ACCEPTABLE POINTS IN GAMES WITH PREFERENCE RELATIONS Текст научной статьи по специальности «Математика»

Contributions to Game Theory and Management, XI, 196-206

Acceptable Points in Games with Preference Relations

Victor V. Rozen

Saratov St,ate University, Astrakhanskaya st. 83, Saratov, 410012, Russia E-mail: rozenvv@info. sgu.ru

Abstract For games with preference relations we introduce an acceptability concept. An outcome of a game is called an acceptable one if no players which have an objection to it in the form of some strategy (all of the required definitions are clarified in the introduction, see section 1). It is easy to show that every outcome at equilibrium point is an acceptable one but the converse is false. An aim of this article is a finding of conditions for existence of acceptable outcomes for games with preference relations (see sections 2 and 3). These conditions relate both to strategies and the preference relations of the players. The main requirements concerning the preference relations are acyclic and transitivity. It is a very important fact, that for game in which the sets of strategies of players are finite, the set of acceptable outcomes is non empty. For the class of games with payoff function acceptability condition is equivalent to individual rationality condition. An example of infinite game in which the set of acceptable outcomes is empty is given in section 4. Keywords: game with preference relations, Nash equilibrium point, general equilibrium point, acceptable point.

1. Introduction

It is known that the equilibrium concept is the main game-theoretic optimality principle. However for realization of this principle we need in the introduction of mixed strategies. This fact is burdensome in terms of the applications of game theory what stimulates an investigation of other solution concepts.

In this article we study the so called acceptability concept for games with preference relations that is games in which a goal structure is given by binary relations on the set of possible outcomes. We consider acceptable situations and acceptable outcomes as optimal solutions in game with preference relations. Let us give precise definitions.

Formally, a game of n players with preference relations in the normal form can be given as a system of the type

G = (N, (Xi)i€N ,A, (pi)ieN ,F> (1)

where N = {1,..., n} is a set of players, n > 2; Xi is a set of strategies of the player i; A is a set of outcomes; pi Ç A2 is a preference relation for player i; F is

X = Xi

ieN

the set of outcomes A. We assume that |Xi| > 2 for all i G N Mid |A| > 2. A game G is called finite one if all sets Xi (i G N) are finite. In general case we suppose the pi

Pi

The correlation a1 < a2 means that the outcome a2 is not less preferable than the outcome a1 for player i. A game G is said to be a game with ordered (or quasi-ordered) outcomes if all (pi)ieN are order (respectively, quasi-order) relations.

Definition 1. A situation x° = (x°) N in the game G of the form (1) is called Nash equilibrium point if for all i € ^^d xj e Xj the correlation

F (x° II Xj) < F (x°)

holds.

In the case when preference relations (pj)j£N not satisfy the linearity condition, we can consider a certain generalization of Nash equilibrium concept in the following manner.

Definition 2. A situation x° = (x°) jeN in game G is called a general equilibrium point if there does not exist i e ^^d xj e Xj such that

F (x° || Xj) > F (x0) .

Obviously, any Nash equilibrium point is a general equilibrium point also but the converse is false. In the case when all binary relations (pj)j£N satisfy the linearity condition these concepts are equivalent to each other.

G

some i e N and put Xw\j = n Xj. It is evident that we can consider XN\j as a set

jew

j=j

of strategies of the complementary coalition N \ i. A pair (xj; xwwhere xj e Xj and xw\j e Xw\.j uniquely defines some outcome in game G which is denoted by

F (Xj,XN\j)-

Definition 3. We say that a strategy x° e Xj is an objection of player i to outcome a e A if for any strategy xw\j e Xw\j of the complementary coalition the correlation

F (x°, xw> a holds. An outcome a e A is called an acceptable one for player i if

aG outcome is acceptable for all players i e N.

Therefore an outcome a e A is an acceptable one in game G if for any i e N and xj e Xj there exists a strategy xw\j e Xw\j of the complementary coalition such

that the condition - ^F (xj,xw>> a^ holds. Indicated strategy xw\j of complementary coalition is called a punishing strategy.

Some strengthening of the acceptability concept is the following.

