CC BY
4XAK: 519.834 MSC2010: 91A12
TO THE PROBLEM OF COALITIONAL EQUILIBRIUM IN MIXED
STRATEGIES
© V. I. Zhukovskiy
Moscow State University named after Lomonosov Faculty of Computational Mathematics and Cybernetics Department of Optimal Control Leninskiye Gory, GSP-1, Moscow, 119991, Russia e-mail: zhkvlad@yandex.ru
© L. V. Zhukovskaya
Federal State Budgetary Institution of Science Central Economic and Mathematical Institute of the
Russian Academy of Sciences (CEMI RAS) Nakhimovskii prosp., 47, Moscow, 117418, Russia e-mail: zhukovskaylv@mail.ru
© L. V. Smirnova
State University of Humanities and Technology Zelenaya, 22, Orekhovo-Zuevo, 142611, Russia e-mail: smirnovalidiya@rambler.ru
To the problem of ooalitional equilibrium in mixed strategies.
Zhukovskiy V. I.,Zhukovskaya L. V.,Smirnova L. V.
Abstract. The Strong Coalitional Equilibrium (SCE), is introduced for normal form games under uncertainty. This concept is based on the synthesis of the notions of individual rationality, collective rationality in normal form games without side payments, and a proposed coalitional rationality. For presentation simplicity, SCE is presented for 4-person games under uncertainty. Sufficient conditions for the existence of SCE in pure strategies are established via the saddle point of the Germeir's convolution function. Finally, following the approach of Borel, von Neumann and Nash, a theorem of existence of SCE in mixed strategies is proved under common minimal mathematical conditions for normal form games (compactness and convexity of players' strategy sets, compactness of uncertainty set and continuity of payoff functions).
Keywords: Normal form game without side payments, uncertainty, guarantee, mixed strategies, Germeier convolution, saddle point, equilibrium.
Introduction
The theory of cooperative games has been developing in three directions as follows. The first direction involves the introduction of equilibrium solutions for normal-form games and their analysis. It is an extension of Nash's theory [18, 19]. The second direction is based on the characteristic function approach. In a characteristic function game, each coalition (subset of players) is associated with a value it can afford. The third and most recent direction considers the coalition formation as a dynamic process.
As the contribution of the present paper is related to the first direction of research, we briefly discuss the second and the third directions.
In characteristic function games, the most prominent solution concept is the core proposed in [12]. The core rests on the idea of blocking: a coalition can block an imputation if it can improve the payoffs of its members by deviating from the current imputation. An imputation is in the core if it cannot be blocked by any coalition. Many other concepts were also introduced, such as the nucleolus, the kernel, and the Shapley value, to name a few. The main drawback of the characteristic function game and its solution concepts is that they do not incorporate the strategic interaction of players.
The obvious limitations of the characteristic form games and their solution concepts led to the appearance of the third direction of research, which considers coalition formation in cooperative games as a dynamic process. The pioneering works in this direction were the publications [7], where players' farsightedness was incorporated into game-theoretic analysis (i.e., the players were assumed to care about long-term outcomes of the game); [22], where «coalition strategies» were introduced to account for the coalitional behavior of players during the game; and [14], where coalition formation was described by a Markov process. For more details, the interested reader is referred to [23].
Now, let us discuss the first direction. Many coalition-related concepts of equilibrium or solutions were introduced for n-player normal-form games. The main motivation for the inception of such investigations was to overcome a well-known drawback of Nash equilibrium (NE): NE is unstable against the deviations of coalitions. A coalition may improve the payoff of all its members by collectively deviating from NE. R. Aumann [1] introduced the strong equilibrium (SE) that is stable against such deviations. As it however turned out, the set of SE is empty for most of the games. Later, Aumann [2] suggested the a-core and fi-core for relaxing the conditions of SE. A strategy profile belongs to the core of a game if no group of players has an incentive to form a coalition and choose a different strategy profile in which each of its members are made better-off, i.e., the strategy profile cannot be blocked by any coalition. The a-core and fi-core differ by the definition of blocking: the a-core requires a blocking coalition to choose a specific
strategy independently of the complementary coalition's choice, whereas the fl-core allows a blocking coalition to vary its blocking strategy as a function of the complementary coalition's choice. C. Berge [3] introduced a very strong equilibrium, called the strong Berge equilibrium (SBE), in the sense that if one of the players chooses his strategy from an SBE, the other players have no choice but to play their strategies from the SBE. The Berge equilibrium (BE), put forward by V. Zhukovskiy [30], is an equilibrium that reflects altruism and mutual support among the players. A BE is a strategy profile in which the payoff function of each player is maximized by all the other players. Recently, research on BE has gained some momentum [33], as more empirical research showed that (besides noncooperative behavior) cooperation, mutual support, reciprocity, and caring about fairness may take place in interactions between individuals; see [9, 10, 13, 26]. Bernheim noticed that in an SE some deviations might not be self-enforcing [8], and therefore cannot be treated as credible threats. This led to the introduction of coalition-proof Nash equilibrium (CPNE). In a CPNE, only self-enforcing deviations are credible threats. A deviation by a coalition is self-enforcing if no subcoalition has an incentive to initiate a new deviation. Finally, some works combined different solution concepts, e.g., the hybrid solution of [29], which assumed a coalition structure to be formed, and the game itself to be noncooperative among coalitions but cooperative within coalitions. As a result, Nash equilibrium was adopted for the former and the core for the latter as solutions.
