a new theory of equilibrium selection for games with complete information

GAMES AND ECONOMIC BEHAVIOR 8, 91-122 (1995)

A New Theory of Equilibrium Selection for Games with Complete Information

JOHN C. HARSANYI I

Haas School of Business, University of California, Berkeley, California 94720

Received October 3, 1993

This paper proposes a new one-point solution concept for noncooperative games, based on a new theory of equilibrium selection. It suggests a mathematical model for measuring the strength of the incentive each player has to use any particular strategy, and then for using these incentive measures to estimate the theoretical probability for any given Nash equilibrium to emerge as the outcome of the game. The solution of the game is then defined as the Nash equilibrium with the highest theoretical probability when this equilibrium is unique. The problems posed by nonuniqueness are also discussed. Journal of Economic Literature Classification Numbers: C7, C71. © 1995 Academic Press, Inc.

1. PRELIMINARY DISCUSSION

1.1. Introduction

In recent years several distinguished game theorists have advocated set-valued, rather than one-point, solution concepts for noncooperative games. Thus, Bernheim (1984) and Pearce (1984) proposed the concept of rationalizable strategies, which usually yields an infinite set of different strategy profiles, most of which are not even equilibria. Likewise, Kohl- berg and Mertens (1986) proposed the concept of strategic stability, which yields a solution typically consisting of several sets of (possibly infinite) sets of equilibria.

t I am indebted for helpful comments to Reinhard Selten, Woody Brock, Bob Aumann, Sergiu Hart, Ken Arrow, Werner Giath, an associate editor, and two referees. My special thanks to the associate editor, who has pointed out that Lemma 4 in Kalai and Samet (1984) ensures that all reduced games will have the regularity property (see Section 2.2. below).

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92 JOHN C. HARSANYI

Yet, even though the Bernheim-Pearce and the Kohlberg-Mertens solution concepts have been widely discussed in the literature, the problem of equilibrium selection, i.e., that of finding rational criteria for choosing one particular Nash equilibrium as the solution of the game from the set of all such equilibria has remained an important focus of research by many game theorists. Indeed, it often was Kohlberg and Mertens's stability theory itself that was used as a choice criterion in choosing among alternative equilibria of many games, particularly in the case of signaling games. (See, e.g., the long list of references in Fudenberg and Tirole, 1991, pp. 475-477. For use of Bernheim and Pearce's theory of rationalizable strategies in a similar way, see Hammond, 1991.)

In this paper I am going to propose a new one-point solution concept for noncooperative games, based on a new theory of equilibrium selection. It will be closely related to the theory that Selten and I proposed in 1988 (see Harsanyi and Selten, 1988, here to be referred to as HS) but will differ from the latter at least in five important respects.

1.2. Multilateral Risk Dominance

First, our 1988 theory used a bilateral risk-dominance concept, involving comparisons between various pairs of equilibria 2 in order to decide which one of the two would be a less risky choice for the players. In contrast, my new theory will use a multilateral risk-dominance concept in order to identify directly the equilibrium representing the least risky choice for the players among all equilibria with suitable stability properties in the game.

A rational player will try to choose a strategy that is a best reply to the other players' strategies. But he will always incur the strategic risk that the strategy he actually chooses will not have this property because his expectations about the other players' strategies will turn out to have been mistaken.

I shall propose a mathematical model for measuring the strength of the incentive each player has to use any particular strategy, and then for employing these incentive measures to estimate the theoreticalprobability that any given equilibrium will emerge as the actual outcome of the game.

Even though the players cannot fully avoid the strategic risk that their strategies will not be best replies to the other players' strategies, they can presumably reduce this risk by using strategies corresponding to that particular equilibrium that is, as judged by the players' incentives in the game, the equilibrium with the highest theoretical realization probability. In all but some exceptional cases (see Section 1.6 below), my theory will choose this particular equilibrium as the solution of the game.

2 Unless otherwise indicated, by the term "equilibrium" I shall refer to a Nash equilibrium.

EQUILIBRIUM SELECTION 93

In my opinion, the concept of the theoretical probabilities associated with the various equilibria of the game--and my new concept of multilateral risk dominance based on these theoretical probabilities--provide my new theory with a much higher degree of theoretical unity and of direct intuitive understandability than possessed by our 1988 theory (which was a rather complicated combination of many mutually more or less independent theoretical assumptions).

1.3. Subgame and Truncation Consistency

Second, our 1988 theory was based on the assumption that further-not- decomposable subgames and truncations are strategically self-contained parts of a game and, therefore, have the subgame and truncation consistency properties. Yet, Kohlberg and Mertens's (1986, p. 1013) discussion of forward induction has now convinced me that moves made by a player before a given subgame has been reached can virtually force some other players to deviate within this subgame from the strategies that the solution of this subgame itself would require them to use. In other words, moves prior to the subga'me may virtually force some players to act contrary to the subgame-consistency assumption.

Thus, I have now concluded that subgames and truncations are not always strategically self-contained parts of the game, and that in general we cannot rationally expect the players to act in accordance with the subgame and truncation consistency assumptions. Accordingly, my new theory will not make these two consistency assumptions.

1.4. Dominance and Inferiority (DAI) Relations between Strategies

Third, our 1988 theory used only those DAI relations that obtain already in further-not-decomposable subgames and truncations. In contrast, my new theory will be based on DAI relations arising in the game as a whole. This will be the case because, as I argued in Section 1.3, we now know that subgames and truncations do not actually possess the extensive strategic self-sufficiency that our 1988 theory attributed to them. On the other hand, Kohlberg and Mertens's theory of forward induction shows the great strategic importance of DAI relations arising in the game as a whole. More particularly, they show that what actually destroys subgame consistency in many games is always the existence of some dominance relation(s) generated by the game considered in its entirety.

1.5. Risk Dominance vs Payoff Dominance

Fourth, our 1988 theory was based on a combination of risk-dominance and payoff-dominance (i.e., Pareto superiority) considerations yet always

94 JOHN C. HARSANYI

gave precedence to the latter over the former. But further reflection on Aumann's (1990) arguments has now convinced me that a solution theory for noncooperative games cannot assume that, in case payoff dominance and risk dominance run in opposite directions between two equilibria, the players will actually choose the payoff-dominant equilibrium over the risk- dominant one? Accordingly, my new theory will use only (multilateral) risk dominance as choice criterion among different equilibria without any use of payoff dominance.

1.6. Probability Mixtures of Nash Equilibria as Possible Solutions

Finally, afifth difference between my new theory and our 1988 theory is this. Our 1988 theory always chose one particular Nash equilibrium as the solution of the game. Therefore, in the case of symmetric games, mathematical consistency required us to choose a symmetric Nash equilibrium (one showing all symmetries of the game itself) as solution (see HS, pp. 73-74 and 228-229). As we shall see, this often meant to use as solution a Nash equilibrium with very poor stability properties and with very low payoffs.

In technical terms, in symmetric games (as well as in some other games), both our 1988 theory and my new theory will give rise to a tie between different Nash equilibria. That is to say, both theories will select a set of two or more equilibria having risk dominance over all equilibria outside this tie, our 1988 theory used the tracing procedure (Harsanyi, 1975) to choose one unique Nash equilibrium as the actual solution of the game. (This approach did satisfy the consistency requirement of always choosing a symmetric Nash equilibrium as solution for any symmetric game.)