Definition 4. An outcome a e A is called quite acceptable one for player i if there exists a strategy xw\j e Xw\j of complementary coalition such that for any xj e Xj

holds the condition - ^F (xj ,xw> ^ ^n outcome a is called ginie acceptable one in game G if it is quite acceptable for all players i e N.

G

Namely, a situation x e X in game G is called acceptable (or quite acceptable) F (x)

Remark 1. A general equilibrium point is a quite acceptable (and hence an ac-

G

Indeed, let x° = (x°)jeN be a general equilibrium point in game G with preference relations. Put x^\j be the projection of situation x° on Xw\j. Using the definition 2, we obtain for any i e N Mid xj e Xj

Hence for each i G N the strategy xN\j of the complementary coalition N \ « is a punishing om and it does not depend on the deviation of player i. Therefore the outcome F (x0) is a quite acceptable one and the situation x0 is a quite acceptable also.

Remark 2. Equilibrium points and acceptable situations are stable situations of game in the following sense. For acceptable situation, any player's deviation from its original strategy could be "punished" by the complementary coalition of other players. In the case of equilibrium point such punishment occurs when the omission of the other players, i.e. automatically. In the general case of acceptable situation the complementary coalition has only "circuit response" to every possible deviation of the player from his initial strategy (that is "stable based on threats" in terminology of H. Moulin, see Moulin, 1981). Finally, if a situation of a game is quite admissible, the choice of "punishment" by complementary coalition does not depend on the deviation of the player. Therefore in this case for complementary coalition it is sufficiently to know only the fact of deviation of a player from its original strategy.

Note that acceptable points in general cooperative n-person games with payoff functions was study by Aumann and Dreze, 1974. See also the monograph of Moulin, 1981.

2. Sufficient conditions for existence of acceptable outcomes 2.1. Games with acyclic preferences

Theorem 1. Let G be a game with preference relations of the form (1) in which the sets of players strategies are finite. If for any i G N the preference relation pj is

G

G

by Wj the set of all outcomes to which player i G N has some objection, i.e.

The case 1: all Wj = 0. Since according to our assumptions the set A is finite and preference relation pj is acyclic then in graph of strict preferences (A, p*} no

A

(see Rozen, 2013). Fix for all i e N some maximal element a* under preference relation pj in the sub set W^ Becau se a* e Wj, we obtain using (2) that for every i e N there exists a strategy x° e Xj satisfying for any strategy xw\j e Xw\j the correlation

Consider the situation x0 = (x°) N. Since i-th component of this situation is the strategy x0 then for situation x0 the correlation (3) hold s for all i G N i.e.

Wj = {a G A : (3xj G Xj) (Vxw\j G Xw\j) F (xj, xw\j) > a}. (2)

F (x°, in\j) > a*.

(3)

(Vi G N) F(x°) > a*. (4)

Because element a* is et maximal one in the subset Wi, it follows from (4) that F (x°) G Wi for all i G N, i.e. the outcome F (x°) is an acceptable one for each player i G N. Hence x° is an acceptable point in game G.

The case 2: Wi = 0 for some i G N. Put N° = {i G N : Wi = 0} and N1 = {i G N : Wi = 0} Like in the case 1 we can fix some maximal element b* under preference relation pi in every non-void subset Wi (i G N1). In accordance with (2) for every i G N1 there exists the strategy x1 G Xi such that for any strategy

, e Pi

xN\i of complementary coalition N \ i the tradition F (x1 ,xN^J > b* holUs. Now for every i G N° fa arbitrary a strategy x1 G X^. Then in situation x1 = (x1) i£N i

N1

F (x1e> b*. (5)

bi* u e u pei Wi follows from (5) the condition F (x1) G Wi i.e. the outcome F (x1) is an acceptable one for all players i G N ^Because for any i G N ° hold s Wi = 0 then every ouUcome in game G is acceptable for an y player i G N Therefore the outcome

F x1 acceptable for all players i G N, i.e. F (x1) is an acceptable outcome and x1 G

It is shown the existence of acceptable situation (and acceptable outcome also)

