Noncooperative games and nontransitive preferences

Mathematical Social Sciences 12 (1986) 1-7 North-Holland

NONCOOPERATIVE GAMES AND NONTRANSITIVE PREFERENCES Peter C. FISHBURN AT&T Bell Laboratories, Murray Hill, NJ 07974, U.S.A.

Robert W. ROSENTHAL 0/ Economics, SUNY, Stony Brook, NY 11794, U.S.A.

Department

Communicated by K.H. Kim Received 15 July 1985 Revised 12 August 1.985

A proof exactly analogous to Nash's proof for the existence of equilibria in finite noncooperative games with von Neumann-Morgenstern utilities shows that such games have Nash equilibria when preferences satisfy the weaker conditions of 'SSB utility theory'. An example illustrates the dual roles of mixed strategies in the SSB game context, namely to disguise a player's actual strategy choice and to resolve the potential intra personal dilemma of cyclic preferences among pure strategies.

Key words: SSB utility theory; noncooperative games.

1. Introduction

This note is concerned with finite noncooperative games in which the players' preferences over mixed-strategy combinations need not satisfy all the von NeumannMorgenstern (1944) axioms. Instead, we assume that preferences satisfy only the. weaker axioms of 'SSB utility theory'. In particular, the standard linearity and transitivity axioms need not hold in SSB theory, which is described and motivated in Fishburn (1982, 1984b). The existence of maximal elements in choice sets is an issue containing additional features when preferences need not be transitive, and in SSB theory randomization plays ~n important role even when there is only one decision-maker (Fishburn, 1984a). Our interest here is to explore the situation in which randomization has, in addition to this role, its customary role in noncooperative game theory. After briefly describing SSB utility theory and extending it in a natural way to noncooperative games having finite sets of players and pure strategies, we prove in Section 2 that Nash equilibria exist in such games. This result is a special case of a theorem in Shafer and Sonnenschein (1975); but our proof, differing from theirs, is exactly analogous to Nash's (1951) existence proof. In Section 3 we analyze a two0165-4896/86/$3.50 © 1986. Elsevier Science Putllishers B.Y. (North-Holland)

P.C. Fishburn, R. W. Rosenthal/Noncooperative games

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person example containing a mixed-strategy equilibrium having the property that one of the player's preferences cycle over the pure strategies in the support of his own equilibrium mixture, when the other player plays his part (respectively) of the equilibrium. At this equilibrium the player's randomization serves both to resolve his own cyclic-preference dilemma (as in one-person SSB theory) and to hide his strategy choice from the opponent (as in standard noncooperative-game theory).

2. SSB games and existence A binary preference/indifference relation t:: on a convex set P of probability distributions satisfies 8SB utility theory whenever there exists a skew-symmetric [¢(q, p) = -¢(p, q)] functional ¢ on P X P that is linear separately in each argument and has p

z. q #

¢(p, q)'~ 0,

for all p, q E P. An axiomatic derivation of 8SB utilities appears in Fishburn (1982), and their relevance to various economic concerns is discussed in Fishburn (1984b). When P is the set of all probability distributions on a finite set X and ¢ is an 8SB functional on P x P, bilinearity means that ¢(p, q) =

1: 1: p(x)q(y)¢(x, y), y

x

where ¢(x, y) == ¢(p', q') whenever p'(x) = q'(y) == 1. . In general, 88B uqlities satisfy neither the linearity property [(p >q, 0< A < 1) => AP +(1 - A)r >Aq + (1 - A)r, where > is the asymmetric part of z.] nor the transitivity property [(p Z. q Z. r)=> p z.r] of the von Neumann-Morgenstern utility theory; in particular,'I' they do not rule out cycles of strict preference such as p >q >r >p. Nevettheless, a simple application of the von Neumann minimax theorem (see Fish~rn, 1984a) shows that an element p* of P exists such that ¢(p*,p) '2!.0 for all peP, and that p*(x»O implies ¢(p*,p') =0 whenever p'(x) = 1 (i.e. p* indifferent to x), but that two such x's need not be indifferent to one another. For our S8B game, let Sj and Z.i be player j's simplex of mixed strategies and his preference/indifference relation over probability distributions on the set of purestrategy n-tuples. We assume that ¢I represents z.; i.n the sense of 8SB utility theory, so that PZ.iq** ¢i(P,q)';?O for such distributions. Following Nash (1951), each Sj in Si is a vector (SiI'"'' Sim) with non-negative components that sum to unity, where mi is the number of strategies in player j's pure strategy set. We let 7l:;a={O, ... , 0, 1a. 0, .... 0) for the pure strategy (1, so that Si = Sia 7l:/a · The set of pure-strategy n-tuples is II, the set of mixed-strategy ntuples is S =. Sl X ... X Sn' and we use (s; ti) to denote the n-tuple (Sl,'''' Si-l' ti' Si+l"",Sn) in S .. It is useful to observe that (s,' t·)=(s· ~ t· 7l:' )=~ f. (s', 7l:') 1 , '"' a la la '"' a l(1 l(1'

g;

r.a

3

P. C. Fishburn, R. W. Rosenthal/Noncooperative games

The n-tuple s = (SI , '" , sn) in S is defined to be a Nash equilibrium if, for all i and all ti in Si' s.£:i (s; ti) or, equivalently, ¢i(S, (s; ti» ~ 0,

n

where each argument in ¢ is interpreted as the probability distribution on induced by the named n-tuple in S under the usual assumption of independence across players. As noted above, the following theorem is implied by the Shafer-Sonnenschein theorem.

