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BY 4.0 license Open Access Published by De Gruyter Open Access December 17, 2020

Weak Berge Equilibrium in Finite Three-person Games: Conception and Computation

Konstantin Kudryavtsev and Ustav Malkov
From the journal Open Computer Science

Abstract

The paper proposes the concept of a weak Berge equilibrium. Unlike the Berge equilibrium, the moral basis of this equilibrium is the Hippocratic Oath “First do no harm”. On the other hand, any Berge equilibrium is a weak Berge equilibrium. But, there are weak Berge equilibria, which are not the Berge equilibria. The properties of the weak Berge equilibrium have been investigated. The existence of the weak Berge equilibrium in mixed strategies has been established for finite games. The weak Berge equilibria for finite three-person non-cooperative games are computed.

1 Introduction

A wide class of economic, social and political processes are well described by the methods of game theory. Often, when making decisons, participants in such processes can not agree among themselves that they are modelled by using non-cooperative games. Certainly, the most well-known concept of a solution in the theory of non-cooperative games was proposed by John Nash in 1950 in [1]. For this work in 1994 he was awarded the Nobel Prize in Economics.

However, the application of the Nash equilibrium concept in the modelling of real socio-economic and political conflicts, in some cases, leads to paradoxical results, such as the “prisoner's dilemma”. One of the first who has noticed this was Claude Berge in [2]. In this book, Berge proposed a new concept of equilibrium, according to which, players are divided into coalitions, while players of one coalition can work together to maximize the payoffs of players of another coalition. Apparently, a crushing review by Martin Shubik [3] on the book of Berge [2], led to the fact that Claude Berge switched his attention from game theory to other areas of mathematics. After decades, based on Berge's ideas, V.I. Zhukovsky [4, 5] and K.S. Weisman [6, 7] suggested a new altruistic concept of equilibrium which is called Berge equilibrium. In this concept, the players act on the principle of “One for all and all for one!” from Alexander Dumas's novel “The Three Musketeers”. Another interpretation of the Berge equilibrium is [8] the Golden Rule of morality: “Treat others the way you want to be treated”. The development of the Berge equilibrium concept is described in details in the review [9]. It is worth noting that the Berge equilibrium solves such well known paradoxes in game theory as the “Prisoner's Dilemma”, “Battle of the sexes” and many others. Also the use of Berge equilibrium is possible to applications in economics [10].

At the same time, the Berge equilibrium concept has some drawbacks. One of these drawbacks is that the Berge equilibrium rarely exists in pure strategies. Moreover, in N-person games (N ≥ 3) with a finite set of strategies, the Berge equilibrium may not exist in the class of mixed strategies. Such example was constructed, in particular, in [11]. The lack of Berge equilibrium might be caused by the fact that it is often impossible to follow the Golden Rule of morality in relation to all players at the same time. For example, if the goals of two players are opposite, then the third player will not be able to apply the Golden Rule to them simultaneously. In this case, increasing the payoff of one player, simultaneously reduces the payoff of the other.

In this paper, we introduce the concept of weak equilibrium according to Berge (the Weak Berge Equilibrium or the WBE). Unlike the Berge equilibrium, this concept is not based on the Gold Rule of morality. The weak Berge equilibrium concept is based on the Hippocratic oath “First do no harm!”. Here, we will assume that, making a decision, each player adheres to the situation, one-sided deviation that can cause harm to one of the other players. Further, in Section 2, the concept of weak Berge equilibrium is formalized, some of its properties are studied and sufficient conditions for the existence of such an equilibrium in N-person games are given. In Section 3, a numerical WBE approximate search method based on [12,13,14] is proposed, and numerical simulation results are given for finite games of three person.

2 The concept of weak Berge equilibrium

Let us consider a non-cooperative N-person game in normal form:

(1) Γ=N,{Xi}iN,{fi(x)}iN,

where N = {1, 2, . . . , N} denotes the set of serial numbers of the players; the set of xi strategies of the i-th player (iN) is denoted by Xi, where XiRni. As a result of the players choosing their strategies, the strategy profile is x = (x1, . . . , xN) ∈ X = X1 × X2 × . . . × XNRn (n = n1 + n2 + . . . + nN). On the set of strategy profiles X for each player i (iN) the scalar payoff function fi(x) : XR was defined. The value of fi(x) realized on the strategy profile chosen by the players xX is called the payoff of the i-th player.