G

G

G° = (N, (Xi)i£N ,A°, (p°ei6N ,F>

where A° is the range of function F and for each i G N the preference relation pi° pi A°

A°

pi° G°

acceptable outcome a* G A°. Let us show that the outcome a* is an acceptable one in game G also. Indeed, in the opposite case there exists a player i G N and a strategy xi G Xi which is its objection to the outcome a* in game G, i.e. holds

(VxN\i G XN\i) F (xi, xN\i) > a*. (6)

Since elements F (xi ,xNand a* belong to the set A° then conditions

F (xi,xN\ie > a* and F (xi,xN\ie > a*

are equivalent. Then using (6) we obtain that the strategy xi G Xi is an objection of player i to outcome a* in game G° which leads to contradiction. □

Corollary 1. An antagonistic game with preference relations G = (X, Y, A, F, p> in which sets of strategies X, Y are finite and the preference relation p of player 1 is acyclic, has an acceptable situation (hence an acceptable outcome also). In particular, any finite antagonistic game with ordered outcomes has an acceptable outcome.

p

implies the acyclic condition for inverse relation p-1.

2.2. Games with quasi-ordered outcomes

In this section we consider n-person game G = (N, (Xj)i£N , A, (pi)ieN , F} with quasi-ordered outcomes. Our aim is a finding of condition for existence of acceptable points in such game. For arbitrary i e N define A-domination of strategies for player i e N in game G by the equivalence

x1 < x? ^ (F (x?,Xn\i))f C (F (x1,Xn\i))f . (7)

Remark 3. We denote by (F (xj, XNthe set of all majorant for subset

(F (xj, xn\j)) = {F (xi, xN\j) : xN\i e XN\j} under quasi-order pj; i.e.

| Pi

(F (xj,XN\i)) = {a e A: (3xw\j e Xw\j) a > F (xj, xw\j)}.

Note that subset (F (xj, XN\.j)) is the dual ideal generated by x.j-row F (xj; XN\j) in quasi-ordered set (A, pj}.

Obviously, ^.¿-domination of strategies for player i e N is an quasi-ordering

Pi

on X^. The strict part and the symmetric part of quasi-order < can be written respectively in the form

x1 < x? ^ F (x?, Xn\j)f C F (x1, Xn\j)f ; (8)

x,1 & x? ^ F (x,1, Xn\j)f = F (x?,Xn\j)f . (9)

Theorem 2. Let G = (N, (Xj)j£N , A, (pj)jeN , F} 6e a proe with quasi-ordered, outcomes. Suppose that every player i e N uses its fa-maximal strategy x° e Xj. Then the situation x° = (x°) j£N is acceptable one and the outcome F (x°) also is

G

A proof of theorem 2 is based on lemma 1 which has some independent interest.

Lemma 1. Let x° e Xj &e fa-maximal strategy of player i. Then for any situation x e X the outcome F (x || x°) is an acceptable one for player i.

Proof (of lemma 1). Fix an arbitrary strategy xN\j e XN\j. We need to show that the outcome F xNis an acceptable one for player i. Indeed, otherwise there exists a strategy x1 e Xj such that for any xN\j e XN\j holds

F (x,,xn\j) > f(x°,xN\j) . (10)

Let us show the inclusion

(F (x,,xn\j))f C (f(x°,xn1. (11)

| Pi

Indeed, assume a G (F (x1, XN\j)) i.e. a > F (x1 ,xN\^) for some xN\j G XN\j.

Using the transitivity of quasi-order pj and (10) we obtain a > F ^xn,xNthen

t t a G ^F ^x0,xN• Moreover since ^F ^xn,xNC (F (xn,XN)T we have

(F (xl, Xn\j))T C (F (x0, Xn\i))T . (12)

We now prove that in (12) the inverse inclusion is false. Indeed, otherwise because

F (x?,xNG F (x0,Xn\j) C (F (x0,Xn\j))T,

we obtain F (xn,xNG (F (x1,Xw\j))T i.e. F(xn,xN> F (x1 ,xNfor some xNG XN\j. On the other hand according with (10) we have the strict

inequality F ^x^xN> F ^xn,xNthat contradicts the previous correlaton.