My new theory will use a different tie-breaking procedure in such cases--at least for games permitting preplay communication among the players. Instead of choosing one Nash equilibrium as solution, it will define the latter as a correlated equilibrium representing a probability

3 Aumann (1990) has shown also that, at least in the example he uses in his paper, even the possibility of preplay communication will not enable the players to attain the payoff- dominant equilibrium--as long as the game is a noncooperative game without enforceable agreements. Even though the players might verbally agree to choose the payoff-dominant equilibrium, they would not keep such an agreement because both of them would know that the other player would expect to benefit by lying about his strategic intentions. But it is not known as yet whether it is generally true that in such cases the players will be unable to reach an effective agreement to opt for the payoff-dominant equilibrium, or whether this depends on the precise nature of the risk-dominance relation to which this latter equilibrium is subject.

EQUILIBRIUM SELECTION

L R

0

1 0

1

0 3

95

FIGURE 1

mixture of all Nash equilibria in this set of equilibria giving rise to the tie in question, assigning the same probability weights to all of them. 4

Of course, in games without preplay communication, correlated equilibria are inaccessible to the players. Thus, if a tie arises in such a game then, for lack of a better alternative, I shall fall back upon the tracing procedure for choosing a unique Nash equilibrium as a "natural outcome" for the game. But since the Nash equilibrium chosen in this way will tend to have mathematical properties undesirable from a game-theoretical point of view, I propose to describe it merely as a "quasi-solution" rather than as a "so lu t ion" , leaving the solution proper formally undefined.

To illustrate the difficulties we run into by trying to use symmetric Nash equilibria as solutions for symmetric games, consider the following Battle of the Sexes game often discussed in the literature (see Fig. 1):

This game has two pure-strategy equilibria Et = (U, L) and Ez = (D, R), as well as the one mixed-strategy equilibrium E 3 -- (4 ~ U + ¼ D, 1 L + ~ R). E 3 is the only symmetric equilibrium of the game. But it would make a very unattractive solution. Whereas both El and E2 are persistent equilibria (see Kalai and Samet, 1984), E 3 itself is nonpersistent. Moreover, the payoffs, ul = u2 = ~ it yields are rather low as compared with the payoffs 3 and 1 provided by each of the two pure-strategy equilibria.

In contrast, in case this game permits prepay communication, my new theory will define its solution as the correlated equilibrium E* = ½ E l +

4 It is now customary to describe a probability mixture of two or more Nash equilibria as a publicly correlated equilibrium. It will be a correlated equilibrium because it will satisfy Aumann 's 0974) definition of such equilibria. On the other hand, the term "publicly correla ted" refers to the fact that, unlike privately correlated equilibria, it coordinates the various players ' strategies by means of publicly observable random signals.

96 JOHN C. HARSANY1

½ E 2. This will be o b v i o u s l y a much m o r e a t t r a c t i v e so lu t ion . F o r it is a p r o b a b i l i t y m ix tu r e o f the two persistent N a s h equ i l ib r i a E I and E 2, and y ie lds the much higher e x p e c t e d payof f s Ul = u2 = 2.

1.7. The Nash Property

A s w e have j u s t seen , m y n e w t h e o r y will d i f fer f rom our 1988 t h e o r y in s o m e i m p o r t a n t r e s p e c t s . But it will have s imi la r m a t h e m a t i c a l p r o p e r - t ies to the l a t t e r in o t h e r i m p o r t a n t r e spec t s . One o f t hem is the fac t tha t bo th theo r i e s have the Nash property (cf. H S , p. 215) for n o n d e g e n e r a t e unanimity games 5 in that bo th o f t hem def ine the so lu t ion o f such g a m e s as the N a s h equ i l i b r ium wi th the highest N a s h p r o d u c t . (My new t h e o r y has a s imi la r N a s h p r o p e r t y a lso for n o n d e g e n e r a t e 2 × 2 games . 6)

This N a s h p r o p e r t y , on the o t h e r hand , m a k e s m y new t h e o r y (as wel l as ou r 1988 theo ry ) into a direct g e n e r a l i z a t i o n o f N a s h ' s (1950) t h e o r y o f simple t w o - p e r s o n ba rga in ing g a m e s and o f its e x t e n s i o n to n - p e r s o n simple barga in ing g a m e s 7 ( H a r s a n y i , 1977, pp. 196-211). F o r , wi th in the c lass of finite g a m e s , the na tu ra l ana logs o f s imple ba rga in ing g a m e s are u n a n i m i t y games .

2. THE REDUCED GAME

2.1. Some Definitions

M y t h e o r y in its p r e s e n t fo rm is r e s t r i c t ed tofinite n o n c o o p e r a t e games . M o r e o v e r , it is d i r ec t l y a p p l i c a b l e on ly to g a m e s with complete information. But I a m ve ry conf iden t tha t it can be e x t e n d e d to ones wi th incom-

-~ By a unanimity game we mean a game in which the players can choose among alternative outcomes, each yielding positive payoffs to all players. If all players choose the same such outcome, then all of them will receive the corresponding positive payoffs. But if their choices are not unanimous, then all of them will receive zero payoffs.

A unanimity game is called nondegenerate if only one of these possible outcomes would yield the highest Nash product available in the game.

6 Our 1988 theory had a more limited Nash property also in dealing with 2 x 2 games, even with those that are not unanimity games (because they do not assign zero values to all four nonequilibrium payoffs in their payoff matrix). Even in such games, if they have two pure-strategy equilibria, then always that one with the higher ?Cash product will have risk dominance over the other.

Yet, under our 1988 theory this equilibrium would not be actually chosen as solution for the game if it were payoff-dominated by the other equilibrium--because our theory gave precedence to payoff-dominance over risk-dominance considerations when there was a conflict between the two (see HS, pp. 88-90). In contrast, my new theory, which makes no use of payoff-dominance relations, will ahvays choose the risk-dominant equilibrium as the solution so that it will have a full Nash property for nondegenerate 2 x 2 games.

7 By a simple bargaining game we mean a bargaining game in which the players cannot increase their payoffs by forming coalitions smaller than the all-player coalition, or by using threats against other players.

E Q U I L I B R I U M SELECTION 97

plete information, in a way consistent with the analytical framework of my paper, (Harsanyi, 1967-1968). Actually, I now have a general idea of how to do this. Yet, this problem still requires some further work so that I have to postpone its discussion to another paper yet to be written.

Let Si be the set of all Ks pure strategies si that player i has in the game, and let S-s be the set of all Ms strategy (n-1)-tuples s-s that the other (n-I) players can use. Of course, Ms = II K i ( j ¢ i). I shall denote specific pure strategies of i as s] . . . . . s/k, . . . , sf~, and shall denote specific pure- strategy profiles s_~ of the other players ~ " S _ i , . . . , S - i , . . . , s M ~ .

Any mixed strategy of i will assign some probability p~ to each of his pure strategies s~. I shall identify any possible mixed strategy of his with the corresponding probability vector Ps = (P] . . . . . p~i). The set Ps of all these probability vectors Pi will be a closed simplex o f ( K / - I) dimensions, defined by the requirements

p~->0 f o r k = I . . . . . Ks (2.1)

and

K i

p~ = 1. (2.2) k=l

By the same token, any correlated mixed strategy of the other (n-l) players will assign some probability q~" to each strategy profile s"_',, and will be identified with the corresponding probability vector qs = (q] . . . . . q~i). The set Qi of these probability vectors qi will again be a closed simplex, this time of (M i - 1) dimensions, and defined by requirements similar to (2.1) and (2.2).

Expectation Vectors. I now propose to characterize any specific expectation that player i may entertain about the other players' strategies, by specifying the subjective probabilities qT' that p layer / - -expl ic i t ly or implicit ly--assigns to use of alternative pure-strategy (n-1)-tuples s"_'i by these other players. The probability vector q; = (q~ . . . . , q~", . . . . q~i) listing these subjective probabilities I shall call player i 's expectation vector.