A common drawback of the coalition equilibria and solutions mentioned above is that their set is often empty; they do not exist under standard assumptions such as the compactness and convexity of strategy sets and the continuity and quasiconcavity of payoff functions [8, 15, 20, 27, 31], except for the a-core and Zhao's hybrid solution. Using the notion of balancedness, Scarf [26] established the non-emptiness of the a-core in the case of compact and convex strategy sets and continuous and quasiconcave payoff functions. However, Scarf's theorem suggests no method for determining an a-core element. Zhao's hybrid solution was obtained under similar hypotheses.
As most of these equilibrium concepts and solutions do not exist in the class of pure strategies under standard assumptions of continuous games, a natural question arises: Do these concepts and solutions exist in mixed strategies? Unfortunately, there are no works dealing with this question and related topics in the existing literature. Moreover, the existence of the concepts and solutions has not been considered in games under uncertainty.
In this paper, we introduce a rational coalitional equilibrium for a game under nonprobabilistic uncertainty as well as establish its existence in mixed strategies. As a matter of fact, this concept generalizes many of the concepts mentioned above.
The mathematical model of cooperation described below is a four-player normal-form game with indeterminate parameters (interval uncertainty). The analysis has been limited to the class of four-player games for the sake of simplicity. Regarding the indeterminate parameters, it is assumed that the players know their range of admissible values only; no probabilistic characteristics are available (for some reasons). The models of game phenomena with a proper consideration of uncertainties yield more adequate results and decisions, which is supported by the numerous publications related to this field of research. (For example, a Google search on the topic «mathematical modeling under uncertainty» returns more than one million links to related works.) The uncertainty appears because of incomplete information about the players' strategy sets, the strategies being chosen by each player, and the related payoffs: «Although our intellect always longs for clarity and certainty, our nature often finds uncertainty fascinating.» (C. von Clausewitz1). One more question arises: How a player can simultaneously consider the game's strategic and cooperation aspects, and the presence of uncertainty when choosing his strategy? In this paper, the following approach to formalize the cooperation aspect of the game is adopted. It is assumed that any non-empty subset of players has the possibility to form a coalition through communication and coordination by agreeing to choose a bundle of strategies to achieve the best possible payoff for all its members. This assumption means that the interests of all possible coalitions are considered. Further, it is also assumed that the game is without side payments or non-transferable utility (NTU). The concept of strong coalitional equilibrium (SCE) is introduced for the game described. A sufficient condition for its existence in pure strategies is provided and its existence in mixed strategies is established under standard assumptions (compact and convex strategy sets of all players, a compact set of uncertainties, and continuous payoff functions of all players).
1. The Game under Uncertainty
In this section we present the normal-form game under uncertainty. For the sake of simplicity, further presentation will be confined to the class of four-player games only. All the results and definitions below can be easily generalized to n-player games in a straightforward way.
1Carl von Clausewitz, in full Carl Philipp Gottlieb von Clausewitz, (1780-1831), was a Prussian general and military thinker, whose work Vom Kriege (1832; On War) has become one of the most respected classics on military strategy.
Consider the four-player normal-form game under uncertainty r = (N = {1, 2, 3,4}, {Xi}i£N, Y, {fi(x, y)}^),
where N = {1,2,3,4} is the set of players; each player i G N chooses his strategy xi from his strategy set Xi C Rni, thereby forming a strategy profile x = (x^x2,x3,x4) G X = 4=1 Xi C Rn, n = ni, an interval uncertainty
y G Y C Rm occurs independently of the players' actions; the payoff function of player i G N is a real-valued function fi(x, y) that depends on the pair (x, y) G X x Y. The goal of each player i G N in the game r is to choose a strategy xi yielding the greatest possible payoff for him. This includes choosing strategies that maximize other players' payoffs if they are beneficial for player i. With this goal in view, the players should consider the possible formation of any coalition and also the possible realization of any uncertainty y G Y. Considering the uncertainty y G Y leads to a multivalued payoff function of the form x ^ fi(x, Y) = [JygY fi(x,y). Such multivalued payoff functions complicate further study of the cooperative games r. To consider the effect of uncertainty on their payoffs, the players need to adopt a principle of decision-making under uncertainty [16], such as the maximin principle [28], the principle of minimax regret [24, 25], etc. Moreover, a reasonable solution concept for the game r must reflect the uncertainty's effect on the players. As uncertainty is considered in equilibria of cooperative games for the first time, we assume that the players adopt a conservative (maximin or risk-averse) approach [28]. Other principles of decision-making under uncertainty in the game r can be studied in future works. Thus, the payoff function of each player i G N will be estimated not by its value fi(x, y), but by its guaranteed level fi[x]. A guarantee over the values fi(xy), y G Y, can be defined as follows:
fi [x] = min fi(x,y).
