 1,1
(${{}_{2}C},{{}_{2}C})_{2}$${}_{3}S^{1}_{2}$ $g_{5}$ |  |
---|---|---
| ${{}_{5}C}$ | ${{}_{5}D}$
${{}_{5}C}$ | 4,4 | 1,3
${{}_{5}D}$ | 3,1 | 2,2
$({{}_{4}C},{{}_{4}C})_{1,2}$(${{}_{5}C},{{}_{5}C})_{1,2}$ $g_{6}$ |  |
---|---|---
| ${{}_{6}C}$ | ${{}_{6}D}$
${{}_{6}C}$ | 4,4 | 2,3
${{}_{6}D}$ | 3,2 | 1,1
Figure 3: Dynamic system of games which how players can reach the satisfactory
conditions for cooperation

Dynamic system of the above games shows that what games players need to
perform to reach a mutual and satisfactory agreement. In many negotiations
between countries such systems as the above system can designed. From the
above example, it is concluded that Nash equilibrium of a game is a pair of
rational actions but pair of rational actions isn’t necessarily Nash
equilibrium.

##  4 Second Persian Gulf War

America entered in Iraq in the shadow of slogans such as struggle with
terrorism, world peace, granting democracy and freedom. Here, using the
dynamic system of games, we will model the clash between the United States and
Iran at the time of American troop presence in Iraq. To this end, we divide
this time interval into five periods. And in each period, we will examine
static games with complete information that has occurred. The first period
before the start of the attack that is shown in the form of game $g_{1}$. The
second period from March 20, 2003 until late 2004, when each player, based on
his forces and facilities, was trying to fulfill his goals that includes games
$g_{2}$ and $g_{3}$. The third period is from the late 2004 to the end of the
Bush administration, which includes games $g_{4}$ and $g_{5}$. The fourth
period with the arrival of the new administration in America until the end of
2010, that including game $G_{6}$. Eventually, on December 15, 2011, with the
total withdrawal of US forces from Iraq, the system stops which includes game
$g_{7}$. For more information about Second War of Persian Gulf, refer to
references ehteshami2003iran ; fawn2006iraq ; kumar2006media .

Invasion of Iraq or Second Persian Gulf War resulted in production of many
games among countries of Middle-East region and other countries including the
game between two countries, for example, one can refer to Iran and America. We
consider America as row player (player 1) and Iran as column player (player 2)
in Fig. 4. The game between these two countries at the beginning of war was
Anti-Chicken game. In this game, America has two actions: either it wouldn’t
attack ${{}_{1}C}$ or attack to Iraq ${{}_{1}D}$. Iran was able to participate
in the attack and make cooperation with America ${{}_{1}C}$ or wouldn’t
participate in the attack to Iraq and make defect ${{}_{1}D}$. In Anti-Chicken
game, the two players have dominant strategy ${}_{1}S^{1}_{i}$, defection and
dominated strategy ${}_{1}S^{2}_{i}$, cooperation. Nash equilibrium of game is
$({{}_{1}D},{{}_{1}D})$.

These countries with different and conflicting purposes and objectives entered
in the conflict arena. Clearly, every country attempts to realize its most and
maximum objectives; but in the way of achieving goals of each player, there is
other player’s objectives and possibilities. The more number and power of
advocate forces of a country, the more its feasibility of purposes and
objectives. America objectives in Iraq can be considered a part of Middle-East
Strategy of this country. Therefore, America totally pursues a government in
Iraq that firstly, doesn’t prevent keeping security and survival of Israel,
secondly, keeping regular energy flow toward west and thirdly, the aim of
prevention of establishment of a anti-American government and anti-western
khalilzad1998sources . Iran totally pursues a government in Iraq that firstly,
America’s withdrawal from Iraq and its undermining in the case of insisting on
presence in Iraq, secondly, prevention of establishment of a regime opposing
Iran and thirdly, promotion of Shia groups position in Iraq scene.

Hence, players don’t desire to cooperate and continue the game based on their
strategic preferences. Dominant strategy for both players is ${}_{1}S^{1}_{i}$
and the pair of rational actions of game is $({{}_{1}D},{{}_{1}D})_{1,2}$.
Based of players being rational, player $1$ chooses dominant strategy
${}_{1}S^{1}_{1}$ and player 2 chooses dominant strategy ${}_{1}S^{1}_{2}$ to
continue.