Obviously, under this representation, any expectation vector q; characterizing i 's expectations about the other players' strategies will have the same mathematical form as a probability vector characterizing a correlated mixed strategy of these players would have. Needless to say, this is not meant to imply that player i will actually expect the other players to use a correlated mixed strategy. Even if he firmly expects them to use an uncorrelated mixed-strategy (n- l)-tuple p_ i = (Pl . . . . . Pi-i, Pi+ 1 . . . . . Pn ) ,

98 JOHN C. HARSANYI

the latter will always necessarily induce a probability vector q; assigning specific probabilities qT to all pure-strategy (n-1)-tuples s"-'i and, for the purposes of my theory, this probability vector qi will be the most conve- nient mathematical characterization of player i's expectations about the other players' strategies.

Stability Sets. Within the simplex Qi, let o(s/0 be the set of all probability vectors qi to which a given pure strategy s~ of player i is a best reply. This set o-(s~3 I shall call the stability set of this strategy s/k (cf. HS, p. 76). For convenience, I shall mainly use the shorter notation tr~ = tr(sik). Obviously, any stability set o-~ will be a closed and convex subset (possibly empty) of simplex Qi and the various stability sets together will cover this entire simplex.

Inferior Strategies. I shall call any pure strategy ski inferior (cf. HS, p. 118) if its stability set tr~ is a set of measure zero, or if it is a proper subset of the stability set o-~" of another pure strategy s~' of i. Clearly, under this definition, any strictly or weakly dominated strategy will always be an inferior strategy. Thus, repeated removal of all inferior strategies from the game (see below) will ensure repeated removal of all dominated strategies.

Duplicate and Semiduplicate Strategies (cf. HS, p. 119). Two pure strategies s~ and si.' of any player i will be called duplicate strategies if, for all strategy combinations s_; of the other players, and for every player j (both j = i and j ¢ i), we have

Uj(s;, s-i) = Uj(s'i', s-i). (2.3)

In contrast, s[ and s'i' will be called semiduplicate strategies if (2.3) holds at least for the special case j = i. Obviously, both duplicateness and semiduplicateness are equivalence relations. Equivalence classes generated by these two relations will be called duplicate classes and semiduplicate classes, respectively.

Duplicate Players. Informally speaking, two or more players will be called duplicates of one another if they could have been created by splitting one player into two or more identical types, possibly followed by renumbering some of these types' pure strategies and/or by positive linear transformations of their utility functions.

Formally, we shall call a set D of two or more players a set of duplicate players if the following two conditions are both satisfied:

(a) For each player i in D his payoff u i will be independent of the strategies used by the other player(s) in D.

(b) Any permutation of the players in D will leave the game invariant--or at least this will be the case after renumbering the pure strategies


of some of these players and/or after subjecting their payoff functions to suitable positive linear transformations.

2.2. Definition of the Reduced Game

I shall assume that before using my theory to compute the solution of a game, the latter will be subjected to the following operations:

1. All inferior pure strategies will be simultaneously removed from the game, repeating this action again and again until any inferior strategies remain in the game. In a finite game this procedure will always end after finitely many iterations because the game will no longer contain any inferior pure strategies, yet each player will be left at least with one noninferior pure strategy.

2. Each set of duplicate pure strategies will be replaced by its centroid. 8

3. Then, each set of semiduplicate pure strategies will be replaced by its centroid.

4. Finally, for each set D of duplicate players, we shall choose one player i as the representative of this set D. Then, we shall assume that i will not only choose his own strategy but will also choose strategies for all other players in D, always choosing for them strategies corresponding 9 to his own strategy. This assumption of course will make all players in D, other than i himself, into dummy players unable to choose their own strategies. Then, we eliminate all these dummy players from the game.~°

When these four operations have been completed the remaining game (which will often be much smaller than the original game) will be called the reduced game. The players and the strategies of this reduced game will be described as admissible players and as admissible strategies.

2.3. A Few More Definitions

Since I shall assume that any game to which my solution theory is applied is already a reduced game, from now on all definitions of Section 2.1 will be interpreted as applying to such a reduced game. Accordingly,

s For any set of (pure and/or mixed) strategies, its centroid is an equal-probability mixture of all these strategies.

9 For two different players in D their pure strategies are corresponding strategies if, after the renumbering of pure strategies described above under (b), the two players' pure strategies will have the same serial number. The mixed strategies of two players in D are corresponding strategies if either mixed strategy can be obtained from the other by replacing each component pure strategy of the latter by a corresponding pure strategy of the other player.

i0 Operation 4 is needed because (as Selten has pointed out to me) my new solution concept is not invariant under splitting one player into two or more identical types. Operation 4 will reverse any such player splitting that has created, or could have created, any given set of duplicate players.

100 J O H N C . H A R S A N Y I

I shall assume that, for each player i, the sets S; and S-i contain only admissible pure strategies, or (n-l)-tuples of such strategies. Likewise, the number n will refer to the number of admissible players. Moreover, I shall reinterpret the numbers Ki and Mi as indicating the numbers of elements in these admissible versions of these sets Si and S-i, respectively.

In view of the fact that all inferior strategies have by now been removed, in a reduced game the stability sets o-~ of any pure strategy s~ of any player i will be nonempty and will also have nonempty interiors.

Indeed, any reduced game will also have the following further regularity property: The various stability sets o-~ of each player i will have disjoint interiors so that each stability set o-~ will overlap with other stability sets o-~', k' ~ k, at most at its boundaries.

To verify this, assume that 4 and o~/' do have an intersection with a nonempty interior. Then, by Lemma 4 of Kalai and Samet (1984), the corresponding strategies s~ and s~' would be semi-duplicates. Yet, if this were the case then both of these strategies would already have been removed from the game (and would have been replaced by the relevant centroid strategy) as a result of operation 3 of section 2.2. (I owe this argument to an anonymous associate editor.)

Let me now propose the following further definitions. I shall denote player i 's payoff function by U;. I shall write

u k m k m = Ui(si, s-i), (2.4)

and

d~ k'm = up" - u~ 'm for i ~ N; k, k' E Ki and k' ~ k; m E Mi. (2.5)

Here N stands for the set of the first n positive integers whereas K-i stands for the set of the first Ki positive integers, and M; for the set of the first M,- positive integers.

Finally, I define the vectors fir and 3~k' as

-k = (u/kl , . . . uikm, uikMi) (2.6) U i , . . . ,

and as

d~k ' = ( d t:k'l . . . . . d t~k'm . . . . . dkk 'Mi) = Ft~ -- ffi k'. ( 2 . 7 )

I shall describe fi/k as the payof f vector associated with the pure strategy


ski, and shall describe di kk' as the payoff-difference vector associated with the two pure strategies si* and ski', with k' ~ k.

3. THE PLAYERS' INCENTIVES TO USE PARTICULAR PURE STRATEGIES

3.1. Structural Incentives vs Strategic Incentives

What incentives does any given player i have to use a particular pure strategy ski? Some of these incentives will be determined simply by his own payoff function U i in the game and will be independent of any information and of any expectations he may have about the other players' likely strategies. Because his payoff function Ui itself is determined by the very nature of the game, such incentives I shall call this player's structural incentives to use specific pure strategies. Since each game under consider- ation will be (or at least will have been transformed into) a game with complete information, the players will know each other's payoff functions and, therefore, will also know each other's structural incentives in the game.

In addition, each player i will also have incentives based precisely on his information and on his expectations about the other players' likely strategies. These I shall describe as his strategic incentives.