yeY
Really, we have fi[x] < fi(x, y), y G Y, therefore, a lower bound on the payoff function of the player i can be given by fi [x]. As it will be demonstrated below, under common conditions the function x ^ fi[x] is well-defined and continuous on X. In this section and also in Sections 2 and 3, the functions x ^ fi[x], i G N are assumed to be well-defined and continuous on X. This leads to the (conservative) game of guarantees
rg = (N = {1, 2,3,4}, {Xi}igN, {fi[x]}igN).
In the next section we will introduce SCE of the game r via the game rg.
2. Coalitional Rationality and Strong Coalitional Equilibrium
First, let us present the main properties of SCE, introducing the con-cept itself later. To define coalitional rationality, the following notations are convenient. For any non-empty subset K of the set N, denote by —K the complement of K, that is, N \ K. In particular, for each i G N, denote by —i the set N \ {i} and for each i, j G N, i = j, denote by — (i, j) the set N \ {i, j}. The notion of partition of a set will be used as well. A partition of a set A is a family of disjoint subsets of A, the union of which equals A. In game theory, a partition of the set of players is called a coalition structure. For a strategy profile x G X, and i G N denote by x = (xj, x-i) and X-i = rijeN\{i} Xj.
In the game r, fifteen coalition structures can be formed as follows {{1}, {2}, {3}, {4}}, {1, 2, 3,4}, K{j} = {{i}, {—i}}, K{j},{j} = {{i}, {j}, { —(i,j)}}, Kj = {{i, j}, { —(i, j)}}, for all i, j G N, i = j. Recall some results from the theory of cooperative games without side payments [16]. For a strategy profile x* G X in the game rg, the following properties are considered:
(a) x* satisfies the individual rationality condition (IRC), if for all i G N,
/i[x*] > / = max min /i[xi,x-i] = min /i[x0,x-ij.
xi t Ai x— itA—i X — i fcA.—i
The value / is the guaranteed payoff of player i G N .If player i chooses his maximin strategy x0, then his payoff satisfies /i[x0,x-i] > /, for all x-i G X-i;
(b) x* satisfies the collective rationality condition (ColRC) if x* is a Pareto-maximal alternative in the multicriteria choice problem rP = (X, /i[x]igN), i.e., for all x G X, the system of inequalities /[x] > /[x*], i G N, with at least one strict inequality, is inconsistent. Note that if ^jgN /[x] < /¿[x*] for all x G X, then x* is a Pareto-maximal alternative in the choice problem rP.
(c) x* satisfies the coalitional rationality condition (CoalRC) if
/k[x*] > /k[x*,x-j], for all x-i G X-i; /k[x*] > /k[x*,x*,x-(i,j)], for all x_(jj) G X-(j,j); /k [x*] > /k [xi ,x_i], for all xi G Xi,
all the three inequalities holding for all i, j, k G N, i = j, where x = (xi,xj, x_(i,j)) and X_(i,j) = nstN\{ij} Xs. This condition means that when a coalition K chooses its strategy profile from x*, then no player can improve his payoff if the countercoalition —K deviates from its strategy profile in x*.
Definition 1. A strategy profile x* G X is called strong coalitional equilibrium (SCE) for the game r if it satisfies IRC, ColRC and CoalRC for the game of guarantees rg.
Remark 1. In accordance with IRC, it makes sense for a player to form coalitions with other players if he gets a payoff not less than what he can guarantee by choosing his maximin strategy. ColRC leads the players to a non-dominated strategy profile in terms of Pareto maximality. Finally, CoalRC means that the payoff of each player is stable against any deviations of individual players or coalitions from a strategy profile satisfying CoalRC. In other words, no player's payoff is increased when any coalition deviates from an SCE. Thus, it is rational for all coalitions not to deviate from x*, because no player in a deviating coalition or outside it will benefit.
