††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Small time delay approximation in replicator dynamics

Jacek Miȩkisz  Institute of Applied Mathematics and Mechanics, University of
Warsaw, Banacha 2, 02-097 Warsaw, Poland  Javad Mohamadichamgavi  Institute of
Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097
Warsaw, Poland  Raffi Vardanyan  Institute of Applied Mathematics and
Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

###### Abstract

We present a microscopic model of replicator dynamics with strategy-dependent
time delays. In such a model, new players are born from parents who interacted
and received payoffs in the past. In the case of small delays, we use Taylor
expansion to get ordinary differential equations for frequencies of strategies
with time delays as parameters. We apply our technique to get analytic
expressions for interior stationary states in two games: Snowdrift and Stag-
hunt. We show that interior stationary states depend continuously upon time
delays. Our analytic formulas for stationary states approximate well exact
numerical results for small time delays.

Keywords: evolutionary game theory, replicator dynamics, time delays, small-
delay approximation, social dilemmas, Stug-hunt game, Snowdrift game.

##  I Introduction

Many biological and socio-economic processes can be modeled by systems of
interacting entities such as animals in ecology and evolutionary biology, and
people in social systems. It was usually assumed that interactions take place
instantaneously and their effects are immediate. In reality, results of
biological interactions between individuals may appear in the future, and in
social models, individuals or players may act, that is choose appropriate
strategies, on the basis of the information concerning events in the past.

It is well known that time delays may cause oscillations in dynamical systems
[1, 2, 3, 4]. One usually expects that equilibrium of evolving populations,
describing coexisting strategies or behaviors, is asymptotically stable for
small-time delays and for big ones it becomes unstable. Here we discuss
replicator dynamics of populations of interacting individuals playing two-
player games with two strategies and unique interior stationary states. It
describes the time evolution of frequencies of strategies played in the
population [5, 6, 7]. Effects of time delays in replicator dynamics were
discussed in [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In [9], the authors
constructed a model, where individuals are born some time after their parents
played and received payoffs. There were constructed two coupled equations: one
for the frequency of the first strategy and the other one for the size of the
population. It was shown that in such a model, oscillations are not possible,
the original stationary state is globally asymptotically stable for any time
delays.

Here we modify the above model by allowing time delays to depend on strategies
played by individuals (strategy-dependent delays were discussed in [14, 15,
16, 17]). We consider small delays and use Taylor expansion to get ordinary
differential equations for frequencies of strategies with time delays as
parameters. We apply our technique to get analytic expressions for interior
stationary states in two games: Snowdrift and Stag-hunt. We show that interior
stationary states depend continuously upon time delays. Our analytic formulas
for stationary states approximate well exact numerical results obtained in
[13] for small time delays.

In Section 2, we construct replicator dynamics with strategy-dependent time
delays. Section 3 contains a derivation of replicator dynamics for small
delays. In Section 4, we apply our technique in two examples: Snowdrift and
Stag-hunt games. Discussion follows in Section 5.

##  II Replicator dynamics with strategy-dependent time delays

We consider games with two pure strategies; denote them by C and D. In
discrete moments of time, individuals compete in pairwise contests and the
outcome is given by the following payoff matrix:

|  | C | D
---|---|---|---
| C | a | b
U = |  |  |
| D | c | d

where the $i,j$ entry, $i,j=C,D$, is the payoff of the first (row) player when
he plays the strategy $i$ and the second (column) player plays the strategy
$j$. We assume that both players are the same and hence payoffs of the column
player are given by the matrix transposed to U; such games are called
symmetric. From now on we will assume that $a<c$ and $d<b$ or $c<a$ and $b<d$
so there is a unique mixed Nash equilibrium,
$\bar{x}=\frac{{b-d}}{{b-d+c-a}}$, an interior stationary state in the
replicator dynamics, respectively asymptotically stable or unstable one. We
assume that each player interacts with all other ones and receive an average
payoff with respect to the structure of the population, i.e. the proportion of
the population playing each strategy. We interpret payoffs as a number of
offspring that an individual has after a contest, the offspring inherits the
strategy of an ancestor.

We assume that during a very small-time interval of length $\varepsilon$, only
an $\varepsilon$-fraction of the population can manage to pair with partners
and play the game. We assume that players do not get payoffs immediately - new
players are born $\tau$ units of time after parents interacted and received
payoffs. We also assume that time delays are strategy dependent, we denote
them by $\tau_{C}$ and $\tau_{D}$.

Let $p_{i}(t)$, $i=C,D,$ be the number of individuals who play strategies $C$
and $D$ respectively at the time $t$. Then the total number of players will be
$p(t)=p_{C}(t)+p_{D}(t)$ and the fraction of the population playing strategy
$C$ will be $x(t)=\frac{p_{C}(t)}{p(t)}$. Average payoffs which players get
when playing the strategies $C$ and $D$, are given by
$U_{C}(t)=ax(t)+b(1-x(t))$ and $U_{D}(t)=cx(t)+d(1-x(t))$ respectively.