Each player's strategic incentives, on the other hand, will be dependent on his information about the other players' structural incentives because he will make use of this information in forming his expectations on these other players' likely strategies.

3.2. A Player's Structural Incentives

We may distinguish between a given player's gross and net structural incentives. Thus, player i 's gross structural incentives to use a particular pure strategy ski are specified by the payoff vector ffk associated with this strategy. But for many purposes more important are his net structural incentives to use this strategy s k rather than some other pure strategy Ski '. These net incentives are specificed by the (Ki - 1) payoff-difference vectors "~tki k' (with a given k and with any k' ~ k and EK-i).

On the other hand, the larger (i.e., the more positive and the less negative) the components d kz'' of each payoff-difference vector ~kk' are

k This is so because the larger will be the size of his stability set trkifor si. k is defined by the (Ki - l) inequalities this stability set or i

dkik'm qm > 0 for all k' in Ki with k' ~ k (3.1) mE"Mi

102 JOHN C. HARSANYI

so that the larger these coeeficients d~ k'm in these inequalities the larger will be this stability set tr/~.

This means that, instead of using all these payoff differences d~ k'm together for measuring the strength of player i 's structural incentives to use a particular pure strategy s~; we can employ simply the size of the stability set tr~ as our overall measure.

This raises the question of what specific quantity to use for measuring the size of this set itself. An obvious possibility of course would be to use the Lebesgue measure h(o-~) of this set itself as our measure. No doubt, this quantity would be a very suitable measure if our only purpose were to measure the size of this set o-~ as such. But our actual objective is to find a quantity that is a suitable measure for the strength of player i 's structural incentives to use his various pure strategies.

I shall argue in Section 3.4 that for this purpose it is preferable not to use the Lebesgue measures of the stability sets tr~themselves, but rather to use the Lebesgue measures of their images under a suitable mapping /x, to be called the inversion mapping, which I shall define presently.

3.3. Inverted Probabilities as Risk Indicators, and the Inversion Mapping tx Defining Them

In everyday life people often make bets yielding high payoffs with rather small probabilities. Such bets are regarded as being riskier the smaller the probabilities of winning these bets. This suggests that people measure the risks involved by the reciprocals of these probabilities of winning.

The same principle is suggested by the way we use the term odds. We are speaking of long odds when the ratio of the possible payoff to the amount one has to risk is very large owing to the very small probability of winning; but this expression is used also to refer to this very small probability itself.

Using this intuitive notion of risk, we may characterize the risks associated with the probability vector

qi (q], '" . . q~') = . . . , q i , . (3.2)

defined in terms of the quantities

~" _ 1/q~" for all mE-M i . (3.3) Z,,,E~- i (1/q~")

Obviously, the quantities x~" are proportional to the reciprocals 1/qT' of the probabilities q~", but are normalized so as to add up to unity. Thus, we can write


xI" -> 0 for all m E M i (3.4)

and

~ " = 1. (3.5) m U - M i

In contrast to the quantities 1/qT' themselves, which may be simply called reciprocal probabilities, the quantities x" just defined I shall call inverted probabilities. I shall decribe them also as risk indicators because, as I have already argued, they may be interpreted as indicators of the risks associated with the probability vector q,.. Accordingly, I shall call the vector

x; ( x ) , . . " x~,) = . , X i , . . . , (3.6)

the risk vector associated with the probability vector qi. In view of (3.4)' and (3.5), the set Xi of all risk vectors associated with

the probability vectors qi in the (M; - 1)-dimensional simplex Q~ will be itself also a simplex of the same number of dimensions.

Now, consider the mapping /z: q~ ~ xi. I shall call it the inversion mapping because it transforms the probability vectors qi into vectors xi of inverted probabilities. Obviously, this mapping/z is well defined for all interior points of the simplex Q;, and, indeed, it is there one-to-one and even analytic. But Eq. (3.3) fails to define it for points q; at the boundary Bi = B(Q~) of Q~ because all such vectors qi contain one or more zero components qm = 0 SO that application of Eq. (3.3) to them would involve dividing by zero.

Yet, we can extend the mapping/z to such vectors q; at the boundary B/by finding some interior vector qi very near to q~, then by applying Eq. (3.3) to this new vector qi, and finally by taking suitable limits. But as we shall see, this extended mapping/z will be highly degenerate at all points q; at the boundary Bi of simplex Qi and will be there neither one-to-one nor analytic.

More specifically, let M*(qi) be the set of all superscripts m for which the components q~" of qi are zero, and let M**(q~) be the set of all superscripts for which the components q~ are positive. Then, we can construct the vector q, as follows.

For each superscript m in M*, we set 01" = hme, where the coefficients h m a r e arbitrarily chosen nonnegative parameters except for the requirement that they must add up to unity. Let m = m ° be the smallest superscript m in set M**. We set 0~ ''° = q~,,o _ e, and set 0~" = q~" for all other super-


scripts m in set M**. (As is easy to verify, all Mi components ~" of the newly constructed vector ~,. will add up to unity.)

Next , we apply Eq. (3.3) to all components ql" of this vector ~¢i but multiply both the numerator and the denominator of the resulting expression by s so as to avoid any division by e. Finally, we compute the limit of the resulting expression when e goes to zero. These limits will give us the components x~" of the image vector xi = /z(qi ) of vector q;.

When we have performed these operations we shall find that all vectors qi having only one zero component q, 'Tr = 0 for the one superscript m = ~ - - a l l the infinitely many vectors satisfying this description for any given superscript m = ~ - - w i l l be mapped into one particular image vector x;

= 1 for this super- that, conversely, has only one positive component x; script m = ~ and has zeros everywhere else.

On the other hand, every vector q,. with two or more zero components , i.e., with two or more superscripts m belonging to the set M*(qi), will be mapped into some image vector xi having nonnegative components x m -- 0 for all superscripts m belonging to this set M*(qi) and having zero components x~" = 0 for all superscripts m belonging to the complementary set M**(qi), which is again the opposite arrangement to that obtaining in the original vector qi itself. More exactly, each vector with two or more zero components will be mapped into infinitely many different image vectors x,---owing to the fact that we could have chosen the arbitrary parameters k,, in infinitely many different ways, provided that they would add up to unity.

In order to restate our results in geometric terms, I shall denote the .vertexes of simplex Q; by ~ and vertexes of simplex X i by ok. I shall write qi(m) and x~(m) for vectors having the number 1 as their mth component and have zeroes as all their other (Mi - I) components . I shall assume that the vertexes °-tt and °k are numbered in such a way that we can write

°It,, = qi(m) and ok,,, = xi(m) for m = 1 . . . . . Mi. (3.7)

On the other hand, the (Mi - 2)-dimensional faces of the (M; - 1)- dimensiofial simplexes Qi and X; I shall denote by ~ and o~, respectively. Let Qi(m) and Xi(m) be the sets of all vectors qi and x,- such that have zero as their mth components . I shall assume that these faces ~ and are numbered in such a way that we can write

~,,, = Qi(m) and ~,,, = Xi(m) for m = 1 . . . . . M i. (3.8)

Obviously, as a result of (3.7) and (3.8), each face ~,, and T~,, will be the face opposite to the vertex ~m and okra, respectively.

What our results show is that the mapping/z will map all points qi of

E Q U I L I B R I U M S E L E C T I O N 105

each face ~,, of simplex Q; into the v e r t e x ° ~ m of simplex Xi; and that it will map each vertex °It,, of simplex Qi into all points of the face °~ m of simplex Xi.