Thus the strict inclusion (F(x1,XN\j))T c (F(xn,XN)T holds and according 1 ft n

with (8) we obtain xl > x0 that contradicts the ^¿-maximality condition of strategy

xn. Lemma 1 is proved. □

()

the situation xn = (xn) j£N in which each player i G N uses its ^¿-maximal strategy

xn is acceptable for all players N that is an acceptable situation in game G and the F xn

We now show some sufficient conditions for an existence of acceptable outcomes in game with quasi-ordered outcomes. These conditions are based on theorem 2.

Let G = (N, (Xj)ieN , A, (p)j£N , F} be a game with quasi-ordered outcomes. Consider the following conditions concerning dual ideals in quasi-ordered set (A, pj}

(i G N).

(CI). For each i G N there exists a strategy xj G Xj such that any strict descending chain of dual ideals of the form

(F (xj,XN\j))T D (F (xl ,XN\j))T D (F (x?, Xn\j))T D ... (13)

is terminated at some finite number.

(C2). For each i G N and strategy xj G Xj any strict descending chain of dual ideals of the form (13) is terminated at some finite number.

(C3). For arbitrary i G N let Xj0 C Xj be some subset of strategies of player

t t i such that for every xj, xj' G Xj0 dual ideals (F (xj,XN\j)) and (F(xj',XN\j))

are comparable under inclusion. Then there exists a strategy x* G Xj satisfying the

condition

p| (F (xj,Xn\0)T = (F(xj. ,Xn\0)T • (14)

XiEX0

Theorem 3. Assume for game G = (N, (Xj)j£N , A, (pj)j£N , F} with quasi-ordered

G

exists an acceptable situation and an acceptable outcome also.

Proof (of theorem 3). Assume the condition (CI) holds. Suppose the chain (13) is terminated at some member (F (x™ , Xwwhere x™ G X.. Then (F (x™, Xwis a maximal dual ideal of the form (F (xj, Xwxj G X. in quasi-ordered set (A, p.}. In accordance with (8) the strategy x™ is ^.-maximal strategy for player i and using theorem 2 we obtain the required statement. In the case when the condition (C2) holds, the proof is similar. Assume now that the con-

/ ft

dition (C3) satisfies. Then it follows from (11) that the quasi-ordered set (X., < is inductive one and according with Zorn's lemma it has a maximal element. It remains to use theorem 2. □

3. Conditions for uniqueness of acceptable outcome

In this section we consider the uniqueness of acceptable outcome problem for games with quasi-ordered outcomes. Firstly note the following

Remark 4. Let G = (N, (Xj)j£N , A, (pj)j£N , F} be a game with quasi-ordered outcomes of the form (1). Consider the so-called a natural equivalence relation e = p| e. where e. = p. n p-1. Since for every i G N the inclusion e C p. holds, the

P e P

conditions a1 > «^d a2 = a2 imply a1 > a2 for all a1; a2, a2 G A. It follows that if some outcome a G A is acceptable one for player i G N then any outcome a' = a is an acceptable one for player i G N also. Therefore the uniqueness of acceptable Ge

Lemma 2. Let x0 be Nash equilibrium point in game G with quasi-ordered, outcomes of the form (1). For any i G N define a set W. consisting of strict guaranteed

i

W = {a G A: (3x. G X.) (VxwG F (x.,xN> a}.

Then the following inclusion holds:

W C {a G A: a< F (x0)}. (15)

Proof (of lemma 2). Assume a G W. i.e. there exists a strategy x* G X. such that

Pi n ( n ) Pi

F (x || x*) > a for any x G X. Set x = x0 and we get F (x° || x*J > a. On the other

Pi

hand, since x° is Nash equilibrium point, the correlation F (x° || x*) < F (x°) holds.

Pi Pi

Because relation < is transitive, it follows from last two correlations that a < F (x°)

which was to be proved. □

Corollary 2. Let x° be Nash equilibrium point in game G with quasi-ordered outcomes of the form (1). Then

U Wj C |J{a G A: a < F (x°)}. (16)

jew .ew

Definition 5. Nash equilibrium point x° in game G is called a special one if in (16) the equality holds, i.e.