With all these notations and assumptions in mind, we propose the following
equations to describe our model:

$p_{i}(t+\varepsilon)=(1-\varepsilon)p_{i}(t)+\varepsilon
p_{i}(t-\tau_{i})U_{i}(t-\tau_{i}),$ (1)

$i~{}=C,D$. Then for the size of the whole population we get

$p(t+\varepsilon)=(1-\varepsilon)p(t)+\varepsilon\Bigl{(}p_{C}(t-\tau_{C})U_{C}(t-\tau_{C})+p_{D}(t-\tau_{D})U_{D}(t-\tau_{D})\Bigr{)}.$
(2)

We divide (1) by (2) for i = C, obtain the equation for $x(t+\varepsilon)$,
subtract $x(t)$, divide the difference by $\varepsilon$, take the limit
$\varepsilon\rightarrow 0$, and get an equation for the frequency of the first
strategy,

$\frac{dx}{dt}=\frac{p_{C}(t-\tau_{C})U_{C}(t-\tau_{C})(1-x(t))-p_{D}(t-\tau_{D})U_{D}(t-\tau_{D})x(t)}{p(t)}.$
(3)

Let us notice that unlike in the standard replicator dynamics, the above
equation for the frequency of the first strategy is not closed, one needs
equations for populations sizes. From (1) and (2) we get

$\frac{dp_{i}(t)}{dt}=-p_{i}(t)+p_{i}(t-\tau_{i})U_{i}(t-\tau_{i}),i~{}=C,D$
(4)

$\frac{dp(t)}{dt}=-p(t)+p_{C}(t-\tau_{C})U_{C}(t-\tau_{C})+p_{D}(t-\tau_{D})U_{D}(t-\tau_{D}).$
(5)

In [13], a transcendental equation for stationary states of (3) was obtained
and then it was solved numerically for various games. Here we derive an
approximate replicator equation for small time delays which enables us to get
an analytical expression for the interior stationary state.

##  III Small time delays approximation

We begin by presenting (3) in a different form. We insert (4) into (3) and
after some simplifications we get the following equation:

$\frac{dx}{dt}=\frac{1}{p(t)}[\frac{dp_{C}(t)}{dt}(1-x(t))-\frac{dp_{D}(t)}{dt}x(t)].$
(6)

Now we Taylor expand the right part of (4), keep first powers of $\tau_{C}$
and $\tau_{D}$, and get the following equation:

$\frac{dp_{i}(t)}{dt}=-p_{i}(t)+p_{i}(t)U_{i}(t)-\tau_{i}[\frac{dp_{i}(t)}{dt}U_{i}(t)+p_{i}(t)\frac{dU_{i}(t)}{dt}],$
(7)

and hence

$\frac{dp_{C}(t)}{dt}=-p_{C}(t)+(a-b)x(t)p_{C}(t)+bp_{C}(t)-\tau_{C}[\frac{dp_{C}(t)}{dt}((a-b)x(t)+b)+(a-b)p_{C}(t)\frac{dx(t)}{dt}],$
(8)
$\frac{dp_{D}(t)}{dt}=-p_{D}(t)+(c-d)x(t)p_{D}(t)+dp_{D}(t)-\tau_{D}[\frac{dp_{D}(t)}{dt}((c-d)x(t)+d)+(c-d)p_{D}(t)\frac{dx(t)}{dt}].$
(9)

We solve (8) and (9) for derivatives of $p_{C}(t)$ and $p_{D}(t)$,

$\frac{dp_{C}(t)}{dt}=px\frac{-1+b+(a-b)x-(a-b)\tau_{C}\frac{dx}{dt}}{1+\tau_{C}(a-b)x+b\tau_{C}},\hskip
30.0pt$ (10)
$\frac{dp_{D}(t)}{dt}=p(1-x)\frac{-1+d+(c-d)x-(c-d)\tau_{D}\frac{dx}{dt}}{1+\tau_{D}(c-d)x+d\tau_{D}}.$
(11)

Now we insert (10) and (11) into (6), solve it for $\frac{dx}{dt}$ and get one
ordinary differential equation for the frequency of the first strategy with
$\tau_{C}$ and $\tau_{D}$ as parameters,

$\frac{dx}{dt}=\frac{x(1-x)}{K}\left[\frac{-1+b+(a-b)x}{1+\tau_{C}(a-b)x+b\tau_{C}}-\frac{-1+d+(c-d)x}{1+\tau_{D}(c-d)x+d\tau_{D}}\right],$
(12)

where

$K=1-x(1-x)\left[\frac{-(a-b)\tau_{C}}{1+\tau_{C}(a-b)x+b\tau_{C}}+\frac{(c-d)\tau_{D}}{1+\tau_{D}(c-d)x+d\tau_{D}}\right].$

Now it is easy to get an equation for the stationary state,

$\alpha x^{2}+\beta x+\gamma=0,$ (13)

where

$\begin{cases}\alpha=(\tau_{D}-\tau_{C})(a-b)(c-d),\\\
\beta=\tau_{D}[(a-b)d+(b-1)(c-d)]+\tau_{C}[(d-c)b-(d-1)(a-b)]+a-b-c+d,\\\
\gamma=\tau_{D}d(b-1)+\tau_{C}b(1-d)+b-d\end{cases}$

We solve (13) on the interval $[0,1]$ and get a formula for the stationary
state,

$\bar{x}=\frac{-\beta\pm\sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}.$ (14)

We see that frequencies of strategies in the stationary state depend
continuously on time delays, $\tau_{C}$ and $\tau_{D}$, not only on their
difference.

It is easy to see that when $\tau_{C}$ is equal to $\tau_{D}$, then there is
only one solution of (13), namely $\bar{x}=\frac{{b-d}}{{b-d+c-a}}$ as in the
replicator dynamics without time delays.

##  IV Examples

Here we will analyze dependence of stationary states on time delays for
Snowdrift and Stag-hunt game. We will also compare our approximate analytical
results with numerical solutions for stationary states obtained in [13].

###  IV.1 Snowdrift game

Snowdrift game describes interactions between two car drives caught in a snow
blizzard. They can cooperate and clear together the road from snow. However
one of the drivers can defect and wait until the second one will do the job.
However if both of them defect they will never go home. We see that it is
profitable for a driver to choose the strategy different from that of the
other one. That leads to a replicator dynamics with a stable interior state -
a stable coexistence of both strategies.