The game Γ is played as follows. Each player i (iN), without entering into a coalition with other players, chooses his strategy xiXi. As a result of this choice, the strategy profile is x = (x1, . . . , xN) ∈ X. After that, each player i gets his payoff fi(x).

Thus, when making a decision, the player is forced to focus not only on his payoff function, but also on the possible choice of the other participants in the game.

Further, (yi, x−i) denotes the strategy profile (xi, . . . , xi−1, yi, xi+1, . . . , xN), which was obtained from the strategy profile x by replacing the strategy of the i-th player xi on yi.

The most popular concept of a solution in the theory of non-cooperative games is the Nash equilibrium.

Definition 1

A strategy profile xe=(x1e,,xNe)X is called a Nash equilibrium (NE) in the game (1) if for every xX the system of inequalities

(2) fi(xe)fi(xi,xie)(iN)

is true.

The Nash equilibrium strategy profile xeX is stable with respect to deviation of an individual player from his strategy which enters in xe. Applying the concept of the Nash equilibrium, the player proceeds from his own selfish motives. He only cares about his payoff, without taking into account the interests of other players. However, this approach leads to a number of paradoxes, such as the Tucker problem in the classic game “Prisoner's Dilemma”.

Example 1

Let us consider the Prisoner's Dilemma game. Two criminals are arrested on suspicion of a crime, but the police do not have direct evidence. Therefore, the police, have isolated them from each other, and offered them the same deal: if one testifies against the other, but he keeps silence, the first one is released for helping the investigation, and the second gets 10 years – the maximum term of imprisonment. If both are silent, their deed goes through a lighter article, and each of them are sentenced to a year in prison. If both testify against each other, each receives a minimum period of 2 years. Every prisoner can choose to keep silence or testify against another. However, none of them knows exactly what the other will do. The Nash equilibrium in this game dictates players to testify against each other, although silence will be more beneficial for them.

Thus, the players’ egoism in the Prisoner's Dilemma leads them to the most unprofitable solution.

The opposite approach to the concept of equilibrium, based on altruism, is called the Berge equilibrium.

Definition 2

A strategy profile xB=(x1B,,xNB)X is called a Berge equilibrium (BE) in the game (1), if for each xX the system of inequalities

(3) fi(xB)fi(xiB,xi)(iN)

is true.

The difference between Nash and Berge equilibria is that, in a Nash equilibrium, each player directs all efforts to increase individual payoff as much as possible. The antipode of (2) is (3), where each player strives to maximize the payoffs of the other players, ignoring its individual interests. Such an altruistic approach is intrinsic to kindred relations and occurs in religious communities. The elements of such altruism show up in charity, sponsorship, and so on.

In Example 1, players receive the best result if they use the Berge equilibrium, thus the Berge equilibrium solves the Tucker problem in the Prisoner's Dilemma.

Property 1. A Berge equilibrium in game (1) with N = = {1, 2} coincides with a Nash equilibrium if both players interchange their payoff functions and then apply the concept of Nash equilibrium to solve the game.

In view of Property 1, all results concerning the Nash equilibrium in the two-player game are automatically transferred to the Berge equilibrium (of course, with an “interchange” of the payoff functions as described by Property 1).

Differences appear when N ≥ 3. So, the Berge equilibrium may not exist in a finite 3-person games. The example of this is given in [11]. The following example is taken from [11].

Example 2

Let us consider the following three-player game in which each of the players has two pure strategies. Pure strategies of the first, the second, and the third player are denoted as A1, A2; B1, B2; C1, C2, respectively.

C1:A1A2B1B2((2,1,0)(1,1,1)(2,0,1)(1,0,2)),

C2:A1A2B1B2((1,2,0)(0,2,1)(1,1,1)(0,1,2)).

The left-hand matrix refers to the pure strategy C1 of the third player, while the right-hand matrix refers to his/her pure strategy C2. Let us note that this game is a very special one: none of the players has any possibility to influence their own payoff, even if they use any of their pure or mixed strategies. On the contrary, players’ payoffs depend exclusively on the choices of the remaining players.