Dominant strategy ${}_{1}S^{1}_{1}$ ends to Bluff game $g_{2}$. In $g_{2}$,
America has two actions: either it takes control of Iraq with synergy of Iran,
that is, it makes cooperation ${{}_{2}C}$ with Iran or take control of Iraq
without synergy of Iran and wouldn’t cooperate ${{}_{2}D}$ with Iran. In this
game, Iran has two actions: either it makes cooperation ${{}_{2}C}$ with
America to control Iraq or wouldn’t make cooperation ${{}_{2}D}$ with America.
$g_{2}$ is strategy maker of order $(2,1)$ and producer of dominant strategy
${}_{2}S^{1}_{1}$, defection and dominated strategy ${}_{2}S^{2}_{1}$,
cooperation. This game isn’t strategy maker for player $2$. Players have pairs
of rational actions $({{}_{2}D},{{}_{2}C})_{1,2}$ and
$({{}_{2}D},{{}_{2}D})_{2}$. Based on players being rational, player $1$
chooses dominant strategy ${}_{2}S^{1}_{1}$ to continue and player $2$ chooses
pair of rational actions $({{}_{2}D},{{}_{2}D})_{2}$ to continue.

America War in Iraq in the classic form lasting for three weeks. By Baghdad
collapse and runaway of Saddam and his boys, Saddam’s regime was ruined, but
despite of American’s initial impression, this was a superficial victory. They
entered in Iraq with an attritional war with rest of Baathist Regime forces
and Sonni groups that gradually its intensity and scope was increased.
Volunteer Sonni forces that mostly had associated with Alqaede, entered in
Iraq from other countries. Their attack against Iraq Shiites provoked an
ethnical war in this country and made its situation more complicated
davis2016presidential .

Dominant strategy ${}_{1}S^{1}_{2}$ ends to Bully game $g_{3}$. In $g_{3}$,
Iran has two actions: both makes attempts to reach its objectives and wouldn’t
make cooperation ${{}_{3}D}$ or withdraw from its objectives and cooperation
with America ${{}_{3}C}$. Also, America has two actions: either want synergy
of Iran to control Iraq ${{}_{3}C}$ or wouldn’t choose cooperation of Iran
${{}_{3}D}$. $g_{3}$ is strategy maker of order $(2,1)$ and producer of
dominant strategy of defect ${}_{3}S^{1}_{2}$ and dominated strategy of
cooperation ${}_{3}S^{2}_{2}$. This game for player $1$ isn’t strategy maker.
Players have pairs of rational actions $({{}_{3}C},{{}_{3}D})_{1,2}$ and
$({{}_{3}D},{{}_{3}D})_{1}$. Based on players being rational, player $2$
chooses dominant strategy ${}_{3}S^{1}_{2}$ and player $1$ chooses pair of
rational actions $({{}_{3}C},{{}_{3}D})_{1}$ to continue.

Iraq situation became more deteriorated daily, violence increased and hate of
America was enhanced. America to prevent this situation attempted to reduce
the violence level by holding elections and transferring power to Iraqis
baker2006iraq . In the new strategy, to confront with threats, America
persuaded increasing American forces in Iraq, increase and reinforcement of
Iraqis forces capabilities to establish stability and peace, increasing
participation of Sonni forces in power, pressure on Iran and Syria to reduce
support of groups and preventing foreign forces into Iraq and attempt to
negotiate with Iraq’s neighbor countries baker2006iraq . By increasing
American forces in Iraq, pressure on militant groups was added and America’
attempt to provide an appropriate base in order to participation of Iraqis
tribes in government and the political process of this country and finally,
negotiation with Iran and Syria to reduce support of competing groups within
Iraq yielded and violence began to decline.

Game $g_{4}$ produced through dominant strategy ${}_{2}S^{1}_{1}$ from $g_{2}$
and pair of rational actions $({{}_{3}C},{{}_{3}D})_{1}$ of $g_{3}$.
Therefore, it can be concluded that player $1$ to continue his play has
engaged in dichotomy between choosing dominant strategy and pair of rational
actions that this dichotomy results in choosing Stag Hunt $g_{4}$ to continue.
In $g_{4}$, America has two actions: either it takes control of Iraq, by
adding forces and negotiation with Iran, that is, making cooperation with Iran
${{}_{4}C}$ or doesn’t cooperate with Iran ${{}_{4}D}$ . Also, Iran also has
two actions: either it makes cooperation to control Iraq ${{}_{4}C}$ or
wouldn’t make cooperation with America ${{}_{4}D}$. $g_{4}$ is strategy maker
of order $(2,0)$. Pair of actions $({{}_{4}C},{{}_{4}C})$ is dominant Pareto
compared to pair of actions $({{}_{4}D},{{}_{4}D})$. Nash equilibria and the
game pairs of rational actions are $({{}_{4}C},{{}_{4}C})_{1,2}$ and
$({{}_{4}D},{{}_{4}D})_{1,2}$.