By Definition 1 a SCE must satisfy all the extremal constraints defining IRC, ColRC, and CoalRC. However, all these constraints can be easily derived from the following seventeen of them:
fi[x1, x2, x3, x4] < fi[x*], for all xk G Xk, k = 2, 3,4 and i = 1, 2, 3,4;
fi[x1, x2, x3, x4] < fi[x*], for all xk G Xk, k = 1, 3,4 and i = 1, 2, 3,4;
fi[x1,x2,x3,x4] < fi[x*], for all xk G Xk, k = 1, 2,4 and i = 1, 2, 3,4; (1)
fi[x1,x2, x3,x4] < fi[x*], for all xk G Xk, k = 1, 2, 3 and i = 1, 2, 3,4;
^ fi[x] < ^ fi[x*], for all x G X,
ieN ieN
where x* = (x*, x*, x*, x*).
From this point onwards, we will use the system of inequalities (1) to establish that a strategy profile is an SCE of the game r instead of the system of inequalities involved in the definitions of IRC, ColRC, and CoalRC (see items (a)-(c) above). From (1) it can be observed that the SCE has two interesting features. First, once the players are in an SCE, they do not have incentive to deviate from it individually, collectively, or in coalitions. Second, if the players are not in an SCE, as soon as one player (or coalition) declares that he (it) will choose his (its) strategy (profile) from an SCE, the other players have no choice but to choose their strategies from the SCE. In other words, any player or coalition can enforce an SCE.
Although SCE does not exist in pure strategies in most of continuous games, in finite games it is not the case. The following example, adapted from [18], shows that an SCE exists in a class of games.
Example 1. Consider a three-player game in which players 1, 2 and 3 choose rows, columns and boxes, respectively and are named accordingly. Let e G [0,1] and also let a, ft, and y be nonnegative numbers such that a + ft + 7 < 9. Each of the players has
two strategies {T, B} for player 1, {T, L} for players 2 and 3. The minimum payoffs, i.e., /¿[xi, x2, x3] i = 1, 2, 3, where x1 = T, B and Xj = T, L; j = 2,3, are given below.
L R L R
T 2, 2, 2 0, 0, e 0, 0, 0 4, 4, 1
B a, ¡3, y 0, 0, e 0, 0, 0 3, 3, 1
L R
The strategy profile (T, R, R) is an SCE. Really, this strategy profile satisfies the last inequality of system (1): the sum of the payoffs in (T, R, R) is higher than the sum of the payoffs in any other strategy profile, including (B, L, L) due to the inequality a + ft + y < 9 and the fact that a, ft, y are nonnegative numbers.Next, the possible deviations corresponding to the inequalities in (1) are (T, L, R), (T, R, L), (T, L, L) when player 1 chooses the SCE strategy T; (B,R, R), (T, R, L) and (B,R, L) when player 2 chooses the SCE strategy R; (B, R, R), (T, L, R) and (B, L, R) when player 3 chooses the SCE strategy R. In all the strategy profiles mentioned, the payoffs of players 1, 2, and 3 are smaller than or equal to their payoffs in (T, R, R), which are 4, 4, and 1, respectively.
2.1. Related Concepts. In this section, we recall the most prominent cooperative solutions of NTU games in normal form and compare them with the SCE. Also, we compare the SCE with the solution concepts defined in dynamic context with respect to coalition deviations.
a) [1] A strategy profile x* G X is a strong equilibrium (SNE) of the game rg if, for all S C N and for all y_s G X_s, the system of inequalities / [x*] < /¿[ys, x_s] is inconsistent for all i G S.
This definition means that no coalition can improve the payoff of all its members by deviating from an SNE when the other players adhere to the SNE.
b) [2] A strategy profile x* G X is in the a-core of the game rg if, for any coalition S C N and for each ys G Xs, there exists z_s G X_s such that the system of inequalities /¿[x*] < /j [ys, z_s] is inconsistent for all i G S.
In other words, if a coalition deviates from a strategy profile x* belonging to the a-core, then the other players have a counterstrategy profile to punish it in such a way that not all members of the coalition are better-off.
c) [2] A strategy profile x* G X is in the ft-core of the game rg if, for each coalition S C N, there exists z_s G Z_s such that for all ys G Xs the system of inequalities /¿[x*] < /¿[ys, z_s] is inconsistent for all i G S.
In other words, for each coalition the other players can use a special strategy profile to punish it for any deviation from a strategy profile x* belonging to the ft-core in such a way that not all members of the coalition are better-off.
d) [3] A strategy profile x* G X is a strong Berge equilibrium (SBE) of the game rg if for all i G N, the inequalities f [x*, z-i] < f [x*] holds for all z-i G Z-i and j G —i.
In other words, no coalition of the form —i can make any of its members betteroff by deviating from an SBE. When a player uses his strategy from an SBE, the other players have no choice but to follow him, simply using their strategies from the SBE.
e) [30] A strategy profile x* G X is a Berge equilibrium (BE) of the game rg if for all i G N, the inequalities fi[x*,z-i] < fi[x*] holds for all z-i G Z-i.