Snowdrift game is usually parameterized as follows:

|  | C | D
---|---|---|---
| C | b-c/2 | b-c
$U_{1}$ = |  |  |
| D | b | 0

In this matrix strategy $C$ stands for cooperation and $D$ for defection, $b$
stands for the benefit and $c$ is the cost that a cooperator pays, we of
course assume that $b>c$.

For such a game we have:

$\begin{cases}\alpha=(\tau_{D}-\tau_{C})bc/2,\\\
\beta=\tau_{D}b(b-c-1)-\tau_{C}(b(b-c)-c/2)+c/2-b,\\\
\gamma=\tau_{C}(b-c)+b-c\end{cases}$

and for $b=6$ and $c=4$

$\begin{split}\bar{x}=\frac{-(6\tau_{D}-10\tau_{C}-4)-\sqrt{(6\tau_{D}-10\tau_{C}-4)^{2}-48(\tau_{D}-\tau_{C})(2\tau_{C}+2)}}{24(\tau_{D}-\tau_{C})}\end{split}$
(15)

Figure 1: Dependence of the stable interior stationary state on time delays in
the Snowdrift game with $b=6,c=4$; we present the small-time approximation
$\bar{x}$ and the numerical solution of the equation for stationary state
[13]. (a) stationary state as a function of $\tau_{C}$, when $\tau_{D}=0$, (b)
stationary state as a function of $\tau_{D}$, when $\tau_{C}=0$, (c)
stationary state as a function of $\tau$, when $\tau_{D}=2\tau$ and
$\tau_{C}=\tau$, (d) stationary state as a function of $\tau$ when
$\tau_{C}=2\tau$ and $\tau_{D}=\tau$.

In Fig. 1 we show how the stationary state, that is the frequency of the
cooperation, depends on time delays. We compare our small-delay approximation
with numerical results obtained in [13], the agreement is quite good. We see
that the bigger is a time delay of a given strategy, smaller is its proportion
in the population. Let us note that delay in the strategy $D$ affects the
stationary state much more dramatically than delay in the strategy $C$. We see
that for $\tau_{C}=0$, the stable interior stationary state increases as
$\tau_{D}$ increases and it disappears as $\tau_{D}>\tau_{D}^{*}=1/9$. In
general, it follows from (15) that a population consisting of just of
C-players becomes globally asymptotically stable for
$\tau_{D}\geq\tau_{D}^{*}=(10\tau_{C}+1)/9$.

###  IV.2 Stag-hunt game

Stag-hunt game describes the competition between two hunters: if they
coordinate their actions, they get a stag, but if one of them defects, he will
get a hare and the other one will stay with nothing. It follows that it is
profitable for hunters to choose the same strategy. The game belongs to a
class of games with two stable Nash equilibria and an unstable interior point
(a mixed Nash equilibrium) in the replicator dynamics. Here we will consider
the following payoff matrix:

|  | C | D
---|---|---|---
| C | s | 0
$U_{2}$ = |  |  |
| D | h | h

Figure 2: Dependence of the unstable interior stationary state on time delays
in the Stag-hunt game with $s=5,h=3$; we present the small-time approximation
$\bar{x}$ and the numerical solution from [13]. (a) stationary state as a
function of $\tau_{C}$, when $\tau_{D}=0$, (b) stationary state as a function
of $\tau_{D}$, when $\tau_{C}=0$, (c) stationary state as a function of
$\tau$, when $\tau_{D}=2\tau$ and $\tau_{C}=\tau$, and (d) stationary state as
a function of $\tau$ when $\tau_{C}=2\tau$ and $\tau_{D}=\tau$.

Now we get

$\begin{cases}\alpha=\alpha=0,\\\ \beta=\beta=\tau_{D}sh-\tau_{C}s(h-1)+s,\\\
\gamma=\gamma=-\tau_{D}h-h.\end{cases}$

Thus the stationary state is given by

$\bar{x}=\frac{h(\tau_{D}+1)}{s+hs(\tau_{D}-\tau_{C})+s\tau_{C}}.$ (16)

Now we choose $s=5$ and $h=3$. In Fig. 2, we show the dependence of the
stationary state on time delays. Again we see that our small-delay
approximation is quite good. Now the bigger is a time delay of a given
strategy, smaller is its basin of attraction. We see that for $\tau_{C}=0$,
the stable interior stationary state increases as $\tau_{D}$ increases and it
disappears as $\tau_{C}>\tau_{C}^{*}=0.2$. In general, it follows from (16)
that a population consisting of just D-players becomes globally asymptotically
stable for $\tau_{C}\geq\tau_{C}^{*}=\frac{12\tau_{D}+2}{10}$. Delay of the
strategy $C$ affects the stationary state more significantly than delays in
the strategy $D$.

##  V Discussion

Replicator dynamics with strategy-dependent time delays was studied in [13].
The authors derived a transcendental equation for a stationary state and
solved it numerically for various games. Here we introduced a small-delay
approximation for time-delayed differential equations. This enabled us to
approximate the system of replicator equations for the strategy frequency and
the population size by an ordinary differential equation for the strategy
frequency with time delays as parameters. In this way we obtained an analytic
expression for the stationary state. We applied our technique to the Snowdrift
and Stag-hunt game. Our analytic formulas approximate well exact numerical
solutions for small time delays.

We showed that stationary frequencies of strategies depend continuously on
time delays. Moreover, for the Snowdrift game, the frequency of a strategy in
the stationary state is a decreasing function of its delay; for the Stag-hunt
game, a basin of attraction of a strategy is a decreasing function of its
delay. Such results were of course already presented in [13].

It would be interesting to analyze time-delayed replicator dynamics taking
into account stochastic effects resulting from mutations and a random
character of interactions. In the latter case, individuals play with
particular opponents and not against the average strategy like in the standard
replicator dynamics. We would also like to study time-delay effects in
evolutionary games in finite populations. Some results were already presented
in [18].