One can easily check that the second and the third players’ best support to any of the first player's (pure or mixed) strategies is a pair of pure strategies (B1, C1); the first and the third players’ best support to any of the second player's (pure or mixed) strategies is a pair of pure strategies (A1, C2); and finally, the first and the second players’ best support to any of (pure or mixed) strategies of the third player is a pair of pure strategies (A2, B2). This game has no Berge equilibria, neither in pure, nor in mixed strategies.

Next, we recall the concept of Pareto optimality, and then formalize the Weak Berge Equilibrium.

Definition 3

The alternative x* is a Pareto-maximal alternative in the N-criteria problem

X,{fi(x)}iN,

if the system of N inequalities

fi(x)fi(x*)(iN),

with at least one strict inequality, is inconsistent.

The moral basis of the following definition is the Hippocratic Oath “First do no harm!”.

Definition 4

Let us call the strategy profile xw=(x1w,,xnw) a weak Berge equilibrium (WBE), if for each player i (iN) strategy xiw is Pareto-maximal alternative in the N − 1-criteria problem

Γi=Xi,{fj(xi,xiw)}jN\{i}.

Let us compare to the game Γ an auxiliary game

(4) Γ˜=N,{Xi}iN,{gi(x)}iN,

where the set of players N and the set of strategies Xi (iN) are the same as in the game (1), and the payoff functions gi(x) have the form

(5) gi(x)=jN\{i}fj(x).

Lemma 1

The Nash equilibrium strategy profile in the game (4) is a weak Berge equilibrium strategy profile in the game (1).

Proof

Let xe is Nash equilibrium strategy profile in the game Γ˜ , i.e.,

(6) gi(x1e,...,xi1e,xi,xi+1e,...,xne)gi(xe)(iN).

With regard to (5), the inequality (6) can be rewritten as

(7) jN\{i}fj(xi,xie)jN\{i}fj(xe)(iN).

Suppose xe is not a WBE strategy profile, then there exists a number i for which the system of inequalities is consistent with

(8) fj(xi,xiefj(xe)(jN\{i}),

of which at least one inequality is strict.

Adding inequalities (8), we obtain

jN\{i}fj(xi,xie)>jN\{i}fj(xe)(iN),

that contradicts (7).

Similarly, the following lemma can be obtained.

Lemma 2

Let us suppose that in the auxiliary game (4) the payoff function is

gi(x)=jN\{i}αjfj(x)αj0.

Then the Nash equilibrium strategy profile in the game (4) is the weak Berge equilibrium strategy profile in the original game (1).

Remark 1

To construct a WBE strategy profile in the game (1), we can use the following algorithm:

  1. compose the auxiliary game Γ˜ ;

  2. construct the strategy profile xe which is the Nash equilibrium strategy profile in the auxiliary game Γ˜ ;

  3. the found strategy profile xe will be the WBE strategy profile in the original game Γ.

As an example, consider the game “Snowdrift” which is proposed in [15].

Example 3

Let us consider the three-player Snowdrift game which is shown in Table 1. The history of the game lies in the fact that A, B and C are the drivers of three cars, that got stuck in a snowdrift at night, each of them has a shovel. If a solution is found for any one car, others can use it. Every driver chooses to dig or wait (in the hope that someone else will dig, or that a snowplow will come to the place of incident). Digging will cost 6 points, which are divided equally between those who perform the work; provided that there is at least one digger. If the players dug out by themselves of a snowdrift, then each player will get 4 points. Thus, if all three players dig, then everyone will get 2 points. If two players dig, they will get one point each, and the third player will earn 4 points. If one player digs, then his payoff will be negative −2, and the payoffs of the remaining two players will be 4 points each. In the case that the players do not dig, but wait until the morning when the utilities arrive and clear the snow, their payoff will be zero.

Table 1

The three-player Snowdrift game.