U Wj =(J {a G A: a< F (x°)}. (17)

jew .ew

Definition 6. Let A be an arbitrary set and a collection (pj)j£N of quasi-orders on A is given. An element c G A is called a centric one if for any a G A holds a = c

Pi

or a < c for some i G N, where e is the natural equivalence relation.

It is easy to show the following statement.

Lemma 3. Let G = (N, (Xj)j£N , A, (pj)j£N , F} be a (ame with quasi-ordered, outcomes which has Nash equilibrium point xn. Then F (xn) is an unique up to the natural equivalence e acceptable ou)come in game G if and only if the situation xn is a special one and element F (xn) is a centric.

Lemma 3 gives a solution of the uniqueness acceptable outcome problem for games having Nash equilibrium point. A main result connecting this problem for class of games with quasi-ordered outcomes is the theorem 4. We need in the following definition.

(A, p}

p p p p

if every strict ascending chain of the form ai < a2 < ... < ak < ... is terminated, i.e. it has a last element.

Theorem 4. Let G = (N, (Xj)j£N , A, (pj)j£N , F} be a game with quasi-ordered outcomes and for every i G N quasi-ordered set (A, pj} (AC) condition satisfies. Then there exists an unique up to the natural equivalence e acceptable outаome if and only if game G has a special Nash equilibrium point xn and outcome F (xn) is a centric.

Proof (of theorem 4). Necessity. Let a* be an unique up to the natural equivalence e acceptable outcome in game G. For any i G N consider the set Wj consisting of strict guaranteed outcomes of player i (see lemma 2). Denote by Nn the set of all i G N satisfying Wj = 0 and by N1 the set of all i G N satisfying Wj = 0. For every i G Nn fix in non-empty set Wj a maximal element a* under quasi-order pj (an existence of maximal element it follows from (AC) condition). Since a* G Wj then

( ) Pi

there exists a strategy x* G Xj such that the correlation F (x*,xN^J > a* holds for any xN\j G XN\j (i G Nn). Moreover for all i G N1 fix an arbitrary strategy

x* G X^. Let us show that the outcome in situation x * = (x*) is an acceptable one

pi

in game G. Indeed, for every i G Nn the correlation F (x *) > a* holds and because aj

maximal element in subset we obtain F (x *) G Wj, that is the outcome F (x *) is an acceptable one for every player i G Nn. Sinee Wj = 0 for all i G Nl5

G i N1

F (x ) i N

G F (x *) = a * e

G

x G

in situation x * some player k G N instead of strategy uses another strategy xk G Xk. In accordance with definition of situation x * we obtain that outcome F (x * || xk) remains to be acceptable for all players i G N, where i = k. It is possible the following two cases.

Case 1. The outcome in situation x * || xk remains to be acceptable for player k. Then outcome F (x * || xk) is an acceptable for all players i G N, hence in accordance

with uniqueness condition we have F (x * || xk) = a * and since F (x *) = a * we obtain e Pfc

F (x * || xk) = F (x *). Because e C ek C pk, in this case we have F (x * || xk) <

F (x *).

Case The outcome in situation x * || xk is not acceptable for player k. Then F (x * || xk) G Wk hence Wk = 0. In accordance with (AC) condition for quasi-ordered set (A, pk}, the subset Wk has a maximal element 6k G Wk such that

Pk

F (x * || xk) < 6k- Let x'k G Xk be a strategy of player k which strict guarantees the

Pk

outcome 6k to him. Then F (x * || x'k) > hence, using a maximality condition for element 6k in subset Wk, we get F (x * || x'k) G Wk, that is outcome F (x * || x'k) is an acceptable for player k. Since the outcome F (x * || x'k) remains to be acceptable for other players i G N where i = k, we get that the outcome F (x * || x'k) is an G

F (x * || x'k) = a * where e is a natural equivalence in game G. Thus we have the following sequence of correlations:

F (x * || xfc) < 6k < F (x * || x'k) = a * = F (x *).

e C ek C pk pk

Pk

get in this case F (x * || xk) < F (x *).

x G

first affirmations of theorem 4. Then other statements of theorem 4 are to be direct

□□

Corollary 3. A game G = (N, (X.).^ , A, (p.).^ , F} with quasi-ordered, outcomes in which for every i G N quasi-ordered set (A, p.} (AC) condition satisfies has an acceptable outcome.