In other words, in a BE the players maximize the payoff functions of each other. This equilibrium reflects mutual support and altruism among the players [33].
Using (1), we can easy to verify that SCE is also an SNE, an SBE and a BE. As is well-known, an SNE is a CPNE; then, an SCE is also a CPNE. Next, an SCE is an element of the a-core and the ft-core. The SCE has similarities with the SBE. However, there are two important differences between these solution concepts. First, in an SBE for each i G N the system of inequalities fj[x*,z-i] < fj[x*] holding for all z-i G Z-i and j G —i does not include the inequality corresponding to player i, fi[x*,z-i] < fi[x*]: the other players do not care about the payoff of player i when choosing their strategies from x*, and his payoff is not maximized. In an SCE, the inequality fi[x*, z-i] < fi[x*] is included, which means that the payoff function of player i is maximized by the other players. This shows that the SCE involves mutual support, whereas the SBE does not. Second, an SBE is generally not Pareto-optimal. The Pareto-maximal SBE was investigated in [33]. The SCE has also some similarities with the BE. However, there are important differences between the two equilibria. The BE expresses mutual support and altruism and ignores the individual interests of players; it is not a refinement of Nash equilibrium as a BE may not satisfy IRC [33]. The SCE differs from the hybrid solution (HS) of [29]: in the latter, it is assumed that a coalition structure is formed and there is no cooperation among the coalitions of this structure; in the former, such assumptions are not made.
Although the coalitional equilibrium [22], the equilibrium binding agreement [23], the equilibrium process of coalition formation [14] and the consistent set [7] were defined in a dynamic context, they can be compared to the SCE based on when a coalition can deviate. In the concepts mentioned, a coalition deviates to another state or strategy profile if and only if all its members are better-off, whereas in the SCE a coalition can deviate if and only if all players of the game are better-off (not only its members).
Moreover, the concepts listed in this section do not consider uncertainty as an exogenous factor, unlike the SCE.
3. Sufficient Conditions for the Existence of SCE in Pure
Strategies
In the previous section we have seen that an SCE is also an SNE and an SBE. Since these equilibria do not exist in pure strategies in most of the continuous games (see the Introduction), the SCE suffers from this drawback too. Nevertheless, we will formulate sufficient conditions for its existence using the approach developed in [32]. The approach used in this section paves the way to the next section, where the main result of this paper will be presented. First, we introduce the convolution [11] related to the SCE
<£i(x,z) = max{/i[zi,x2,x3,x4] - /[z]}, <^2(x,z) = max{/j[xi,Z2,x3,x4] - /[z]},
igN
<£3(x,z) = max{/i[xi,x2,Z3,x4] - /[z]}, (2)
igN
<P4(x,z) = max{/i[xi,x2,x3,Z4] - /i[z]},
jgN
^5(x,z) = ^/i[x] - Y1^^
igN igN
<£>(x,z) = max {<£>r(x, z)}, r=i,...,5
where x = (xi, x2, x3, x4) and z = (zi, z2, z3, z4) G X = igN Xj.
A saddle point (x0,z*) G X x X of the real-valued function <£>(x,z) in (2) is defined by the chain of inequalities
^(x, z*) < ^(x°, z*) < ^(x°, z), for all x, z G X. (3)
Proposition 1. If (x°,z*) G X x X is a saddle point of the function <£>(x, z), then the minimax strategy z* is an SCE of the game r.
Proof. Letting z = x° in (3), from (2) we obtain <£>(x°,x°) = 0. Then by transitivity, from (3) it follows that
^(x°, z*) < ^(x°, x°) = 0 ^ ^(x, z*) < 0, for all x G X, which implies (1).
□
Remark 2. In according with Proposition 1, the determination of an SCE reduces to the determination of a saddle point (x0, z*) of the Germeier convolution <£>(x, z) from (2). We obtain the following procedure for calculating an SCE in the game r.
Step 1. Construct the function <£>(x, z) by (2).
Step 2. Find a saddle point (x0, z*) G X x X of the function <£>(x, z).
Step 3. Compute the four values fi[z*], i G N.
Then the pair (z*,f [z*] = (f1[z*],f2[z*], f3[z*], f4[z*])) G X xR4 consists of the SCE z* and the corresponding payoffs of the four players. When the players choose their strategies from the SCE z*, they gain the payoffs fi[z*], i G N, respectively.
Thus, if the function of two variables <£>(x, z) has a saddle point, the well-known numerical methods can be used for computing saddle points.