Acknowledgments: This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 955708. J. Miȩkisz and R. Vardanyan thank the
National Science Centre, Poland, for a financial support under Grant No.
2015/17/B/ST1/00693.

## References

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# A potential mechanism of gas supply-security cooperation based on a game-
theoretic model

Dávid Csercsik  Centre for Economic and Regional Studies,
Tóth Kálmán u. 4., H-1097 Budapest, Hungary
Tel.: +36-1 309 26 52 Fax: +36-1 319 31 36 and
Pázmány Péter Catholic University, Faculty of Information Technology and
Bionics, Práter u. 50/A 1083 Budapest, Hungary
Tel.: +36-1 886 47 00 Fax: +36-1 886 47 24
<EMAIL_ADDRESS>
###### Abstract

The escalating gas-supply crisis in the EU calls for immediate political and
regulatory actions to improve the energy security. Currently, all member
states are aiming to fill their reservoirs independently, while it is not
clear how solidarity will or even could be put into practice in the future,
i.e. how the accumulated reserves of one or more members may be potentially
redistributed to help others in need. In this paper we aim to formalize a
simple game-theoretic model in order to capture the basic features of the
problem, considering the related uncertainty of the future conditions related
to reservoir levels and possible transmission bottlenecks as well. We propose
a mechanism for supply-security related cooperation, which is based on
voluntary participation, and may contribute to the more efficient utilization
of storage capacities if its principles may be later implemented. We
demonstrate the operation of the proposed framework on a simple example and
show that under the assumption of risk-averse participants, the concept
exhibits potential.

##  1 Introduction

###  1.1 Abbreviations and notations used in the paper

Table 1 summarizes the abbreviations and notations used throughout the
mathematical formalisms of the paper.

abbreviation / notation | meaning
---|---
TCRRGU |  transaction-constrained resource-redistribution games under uncertainty
$q^{P}(n)$ |  level (amount) of participation of player $n$
$N$ | number of players / nodes of the network
$M$ | number of edges (pipelines) of the network
$q^{+}(m)$ / $q^{-}(m)$ |  signed maximal capacity of edge $m$ in the positive and negative direction ($q^{+}(m)\geq 0,~{}q^{-}(m)\leq 0$)
$W$ | number of steps in the inverse demand function of players
$p^{c}_{n,w}$ |  price parameter of the $w$-th step of the inverse demand function of player $n$
$q^{c}_{n,w}$ |  quantity parameter of the $w$-th step of the inverse demand function of player $n$
$S$ | number of scenarios
$p_{s}$ |  probability of scenario $s$
$q^{+}_{s}(m)$ / $q^{-}_{s}(m)$ |  signed edge capacities in the case of scenario $s$
$r_{s}(n)$ |  the amount of resources available for player $n$ in the case of scenario $s$
$f_{m}$ |  flow on edge $m$
$c_{n,w}$ |  consumption related to the $w$-th step of the inverse demand function of player $n$
$U^{c}(n)$ |  consumption utility of player $n$
$RCP_{s}$ |  redistribution clearing price in the case of scenario $s$
$\mu_{s}(n)$ |  marginal utility increment of player $n$ in the case of the scenario $s$
$U^{f}(n)$ |  financial utility of player $n$
ES | expected shortfall
Table 1: Abbreviations and notations used in the paper

###  1.2 Gas supply security concerns in the EU

The impacts of the invasion of Ukraine by Russia, mainly through the implied
sanctions and countermeasures [20] are evolving towards a long-term EU-wide
energy crisis. According to the recent report of ENTSOG [22], it is possible
(if not likely) that most of the European gas storages would be depleted
during the winter period and several European countries will not be able to
fill storages during Summer 2023 to levels necessary to ensure security of gas
supply for Winter 2023/2024. One key conclusion of the report is that is
advised to ’ _improve the cooperation possibilities between Member States and
allow for more efficient storage utilization_ ’ in order to improve the
security of gas supply. However, answering the related ’How?’ question is not
straightforward. As it has been recently discussed [14], the lack of a common
external energy security policy does not allow the EU to represent itself as a
single and powerful actor in international gas-trades. Similarly, the
decisions connected to internal energy security policy of the EU are also
mainly in national hands, and since national governments are typically not
willing to give up regulatory rights, it is problematic to even consider a
common gas-security policy aiming at the possibility of centralized need-based
redistribution of resources. There are even EU prime ministers who, to argue
against centralized initiatives, simplify the question in their rhetoric as ’
_the plan (of the EU) is for those who run out of gas to take it away from
those who have it_ ’ [7].

In such a challenging environment, the EU might be motivated to establish
centralized mechanisms based on the optional participation of member states,
which are transparent and attractive enough to convince decision makers about
the benefits of joining such a cooperation. In the current paper, we aim to
formulate a very simplified model of how the fundamental elements of such a
cooperation might look like. In order to do this, we define the class of
transaction-constrained resource-redistribution games under uncertainty
(TCRRGU), and demonstrate the operation of the ideas on an example. The
proposed TCRRGU framework is based on the following key points. The bargaining
process takes place as the first phase of events. In this phase (1) every
player quantitatively defines its gas demand in the form of parametrized
inverse-demand functions, based on which the latter redistribution processes
take place, and (2) every player quantitatively defines its level of
participation ($q^{P}$) in the cooperation mechanism – this quantity will
limit the amount which may be redistributed among other players, but also
limits the quantity the player in question may receive in the redistribution
process if in need. As we will see later, since the expected benefit of
participation with a given quantity in the mechanism strongly depends on the
defined participation quantities of other players, an iterative scheme is
proposed for the definition of the $q^{P}$ values, which potentially allows
the players to reach a Nash equilibrium. The latter redistribution process (in
the second phase) also depends on the accumulated resources and available
transport paths, which are subjects to uncertainty. We assume that the nature
and parameters of uncertainty are known for every player during the
bargaining.

