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14удк: 519.83 msc2010: 91a35
HYBRID EQUILIBRIUM IN N-PERSON GAMES © K. N. Kudryavtsev
South Ural State University 76, Lenin prospekt, Chelyabinsk, 45480, Russia Chelyabinsk State University 129, Bratiev Kashirinykh st., Chelyabinsk, 454001, Russia e-mail: kudrkn@gmail.com
© V. I. Zhukovskiy
Lomonosov Moscow State University Faculty of Computational Mathematics and Cybernetics 1, Leninskiye Gory, Moscow, Russia, 119991, e-mail: zhkvlad@yandex.ru
© L. V. Zhukovskaya
Central Economics and Mathematics Institute of Russian Academy of Science 32, Nakhimovsky prospect, Moscow, 117418, Russia e-mail: Zhukovskaylv@mail.ru
Hybrid Equilibrium in N-person Games.
Kudryavtsev K. N., Zhukovskiy V. I., Zhukovskaya L. V
Abstract. How can we combine altruism of Berge equilibrium with selfishness of Nash equilibrium? The positive answer to this question will be given below. In short, they can be combined but in the class of mixed strategies. For a noncooperative N-player normal form game, we introduce the concept of hybrid equilibrium (HE) by synthesizing the concepts of Nash and Berge equilibria and Pareto maximum. Some properties of this equilibrium are explored and its existence in mixed strategies is established under standard assumptions of mathematical game theory (convex and compact strategy sets and continuous payoff functions).
Keywords: Berge equilibrium, Nash equilibrium, Pareto optimum, Germeier convolution, noncooperative game
Introduction
In 1949 twenty-one years old Princeton University postgraduate J.F. Nash suggested and proved the existence of a solution [1, 2], which subsequently became known as Nash equilibrium (NE). Nash equilibrium has been widely used in economics, military science, policy, and sociology. After 45 years, J. Nash together with R. Selten and J. Harsanyi were awarded the Nobel Prize in Economic Sciences "for their pioneering analysis of equilibria
in the theory of non-cooperative games." The whole point is that NE has stability against any unilateral deviations of a single player, which explains its success in economic and political applications.
Almost every issue of modern journals on operations research, systems analysis or game theory contains papers with the concept of Nash equilibrium. However, there are spots on the sun: an obvious drawback of NE consists in pronounced selfishness as each player seeks to increase his/her/its own payoff only.
The antipode of NE is the concept of Berge equilibrium (BE): each player applies every effort to maximize the payoffs of the other players, neglecting his/her/its individual interests. BE was formalized in 1985 by V. Zhukovskiy [3] as a possible solution of noncooperative N-player games, after perusal of C. Berge's book Théorie générale des jeux a n personnes [4] published in 1957 (which explains the term "Berge equilibrium"). In 1995 Russian mathematician K. Vaisman defended his Candidate of Sciences Dissertation entitled "Berge equilibrium" at Department of Applied Mathematics and Control Processes (St. Petersburg State University) under scientific supervision of Zhukovskiy. This dissertation and Vaisman's early papers [5, 6] attracted the attention of researchers, in Russia and then abroad. As of today, the number of publications related to this equilibrium has exceeded three hundreds. BE is a good mathematical model for the Golden Rule of ethics ("Behave to others as you would like them to behave to you.") [7, 8]. BE is famed for its altruism.
Obviously, these streaks - selfishness and altruism are intrinsic (in some proportion) to any individual, including a conflicting party. However, it seems delusive to expect that such a combined solution exists in pure strategies. Therefore, again following the approach of E. Borel [9], J. von Neumann [10], J. Nash [1] and their scholars, we will establish the existence of combined Nash-Berge equilibrium in mixed strategies. This solution is called hybrid equilibrium (HE). The main goal of this paper is to prove the existence of HE in mixed strategies. Also note a negative property of NE [11] and BE: the sets of both types of equilibria are internally unstable, i.e., there may exist two (NE or BE) profiles such that the payoff of each player in one of them is strictly greater than in the other. We will remove this "negativity" by adding the Pareto maximality of HE with respect to all other equilibria. Well our formalization unites three properties, namely, a HE is first, a Nash equilibrium,, second, a Berge equilibrium,, third, Pareto maximal with respect to the other equilibria.