C – to wait
A\B to wait to dig
to wait (0,0,0) (4,–2,4)
to dig (–2,4,4) (1,1,4)

C – to dig
A\B to wait to dig
to wait (4,4,–2) (4,1,1)
to dig (1,4,1) (2,2,2)

Here, A, B, C are the 3-dimensional matrices, which determine the payoffs of the players will be

A:A1=(0421),A2=(4412);B:B1=(0241),B2=(4142);C:C1=(0444),C2=(2112).

The Nash equilibrium (NE) here will be (wait, wait, wait) with payoffs (0, 0, 0).

We will now compile an auxiliary game, the payoff matrices will be:

  • for the first player

    A*=B+C:A1*=(0285),A2*=(2254);
  • for the second player

    B*=A+C:B1*=(0825),B2*=(2524);
  • for the third player

    C*=A+B:C1*=(0222),C2*=(8554).
The Nash equilibrium in the auxiliary game with matrices A*, B*, C* will be (dig, dig, dig), respectively, the weak Berge equilibrium (WBE) in the original game will also be (dig, dig, dig) with payoffs (2, 2, 2).

Obviously, in this example, the WBE is more profitable for all players than the NE.

Remark 2

In this game, the Berge equilibrium (BE) coincides with the WBE.

Following to Lemma1, it is possible to obtain sufficient conditions for the existence of the WBE under the usual restrictions for the game theory.

Theorem 1

In a non-cooperative N-person game Γ with a finite set of strategies, a weak Berge equilibrium strategy profile in mixed strategies exists.

Theorem 2

If in a non-cooperative N-person game Γ, the sets of strategies Xi are convex compacts, and the payoff functions fi(x) are continuous in the aggregate of variables, then in the game Γ a weak Berge equilibrium strategy profile in mixed strategies exists.

3 The WBE in a finite three-person game

This section consists of two parts. In subsection 3.1, a finite 3-person game is formalized and the 3LP-method for solving the finite 3-persons game is described. In subsection 3.2, the test results for computing the WBE in the finite 3-person game are presented.

3.1 The 3LP-method for solving the finite 3-person game

Let us consider a non-cooperative three-person game.

Γ3={1,2,3},{Xi}i=1,2,3,{fi(x)}i=1,2,3.

The strategy profile xw=(x1w,x2w,x3w) is the WBE strategy profile, if and only if

  1. 1)

    the strategy x1w is the Pareto maximum alternative in the two-criterial problem

    X1,{f2(x1,x2w,x3w),f3(x1,x2w,x3w)};
  2. 2)

    the strategy x2w is the Pareto maximum alternative in the two-criterial problem

    X2,{f1(x1w,x2,x3w),f3(x1w,x2,x3w)};
  3. 3)

    the strategy x3w is the Pareto maximum alternative in the two-criterial problem

    X3,{f1(x1w,x2w,x3),f2(x1w,x2w,x3)}.

Let us compose an axillary game for the game Γ3

Γ˜3={1,2,3},{Xi}i=1,2,3,{gi(x)}i=1,2,3,

where, according to (5)

(9) g1(x)=f2(x)+f3(x),g2(x)=f1(x)+f3(x),g3(x)=f1(x)+f2(x).

The Nash equilibrium strategy profile in Γ˜3 will be the WBE strategy profile in the original game, Γ3.

Below, a finite non-cooperative 3-person game Γ3 is defined with three sets X, Y, Z of strategies of the first, second, and third player respectively, where X = {x = (x1, . . . , xm)TRm : xT em = 1, x ≥ 0m}, Y = {y = (y1, . . . , yn)TRn : yT en = 1, y ≥ 0n}, Z = {z = (z1, . . . , zl)TRl : zT el = 1, z ≥ 0l}, ω = (x, y, z) ∈ Rm+n+l, together with their payoff functions as follows

fx(ω)=i=1mj=1nk=1laijkxiyjzk,fy(ω)=i=1mj=1nk=1lbijkxiyjzk,fz(ω)=i=1mj=1nk=1lcijkxiyjzk.

Here, one has (aijk), (bijk), (cijk) is the players’ 3-dimensional payoff tables (without any loss of generality one can assume that all the entries of those tables are positive real numbers); the vector ωT = (xT , yT , zT), ωΩ=X×Y×ZR+m+n+l . Next, for p = m, n, l, we define the vectors 0p=(0,,0)TR+p , ep = (1, . . . , 1)TRp, as well as R+p is the nonnegative orthant of the Euclidean space Rp. The symbol T denotes the operation of transposition of a vector (matrix).