Indeed, in theorem 4, a proof of the fact that outcome F (x *) is an acceptable G

condition. 4. Examples

4.1. Antagonistic games with payoff functions

Consider an antagonistic game with payoff function r = (X, Y, u} where X is a set of strategies of player 1, Y is a set of strategies of player 2, u is a payoff function. We can mean r a game with ordered outcomes, in which the set of strategies of players are the same, a set of outcomes is real numbers R, realization function is the function u (x, y) and preference relation is determined by the value of payoff. Put v1 = sup inf u (x, y) be the lower value and v2 = inf sup u (x, y) the upper value

of game r. Consider now the following condition.

(C) If the external extremum of sup inf u (x, y) is realized at the point x° G X

xex yeY

then the inner extremum of inf u (x°, y) must be realized at some point y° G Y.

yeY

It is easy to show that for game r considered as a game with ordered outcomes, the set of all acceptable outcomes for player 1 is the interval (v1, to) and possibly the point V1 Moreover, the out come v1 is an acceptable one for player 1 if and only if the condition (C) holds. For finding of all acceptable outcomes for player 2 we can

use a dual condition (C*). Thus the set Acr consisting of all acceptable outcomes of game r is the interval (v1; v2) and possibly points v^d v2. In particular let the sets X, Y be compact topological spaces and the function u is continuous on X x Y. Then the conditions (C) and (C*) hold, hence in this case we have Acr = [v1; v2].

4.2. n-person games with payoff functions

A finding the set of acceptable outcomes in n-person game with payoff functions can be reduced to this problem for antagonistic game. Namely let G = ((Xj)i£N , (uj)i£N} be a game of players N = {1,..., n} where Xj is a set of strategies and uj is a payoff function of player i. We can consider G as a game with quasi-ordered outcomes in which RN is a set of outcomes and for any two vectors (yi,..., yn), (yi,..., yn) G RN put

Pi

(yi,. .., yn) < (y'i, ..., yn) ^ yi < yj. G

Xj G

¿eN

are exactly vectors (y0,..., y^ G RN such that for any i G N the tradition y0 > Vj holds, where Vj is the lower value of antagonistic game of player i against the complementary coalition N \ i.

4.3. An example of game which has not of acceptable outcomes

Consider an antagonistic game r1 with payoff function given by table 1.

Table 1. Payoff function of game A

Y X yi y2 ys Vn inf

Xl 1 1/2 1/3 1/n 0

X2 2 -1/2 -1/3 — 1 / n — 1/2

sup 2 1/2 1/3 1/n v1 = = v = 0

In this game a set of outcomes is real numbers R. It follows from table 1 that any outcome r < 0 is not acceptable for player 1 since the strategy x1 is an objection of player 1 to such outcome. Moreover, any outcome r > 0 is not acceptable for player 2: an objection of player 2 to such outcome r > 0 is its strategy yn where n = [1/r] + 1. Therefore in game r1 the set of acceptable outcomes is empty.

References

Aumann, R. J. (1959). Accept,able points in general cooperative n-person games. In: Contributions to the Theory of Games, vol.IV (Annals of Mathematics Studies, 40) (Tucker, A.W. and R. D. Luce, eds.), pp. 287-324. Princeton University Press: Princeton. Moulin, H. (1981). Theory of games for economics and politics. Hermann: Paris

Rozen, V. V. (2010). Equilibrium points in games with ordered outcomes. Contributions to

Game Theory and Management, 3, 368-386. Rozen, V.V. (2014). Decision making under qualitative criteria. Mathematical models.

Palmarium Academic Publishing: Saarbrucken, Deutschland (in Russian). Rozen, V.V. (2014). Ordered vector spaces and its applications. Saratov State University: Saratov (in Russian).