This article proves the following result: if a noncooperative N-player normal form game has bounded convex and closed strategy sets of players and also continuous payoff functions, then there exists a HE in mixed strategies in this game.
In addition, we obtain sufficient conditions for the existence of HE that are reduced to saddle point calculation for a special Germeier convolution of payoff functions.
1. Formalization of hybrid equilibrium
Consider a mathematical model of a conflict as a noncooperative N-player normal form game described by an ordered triplet
r = <N, (Xi}i€n, {/i(x)}ien>.
Here N = {1, 2,..., N} denotes the set of serial numbers (indexes) of players (N > 1); each of N players chooses his/her/its strategy xi E Xi C Rni, thereby forming a strategy profile
x =(xi,...,xN) e X=nx C Rn (n = ^2 ni)
ie n ien
in this game; a payoff function /¿(x) is defined over the set X, which gives the payoff of player i (i E N). At conceptual level, each player i in the game r seeks for choosing a strategy xi that would maximize his/her/its payoff.
A natural approach is the define a solution of the game r using a pair
(x*, /(x*) = /i(x*),..., /n(x*)) E X x RN,
where the strategies of a profile x* = (x*,..., xN) E X1 x ... x XN = X are determined by an optimality principle while the elements of a vector / (x*) specify the corresponding payoffs of players under these strategies. As noted by N. Vorobiev, the founder of the largest national scientific school on game theory, ".. .the practice of games shows that all the optimality principles developed so far directly or indirectly reflect the idea of a stable strategy profile that satisfies these principles..." [12, p. 94]. For introducing the concept of hybrid equilibrium, we will adopt three optimality principles, namely, Nash equilibrium, Berge equilibrium (from theory of noncooperative games) and Pareto maximum (PM, from theory of multicriteria choice problems). Interestingly, each of these principles has its own type of stability: NE is stable against the unilateral deviations of any player i (i.e., the deviations of xi from x*); BE is stable against the deviations of all players except given player i subject to the payoff function /¿(x) (i.e., the deviations of (x1,..., xi-1,xi+1,..., xN) from (xi,..., x*-1,x*+1,..., xN)); PM is stable against the deviations of all players (i.e., the deviation of the whole current profile x from the optimal solution x*). Using the standard notation (x|zi) = (x1,..., xi-1, zi, xi+1,... ,xN) of noncooperative games, we proceed with the following notions.
Definition 1. A strategy profile xe = (xl,..., xf,..., xeN) G X is called a Nash equilibrium in the game r if
max fi(xe\\xi) = /¿(xe) (i G N). (1)
XiGXi
Definition 2. A strategy profile xB = (xf , ...,xB, ...,xN) G X is called a Berge
equilibrium in the game Г if
max/¿(x\\xf ) = /¿(xB) (i G N). (2)
Let us associate the game r with the N-criteria choice problem
rv = <X, /(x)>,
where the set of alternatives X coincides with the set of strategy profiles X in the game r and the vector criterion has the form /(x) = (/i(x),..., /N(x)), consisting of the payoff functions /¿(x) of all players i G N in the game r.
Definition 3. An alternative (here a strategy profile x G X) is Slater (Pareto) maximal in the problem rv if, for all x G X, the system of inequalities
/¿(x) >/i (x*) (i G N)
(/¿(x) > /¿(xP) (i G N), respectively), with at least one strict inequality, is inconsistent.
Corollary 1. It is possible to suggest the following sufficient condition of Pareto m,axim,ality: if
max V^ /¿(x) = V^ /¿(x*^
_Ef(x) = E Vx G X, (3)
¿en ¿en
then the strategy profile x* is Pareto maximal in the problem rv.
Now, we introduce the central concept of this paper.
Definition 4. A pair (x*, f (x*)) G X x RN is called a Pareto hybrid equilibrium (PHE) in the game Г if the strategy profile x* is simultaneously a Nash equilibrium and a Berge equilibrium in this game and also a Pareto maximal alternative in the multicriteria choice problem rv, i.e., the PHE x* satisfies three conditions as follows:
max fi(x*||xi) = fi(x*) (i G N),
xieXi (4)
maxfi(x||x*) = fi(x*) (i G N), xex
x* is Pareto maximal in .