###  1.3 Related literature

While there are several examples for the application of game theoretic methods
for oil and gas related problems (see e.g. [17, 10, 3, 11, 21]), quantitative
methods related to the economics of reservoir-usage are not prevalent.
Although there are models which aim to describe reservoir operation in the
context of economic implications of CO2 storage [23, 18], the literature on
the concept of reservoir sharing is very scarce. There are valuable results,
which consider the sharing of a single reservoir, while taking into account
characteristic technological features of the problem [8, 9], but according to
our current knowledge, no literature results discuss the sharing of multiple
reservoirs in a networked setting under the assumption of uncertainty.

Further preliminary results, which are related to the question of accumulation
and redistribution are the articles discussing risk-pricing related to gas-
storage, as [13]. Regarding the question of redistribution, an asymmetric
hybrid bankruptcy and Nash bargaining model for gas distribution is presented
in [12]. Approaches which aim to capture the simultaneously competitive and
cooperative aspects of gas-related issues are discussed in [19] and [16].

###  1.4 Motivation and aim

Our aim in this article is to provide a simple formal model describing how a
possible centralized reservoir and network-based supply security cooperation
mechanism assuming voluntary participation might look like. According to the
principles of the proposed framework, each participant reports (1) an inverse
demand function, describing its price-flexibility and (2) a value determining
the level of cooperation in the redistribution process. After the resolution
of uncertainty regarding the levels of accumulated gas in reservoirs and
availability of transfer capacities, gas is redistributed among participants
in order to supply those, who need it most according to the reported inverse
demand function. Participants from whom gas is relocated are compensated
according to a redistribution-clearing price. The redistribution process takes
into account the actual transmission constraints of the network and the levels
of cooperation of participants. The level of participation limits the
redistribution quantity of the participant in both directions: No participant
may receive more gas than this quantity, and the quantity which is reallocated
at the expense of the participant, is also limited by this quantity.

We demonstrate the operation of the proposed mechanism via a simple example
and show that under the assumption of risk-averse participants, the concept
shows potential. In addition, we discuss some issues implied by the proposed
framework with regard to its latter implementation possibilities.

##  2 Model

In this section we describe the model used for the description of the supply-
security related accumulation-redistribution process.

###  2.1 Network model

The natural gas network is represented by a directed graph with $N$ nodes and
$M$ edges. Individual nodes are representing the players of the game, with
potentially different parameters (see later). Each edge $m$ is characterized
by direction-dependent capacity values. $q^{+}\in\mathcal{R}^{m}>0$ and
$q^{-}\in\mathcal{R}^{m}<0$ denote the maximal transfer capacity vectors of
edges in the positive and negative direction respectively (the maximal
transferable quantity of edge $m$ in the negative direction is equal to
$-(q^{-}(m))$). This sign convention will allow us later to use a single
variable for the description of the flow on the edge, the sign of which
defines the direction of the flow.

Furthermore, the differentiation of transfer capacity over edge directions
enables to describe limited directionality of pipelines, which may e.g. depend
on the presence of compressor stations in various parts of the pipeline.
Figure 1 depicts such a network, with $N=M=3$, where the edge labels include
the index of the edge ($m$), and the $(q^{-}(m),q^{+}(m))$ values. Nodes are
labelled with their indices ($n$).

Figure 1: Simple example network network. Edges are labelled with their index
$m$, and $q^{-}(m)$ and $q^{+}(m)$ in parentheses.

###  2.2 Consumer demand

We assume that players describe their elasticity of demand via piecewise
constant inverse-demand functions composed of $W$ steps as depicted in Fig. 2
($W=3$). Each piecewise constant part is characterized by two parameters: A
price ($p^{c}$) and a consumption quantity ($q^{c}$). In this formalism
$p^{c}_{n,j}$ denotes the price (per unit) level of the $j$-th step of the
inverse demand function of player $n$, while $q^{c}_{n,j}$ defines the
quantity (width) of the j-th step. The parameters of the demand functions used
in the example and depicted ion Fig. 2 are summarized in Table 2.

Figure 2: Inverse demand functions of consumers. $p^{c}_{1,1}$ | 25 | $p^{c}_{2,1}$ | 20 | $p^{c}_{3,1}$ | 25 | $q^{c}_{1,1}$ | 3 | $q^{c}_{2,1}$ | 5 | $q^{c}_{3,1}$ | 7
---|---|---|---|---|---|---|---|---|---|---|---
$p^{c}_{1,2}$ | 21 | $p^{c}_{2,2}$ | 17 | $p^{c}_{3,2}$ | 18 | $q^{c}_{1,2}$ | 4 | $q^{c}_{2,2}$ | 9 | $q^{c}_{3,2}$ | 4
$p^{c}_{1,3}$ | 13 | $p^{c}_{2,3}$ | 11 | $p^{c}_{3,3}$ | 14 | $q^{c}_{1,3}$ | 5 | $q^{c}_{2,3}$ | 10 | $q^{c}_{3,3}$ | 4
Table 2: Parameters of the inverse demand functions considered in the example.

###  2.3 Uncertainty

We take uncertainty into account on two levels. First, we assume that it is
uncertain, how successful the players are in accumulating resources (i.e.
filling their reservoirs to fully cover their demand). Second we assume that
the state of the pipeline network is also subject to uncertainty in the sense
that external flows (’external’ in the sense that they are not related to the
supply-security cooperation) might limit the redistribution of resources.