Suppose that x; =/z(q;), i.e., x; is the image of q; under the mapping/x. Suppose also that all Mi components of x; are nonzero. Then, in view of (3.3), we can compute the M; components q~ of q; by using the equation

I/xT' for all m • Mi. (3.9) q[" - ~'i~-ff, (1/x[" )

In other words, the inverse mapping tz-~ has the same ma themat i ca l f o r m as the mapping/z itself has and, therefore, also has the same mathematical properties as the latter does. Thus, the vectors qi and the mapping ~ on the one hand, and the vectors x; and the inverse mapping tz- ~ on the other hand, are connected by an interesting duality relation.

3.4. The Players ' S tructural Incent ives in Unanimi ty Games .

Let G be now an n-person unanimity game, in which the n players can choose among K possible agreements A ~ . . . . . A ~ . . . . . A r. Accordingly, each player i will have the same number, viz. K; = K, different pure strategies, with strategy s~ involving a vote by him for agreement Ak. If all players vote for the same agreement Ak then all of them will receive posit ive payoffs as specified by the payoff vector

u . , . , U.q). (3.10)

But if they vote for two or more different agreements then all of them will receive zero payoffs. As is easy to verify, any unanimity game G will be automatically already a reduced game.

As each player i has K different pure strategies, the number of pure- strategy combinations (i.e., pure-strategy (n-1)-tuples) s_; available to the (n-l) players other than player i will be M = K n-~. Strategy combinations s_; in which all of these (n-l) players vote for the same agreement A ~ I shall call concordant strategy combinations. The other strategy combinations s_i I shall call discordant strategy combinations. Obviously, the former are characterized by the fact that they will contain only pure strategies with the same superscript for all these (n-l) players whereas the latter will contain pure strategies with two or more different superscripts.

I shall assume that these strategy combinations s_; are numbered in such a way that any strategy combination st_; with k -< K will be a concordant combination in which all pure strategies have this particular super-


script k whereas all strategy combinations s'_"; with m > K will be discordant combinations.

Under this numbering convention, any given agreement A ~ will be obtained only if player i uses his pure strategy s~ and if the other (n-I) players use the corresponding strategy combination s~_,..

Furthermore, each expectation vector q~ of any player i will have the form

q; = (q] . . . . . q/K, q~+, . . . . . q~), (3.11)

where q] . . . . qff will be the probabilities that player i assigns to various concordant strategy combinations of the other players whereas q~+~ . . . . . qff will be the probabilities he assigns to various discordant strategy combinations of these players.

Accordingly, if player i 's expectations correspond to a given expectation vector qi then he will have to assess his expected payoff by use of his pure strategy s~ as

Ui(s ~ , q,) = q~u~, (3.12)

where u~ is the payoff specified by (3.10). By the same token, any risk vector xi = Iz(q;) of player i will have the

form

X i ' : X , . . . , A i , a i , . . . , X , (3.13)

where again the first K components of x; will refer to concordant strategy combinations of the other players whereas the (M-K) remaining components will refer to discordant strategy combinations of these players.

Among games in normal form, unanimity games are the simplest possible ones. In all other games, the gross structural incentives for any player i to use a particular pure strategy s~ are typically specified by the M; different payoffs u~" listed by the payoff vector/i/~ [cf. Eqs. (2.4) and (2.6)].

In contrast, in a unanimity game, these incentives are fully specified by the one payoff u~, which (under our numbering convention) is the kth component, and is the only nonzero component, of this vector ~ . (It is of course the payoff that player i would obtain if all other players joined him in voting for agreement A~.) In view of the fact that in a unanimity game this one payoff fully describes player i 's gross structural incentives to use his pure strategy s~, I propose the following principle:

Proportionality Requirement for Unanimity Games. In a unanimity game, the gross structural incentives for any player i to use a particular


pure strategy s~ should be measured by a quantity proportional to the payoff u~, i.e., to the payoff he would receive if all players likewise voted for agreement Ag.

I shall now need two lemmas:

LEMMA I. Use o f the Lebesgue measures h(o'~) o f each player i's stability sets o-~ themselves in unanimity games as measures o f i's structural incentives he has to use his alternative pure strategies s~ would be inconsistent without our Proportionality Requirement.

LEMMA II. Let p~ = tz(o-~) denote the image set o f each stability set o'~ o f player i under the inversion mapping tz. Then, use o f the Lebesgue measures h(p~) o f these image sets p~ as measures o f i's structural incentives in unanimity games will be fully consistent with our Proportionality Requirement.

As Lemma I can be established by one counterexample, I shall discuss a simple counterexample in the remaining part of this section. I shall use the same exampl.e also to illustrate the correctness of Lemma II in that particular case.

On the other hand, as actual proof of Lemma II in the general case takes up more space, I shall present it in an Appendix at the end of this paper.

Consider a two-person unanimity game G ° in which the players can choose among three different agreements A t, A 2, and A 3, so that each player i(i = 1, 2) has three pure strategies s), s/z, and s~. To save space, I shall at first consider this game G ° only from player l ' s point of view.

As this is a two-person game, each expectation vector ql = (ql, q~, q~) will simply list the three probabilities q~ that player I assigns to the three pure strategies s~ of player 2. Consequently the set Qu of all of player l ' s expectation vectors qt will be a two-dimensional simplex, which can be represented by an equilateral triangle (see below).

Each stability set o-] (k = 1, 2, 3) of player 1 will be the set of all expectation vectors qt to which his pure strategy s~ is a best reply. Hence, in view of(3.12), cr~ will be the set of all vectors ql whose three components satisfy two inequalities of the form

k k k ' k ' k t qltq >-- ql ul for # k (3.14)

(as k, k' = 1, 2, 3, there are only two k' values with k' # k). Moreover, the boundaries of each stability set o-] will be two straight lines whose equations are of the form

k k k' k' qlul = ql ul with k' ~ k. (3.15)


U3

Ol U2

FIGURE 2

In Fig. 2 above, the equilateral triangle represents the two-dimensional simplex Qt. Using the probabilities ql, q~, and q~ as barycentric coordinates, the three lines inside this triangle are defined by equations of form (3.15). (I have drawn them as they would be in the special case of ul = 1, u~ = 2, and u~ = 3.) In view of (3.14), each of the three stability sets cr I, o-~, and o-~ is represented by two adjacent small triangles, both marked by 1, or by 2, or by 3, respectively. (See Fig. 2.)

Simple computation shows that the areas and, therefore, the Lebesgue measures of the two stability sets o-'1 and o-~ are defined by expressions of the form

( ,,I ) h(° I ) = Y\u~(ul + u~) ~- u~(u I + u~) (3.16)

and

(3.17)

where 7 is the same constant in both equations. If it were consistent with our Proportionality Requirement to use the

Lebesgue measures h(cr~) of the various stability sets cr~ themselves as

EQUILIBRIUM SELECTION

FIGURE 3

109

measures of the relevant player's structural incentives in unanimity games then we would have to have

(3.18)

for all possible values of ul, u~, and u~. Obviously, this is not the case. This establishes the truth of Lemma I.

I now propose to show that, at least in the special case of a 3 × 3 unanimity game like G ° Lemma II is likewise true.

In view of (3.3) and (3.14), each image set p~ of player 1 will be the set of all risk vectors X l =/x (q t ) whose three components satisfy two inequalities of the form

k k< k' k' k' x l / u l - - X l / U l for ~ k . (3.19)

Moreover, the boundaries of each image set pk will be two straight lines with equations of the form

k k ~' k' k' xl/ul = x l / u l for ~ k. (3.20)

In Fig. 3 (see above), the two-dimensional simplex XI is represented as an equilateral triangle, together with the three lines separating player

110 J O H N C. H A R S A N Y I

l ' s three image sets Pl, P~, and p~. (Again, I have drawn these lines so as they would be in the special case of ul = 1, ul z = 2 and u~ = 3.)