Remark 1. In accordance with Corollary 1, a strategy profile x* is a PHE in the game Г if it simultaneously satisfies the three optimality conditions (1)-(3).
Remark 2. By analogy with Definition 4, we may easily introduce the concept of Slater hybrid equilibrium (SHE), just replacing the Pareto maximality of x* with its Slater maximality in the problem .
2. Properties of hybrid equilibria
Hereinafter, the notation cocomp Rn is used for the set of convex and compact subsets from space Rn while <^(-) E C(X) for a continuous scalar function <^(x) defined over X. In this section, let the game r satisfy the assumptions
Xi E cocompRni, /i(-) E C(X) (i E N). (5)
Property 1. Under conditions (5), any PHE in the game r is simultaneously a SHE; the set of all SHE is compact in X x RN (perhaps, empty).
Property 1 directly follows from the fact that a Pareto maximal alternative in the choice problem rv is also Slater maximal (in general, the converse fails) while the set of Slater maximal alternatives Xs in rv is a nonempty compact set in X [13, p. 142].
The sets of Nash and Berge equilibria, Xe and XB, in the game r are also compact in X (perhaps, empty) if assumptions (5) hold. In this case, the intersection of the three compact sets (Xs P| Xe P| XB) = X* is also a compact set in X (again, can be empty). The compactness of /(X*) = {/(x)|x E X*} is immediate from the continuity of the payoff functions /¿(x) over X (i E N).
Note that the set of PHE can be noncompact, generally speaking, due to the noncompactness of the set of all Pareto maximal alternatives XP in the choice problem rv. Also keep in mind the inclusion /(XP) C /(Xs).
Property 2. Under assumptions (5), the PHE x* satisfies the individual rationality condition, i.e.,
/¿(x*) > max min /¿(xi,in\|t}) =
xieXi xN\{i}eXN\{i} (6)
= min /i(x0,xn\{i}) = (i E N)
xN\{i}eXN\{i}
where x = (x1,..., xi,..., xN) = (xi,xN\{i}), xN\{i} = (x1,..., xi-1, xi+1,..., xN) and Xn\{i} = n Xj (N\{i} = 1,...,i - 1,i + 1,..., N).
jen\{i}
Really, each Nash equilibrium x* in the game r has property 2 (individual rationality), i.e., /¿(x*) > /¿q (i E N), where x0 and /¿0 are the maximin strategy and payoff of player i, respectively.
Remark 3. As illustrated by Vaisman's counter-example [14, pp. 68-69], individual rationality generally fails for a Berge equilibrium xB in the game r.
Property 3. A PHE x* is collectively rational in a cooperative N -player game without side payments, which appears from the Pareto maximality of the alternative x* in the choice problem rv.
Remark 4. Individual rationality applies certain requirements to alliances (coalitions) with other players: player i joins a coalition only if his/her/its payoff becomes not smaller than the maximin value f0, which can be achieved by this player independently using the maximin strategy x0.
Collective rationality drives all players to the largest payoffs (in the vector sense!) -the Pareto maximums.
As x is a Nash equilibrium, each player seeks to maximize his/her/its payoff.
Berge equilibrium matches an altruistic aspiration of each player for the maximal payoffs of all other players.
Actually, the first two requirements (individual and collective rationality) are among the standard criteria of "good" solutions for cooperative N-player games without side payments. At the same time, Nash and Berge equilibria are new properties for such games, which (we believe) makes the novel concept of PHE an efficient solution for the game r.
3. Sufficient conditions
To formulate sufficient conditions for the existence of PHE in the game Г, we will ensure Pareto maximality in terms of Definition 3 by satisfying equality (3). The sufficient conditions will be based on the original approach from [15]. Introduce an N -dimensional vector z = (zi,..., zN) G X and the Germeier convolution [16] of the form
^¿(x,z) = fi(z||xi) - fi(z) (i G N),
^i+N(x z) = fi(x|zi) - fi(z) (i G N) (7)
+i(x,z) = Ejen fj(x) - Ejen fj(z)> ^(x, z) = maxr=i.....2N+i ^r(x, z).
A saddle point (x0, z*) G X x X of the scalar function ^(x, z) (7) is given by the chain of inequalities
^(x, z*) < ^(x0, z*) < ^(x0, z) Vx G X, z G X. (8)
Theorem 1. If (x0,z*) is a saddle point of the function <^(x, y) (8) in the zero-sum, two-player game
Га = (X, Z = X,^(x,z)), then the maximin strategy z* G X is a PHE of the game Г.