Considering the power and forces had in Iraq, Iran, started to extort from
America to reach its objectives. The Iraq National Parliament election was
held in January 30, 2005 throughout the country. Most of Sonni groups
boycotted it. Shiites Union and Kurds obtained 140 and 75 seats, respectively.
Premiership of Iraq was devoted to Shiites, Presidency office to Kurds and
Parliament administration to Sonni group katzman2006iraq .

Game $g_{5}$ produced through dominant strategy ${}_{3}S^{1}_{2}$ from $g_{3}$
and pair of rational actions $({{}_{2}D},{{}_{2}D})_{2}$ of $g_{2}$. In
Blackmailer game $g_{5}$, Iran has two actions: either intends to prevent
establishment of a government opposing with itself and promotion of Shiite
position and wouldn’t make cooperation ${{}_{5}D}$ or make cooperation
${{}_{5}C}$ with America for establishment of a government aligned with
America. Also, America also has two actions: either it supports establishment
of a aligned government with Iran and makes cooperation to control violence in
Iraq ${{}_{5}C}$ with Iran or oppose an aligned government with Iran and
wouldn’t make cooperation ${{}_{5}D}$. $g_{5}$ is strategy maker of order
$(2,2)$. In this game, dominant strategy ${}_{5}S^{1}_{1}$ for player 1 is
cooperation and dominated strategy ${}_{5}S^{2}_{1}$, is defection. Also for
player $2$, the dominant strategy ${}_{5}S^{1}_{2}$ is defect and dominated
strategy ${}_{5}S^{2}_{2}$ is cooperate. The game Nash equilibrium and the
only pair of rational actions for both players is
$({{}_{5}C},{{}_{5}D})_{1,2}$.

In Iraq, the process of changes in the final years of Bush and the first year
of Obama showed that America’s new strategy was effective in Iraq, violence
has been controlled partly and tensions are abating. Continuation of this
trend provides more ground for US forces withdrawal from Iraq. So, Obama
government and America Congress have explicitly announced that America doesn’t
need establishing permanent military bases in Iraq. According to the reached
agreements and policies announced by the American government, withdrawal of
American military forces until August 2010 is realized and ended until end of
year 2011. In December 15, 2011, America terminated officially its military
presence in Iraq by holding an official ceremony in Baghdad airport located in
Baghdad Green Region, in presence of America then Secretary of Defense.

According to players being rational, pair of rational actions
$({{}_{1}D},{{}_{1}D})_{1,2}$, $({{}_{5}C},{{}_{5}D})_{1,2}$ and
$({{}_{4}C},{{}_{4}C})_{1,2}$ chosen by players ends to Trickery game $g_{6}$.
In other words, $g_{4}$ and $g_{5}$ are producer of $g_{6}$. In $g_{6}$,
America has two actions: either it reduces its forces ${{}_{6}C}$ and
gradually withdraws from Iraq or increases its forces in Iraq ${{}_{6}D}$ and
imports more military equipments into Iraq. Also, Iran has two actions: either
it puts pressure on America in assigning control of Iraq to a new government
${{}_{6}D}$ or cooperates with America ${{}_{6}C}$. The game Nash equilibria
are $({{}_{6}C},{{}_{6}D})$ and $({{}_{6}D},{{}_{6}D})$. In $g_{6}$, weakly
dominant strategy ${}_{6}S^{1}_{1}$ for player $1$ is reducing forces and
weakly dominated strategy ${}_{6}S^{2}_{1}$ is increasing forces. Also, for
player $2$ weakly dominant strategy ${}_{6}S^{1}_{2}$ is to put pressure on
America and weakly dominated strategy ${}_{6}S^{2}_{2}$ is cooperation with
America. The players’ pairs of rational actions are
$({{}_{6}C},{{}_{6}D})_{1,2}$, $({{}_{6}D},{{}_{6}D})_{1,2}$ and
$({{}_{6}C},{{}_{6}C})_{2}$.