Theorem. Every SSB game has a Nash equilibrium. Proof. For each S E S define 'ia(S) = max{O, ¢i«S; 7ria), s)} for all i and a. The ria are continuous in s. Define T: S~S by T(s)=s', where

S; = [ Si + ~ ria (S)7ria] /

[ 1+

~ 'ia(S)

l

Since T is continuous, it has a fixed point by Brouwer's theorem. We show next that the fixed points of T are the Nash equilibria of the SSB game. (1) Suppose s is a Nash equilibrium. Then, for all i and (7, ifJi(S, (s; 7ria»~O, so ¢i«S; 7r/a ),s):s;O by skew-symmetry, so '1<1(S) =0. Hence s'=T(s)=s. (2) Given any s E S, for each i we have

Therefore, there is a a such that SI<1>O and ¢I«S; 7ri(1)'S):S;O, or Sia>O and 'la(S) == 0. Now suppose S is a fixed point of T, so s' =s. Then, for the a just noted, Sia=sia=Sia/[ 1 +

~ Lie(S)].

°

which forces Lie(S) = for all e. This is true for each i. Hence, for all i and a,
L lirp ¢i(S, (s; 7ria», a

it follows that ¢i(S, (s; Ii» ~

°

for all i and tl , so

S

is a Nash equilibrium.

0

3. Example Mixed (or randomized) strategies in SSB games serve two purposes. First, as in games with von Neumann-Morgenstern utilities, they have the interpersonal strategic aspect that leads to the existence of Nash equilibria; namely that of disguis-

P.e. Fishburn, R. W. Rosenthal/Noncooperative games

4

ing a player's actual strategy choice. Second, they offer an intrapersonal resolution of the potential dilemma of cyclic preferences among pure strategies. In the example below (for the case k] defined on the basis of payoff pairs (x, y) by f{J(x,y)

= (x- y)(5 - min {x, y}f

Player 2 is assumed to have a von Neumann-Morgenstern utility function u determined by the negatives of the payoffs x, y, ... in the matrix, i.e. u(x) = -x.

Hence, the expected utility for player 2 at SI "" (ai' a2, 1 U(SI'

S2)::;; 2/3(a2 - al) - 2a] - 3a2 - k(1 -

al -

al -

a2) and

S2 =

(/3, 1 - /3)

a2)'

Since (SI> S2) is a Nash equilibrium only if u(Sj, S2)::::: u(Sj, t2) for all 12 E S2' it follows from the preceding equation and the SSB function for player 1 that equilibrium requires aj = a2' Thus, at equilibrium, Sj must have the form (a, a, 1- 2a). It then follows that «a, a, 1- 2a), (/3, 1 - /3» is a Nash eqUilibrium if, and only if, f{J«(a, a, 1- 2a), ([3,1- /3», «a, b, 1 - a - b), (/3, 1 -

/3»)::::: 0

PLAYER 2

/3

PLAYER 1

1-/3

Cl1r:2

Cl 2

1

3

1-Cl 1 -Cl 2

k

k

A1 = (Cl1, Cl2, 1-Cl1- Cl 2) 42 = ({3,

1-/3)

Fig. '1. Payoff matrix for a two-person game.

P. C. Fishburn, R. W. Rosenthal/Noncooperative games

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for all (a, b, 1 - a - b) E SI' For convenience, let !fI(a, /3, a, b, k) denote the left-hand side of this inequality. Then, using bilinearity and a little algebra, we obtain: !fI(a, /3, a, b, k) = a(b - a)C + (1 - 2a)[aA + bB] - a(1- a - b)D,

where

C= 19/32+38/3-9, A

= 9(k -

2) - /3[9(k - 2) + (4 - k)(5 - k)2],

B = /3[(3 - k)(5 - k)2 + 16(k-1)] - (3 - k)(5 - k)2, D=/3{17k-k 2 - 23) +k3 -13k 2 + 64k-93.

Given k, we want to choose a and /3 so that !fI(a, /3,a, b, k);::O over V= {(a, b): a;:: 0, b;::0,a+b:51}.

Our analysis of !fI(a, /3, a, b, k) depends in large measure on the signs of A, B, C, and D at different k and /3, and on the extremes of (a, b) in V. We omit most of the details but summarize the results. For 2:5k:5kl' where kl is approximately 2.30396 and equals the largest k for which D:5 when /3 is the root between and 1 of the quadratic equation C = 0, there is a unique Nash equilibrium given by

°

°

a""t and

/3=/3*=Y28/19-1.