In view of (3.19), the three image sets Pl, P~, and p~ are represented by the triangles T I = °l~5q°l~3, T 2 = ~lJq°l~, and T 3 = ~ lJq°~ , respectively. These three triangles meet at point 5c I , which is the point where the three boundary lines separating the various image sets intersect. Thus, the

"1 "'~ barycentric coordinates x~, x T and 5c] of Jq must satisfy all three equations of form (3.17), so that we must have

= ul: u?: u?. (3.21)

On the other hand, each triangle T k can be regarded as a triangle with one of the three equal sides of the equilateral triangle ~° l~°g 3 as its base, and with the quantity J:] as its altitude. Therefore, in view of (3.21) we can write

~.(T I):X(T2):,k(T 3)=u[:u~'u~. (3.22)

Yet, T 1, T 2 and T 3 represent the three sets p], p~, and p~. Therefore, the Lebesgue measures of the former will be proportional to the Lebesgue measures of the latter. Thus, we can conclude that

X(pl) : = u l : u?: u?. (3.23)

Of course, a similar statement will hold for player 2's image sets p~ and his payoffs u~.

Consequently, Lemma II is t rue- -a t least in the special case of unanimity games of size 3 × 3. As I have already indicated, proof of this lemma in the general case can be found in the Appendix at the end of this paper.

3.5. Measuring the Players' Structural Incentives in the General Case

3.5.1. Incentive Measures for Pure Strategies. I have tried to show in Section 3.4 that if we want to satisfy the Proportionality Requirement I proposed for unanimity games then we must define our measures for the players' structural incentives, not in terms of the Lebesgue measures X(o-~) of the stability sets o-~ themselves, but rather in terms of the Leb- esgue measures X(p~) of the latter's image sets p~ = lz(o-~).

Accordingly, I am now going to define incentive measures qJi(S*) in order to measure the strength of the structural incentive each player i has to use a particular pure or mixed strategy s* in the game. First I shall define such incentive measures tOi(s~) for pure strategies s~ in terms of


the Lebesgue measures h(O/~) already discussed. Later I shall extend my definition of incentive measures to mixed strategies.

Thus, let s,. k be one of player i 's pure strategies. Then, I propose to define the incentive measure for s~ as

tki(s~) = h(p~). (3.24)

Next, before extending my incentive measures to mixed strategies, I shall try to highlight the mathematical meaning of my proposal to base our incentive measures on the Lebesgue measures h(p~) rather than on the Lebesgue measures h(o-k). I shall do this by expressing both of these Lebesgue measures as multiple integrals over the same region of integration, viz.

d-~ = o-~\Bi, (3.25)

where Bi is the boundary of simplex Qi. (My reasons for excluding points of this boundary B i from the region of integration I shall state presently.) Moreover, for convenience, I shall write M[ = M,. - I. The two integrals are :

~(o'~) = ~6-~"" f 1 dq] . . .dq M; (3.26)

and

fa . f O(x],_... ,_x~;) dq] . . , dqi~;. h(P/k)= ~'" O(q], ,q~;) (3.27)

Note 1. Each vector q,. and each vector xi has M,. components but, out of these, only M[ = Mi - 1 components are independent. For this reason, I have omitted the last components q~i and xffi as independent variables from (3.26) and (3.27).

Note 2. In (3.27) I had to exclude the boundary points of simplex Q,. from the region of integration because at such boundary points the Jacob- ian of the inversion mapping/x, J = (ax~ . . . . . xiM;)/a(q~ . . . . . qiM;) is not well-defined. In view of this, for the sake of greater comparability, I have excluded these boundary points from the region of integration also in (3.26).


When we compare the two integrals in (3.26) and in (3.27), we shall find that the first integral assigns the same weight to every vector qi in 6 -k whereas the second assigns greater weight to those vectors q; at which the Jacobian J takes relatively high values. This, however, will be the case in those areas of 6/k where very small changes in any vector qi can give rise to rather large changes in its image vector x; =/z(q;).Jl

Thus, our decision to base our theory on the Lebesgue measures h(pki) rather than on the Lebesgue measures h(o-/k) amounts to assuming that, within the entire simplex Q;, player i will pay special attention to those areas where the Jacobian J takes relatively high values, i.e., to those areas where small changes in any expectation vector q,- may give rise to large changes in the corresponding risk vector xi =/z(qi) .

3.5.2. Incentive Measures for Mixed Strategies. Let s* be a mixed strategy of player i. Let o-* denote the stability set of s*, i.e., the set of all expectation vectors q~ to which s* is a best reply. Let p* denote the image set of tr* under the inversion mapping/z.

We cannot base our incentive measures ~ki(s*) for such mixed strategies s* on these sets o-* or p* because in all reduced games both sets tr* and 19" tend to be sets of measure zero. For this reason, I shall define these incentive measures ~b,-(s*) in terms of the incentive measures for their component pure strategies.

Let s* be a mixed strategy in any reduced game. Suppose that it is characterized by the probability vector

= (pl . . . . . p f . . . . . p F , ) , (3.28)

where p/k is the probability that s* assigns to the pure strategy sk(k = I, . . . . Ki). Then, I propose to define the incentive measure for s* as

K i

q,i(s,) = p f q,,(sf). k=l

(3.29)

i, As we have seen in Section 3.3, this will happen in areas very close to some vertex ~m of simplex Qi. We constructed an expectation vector qi very near to such a vertex ~m by making use of some arbitrarily chosen parameters h m. We found that even a very different choice of these parameters would have made very little difference to the resulting expectation vector t) i (because in constructing qi each parameter h,, was multiplied by a very small positive number e). But a very different choice of these parameters ~.m would often have had major effects on the image vector xi = /x(~i) corresponding to this vector qi.


4. DEFINITION OF THE SOLUTION OF THE GAME IN TERMS OF

THEORETICAL PROBABILITIES

I shall assume that the players will restrict their choices to equilibria with acceptable stability properties and, more particularly, to proper and persistent equilibria, t2 which I shall call eligible equilibria. (On proper equilibria, see Myerson, 1978. On persistent equilibria, see Kalai and Samet, 1984.) The set of all such eligible equilibria will be called S*. My theory defines a solution only for games in which S* is afinite set.

For any player i, any strategy s*, whether pure or mixed, that is i 's equilibrium strategy in some eligible equilibrium s* I shall call an eligible strategy for i. The set of all his eligible strategies s* will be called S*.

The basic assumption of my solution theory will be that, other things being equal, the probability that player i will use one of his eligible strategies s* will tend to be proportional to the strength of his structural incentive to use this strategy s* as assessed by the incentive measure ~bi(s*). Thus, I shall write

Prob(s*) = ci~i(s*), (4.1)

where ci is a constant such that

P r o b ( s * ) = I. (4.2)

Moreover, if the various players' strategy choices are statistically independent then the probability for a given eligible equilibrium s* = (s~ . . . . . s*) to become the outcome of the game will be

t !

Prob(s*) = I-[ Prob(s*). (4.3) i=1

The probabilities defined by (4.1) to (4.3) I shall call theoretical probabilities.

The solution of the game I propose to define as the eligible equilibrium

12 By my eligibility requirement, I want to ensure that the solution of the game will be a perfect equilibrium (as defined by Selten, 1975) as well as a persistent one. But actually I require properness rather than perfectness as such because I need a property ascertainable without going beyond the normal form, yet implying perfectness. This is so because my theory is a theory of noncooperative games in normal form.