4. The Existence of SCE in Mixed Strategies
Like the SNE and SBE, the SCE does not exist in pure strategies in the majority of continuous games. Hence, we can naturally employ the strategy randomization approach, which was used in [4-6, 18, 19, 21] to establish the existence of Nash equilibrium in mixed strategies. Following these great scholars, we will establish the existence of an SCE in mixed strategies. For this purpose, some preliminary results are needed, which will help in proving the main existence theorem.
4.1. Preliminaries. First, we introduce some auxiliary notations. Denote by comp Rni and cocomp Rni the set of compact subsets of Rni and the set of convex and compact subsets of Rni, respectively, and also by C(X x Y) the set of real-valued and continuous functions with a domain of definition X x Y.
Assume that the elements of the game r satisfy the following condition.
Condition 1.
Xi G cocomp Rni, Y G cocomp Rm, fi(-) G C(X x Y), for all i G N. (4)
Then, in accordance with Berge' maximum theorem [17] , the function x ^ fi [x] is well-defined and continuous on X for all i G N.
Next, we construct the mixed extension of the game rg, which includes the sets of mixed strategies and mixed strategy profiles as well as the expected value of the players' payoff functions.
First, we associate with each strategy set Xi G cocomp Rni the Borel ^-algebra B(Xi), which consists of subsets Q(i) of Xi such that the intersection and union of a countable set of elements of B (Xi) belong to B (Xi); moreover, B (Xi) is the minimal ^-algebra that contains all closed subsets of Xi. In game theory, a mixed strategy vi(-) of player i
can be identified with a probability measure on the compact set of pure strategies Xi. A probability measure is a nonnegative function vi(■) defined on the Borel ^-algebra B(Xi) and satisfies the two conditions:
(C.1) vi ^U Qfc^ = S vi ^Qfc^ for any sequence of disjoint elements {Q£i)} of B(Xi)
(countable additivity); (C.2) vi(Xi) = 1 (normality).
Note that (C.2) implies the inequality vi (Q(i)) < 1 for all Q(i) G B(Xi). Denote by {vi} the set of mixed strategies of player i G N. Then a mixed strategy profile of the game rg can be formulated as a product-measure
v (dx) = v1(dx1 )v2(dx2)v3(dx3)v4(dx4);
the set of such measures will be denoted by {v}. The payoff of player i corresponding to his payoff function in the game rg is defined by fi[v] = Jfi[x]v(dx). Then the mixed
extension of the game rg has the form
rg = (N = {1, 2, 3,4}, {Vi}iSN, {fi[v]}igN>. (5)
Here we have committed an abuse of notations, denoting the expected value of the function fi[x] by fi[v]. The reader can distinguish between the two functions by the variable involved.
Now, we suggest the following definition of equilibrium, using Definition 1 and (1).
Definition 2. A mixed strategy profile v*(■) G {v} is called a mixed strategy coalitional equilibrium (MSCE) of the game r if it is an SCE of the mixed-extension game (5), that is,
(i) v*(■) satisfies individual rationality and coalitional rationality (IRC and CoalRC), which can be derived from the following inequalities
fi[v*, v2, v3, v4] < fi[v*], for all vfc(■) G {vfc}, k = 2, 3,4 and i = 1,2, 3,4; fi[v1, v*, v3, v4] < fi[v*], for all vfc(■) G {vfc}, k = 1, 3,4 and i = 1,2, 3,4; fi[v1, v2, v*, v4] < fi[v*], for all vk(■) G {vk}, k = 1, 2,4 and i = 1, 2, 3,4; (6) fi[v1, v2, v3, v4*] < fi[v*], for all vfc(■) G {vfc}, k = 1, 2,3 and i = 1,2, 3,4;
where v* = (v*, v*, v3*, v*).
(ii) v*(■) satisfies the collective rationality (CLRC), or it is Pareto maximal alternative in the quad-criteria choice problem
= ({v}, {fi[v]}igN>,
that is, for all v(■) G {v} the system of inequalities
/¿[v] > /¿[V], i = 1, 2, 3,4, with at least one strict inequality, is inconsistent.
A sufficient condition for Pareto optimality (see item (ii)) is as follows.
Remark 3. A mixed strategy profile (alternative) v*(•) G {v} is Pareto-optimal in the multicriteria choice problem = ({v}, {/¿[v]}igN) if
max > /¿[v ] = > /¿[v *].
v(-)eM tN tN
îgN îgN
Consider the function ^¿(x, z), i = 1, 2, 3,4 and also the function
<£>(x,z) = max {<£>r(x,z)} (7)
r=1,...,5
introduced in (2).
Proposition 2. The inequality
max / <£>r(x, z)^(dx)v(dz) < / max <£>r(x,z)^(dx)v(dz) (8)
r=1,...,5 J J r=1,...,5
XxX XxX
holds for all v(■) G {v} and G {v}.