The value fJ*, which is approximately 0.21395, is the positive root of C= 0. At this Nash equilibrium, the first player's strategy is (112, 112,0), and he is indifferent between his first two pure strategies at fJ*. 0.7 0.6 0.5 0.4

co.. 0.3

{3~

EVERY {3 HERE IS AT EQUILIBRIUM WHEN a =0

0.2 0.1 0

.2.0

2.1

2.3

2.4

2.5

2.7

2.8

2.9

3.0

k Fig. 2. Equilibria for example: (a, 11, 1-2(1) for row player, (P,I-{J) for column player.

p.e.

6

Fishburn, R. W. Rosenthal/Noncooperative games

For k 2 :5k:53, where k2 is approximately 2.31576 and is the smallest k for which there exists a feasible P that has A;::: and B;::: 0, the only equilibrium strategy for player 1 is a=O, i.e. Sl =(0,0,1). If we take a>O for any such k, then there is no P that makes I/f(a, p, a, b, k);::: 0 for all feasible (a, b). Given a = 0, I/f reduces to aA + bB. Hence, given k, all P for which A;::: and B;::: are Nash-equilibrium strategies for player 2. Except at k =k2' there is a range of such P as shown on the right-hand part of Fig. 2. The left-hand part of the figure shows the (a, P) =:; (1/2, P*) equilibria for ks k 1 • Finally, for each k strictly between kl and k2' there is a unique Nash equilibrium that assigns positive probabilities to all three of player l's pure strategies. Given k E (kl' k 2), the equilibrium values of a and p, say a(k) and P(k), are determined as follows. First set a=:; 1 and then b = 1 in I/f(a, p, a, b, k);::: O. These two inequalities give upper and lower bounds on a(k) in t~rms of P and k, say a*(k) sa(k) sa*(k).

°

°

°

0.22

~*~----------~I

I I I I I

0.21

I I I I I I


0.19

I

I.

I 1

0.18

Ia

DECREASES I a::: 2"--' FROM 1/_2 TO ~a = 0 I O. AT~k-2.31, I I a =0.226

I

I 2.29

I 2.31

k Fig. 3. Area of 'Fig. 2 detailed. Equilibria for k from 2.29 to 2.33.

2.33

P. C. Fishburn, R. W. Rosenthal I Noncooperative games

7

The necessary inequality a *(k)::s; a *(k) is then solved for 13 to yield 13 ::S;f(k). Next, when a = b == in III?:. 0, we obtain another inequality on 13 which turns out to be 13 ?:.f(k). Hence f3(k) =f(k) , where (see D)

°

f3(k) == [_k 3 + 13k2 - 64k + 93]1[17 k - k 2 - 23].

When 13 = f3(k) , however, a*(k)==a*(k), and the unique solution for a is A/[2A +C]. The Nash equilibria near kl and k2 are illustrated in Fig. 3. f3(k) decreases approximately linearly from kl to k2' and a(k) decreases from a(k1) = 1/2 to a(kz) =0. As indicated above, the preferences of player 1 cycle on his three pure strategies at the equilibria for k2 < k < k 3 • In particular, when 13:::: f3(k) , player 1 prefers 7! 12 to 7! II, 7! II to 7! 13, and 7! 13 to 7! 12' These three preferences correspond respectively to CO. For example, he prefers 7!12 to 7!11 if and only if 1II«7!12' (f3(k), 1- f3(k»), (7!II' (f3(k), 1 - f3(k»» >0. This inequality reduces to 9 > 1913 2+ 3813, or 0> C. Since f3(k) < 13* between kl and k2' and C= Oat 13*, we obtain C
Acknowledgments

Research supported by National Science Foundation Grant SES-8317924 to R.W.R. We are indebted to Denis Bouyssou for noting that our proof of the theorem in Section 2 is the same as a proof by G. Kreweras in 'Sur une possibilite de rationaliser les intransitivites' (Colloq. Internat. du CNRS, Paris, 1961) and we are pleased to acknowledge Kreweras' priority for the result and proof.

References R.J. Aumann and M. Maschler, Some thoughts on the minimax principle, Management Sci. 18 (1972) (part 2) 54-63. P.C. Fishburn, Nontransitive measurable utility, J. Math. Psycho!. 26 (1982) 31-67. P.C. Fishburn, Dominance in SSB utility theory, J. Econom. Theory 34 (1984a) 130-148. P .C. Fishburn, SSB utility theory: an economic perspective, Math. Soc. Sci. 8 (1984b) 63-94. J. Nash, Non-cooperative games, Ann. Math. 54 (1951) 286-295. W. Shafer and H. Sonnenschein, Equilibrium in abstract economies without ordered preferences, J. Math. Econom. 2 (1975) 345-348. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (Princeton University Press, Princeton, N.J., 1944).