I want to ensure that the solution will be a persistent equilibrium because Kalal and Samet (1984) have pointed out that nonpersistent equilibria have extremely poor stability properties.


s* with the highest theoretical probability, provided that s* is the only eligible equilibrium with this probability. More generally, consider the equation

zr* = max Prob(s) = Prob(s*), (4.4) sES*

and let S** be the set of all eligible equilibria s* satisfying (4.4). We can distinguish three possible cases:

Case I. S** contains only one equilibrium s*. In this case, s* will be defined as the solution.

Case 2. S** contains two or more equilibria, and the game does permit preplay communication among the players. In this case, the solution will be defined as the correlated equilibrium representing an equal-probability mixture s** of all equilibria s* in S**.

Case 3. S** contains two or more equilibria, yet the game does not permit preplay communication. In this case, the solution of the game will be left undefined but the unique Nash equilibrium s ° chosen by the tracing procedure when the centroid j3 of set S** is used as the latter's starting point will be defined as a quasi-solution for the game. (On the term "quasi- solution", see Section 1.4 above.)

5. THE NASH PROPERTY OF MY N E W THEORY FOR

NONDEGENERATE UNANIMITY GAMES

In a unanimity game G, the equilibrium corresponding to unanimous acceptance of an agreement A k (k = 1 . . . . . K) will have the form

s k = ( s , . . . . . s,.* . . . . . ( 5 . l )

and will yield player i (i = 1 . . . . . n) a positive payoff

u~ = Ui(s~). (5.2)

The Nash product for this equilibrium s k will be defined as

13 The c e n t r o i d of a set of s t r a t e g i e s was defined in Footnote 8 but here we need the concept of the centroid of a set S** o f s t r a t e g y c o m b i n a t i o n s s* = (s~ . . . . . s* ) . This centroid is defined as a strategy combination s + = ( s { . . . . . s 2 ) in which the strategy s/+ of each player i is the centroid of all strategies s* assigned to him by the various strategy combinations s* in the set S**.

EQUILIBRIUM SELECTION 1 15

zr ~ = IZl u~. (5.3) i=1

This unanimity game G will be called nondegenerate if it contains some equilibrium s k" such that

zr ~* > zr ~ for all k ~ k*. (5.4)

By Proposition 13 of the Appendix, we can write

+ ; ( s ~ . % ( s ~ ) : . . . : + , ( s f ) = u~: u ~ : . . . :u~. (5.5)

In view of (3.24), (4.1), and (4.2), this implies that

Prob(s~) = u~/-ffi, (5.6)

where

K

i=E uf. k=l

On the other hand, by (4.3) we have

Prob(sk) = (i=I~ u')/(-ai) n.

In view of (5.3), this is equivalent to the equation

Prob(s k) = ~rk/(-ffi) ".

By similar reasoning

Hence, by (5.4)

Prob(s ~') = 7.rk*/(-~i) n.

Prob(s ~'*) > Prob(s k) for all k # k*.

(5.7)

(5.8)

(5.9)

(5. lO)

Thus, the solution of a nondegenerate unanimity game will always be the equilibrium s k* with the highest Nash product in the game so that our new theory does have the Nash property for such games.

(5.11)


Note . It is easy to show that my theory does have the Nash property also for nondegenerate 2 × 2 games , even those that are not unanimity games. But to save space I shall omit the proof here.~4

6. THE NEED FOR A THEORY OF EQUILIBRIUM SELECTION

The theory of equilibrium selection I have now proposed is obviously much simpler than the theory Selten and I proposed in 1988. It shows that such a theory can take a reasonably simple and (I hope) intuitively attractive form and still represent a generalization of Nash's bargaining solution and of its n-person generalization, in the sense of having the Nash property for unanimity games and for 2 x 2 games (subject to the nondegeneracy requirement). Of course, future research may give rise to further advances in theories of equilibrium selection.

I am now going to argue that a theory of equilibrium selection can play an important role within a general theory of strategic rationality. Cooperative solution concepts are outcome theories. They define what they consider to be desirable outcomes for cooperative games of certain types by some axioms specifying the mathematical properties these outcomes ought to have on the basis of some rationality requirments and possibly also some "fairness" criteria or other moral considerations.

In contrast, the standard theory of noncooperative games is a process theory, based on detailed specification of the process of strategic interaction among the players regarded as a noncooperative game in extensive or in normal form.

Both approaches seem to be open to some objections. Cooperative solution concepts usually fail to indicate the actual process by which the players are expected to reach an outcome satisfying the relevant axioms. Moreover, they fail to tell us how to decide when to use one particular cooperative solution concept and when to use another for analysis of various game situations.

On the other hand, many noncooperative models make their outcomes unduly dependent on very minor details of the assumed process of strategic interaction--often in a way very contrary to our intuition and to our common experience. For instance, many noncooperative bargaining models make the outcome highly dependent on the minutiae of the "rules" assumed to govern the bargaining process among the players--even though bargaining negotiations in real life are seldom subject to precise "rules" and, in those rare cases where they are, these rules are usually

J4 In 2 x 2 games my new theory uses the same definition for Nash products as our 1988 theory did (see HS pp. 86-88).

EQUILIBRIUM SELECTION 1 17

chosen by the participants themselves rather than being imposed from the outside (cf. Kreps, 1990, especially pp. 128-132). Even otherwise quite well-behaved noncooperative bargaining models, like Rubinstein's (1982) ingenious alternating-offer model, may turn out to have their outcomes dependent on trivial details, such as whether money is assumed to be a continuous or a discrete quantity (see van Damme et al. 1990).

In analysis of duopoly situations, standard noncooperative theory, if taken literally, seems to imply that Cournot-type outcomes will arise if the two firms move simultaneously whereas von Stackelberg-type outcomes will arise if they move one after the other--even though the actual outcome will presumably depend on the two firms' relative economic power rather than on the time order of their moves as such.

More fundamentally, for most noncooperative games the standard theory fails to yield any well-defined outcome at all, or will yield such an outcome only as a result of fairly arbitrary modeling assumptions.

In contrast, the theory of equilibrium selection I am proposing in this paper (just as the theory Selten and I proposed in 1988) has something like an intermediate status between cooperative solution theories and the standard theory of noncooperative games--much in the same way as the Nash solution does, of which, in view of its Nash property, my own theory is a generalization.

Like the standard theory of noncooperative games, my own theory as well is a theory of noncooperative games, and always selects one of the Nash equilibria of the game, or some probability mixture of such equilibria, as solution. But it differs from the former theory by use of a much stronger concept of strategic rationality--one strong enough always to select one specific equilibrium as solution. At the same time, like the Nash solution, my theory does have also some affinity to cooperative solution theories in that it tries to select a solution on the basis of some very fundamental parameters of the game as a whole, such as the basic interests and incentives of the players in the game, and the basic balance o f power among these players, just as many cooperative solution theories try to do--rather than an outcome strongly dependent on minor details of the assumed process of interaction among the players.

It seems to me that my theory has some important advantages over both cooperative solution theories and the standard theory of noncooperative games. Unlike cooperative solution theories, it does involve choosing a specific noncooperative-game model to characterize the process of strategic interaction among the players. On the other hand, by always selecting one specific Nash equilibrium (or correlated equilibrium) as solution for any noncooperative game, it gives us a much freer hand in choosing among alternative noncooperative models for analysis of any given game situation, without any need for special modeling assumptions in order to


ensure a well-defined outcome--and without making the outcome undesir- ably dependent on some arbitrary details of o u r o w n modeling assumptions.