Proof. Really, formula (7) with z = x0 gives -0(x°,x0) = 0. Then, by transitivity,
#z,z*) < 0 Vx E X.
Using max (x, z*) < 0 Vx E X and (7), we arrive at 2N + 1 inequalities of the form
r=1,...,2N+1
/i(z*|xi) < /¿(z*) Vxi E Xi (i E N), /¿(x|k*) < /¿(z*) Vx E X (i E N), E /j(x) < E /j(z*) Vx E X.
jen jen
Here the first N inequalities make z* E X a Nash equilibrium in the game r (see (1)); the second group of inequalities ensures that z* is a Berge equilibrium as dictated by (2); finally, the last (2N + 1)th inequality means that z* is a Pareto maximal alternative in the choice problem rv. □
Remark 5. In accordance with Theorem 1, PHE design is reduced to calculation of a saddle point (x°,z*) for the Germeier convolution ^(x,z) (7). Thus, we have developed a constructive method of PHE design in the game r, which includes the following steps: first, define the scalar function ^(x,z) using formulas (7);
second, find a saddle point (x°,z*) of the function ^(x,z) (see the chain of inequalities (8));
third, calculate the values /¿(z*) (i E N).
Then the pair (z*, /(z*) = (/1(z*),..., /N(z*))) is a PHE in the game T: each player i E N should apply his/her/its strategy from the profile z*, thereby obtaining the payoff
/¿(z*).
Remark 6. The whole complexity of PHE design in the game r lies in calculation of the saddle point (x°,z*) for the Germeier convolution (7). The fact is that maximization of a finite number of functions (x,z) (r = 1,..., 2N + 1) spoils the differentiability and concavity (or convexity) of (x,z), although preserving the continuity of this function over the product X x Z of the compact sets X and Z, see [17, p. 54]. Here we face a situation well described by C. Hermite: "I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives". So it is necessary to develop numerical calculation methods for the saddle point (x°,z*) of the Germeier convolution jm&X (x,z). Unfortunately, no literature has been found in this field of research to date. In particular, the saddle point calculation problem was not solved at the International Conference on Constructive Nonsmooth Analysis and Related Topics (CNSA-2017, St. Petersburg, May 22-27, 2017) dedicated to the Memory of Professor V. Demyanov.
4. Existence of Pareto hybrid equilibrium in mixed strategies
One must be a reckless optimist for considering the game Г (especially an explicit form of the payoff function) with PHE in pure strategies x* E Xi (i E N) (by Definition 4, the desired strategy profile x* is simultaneously a Nash equilibrium and a Berge equilibrium in the game Г and also a Pareto maximal alternative in the corresponding choice problem). Thus, employing the approach of E. Borel [9], J. von Neumann [10], J. Nash [1] and their followers, we will extend the set Xj of pure strategies xj to the mixed ones. Then we will establish the existence of appropriately formalized mixed strategy profiles in the game Г that satisfy the three requirements of hybrid equilibrium.
As before, cocomp Rni stands for the set of all convex and compact (closed and bounded) subsets of the Euclidean ^-dimensional space Rni while /¿(-) E C(X) means that a scalar function fj(x) is continuous over X.
Again consider the noncooperative N-player game Г without side payments. Without special mention, assume the elements of the ordered triplet Г satisfy requirements (5), i.e.,
Xj E cocompRni, fj(-) E C(X) (i E N).
For each compact set Xj С Rni (i E N), construct the Borel a-algebra B(Xj), i.e., the set of all subsets of Xj such that Xj E B(Xj), and B(Xj) is closed with respect to the complement and union of a countable number of sets from B(Xj); in addition, B(Xj) is the minimal a-algebra that contains all closed subsets of the compact set Xj.
Therefore, construct the Borel a-algebra B(Xj) for each compact set Xj (i E N) and
the Borel a-algebra B(X) for the set X = П Xj of all strategy profiles, assuming that
ien
B(X) contains all Cartesian products of the elements from the Borel a-algebras B(Xj) (i E N).