Based on rationality of players and strategic preferences, players selected
pair of rational actions $({{}_{6}C},{{}_{6}D})_{1,2}$ that ends to Hegemony
game $g_{7}$. In $g_{7}$, America has two actions: either withdraw its forces
from Iraq ${{}_{7}C}$ or maintain its forces in Iraq ${{}_{7}D}$. Also, Iran
has two actions: either cooperate with America ${{}_{7}C}$ or wouldn’t
cooperate with America ${{}_{7}D}$. The game Nash equilibrium is
$({{}_{7}C},{{}_{7}D})$. In this step, the players have no appetence to
continue. According to our definition, $g_{7}$ is a final node and the system
is completed. Dynamic system of strategic games between Iran and America is
represented by graphs in Fig. 4. History of system is as follows:

$\displaystyle
H=\Big{\\{}\emptyset,\big{\\{}g_{1},{{}_{1}S^{1}_{1}},{{}_{1}S^{1}_{2}}\big{\\}},\big{\\{}{{}_{1}S^{1}_{1}},\\{g_{2},{{}_{2}S^{1}_{1}},({{}_{2}D},{{}_{2}D})_{2}\\}\big{\\}},\big{\\{}{{}_{1}S^{1}_{2}},\\{g_{3},{{}_{3}S^{1}_{2}},({{}_{3}C},{{}_{3}D})_{1}\\}\big{\\}},~{}~{}$
$\displaystyle\;\;\;\;\;\;\;\;\;\
\big{\\{}{{}_{2}S^{1}_{1}},({{}_{3}C},{{}_{3}D})_{1},\\{g_{4},({{}_{4}C},{{}_{4}C})_{1,2}\\}\big{\\}},\big{\\{}{{}_{3}S^{1}_{2}},({{}_{2}D},{{}_{2}D})_{2},\\{g_{5},({{}_{5}C},{{}_{5}D})_{1,2}\\}\big{\\}},$
$\displaystyle\;\;\;\;\;\;\;\;\;\
\big{\\{}({{}_{4}C},{{}_{4}C})_{1,2},({{}_{5}C},{{}_{5}D})_{1,2},({{}_{1}D},{{}_{1}D})_{1,2},\\{g_{6},({{}_{6}C},{{}_{6}D})_{1,2}\\}\big{\\}},$
$\displaystyle\;\;\;\;\;\;\;\;\;\
\big{\\{}({{}_{6}C},{{}_{6}D})_{1,2},\\{g_{7}\\}\big{\\}}\Big{\\}}.$

$g_{1}$ |  |
---|---|---
| ${{}_{1}C}$ | ${{}_{1}D}$
${{}_{1}C}$ | 1,1 | 2,4
${{}_{1}D}$ | 4,2 | 3,3
${}_{1}S^{1}_{1}$ $g_{2}$ |  |
---|---|---
| ${{}_{2}C}$ | ${{}_{2}D}$
${{}_{2}C}$ | 3,3 | 1,4
${{}_{2}D}$ | 4,2 | 2,1
${}_{1}S^{1}_{2}$ $g_{3}$ |  |
---|---|---
| ${{}_{3}C}$ | ${{}_{3}D}$
${{}_{3}C}$ | 3,2 | 2,4
${{}_{3}D}$ | 4,1 | 1,3
${}_{2}S^{1}_{1}$ $g_{4}$ |  |
---|---|---
| ${{}_{4}C}$ | ${{}_{4}D}$
${{}_{4}C}$ | 4,4 | 1,3
${{}_{4}D}$ | 3,1 | 2,2
(${{}_{2}D},{{}_{2}D})_{2}$(${{}_{3}C},{{}_{3}D})_{1}$${}_{3}S^{1}_{2}$ $g_{5}$ |  |
---|---|---
| ${{}_{5}C}$ | ${{}_{5}D}$
${{}_{5}C}$ | 4,2 | 2,4
${{}_{5}D}$ | 3,1 | 1,3
$({{}_{1}D},{{}_{1}D})_{1,2}$$({{}_{4}C},{{}_{4}C})_{1,2}$$({{}_{5}C},{{}_{5}D})_{1,2}$ $g_{6}$ |  |
---|---|---
| ${{}_{6}C}$ | ${{}_{6}D}$
${{}_{6}C}$ | 4,3 | 2,4
${{}_{6}D}$ | 3,1 | 2,1
(${{}_{6}C},{{}_{6}D})_{1,2}$ $g_{7}$ |  |
---|---|---
| ${{}_{7}C}$ | ${{}_{7}D}$
${{}_{7}C}$ | 4,3 | 2,4
${{}_{7}D}$ | 3,1 | 1,2
Figure 4: Dynamic system of games between America and Iran

The above modeling shows that complete withdrawal of American forces from Iraq
lead to more influential of Iran in the region. The Nash equilibrium
$({{}_{6}D},{{}_{6}D})$ in game $g_{6}$ shows if America in this game prefer
tactical preferences over strategic preferences, obtain better result and
conditions to continue this system was changed.