To model the uncertainty-related phenomena, we consider a two-stage timeline
($t=0$ and $t=1$). Uncertainty is represented in the model by a finite number
of ’ _states of nature_ ’ or ’ _scenarios_ ’, one of which is randomly
realized at stage 2. The total number of scenarios is equal to $S$. State of
nature $s$ occurs with probability $p_{s}$ at the second stage and
$\sum_{s=1}^{S}p_{s}=1$. This representation of uncertainty modelling is used
in the financial literature discussing risk measures and allocations (see e.g.
[6]). In our model each scenario $s$ is defined by an ordered tuple
$(R_{s},q^{-}_{s},q^{+}_{s})$, where $R_{s}\in\mathcal{R}^{n}$ defines the
amount of resources available for the players in the case of the scenario $s$,
while $q^{-}\leq q^{-}_{s}\leq 0$ and $0\leq q^{+}_{s}\leq q^{+}$ define the
available transfer capacities of edges in the case of the scenario $s$.

In the case of our simple example, let us assume $S=4$ and
$p_{s}=0.25~{}~{}\forall s\in\\{1,...,S\\}$. Furthermore let the scenario
parameters be as described in equations (26) and (27), where $q^{+}_{s}$ and
$q^{-}_{s}$ are describing the available edge capacities in the case of
scenario $s$ in positive and negative directions respectively, and $r_{s}$ is
the vector of available resources (gas amount in the reservoir) in the case of
the scenario $s$ ($r_{s}(n)$ corresponds to the resource of player $n$).

$\displaystyle q^{-}_{1}=\left(\begin{array}[]{c}-7\\\ -3\\\ -4\\\
\end{array}\right)~{}~{}q^{+}_{1}=\left(\begin{array}[]{c}7\\\ 5\\\ 12\\\
\end{array}\right)~{}~{}q^{-}_{2}=\left(\begin{array}[]{c}-5\\\ -12\\\ -7\\\
\end{array}\right)~{}~{}q^{+}_{2}=\left(\begin{array}[]{c}12\\\ 0\\\ 12\\\
\end{array}\right)$ (13) $\displaystyle
q^{-}_{3}=\left(\begin{array}[]{c}-1\\\ -12\\\ -1\\\
\end{array}\right)~{}~{}q^{+}_{3}=\left(\begin{array}[]{c}12\\\ 1\\\ 12\\\
\end{array}\right)~{}~{}q^{-}_{4}=\left(\begin{array}[]{c}-5\\\ -12\\\ -6\\\
\end{array}\right)~{}~{}q^{+}_{4}=\left(\begin{array}[]{c}12\\\ 0\\\ 12\\\
\end{array}\right)$ (26) $\displaystyle
r_{1}=[12~{}~{}12~{}~{}12]~{}~{}~{}r_{2}=[6~{}~{}21~{}~{}13]~{}~{}~{}r_{3}=[10~{}~{}16~{}~{}8]~{}~{}~{}r_{4}=[6~{}~{}17~{}~{}13]$
(27)

###  2.4 The redistribution process

Let us assume that the vector describing the levels of participation, denoted
by $q^{P}$ is defined ($q^{P}(n)$ is the participation level of player $n$) –
later we will return to the question of how these values are determined. The
redistribution process is described by an optimization problem, where the
resources available in the particular scenario ($r_{S}$) are redistributed to
maximize the total utility of gas consumption (i.e. gas is routed to those who
need it most), while considering the participation levels and transmission
constraints of the scenario. The linear program of the redistribution process
in the case of scenario $s$ is described by the formulas (28)-(30).

$\displaystyle\max_{x}Ux~{}~{}~{}\text{w.r.t.}$ (28)
$\displaystyle-q^{P}(n)\leq\sum_{i\in E^{in}_{n}}f_{i}-\sum_{i\in
E^{out}_{n}}f_{i}\leq q^{P}(n)~{}~{}~{}~{}\forall~{}n$ (29)
$\displaystyle\sum_{i\in E^{in}_{n}}f_{i}-\sum_{i\in
E^{out}_{n}}f_{i}-\sum_{w}c_{n,w}+r_{s}(n)=0~{}~{}~{}~{}\forall~{}n$ (30)

The variable vector $x\in\mathcal{R}^{M+nW}$ of the problem is composed as

$x=\left(\begin{array}[]{c}f\\\ c\\\ \end{array}\right).$ (31)

$f\in\mathcal{R}^{M}$ denotes the vector of edge flows
$f=[f_{1},~{}...,f_{M}]^{T}$, where $f_{m}$ is the signed flow of edge $m$
($q^{-}_{s}(m)\leq f_{m}\leq q^{+}_{s}(m)$), and $c\in\mathcal{R}^{nW}$ is the
vector of consumptions $c=[c_{1,1},~{}c_{1,2},~{}...,c_{N,W}]$, where
$c_{n,w}$ is the consumption related to the $w$-th step of the inverse demand
function of player $n$ ($0\leq c_{n,w}\leq q^{c}_{n,w}$).

The objective function described by (28), where $U$ is composed as
$U=[0^{1\times M},~{}p^{c}_{1,1},~{}...,~{}p^{c}_{N,W}]$. Constraint (29)
describes that the difference of the total inflow and outflow of any node is
constrained by the respective participation level ($q^{P}$) from both above
and below for each node $n$. In this formalism, $E^{in}_{n}$ and $E^{out}_{n}$
denotes the set of incoming and outgoing edge indices of node $n$
respectively. The conservation constraint (30) describes that the sum of
inflows minus the sum of outflows must be equal to the consumption minus the
available resources for each node $n$.

###  2.5 Compensation

During the redistribution process (if the outcome is non-trivial, i.e. a
nonzero redistribution takes place), the reserves (thus the consumption
utility) of some participants are decreased, while those to whom the gas is
redistributed gain additional consumption utility. The resulting consumption
utility ($U^{c}$) of each player may be easily derived based on the respective
inverse demand function, and the (resulting) available amount of resource.