Within the framework of mathematical game theory, a mixed strategy vj(-) of player i is identified with a probability measure over the compact set Xj. By definition [18, p. 271], in the notations of [19, p. 284] a probability measure is a nonnegative scalar function vj(-) defined on the Borel a-algebra B(Xj) of all subsets of the compact set Xj С Rni that satisfies two conditions as follows:
(1) Vi ^U = U V for any sequence (Q^I^Li of pairwise disjoint elements
from B(Xj) (countable additivity);
(2) vj(Xj) = 1 (normalization), which implies vj (Q(j)) < 1 for all Q(j) E B(Xj).
Designate as (vj| the set of all mixed strategies of player i (i E N).
Also note that the product measures v(dx) = v1(dx1).. (dxN), treated in the sense of the well-known definitions of [18, p. 370] (and the notations of [19, p. 123]), are probability
measures over the strategy profile set X. Let {v} be the set of such probability measures (strategy profiles). Once again, we emphasize that in the design of the product measure v(dx) the role of the a-algebra of all subsets of the set Xi x ... x XN = X is played by the smallest a-algebra B(X) that contains all Cartesian products Q(1) x ... x ), where G B(Xj) (i G N). The well-known properties of probability measures [20, p. 288], [18, p. 254] imply that the sets of all possible measures Vj(dxj) (i G N) and v(dx) are weakly closed and weakly self-compact (see [18, pp. 212, 254], and [21, pp. 48, 49]). As applied, e.g., to {v}, this means that from any infinite sequence {v(k)} (k = 1, 2,...) it is possible to extract a subsequence {v)} (j = 1, 2,...) with a weak convergence to a measure v(0)(-) G {v}. In other words, for any scalar function ^(x) that is continuous over X, we have
lim / ^(x)v)(dx) = ^(x)v(0)(dx)
J J
X X
and v(0)(-) G {v}. Owing to the continuity of ^(x), the integrals f -0(x)v(dx) (the
X
expectations) are well-defined; by Fubini's theorem,
J <^(x)v (dx) = J - J ^(x)vN (dxN) ...v1(dx1),
X Xi Xn
and the order of integration can be interchanged.
Let us associate the game r in pure strategies with its mixed extension
r = (N, {vt}i€N, {fi[v] = J f [x]v(dx)}i€n>, (9)
X
where, like in r, the set N consists of all serial numbers (indexes) of players while {v^} is the set of mixed strategies vj(-) of player i; in game (9), each conflicting party i G N chooses its mixed strategy vj(-) G {v^}, thereby forming a mixed strategy profile v(■) G {v}; the payoff function of each player i, i.e., the expectation
fi[v] = J fi[x]v(dx^
X
is defined over the set {v}.
For game (9), the notion of a PHE x* (see Definition 4) has the following analog.
Definition 5. A mixed strategy profile v*(■) G {v} is called a hybrid equilibrium (HE) in the mixed extension (9) (equivalently, a hybrid equilibrium in mixed strategies in the game r) if
first, the profile v*(•) is a Nash equilibrium in the game r, i.e
max /¿(v*||vj) = /¿(v*) (i G N);
• i
(10)
max
vN\{i} (•)€(vN\{i}}
(11)
and third, v*(•) is a Pareto maximal alternative in the N-criteria choice problem
in addition, denote by {v*} the set of hybrid equilibria v*(•), i.e., the strategy profiles that satisfy the three requirements of Definition 5.
Consider several results used below for proving the existence of HE in mixed strategies. The following sufficient condition of Pareto maximality is obvious, see the statement below.
Proposition 1. A mixed strategy profile v*(•) G {v} is a Pareto maximal alternative in the choice problem rv = ({v}, {/¿(v)}ieN) if
Proposition 2. Consider the game r under conditions (5), i.e., the sets Xj are convex and compact while the payoff functions /¿(x) are continuous over X = Xi x ... x XN. Let {ve} be the set of Nash equilibria ve(-) that satisfy (10) with v*(-) replaced by ve(-); {vB} be the set of Berge equilibria vB(•) that satisfy (11) with v*(•) replaced by vB(•); {vP} be the set of alternatives vP(•) that satisfy (12) with v*(-) replaced by vP(•) (i.e., vP is a Pareto maximal alternative in mixed strategies in the N-criteria choice problem
vN\{i}(dxn\{i}) = vi(dxi)... vi-i(dxi-i)vi+i(dxi+i)... vw(dxw), (v||v*) = vi(dxi)... vi-i(dxi-i)v*(dxj)vj+i(dxj+i)... vw(dxw), v*(dx) = vj:(dxi)... vN(dxN);
(12)
({v}, {/i(v)}i€n)).