Following the algorithm in remark 1, we construct the functions (9).

gx(ω)=fy(ω)+fz(ω)=i=1mj=1nk=1l(bijk+cijk)xiyjzk,gy(ω)=fx(ω)+fz(ω)=i=1mj=1nk=1l(aijk+cijk)xiyjzk,gz(ω)=fx(ω)+fy(ω)=i=1mj=1nk=1l(aijk+bijk)xiyjzk.

Introduce the Nash function G(ω) = δx(ω) + δy(ω) + δz(ω), where

δx(ω)=maxxXg(x,y,z)g(ω),δy(ω)=maxyYg(x,y,z)g(ω),δz(ω)=maxzZg(x,y,z)g(ω).

The function G(ω) is an analogue of the Nash function defined for the bi-matrix games [16]. As the above–defined payoff functions are linear with the respect to each variable x, y, z (when the other two variables are fixed), the auxiliary game Γ˜3 is convex, hence the set of Nash points Ω* is non-empty (but not necessarily convex).

Since G(ω) ≥ 0 for all ωΩ, and G(ω) = 0 if, and only if ω is the Nash equilibrium of the game Γ˜3 , one can find the Nash equilibrium strategy profile of game Γ˜3 as the global minimum (equalling zero) of the function G(ω) on Ω.

Now we turn to the approximately numerical method for the construction of the WBE in Γ3. In [12] the algorithm for approximately solving finite non-cooperative three-person games (3LP) was proposed. The testing results illustrating the efficiency of applying this algorithm can be found in [13, 14].

Next we denote ãijk = bijk + cijk, ijk = aijk + cijk, ijk = aijk + bijk and dijk = ãijk + ijk + ijk = 2(aijk + bijk + cijk).

Set the iteration counter t = 0. As a starting strategy, one can use any pair of the players’ pure strategies (the total number of such pairs is mn + ml + nl); for example, fix a pair of strategies {y(0), z(0)} with the components y1(0)=1,yj(0)=0(j=2,,n) , z1(0)=1,zk(0)=0(k=2,,l) , and solve successively (for t = 0, 1, . . .) the triple problem Px(x(t+1), y(t), z(t)), Py(x(t+1), y(t+1), z(t)), Pz(x(t+1), y(t+1), z(t+1)), where

Px(x,y,z):i=1m(j=1nk=1ldijkyjzk)xiβγmaxx,β,γ,i=1m(k=1lb˜ijkzk)xiβ0,j=1,,n,i=1m(j=1nc˜ijkyj)xiγ0,k=1,,l,xTem=1,x0m,β,γR+1.

If x* is an optimal solution of this problem, then we set x′ := x*; next we solve:

Py(x,y,z):j=1n(i=1mk=1ldijkxizk)yjαγmaxy,α,γ,j=1n(k=1la˜ijkzk)yjα0,i=1,,m,j=1n(i=1mc˜ijkxi)yjγ0,k=1,,l,yTen=1,y0n,α,γR+1.

Again, if y* is an optimal plan for the above problem, then put y′ := y*, and continue solving:

Pz(x,y,z):k=1l(i=1mj=1ndijkxiyj)zkαβmaxz,α,β,k=1l(i=1mb˜ijkxk)zkα0,j=1,,n,k=1l(j=1nc˜ijkyj)zkβ0,i=1,,m,zTel=1,z0l,α,βR+1.

Now that z* is an optimal solution of that problem, we denote z′ := z*.

The optimal objective function values Gt = G(ω(t+1)) are monotone non-increasing by t. The iteration process continues until the value Gt stabilizes, that is, for some t*, the difference Gt*Gt*+1 becomes small enough. In addition, if Gt* = 0, it means that an (exact) Nash point has been found. If the value Gt* is positive but small enough, an approximate solution of the game is reported. Otherwise, select a new pair of the initial strategies and start the process again (probably, having altered the order of the solved problems Px, Py, Pz).