In the proposed framework we assume that those players who receive additional
gas during the redistribution have to financially compensate those, who suffer
a decrease in their reservoir levels. The compensation is based on the
redistribution clearing price ($RCP_{s}$ denotes the redistribution clearing
price in the case of scenario $s$). $RCP_{s}$ is calculated as follows. If a
nonzero redistribution takes place, the resulting consumption for at least a
subset of players is not equal to the resources originally available for them.
The marginal utility increment of player $n$ in the case of the scenario $s$
($\mu_{s}(n)$) is defined as the utility increment (or decrement) implied by
the last unit of gas redistributed to (or from) $n$.

To give an example, assuming the inverse demand functions depicted in Fig. 2
and summarized in Table 2, if player 3 had originally 5 units of gas in the
case of scenario $s$ ($r_{s}(3)=5$) and during the redistribution it receives
an additional 5 units ($\sum_{w}c_{3,w}=10$), then $\mu_{s}(3)=18$, since the
last unit of received gas implied an utility increase of 18 units. According
to this, $RCP_{s}$ is defined as described in eq. (32).

$RCP_{s}=\frac{\min_{n:\mu_{s}(n)>0}\mu_{s}(n)+\max_{n:\mu_{s}(n)<0}(-\mu_{s}(n))}{2}$
(32)

The financial utilities of players, denoted by $U^{f}$ is defined by
transactions related to redistribution, cleared on the price of $RCP_{s}$. As
we will see shortly, the sum of the change in the consumption utility due to
redistribution ($\Delta U^{c}$) and the financial utility is always
nonnegative for each player by construction, since gas is redistributed to
those players, who value it the most, according to the reported inverse demand
functions.

###  2.6 Example scenario calculation

Before we proceed with the definition of additional elements of the framework,
connected to the choice of the $q^{P}$ values, to give an example for the
application of the assumptions defined above, let us calculate the outcome of
scenario 1, assuming $q^{P}=[2~{}~{}2~{}~{}4]$. According to eq. (27), the
available resources are equal to 12 units for each of the players in the case
of this scenario. If we solve the optimization problem (28)-(30) taking into
consideration the inverse demand functions defined in table 2 (and depicted in
Figs. 2), and the pipeline capacity constraints defined in eq. (26), we obtain
the solution, in which 2 units of gas from player 1 are redistributed to
player 2, via edge 1 (implying a flow of -2 because of counter-
directionality). This implies that the vector of consumption utilities
($U^{c}$) changes from $[224~{}~{}219~{}~{}261]$ (the reference $U^{c}$ vector
of the scenario, implied by $r_{1}$ without redistribution) to
$[198~{}~{}253~{}~{}261]$ thanks to the redistribution process. In this case
$RCP_{1}=15$, implying $U^{f}=[30~{}~{}-30~{}~{}0]$.

We may calculate the other scenarios similarly. In scenario 2, two units of
gas are redistributed from player 2 to players 1 and 3 (1 unit for each),
implying a resulting $U^{c}=[159~{}~{}308~{}~{}289]$ and
$U^{f}=[-12.5~{}~{}25~{}~{}-12.5]$. Regarding scenario 3, players 1 and 2 both
provide 2 units of gas for player 3, resulting in
$U^{c}=[172~{}~{}253~{}~{}261]$ and $U^{f}=[27~{}~{}27~{}~{}-54]$. Finally, in
scenario 4, two units of gas are redistributed from player 2 to players 1 and
3 (1 unit for each), implying a resulting $U^{c}=[159~{}~{}264~{}~{}289]$ and
$U^{f}=[-12.5~{}~{}25~{}~{}-12.5]$. Table 3 summarizes the utility results for
the various scenarios.

scenario | 1 | 2 | 3 | 4
---|---|---|---|---
$U^{c}(1)$ | 224 (198) | 159 (138) | 172 (198) | 159 (138)
$U^{c}(2)$ | 219 (253) | 308 (330) | 253 (275) | 264 (286)
$U^{c}(3)$ | 261 (261) | 289 (275) | 261 (193) | 289 (275)
$U^{f}(1)$ | 30 | -12.5 | 27 | -12.5
$U^{f}(2)$ | -30 | 25 | 27 | 25
$U^{f}(3)$ | 0 | -12.5 | -54 | -12.5
$RCP$ | 15 | 12.5 | 13.5 | 12.5
Table 3: Resulting consumption and financial utilities and $RCP$ values in the
example. In the rows of $U^{c}$ the original $U^{c}$ values without
redistribution is indicated in parentheses.

###  2.7 Risk measure and aversion

Let us emphasize that the model elements described up to this point are the
principles, which may serve as basis for a voluntary redistribution mechanism
(based on the choice of $q^{P}$ for each player). In the following, we will be
interested in considerations, based on which players may determine their level
of participation ($p^{P}$), and show that assuming risk-averse players, the
framework indeed motivates players for participation.

First, we have to define how players of the game measure the risk in this
context. For this aim, we will use the concept of expected shortfall (ES) [1,
2], which is a coherent measure of risk [4]. The $\alpha$-expected shortfall
is calculated as the expected value of the worst $\alpha~{}\%$ of the
scenarios. Here, we assume that the players consider the value of $U^{c}$ as
basis of their calculations, and in this simple example we use the
$\alpha=25\%$ expected shortfall. In this very case, the ES value of each
player may be calculated as taking the worst $U^{c}$ value over the possible
(4) scenarios.