«TaepunecKuu eecmuuK urnfiopMamuKU u MameMamuKU», № 3 (44)' 2019
Then the set {v*} of hybrid equilibria v*(•) in the mixed extension r of the game r is a weakly compact-in-itself subset of the set of mixed strategy profiles {v} in the game r (although, {v*} can be empty).
Proof. Under conditions (5), we have {ve} = 0 in accordance with the Gliksberg theorem [22]. Next, the fact {vB} = 0 has been established in the preceding sections of our book. The non-emptiness of the set of Pareto maximal alternatives, {vP} = 0, can be demonstrated by analogy. The intersection of a finite number of weakly compact sets (in our case, three ones) is also weakly compact, perhaps empty. □
Corollary 2. Under conditions (5), the set
/ ({v *})= U f (v ),/ = (/1,..., /n ),
is compact (bounded and closed) in the N -dimensional Euclidean criterion space RN.
Theorem 2 below proves the implication (5) ^ {v*} = 0, which is the central result of this article.
Proposition 3. Consider game (9) under conditions (5). Then the function (x,z) from
-0(x,z) = max (x,z ) (13)
r=1,...,2N,2N+1
satisfies the inequality
max f (x,zWdx)v(dz) < r=1,...,2N,2N+1x^x
< f max (x,z Wdx)v (dz)
XxX r=1,...,2N,2N
(14)
for any G {v} and v(■) G {v}, where
^i(x, z) = /¿(x||zi) - fi(z) (i G N),
Pj (x,z) = fj (z|xi) - fj (z) (j G{N +1,..., 2N}), (15)
^2N+1(x, z) = E [fi(x) - fi(z)].
¿en
This proposition was argued in [11].
Remark 7. In fact, formula (14) generalizes the well-known property of maximization: the maximum of a sum does not exceed the sum of the maximums.
Now, proceed with an interesting fact from operations research, which will be vital for proving Theorem 2. Consider (2N + 1) scalar functions (x, z) (r = 1,..., 2N, 2N + 1), where z = (z1,..., zN) G Z = X and Pj(x, z) (j = = 1,..., 2N + 1) are defined by (15).
Proposition 4. If (2N + 1) scalar functions Wj(x,z) (j = 1,..., 2N +1) are continuous over the product X x (Z = X) of compact sets, then the function
-0(x,z) = max Wj (x,z)
j=1,...,2N +ij
is also continuous over X x Z.
The proof of a more general result can be found in many textbooks on operations research, e.g., [17] and [23].
Finally, let us establish the central result of this article - the existence of a hybrid equilibrium (HE) in mixed strategies under conditions (5).
Theorem 2. If in the game Г the sets Xj E cocomp Rni and /¿(-) E C(X) (i E N), then there exists a hybrid equilibrium in mixed strategies in this game.
Proof. Consider an auxiliary zero-sum two-player game
Га = ((1, 2|, (X, Z = X|,#r,z)>.
In the game Г", the set X of strategies x chosen by player 1 (which seeks to maximize -0(x,z)) coincides with the set of strategy profiles of the game Г; the set Z of strategies z chosen by player 2 (which seeks to minimize -0(x,z)) coincides with the same set X. A solution of the game Г" is a saddle point (x0, zB) E X x X; for all x E X and each z E X, it satisfies the chain of inequalities
^(x,zB) < ^(x°,zB) < ^(x°,z).
Now, associate the game Г" with its mixed extension
Га = ((1, 2|, M, (v )>,
where (v} and = (v} denote the sets of mixed strategies v(■) and of players 1 and 2, respectively. The payoff function of player 1 is the expectation
)= J ^(x,z)^(dx)v(dz).
XxX
The solution of the game Г" (the mixed extension of the game Г") is also a saddle point v*) defined by the two sequential inequalities
*) < ^°,v*) < ^°,v) (16)
for any v(■) E (v| and E (v}.
Sometimes, this pair v*) is called the solution of the game Г" in mixed strategies.