Proof. From (7), for all x, z G X, we obtain the five inequalities
<£V(x,z) < <£>(x, z) = max (x,z), r = 1,..., 5.
r=1,...,5
Integrating both sides of these inequalities with an arbitrary product-measure ^(dx)v(dz) yields
/<£V(x, z)^(dx)v(dz) < / max <£>r(x,z)^(dx)v(dz),
J r=1,...,5
Xx X Xx X
for all v(■) G {v} and r = 1,..., 5. Therefore,
max / <£>r(x,z)^(dx)v(dz) < / max <£>r(x, z)^(dx)v(dz), r=1,...,5 J J r=1,...,5
Xx X Xx X
for all v(■) G {v}. Hence, (8) is satisfied.
□
Remark 4. In fact, Proposition 2 generalizes the well-known property of maximization: the maximum of a sum of some functions does not exceed the sum of their maxima.
Proposition 3. The function <£>(x, z) defined in (7) is continuous on X x Z, where Z = X.
The proof of a more general result (the continuity of the maximum of a finite number of continuous functions on a compact set) can be found in many textbooks, e.g., in [17].
4.2. Existence Theorem. In this subsection, we prove the main result of this paper, that is, the existence of an MSCE in the game r.
Theorem 1. Under the Condition 1, the game r has an MSCE.
Proof. Consider the two-player zero-sum game
where X = Z .In the game Ta, the maximizing and minimizing players choose their strategies from the same set X; <£>(x, z) is the payoff function of the maximizing player and — <£>(x, z) is the payoff function of the minimizing player. Any saddle point (x°, z*) of the function <£>(x,z) is an NE in the game Ta. Really, by the saddle point definition
the strategy profile (x0, z*) is an NE of the game Ta. Now, we associate with the game ra its mixed strategy extension
where is the set of all strategies of the maximizing player; {v} = is the set of all strategies of the minimizing player; v) is the payoff (expected utility) of the maximizing player,
Here, we have committed another abuse of notations, denoting the expected value of the function <£>(x,z) by v). The reader can distinguish between the two by the variables involved. In a similar fashion, any saddle point v*) of the function v) is an NE in the game Ta. Really, by the saddle point definition
ra = ({1, 2},X, Z, ^(x,z)),
^(x,z*) < ^(x°,z*) < ^(x°,z), for all (x,z) G X x Z,
ra = ({1,2}, M, {v}, v)),
(9)
X x!
v*) < v*) < v), for all (u, v) G {v} x {v} (10)
the strategy profile , v*) is an NE in the game Ta.
In 1952 I. Gliksberg established the existence of an NE in mixed strategies for the N-player games with N > 1; see the original paper [12]. Using his result for the two-player zero-sum game ra as a special case, we obtain the following statement. Since the set of all strategy profiles X C Rn is convex and compact and the function <£>(x, z) is continuous on X x X (Proposition 3), the game ra has a mixed-strategy NE v*) satisfying (10). In view of (7) and (9), inequalities (10) take the form
/max <£>r (x, z )^(dx)v*(dz) < r=1,...,5
X xX
< / max <£>r (x, z)^°(dx)v*(dz) < J r=1,...,5
Xx X
< / max <£>r(x,z)^°(dx)v(dz), J r=1,...,5
Xx X
for all v) G {v} x {v}. Letting vj(dzj) = (dxj), i G N, in
<£>(^°, v)= / max (x,z)^°(dx)v(dz), J r=1,...,5
Xx X
we obtain
<£>(^°,^°) = / max <£>r(x, x)^°(dx)^°(dx). J r=1,...,5
Xx X
(In this case, v(dz) = (dx)) From (2) it follows that <£>r(x, x), r = 1,..., 5 for all x G X. Then the previous integral yields <£>(^°, = 0. A similar reasoning leads to <£>(v*, v*) = 0. From (10) we obtain
¥>(/, v*) = 0. (11)
Using (11) and inequalities (10), by transitivity we write
v*) = max <£>r(x,z)^(dx)v*(dz) < 0, for all ^ G {v}.
J r=1,...,5 XX
In accordance with Proposition 2,
max / <£>r(x,z)^(dx)v*(dz) < / max <£>r(x, z)^(dx)v*(dz) < 0, r=1,...,5 j j r=1,...,5
XxX XxX
for all ^ G {v}. Therefore,
J <£>r(x, z)^(dx)v*(dz) < 0, for all ^ G {v} and for all r = 1,..., 5. (12)
XxX
We will distinguish two cases as follows.