3.2 Test results for the 3LP-algorithms for finding the WBE

We tested the algorithms finding the weak Berge equilibrium in the finite 3-persons games by making use of the personal computer with the processor Intel(R) Core(TM) i5-3427U (CPU @ 1.80GHz 2.300 GHz, memory 4.00 GB, 4 cores). The test codes were written in the MatLab. A series of 10 games was solved for each triple n, m, l.

We investigated 2 cases: an independent matrices and a mutually dependent matrices. In the first case (independent matrices) we used pseudo-random counters to generate independently the elements of the tables aijk, bijk, cijk (1 ≤ im, 1 ≤ jn, 1 ≤ kl).

For the game with mutually dependent matrices, we first used a pseudo-random counters to generate independently the elements of the auxiliary tables aijk , bijk , cijk (1 ≤ im, 1 ≤ jn, 1 ≤ kl). At the second stage, we constructed the mutually dependent payoff tables by the formulas

aijk=aijkλbijk+cijk2+1,bijk=bijkλaijk+cijk2+1,cijk=cijkλaijk+bijk2+1

for all 1 ≤ im, 1 ≤ jn, 1 ≤ kl, where 0<λ12 is a covariance coefficient.

We solved games up to the dimension dim = m = n = k = 100. For comparison, using 3LP-algorithm, we calculated the Nash equilibrium for the same games.

Table 2 reports the results of the 3LP-algorithm solving the set of test games (5 series with 10 instances in each) with independent matrices. The algorithm switched to the next initial pair of strategies after having made dim iterations.

Table 2

The results of solving 5 series of games of ten problems with independent matrices

dim NE WBE tNE tWBE
20 327 85 745.85 129.88
40 230 59 539.28 99.34
60 169 40 404.43 88.1
80 129 28 373.14 92.03
100 159 41 904.85 162.62

Here in Table 2, the following notation is used: dim = m = n = k are the game's sizes (dimension); NE — the number of initial (starting) point when searching for a Nash equilibrium; WBE — number of starting points when searching for a weak Berge equilibrium; tNE — the total amount of time to search a Nash equilibrium for the series of 10 games (sec); tWBE — the total amount of time to search a weak Berge equilibrium for the series of 10 games (sec).

Also in Table 3, for mutually dependent cases, the following notation is used: dim = m = n = k are the game's sizes (dimension); WBE—number of starting points when searching for a weak Berge equilibrium; tWBE — the total amount of time to search a weak Berge equilibrium for the series of 10 games (sec); itn - the total number of steps of 3LP algorithm. Covariance coefficient λ = 0, 4 was used in the calculation of Table 3.

Table 3

The results of solving 5 series of games of ten problems with mutually dependent matrices

dim itn WBE tWBE
20 3173 479 1106.49
40 6532 814 2101.15
60 12826 1415 5134.66
80 10306 1017 5564.54
100 16725 1527 13049.09

For mutually dependent cases, we have given the results only for the WBE, so when calculating the NE for these problems took an unacceptable time or they were not solved at all.

It is easy to see from the reported results (see Table 2 and Table 3), that the reciprocal dependence of the payoff matrices affect much to solve a problem by the 3LP-algorithm. The reciprocal dependence sufficiently increases the complexity of problems.

It is also clear that, the search WBE is much faster than the searching NE. This is most likely due to a pure strategy weak Berge equilibrium existing more often than a pure strategy Nash equilibrium.

4 Conclusion

In this paper, we formalized conception of the weak Berge equilibrium. The WBE follows the Hippocratic oath “First do no harm!”. This conception is more often realized in practice than the Berge equilibrium conception. In contrast to the NE, the WBE always exists for every finite N-person games. As an example, we found the WBE in the finite three-person games using 3LP-algorithm.

In our opinion, the properties of the weak Berge equilibrium should be studied in more detail. In the future, the authors plan to study equilibrium in linear-quadratic games. Also we plan to develop a numerical method for finite 3-person games based on the Germeier convolution [17].

Acknowledgement

The work was supported by Act 211 Government of the Russian Federation, contract N 02.A03.21.0011.

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Received: 2019-10-29
Accepted: 2020-08-05
Published Online: 2020-12-17

© 2021 Konstantin Kudryavtsev et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.