First, we may compare the ES values of players, without or with cooperation,
assuming $q^{P}=[0~{}~{}0~{}~{}0]$ and $q^{P}=[2~{}~{}2~{}~{}4]$ respectively.
Based on table 3 we may easily conclude, that the ES values are increased with
the cooperation: From 138 to 159, from 219 to 253 and from 193 to 261 in the
case of player 1, 2 and 3 respectively. This increase in the ES values of
players clearly shows that cooperation (with $q^{P}=[2~{}~{}2~{}~{}4]$ in this
case) reduces the risk for the players.

###  2.8 The implied nonzero-sum matrix game

Are the values $q^{P}=[2~{}~{}2~{}~{}4]$ the most efficient choices for
players to reduce risk, i.e. to maximize their ES values in the current
context? In general, the resulting ES value of player $n$ in the case of
choosing $q^{P}(n)$ clearly depends on the choice of $q^{P}$ of other players
(e.g. if other players choose $q^{P}=0$, it is sure that the cooperation will
not bring any benefit for player $i$). Thus, the framework defines a nonzero-
sum matrix game, where the strategy space of the players is given by their
possible choices of $q^{P}$, and their payoff is their respective ES value in
the resulting multi-player strategy space. If we constrain the possible set of
choices for $q^{P}$ for integer values for simplicity, we can also state that
$q^{P}=[2~{}~{}2~{}~{}4]$ is a Nash-equilibrium of this non-cooperative game.

##  3 Discussion

###  3.1 Equilibrium aspects

The reader may ask, how the equilibrium point of the example ha been
determined, or in general, how is it possible to determine equilibrium
point(s) of the implied non-cooperative game. Although the main aims of this
article do not include to deeply discuss the equilibrium properties of the
non-cooperative game class implied by the TCRRGU problem, let us formulate
some observations.

The iterative application of best-response strategies (see e.g. [5])
potentially leads to equilibrium. In the case of the proposed example, the
initial $q^{P}$ vector was determined based on the parameters of the inverse-
demand functions, as $q^{P}(0)=[5~{}~{}10~{}~{}4]$, i.e. all players defined
their initial $q^{P}$ value as the quantity parameter of the last step of
their inverse demand function. Iterating the best response functions in this
case led to an equilibrium after 3 steps.

Even in the case of the proposed simple example, this equilibrium is not
unique. All of the vectors $q^{P}=[1~{}~{}1~{}~{}1]$,
$q^{P}=[1~{}~{}2~{}~{}1]$ $q^{P}=[2~{}~{}2~{}~{}4]$ $q^{P}=[1~{}~{}2~{}~{}3]$
show the equilibrium property, i.e. no player $n$ is able to increase its own
$ES$ value by unilaterally changing $q^{P}(n)$ (if we restrain ourselves on
integer values regarding the level of participation).

###  3.2 On incentive compatibility

Let us return to a key assumption of the proposed framework for some
discussion. The cooperation framework is based on the reported inverse demand
functions, which are used in the objective function of the optimization steps.
It is easy to see that if a player submits an inverse demand function with
high price-parameters, it is likely that the gas will be redistributed for
him/her in most of the scenarios. However, as we have seen the $RCP$ is also
determined by marginal utilities, and such a strategy will likely result in
the increase of the $RCP$, thus it implies more loss for the player through
$U^{f}$ (since the gas is redistributed to him/her, he/she will have to pay
for it). Let us emphasize that this does not mean at all that the proposed
framework necessary motivates players to reveal the parameters of their true
inverse-demand functions – this, and more details related to incentive-
compatibility [15] may be the subject of later studies.

###  3.3 Potential practical implementation

A further simplification of the proposed model is that it has been assumed
that the possible scenarios and their realization probabilities are determined
and known for each player. In practice, the uncertainty is much less defined,
and various players potentially have different beliefs about it, and they
calculate their risk measures and strategies according to these individual
considerations. In other words, an abstract game such as the one proposed will
be never realized in practice, which also implies that the theoretical
properties of the proposed formal model shortly discussed above (equilibrium
aspects and incentive compatibility) would have moderate significance in the
case of a potential realized application. Nevertheless, some elements of the
proposed model and its solution concepts (like reporting of inverse demand
functions, iterative determination of the levels of participation) may
represent useful approaches in the process of designing realistic mechanisms
in the future.

Altogether, our main aim in this article was to show that under textbook-like
simplifying assumptions, such a cooperative model assuming a transparent
mechanism and voluntary participation may indeed work, and such approaches may
exhibit potential, when one would like to answer the question ’ _How to
improve supply-security related cooperation of the EU-member countries, and
improve the more (internationally) efficient usage of storage facilities_ ’.

##  4 Conclusions

Regarding the usage of storage facilities, there are two possible ways to
enhance the supply security of the EU for the following years. (1) To
constitute EU-level reserves and redistribution mechanisms, which aim to help
the member states in potential future need. Since under the current
circumstances, the construction/exploitation of new reservoirs and gas for
this aim seems not realistic in the short term, this approach would require
the partial expropriation of national gas reserves and/or reservoir
capacities. Such centralized approaches are likely to meet resistance by
countries who consider that they made more sacrifices and effort for their own
supply security in the past than others. We do not argue that such initiatives
are necessary doomed, but it is possible that obtaining a sufficient political
support for such a regulation framework will be challenging. (2) The EU might
also act as a catalyst of voluntary supply security cooperations by defining
the appropriate transparent and predictable regulatory frameworks. Such
approaches may complement or maybe even partially substitute the initiatives
of the first type to further enhance the dynamism and flexibility of the
reaction of the Union in the case of an emergency event. Based on simple
computational modelling studies, this paper argues that multilateral voluntary
supply security cooperation mechanisms may have significant potential in
reducing the individual risk of participants.

##  5 Acknowledgments

This work has been supported by the Hungarian Academy of Sciences under its
Momentum Programme LP2021-2 and by the János Bolyai Research Scholarship of
the Hungarian Academy of Sciences

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