##  5 Trickery game

We introduced a new game $g_{6}$ in Fig. 4, called Trickery game. The trickery
game is a asymmetric game $2\times 2$ that examines difficult conditions of
decision making between players. Consider two companies which compete for
achieving a common source. Two players seek for more benefit from common
source. Therefore, players request for a portion of source considering their
potential and capability that determines bargaining power of players. One of
two players has fewer capability to use from common source. Each player can
choose or cooperation or defect to resolve conflict. So, set of actions of the
players includes cooperation $C$ and defect $D$. We consider row player as
player $1$ and column player as player $2$. player $1$ has fewer capability to
use from common source. The trickery game table is given in Fig. 5.

$G$ |  |
---|---|---
| C | D
C | 4,3 | 2,4
D | 3,1 | 2,1
Figure 5:  Payoff table for the Trickery game

The game has two Nash equilibrium $(C,D)$ and $(D,D)$. We call this game,
Trickery game; because player $1$ can choose cooperation with player $2$ until
the last moment and finally changes his action to defect by trickery. While,
player 1 has weakly dominant action of cooperation, but assuming the choice of
non-cooperation by player 2, player 1 can with cunning change his action that
Reduces the payoff of player 2. This game is strategy maker of order $(2,0)$.
It has three pairs of rational actions $(C,C)$, $(D,D)$ and $(C,D)$.

##  6 Discussion

The majority of results in game theory concern simple games with a few players
and a few possible actions, characterizing them in terms of their equilibria.
Game theory in static state use one game, but dynamic system of strategic
games use several games to model an event. With emphasis on players’
rationality, we present new properties of strategic games, which result in the
dynamics existing in interactions among players. Since the classic theory of
games lacks an explicit treatment of the dynamics of rationadeliberation,
dynamic system of strategic games can be seen, as filling an important lacuna
of classic game theory.

In this study, with a new attitude toward $2\times 2$ games that Nash
equilibrium is one of the most important properties of these games, we
achieved new properties such as strategies generated by a game and pairs of
rational actions. According to this feature, strategic games was divided into
two classes, strategy maker games and games that aren’t strategy maker.
Strategy maker games itself are of two groups: either strategy maker of order
$(2,2)$, that is, the game is strategy maker for both players or strategy
maker of order $(2,1)$, that is, game just is strategy maker for one player
and not for other one. Also, property of pair of rational actions for a game
was raised. A $2\times 2$ game based on payoff of players has one, two, three
or four pairs of rational actions or doesn’t have. Also, games that aren’t
strategy maker itself are of two groups: either has pair of rational actions
or does’nt have. The $2\times 2$ games can be classified based on number of
pair of rational actions as well.

With emphasis on players’ rationality and with help of strategy maker being
property of a game and pairs of rational actions of a game, how new games are
produced was discussed through a game. This choice that is taken place based
on player’s rationality is called the player’s strategic preferences. Player’s
action selection in the game conditions is called tactical preferences.
Depending on present and future conditions, a player may prefer strategic
preferences on tactical preferences and vice versa. Also, this attitude
enables players to find in negotiations and conflicts a solution to reach a
mutual agreement. Therefore, to reach this agreement, we outlined dynamic
system of games with strategic games. In this system players according their
rationality determine the path of achieving agreement. We use graph tools to
display this system. In each node of the graph, there is a strategic $2\times
2$ game and both players are able to decide. Moves or edges of this graph are
generated strategies or pairs of rational actions. The system history includes
games and moves chosen by players. Each player selects a move which has more
benefit for him according to his rationality. These stages continue by moving
from initial node until players reach to agreement in one node that this node
is called final node of system. By cutting periods and environmental
conditions, dynamic system of games can applies for modeling past events.
Also, this system can be used to model present and future conditions. With
this system, relations between players (countries, companies, humans,…) can be
modeled from the first relationship to the last one. For example, e will
examine the clash between the United States and Iran from the time of start of
the invasion of Iraq until the complete withdrawal of U.S. troops using
dynamic system of strategic games. The result show that, whit withdrawal of
American forces from Iraq results in Iran hegemony in the region. Another
application of this system is to relate games and examine their impact on each
other. Moreover, we introduce an asymmetric game $2\times 2$ which one of
players can with cunning change his action that reduces the payoff other
player.

Dynamic system of games says that in most of the cases, player must wait for
other players’ move. The more limitations a game theory use, the more unlikely
its prediction for future is and the less limitations we need to heavy
reasoning and its analysis would be. We believe that maybe complete modeling
of an event through one game wouldn’t be possible but there is more chance
with several games.

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