Case 1. (r = 1,..., 4) Due to (2), (12) and the fact that is normalized (i.e. J ^(dx) = 1), for example, for r = 1 we write
X
/¿[v*,^2- /¿[v*] =
= J / [zi,x2,x3,x4]^(dx)v*(dz) ^y /¿[z]v*(dz) J ^(dx) =
Xx X X X
= J /¿[z1,x2, x3, x4]^(dx)v*(dz) — J fj[z]^(dx)v*(dz) =
Xx X Xx X
= J (fi[zi,x2,x3,x4] — /¿[z])^(dx)v*(dz) <
Xx X
< max{/i[zi,x2 ,x3,x4] — /¿[z]}^(dx)v*(dz) = J
Xx X
= J ^1(x, z)^(dx)v*(dz) < 0,
Xx X
which holds for all i G N. Thus, /¿[v*, ^4]—/¿[v*] < 0, for all i G N and ^ G {vk(■)},
k = 2, 3,4.
Similar considerations can be used to establish the following three inequalities for r = 2, 3,4:
/¿[^i, v*,^3,^4] — /¿[v*] < 0, for all i G N and ^ G {vfc(■)}, k = 1, 3,4, /¿[^1,^2, v3*, ^4] — /¿[v*] < 0, for all i G N and ^ G {vfc(-)},k = 1, 2,4, /¿[^1,^2,^3, v*] — /¿[v*] < 0, for all i G N and ^ G {vfc(■)}, k = 1, 2, 3.
Therefore, the mixed strategy profile v*(■) satisfies the four inequalities (6) from condition (i) of Definition 2. It remains to prove that v*(■) also satisfies condition (ii)
of Definition 2, i.e., that it has the property of Pareto optimality or collective rationality. For this purpose, we will use Remark 3.
Case 2. (r = 5) Due to (2), (12) and the fact that and v*(-) are both normalized (J ^(dx) = J v-(dx) = 1), we write
" " v-1 = V I /¿[xWdx) - V I fjzlv*(dz) =
£ /¿M - £ fi[v= £ / Zi[x1^(dx) - W /¿[z]v-(dz) ¿eN ¿eN is^X i£NX
= f £ /iMMdx^ v -(dz) -J £ /¿[z]v-(dz) ^(dx)
X X ¿e
£ /¿[x] - £ /¿[z]
.ieN ¿eN
X ¿eN X X ¿eN X
XX
^(dx)v*(dz) =
^5(x, z)^(dx)v*(dz) < 0.
X xX
Therefore, ¿gN /¿[^] - ¿gN /¿[v-] < 0 for all ^ G {v}. Then, in accordance with Remark 3, the mixed strategy profile v*(•) is Pareto-optimal alternative in the multicriteria choice problem
fv = <{v}, {/¿[v]}teN>.
Thus, we have established that the mixed strategy profile v*(•) is an SCE in the game
rg. By Definition 2, v*(-) is an MSCE in the game r and /[v-] is the players' payoff vector.
□
Conclusion
This paper has contributed to the theory of cooperative normal-form games in the following way. First, the concept of the strong coalitional equilibrium (SCE) in normalform games under uncertainty has been formalized. This concept considers the interests of all coalitions. Second, a constructive procedure for determining a pure-strategy SCE has been provided; this procedure reduces to saddle point calculation for a function of two variables. Third, the existence of SCE in mixed strategies has been proved under standard assumptions of cooperative came theory (continuous payoff functions of the players, compact and convex strategy sets of the players, and a compact set of uncertainty).
In our view, the following qualitative results of this paper are important.
1. The approach presented here can be extended to the games with any finite number of players (more than four players).
2. An SCE x* G X is stable against any deviation of any coalition of players and is attractive, because when a coalition chooses its strategies from x*, all other players will have incentive to choose their strategies from x* as well.
3. The SCE could be applied even if the coalition structure changes over time.
4. The SCE could be used for the formation of stable alliances.
5. Game theory has been focusing on individual rationality and collective rationality so far. On the one hand, the individual interests of players are represented by the prominent Nash equilibrium with its selfish character (each player acts for himself only). On the other, the collective interests of players are represented by the concept of Berge equilibrium with its altruism (each player helps others, neglecting his own interests). Such an omission is not rooted in the human nature of players. The SCE partially addresses the incomplete representation of human behavior in the two concepts mentioned. In the game r, when player 1 chooses his SCE strategy, he does not neglect his own interests as an SCE is also a Nash equilibrium; moreover, in accordance with (1), he also helps (maximizes the payoffs of) all other players, which is the inherent property of a Berge equilibrium. The other players act in a similar way. Thus, the SCE fills the gap between the concepts of Nash equilibrium and Berge equilibrium, completing the former by «caring about others» and the latter by «caring about oneself».
Finally, we suggest two possible ways of extending this research. The first is to investigate the SCE in finite games. The second is to consider other approaches to manage uncertainty, such as the maximin regret principle.
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