In 1952, I. Gliksberg [22] established the existence theorem of a mixed strategy Nash equilibrium for a noncooperative game of N > 2 players. Applying this theorem to the zero-sum two-player game r" as a special case, we obtain the following result. In the game ra, let the set X C Rn be nonempty, convex and compact while the payoff function ^(x, z) of player 1 be continuous over X x X (note that the continuity of ^(x,z) is assumed in Proposition 4). Then the game T" has a solution v*) defined by (16), i.e., there exists a saddle point in mixed strategies in this game.
Taking into account (13), inequalities (16) can be written as
f max p,(x,zWdx)v*(dz) <
XxX j=1,...,2N,2N+1 JX
< f max p, (x, z)u°(dx)v*(dz) < (17)
XxX j=1,...,2N,2N+1 j (1()
< f max p, (x,z)u°(dx)v (dz) XxX j'=1,...,2N,2N + 1 j
for all v(■) G {v} and G {v}. Using the measure vj(dzj) = ^°(dxi) (i G N) (and hence v(dz) = ^°(dx)) in the expression
-0(u°,v )= / max p, (x,z )u°(dx)v (dz), J j=1,...,2N,2N+1 j
XxX
we obtain ) = 0 on the strength of (13). Similarly, ^(v*,v*) = 0, and then it
appears from (16) that
^°,v *) = 0.
The condition = 0 and the chain of inequalities (16) by transitivity give
*) = J j=1 inax^+1 Pj(x,z)Mdx)v*(dz) < 0 VM0 G {v}.
Xx X
In accordance with Proposition 3, then we have
0> f max p,(x,zWdx)v*(dz) > XxX j=1,...,2N,2N+1 j
> max f p, (x,zWdx)v * (dz). j=1,...,2N,2N+1X^X j
Therefore, for all j = 1,..., 2N, 2N +1,
J pj(x,z)^(dx)v*(dz) < 0 tyu(-) G {v}. (18)
Xx X
Consider three cases as follows.
Case I (j = N,..., 2N). Here, by (18), (15) and the normalization of we arrive
at
0 > J ^n+i(x,z)v°(dx)v(dz) = J [fi(z\\xi) - fi(z)]^°(dx)v(dz) =
XxX XxX
= f fi(z\\xi)^°(dx)y(dz) - f fi(z)^°(dx) f v(dz) =
Xx X X X
= fi(v°\\vi) - ) Vv(■) e{v} (i e N).
By (10), is a Nash equilibrium in the game Г (equivalently, a Nash equilibrium in mixed strategies in the game Г).
Case II (j = 1,..., N). Again, using (18), (15) and the normalization of v(■),
0 > J <^i(x,z)^(dx)v*(dz)= J [fi(x\\zi) - fi(z)]y(dx)v*(dz) =
XxX XxX
= f fi(x\\zi)^(dx)v*(dz) - f fi(z)y(dz) J v*(dz) =
X xXi X X
= fMv*) - fi(v*) V»(-) e{v} (i e N).
On the strength of (11), the mixed strategy profile v*(■) is a Berge equilibrium in the game Г by Definition 5.
Case III (j = 2N + 1). Again, using (18), (15) and the normalization of v(■) and ^(■), we have
0 > J [Eren fr (x) - Eren fr (z)] V(dx)v*(dz) =
XxX
= I Eren fr(x)^(dx)lX v*(dz) - I V(dx)J Eren fr(z)v*(dz) =
X XX
= Eren fr(p) - Eren fr(v*) Vrt) e {v}. In accordance with Proposition 1 and (12), the mixed strategy profile v*(■) e{v} of the game Г is a Pareto maximal alternative in the multicriteria choice problem
Г = {{v}, {fi(v)}ien).
Thus, we have proved that the mixed strategy profile v*(■) in the game Г is simultaneously a Nash equilibrium and a Berge equilibrium that satisfies Pareto maximality. Hence, by Definition 5, the mixed strategy profile v*(■) is a hybrid equilibrium in the game Г. □
Remark 8. Note that a bimatrix games do not satisfy the conditions of Theorem 2.
Conclusion
In this work, we propose a new concept of equilibrium. It combines the concepts of Nash and Berge equilibria and Pareto maximum. In the future, we hope to develop constructive methods for constructing this equilibrium.
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