APPENDIX: PROOF OF LEMMA I I OF SECTION 3 .4

In order to prove this lemma, I shall have to continue analysis of the n-person unanimity game G, which I started discussing in Section 3.4. As will be recalled, in this game the players can choose among K alternative agreements. As a result, each player i has K different pure strategies whereas the (n-l) players other and players i have M = K n-I different pure- strategy combinations (strategy (n-l)-tuples) s_ i. The latter are numbered in such a way that the first K strategy combinations are concordant combinations whereas the remaining ( M - K ) strategy combinations are discordant ones.

Here, in my analysis of game G, I shall focus on the K stability sets o'~ of each player i, and on their image sets p/k =/z(tr/k) under the inversion mapping/z.

PROPOSITION 1. Tile stability set o'~ f o r each pure strategy s~ o f any player i will be the set o f all expectation vectors qi whose f irst K components satisfy the (K - 1) inequalities:

qku~> k' k' f o r a l l k ' - - ql ui in K with k' # k, (A.I)

m

where K is the set o f the f irst K posit ive integers.

Proof. By the definition of a stability set, o-~ is the set of all vectors qi to which strategy s~ is a best reply so that

Ui(sk, qi) >- Ui(s~', qi) for all k' in -K with k' # k. (A.2)

Yet, in view of (3.12), this implies (A.1).

PROPOSITION 2. Any image set p~ = ix(o'S) will be the set o f all risk vectors xi = tz(qi) whose f r s t K components satisfy the (K - 1) inequalities

xk/u~ <-- x~'/uki ' f o r all k' in -K with k' -'~ k. (A.3)

Proof. In view of (3.3), these (K - 1) inequalities are implied by the (K - 1) inequalities under (A. 1).

PROPOSITION 3. Let the set I be defined as

I = N p/k. (A.4)

Then, I will be the set o f all risk vectors xi whose first K components satisfy the inequalities

E Q U I L I B R I U M SELECTION

xki/uk i = xf ' luf ' fora l l k, k' in K .

119

(A.5)

Proof. In view of (A.4), for each pair k and k' in K , both inequality (A.3) and the reverse inequality mus t be satisfied. This implies (A.5).

Le t me define a sequence o f sets Yi(r) for r = K, K + 1 . . . . . M, such that Yi(r) is the set of all risk vectors xl whose componen t s x m satisfy:

x~ = 0 for all m > r. (A.6)

Of course

Yi(M) = X i (A.7)

as all vectors xl in Xi will vacuous ly satisfy (A.6) for r = M because they simply have no

componen t x~ with m > M.

PROPOSITION 4. For any vector x i in a given set Yi(r), we have

M

x~'= E x~ = 1. (A.8) m=l m=l

This follows f rom (A.6).

PROPOSITION 5. Any vector x iin a given set Yi(r) can be obtained as a convex combination

o f the vertexes ~1 . . . . . o~ o f simplex Xi. This likewise follows f rom (A.6)

PROPOSITION 6. Let the set I* be defined as

I* = I N Y~(K), (A.9)

where I is the set defined by (A.4). Then, this set I* will contain only one vector gi whose

components are

~c~ = u~l'ff for all k <- K, (A. 10)

with

and

xrfl = 0 for all m > K. (A. 12)

Proof. In view of (A.5) and (A.9) we can write


3:I/ul = ~/u~ . . . . . ~f/a~" = c,

where c is some constant. Hence

k~ = cu~ for all k in K.

Consequently,

so that

(A.13)

(A. 14)

c = ~,,_ 3:/*/~_ u~. (A.16) k E K t k E K

In view of (A.8) and (A. II), this can be written also as

C = l / - f f i . (A.17)

Yet, (A.14) and (A.17) imply (A.10). On the other hand, (A.6) and (A.9) imply (A.12). Finally, uniqueness of vector k i follows from the fact that (A. 10), (A. 11), and (A. 12) together fully define 3:i'

Let me now define the family of sets r~(r) such that

r~(r) =p~ ("1 Yi(r) f o r k = 1 . . . . . K a n d f o r r = K . . . . . M. (A.18)

Note that in view of (A.7) we can write

r ~ ( M ) = p ~ forallk ~ K. (A.19)

PROPOSITION 7. Any given set z~(r) will consist o f all vectors xi whose components satisfy the (K - 1) inequalities

x~/u~ <- x~'/u~' f o r all k' in K with k' # k (A.20)

as well as the additional requirement that

x 7' = 0 f o r a l l m > r. (A.21)

Proof. In view of (A.18), x i will belong to set z~(r) if and only if it belongs both to set p,.* and to set Yi(r). Yet, by Proposition 2, it will belong to pg* if it satisfies (A.20), and it will belong to Yi(r) if it satisfies (A.21).

PROPOSITION 8. Any g.iven set r~(r) will be the convex hull o f the (r - 1) vectors o-(l I . . . . . o~_ ~, ~.+ t . . . . . ~ and o f the additional vector .ri defined by Proposit ion 6.

Y~ 3:f=c Y, u~ (A.15)


Proof. In view of (A.18), r~(r) is the intersection of the two convex sets p/k and Yi(r) and, therefore, is itself a convex set. Moreover, as is easy to verify, all r vectors mentioned in Proposition 8 satisfy both (A.20) and (A.21) and, therefore, all of them belong to set r~ (r). As the latter is a convex set, all vectors xi in the convex hull H of these r vectors likewise belong to this set. On the other hand, as is equally easy to verify, none of the vectors x i outside the boundaries of H can satisfy both (A.20) and (A.21), so that none of these vectors can belong to ~'/k(r).

Let me now state the two rather obvious Propositions 9 and 10:

PROPOSITION 9. Suppose that the sets R t . . . . . R k, . . . , R r are pyramids whose bases are congruent to one another. Then, the Lebesgue measures o f these pyramids will be proportional to their altitudes.

PROPOSITION 10. Suppose that the sets T I, . . . , T ~ . . . . . T ~ are pyramids whose altitudes are equal. Then, the Lebesgue measures o f these pyramids will be proportional to the Lebesgue measures o f their bases.

PROPOSITION 11. The Lebesgue measures h[r/~(r)] o f the sets "r~(r) will be proportional to the payoffs u~ for k = I . . . . . K.

Proof. By Proposition 5, the set Yi(K) is a convex hull of the K vertexes ~ t . . . . . ~K of the (M - 1)-dimensional simplex Xi and, therefore, is itself a (K - 1)-dimensional simplex. By (A.18), each set r/k(K) is a subset of this simplex Yi(K). More particularly, each set ~'~(K) is a pyramid whose base is one of the (K - 2)-dimensional faces of this (K - l)-dimensional simplex Yi(K), whereas its apex is the vector xi defined by Proposi- tion 6, and its altitude is the kth component k/~ of this vector k i. Therefore, by Proposi- tion 9, the Lebesgue measures h[~'~(K)] of these K pyramids will be proportional to the K quantities .~/k and, in view of (A.10), also to the K payoffs u/k.

PROPOSITION 12. Sappose the Lebesgue measures h[r/k(r)] o f the sets "r~(r) are proportional to the payoffs u~ for k = i . . . . . K. Then, the same will be true for the Lebesgue measures X[r/k(r + 1)] o f the sets r~(r + l) for k = 1 . . . . . K.

Proof. In view of Proposition 5, set Yi(r + 1) is an r-dimensional simplex whereas Yi(r) is an (r - l)-dimensional face of this simplex. Moreover, by (A.9), for k = 1 . . . . . K, each set r~(r + 1) is a subset of Yi(r + 1), and each set ~-/k(r) is a subset of Yi(r). Furthermore, each set r~(r + 1) is apyramid whose base is the set r/k(r) whereas its apex is vertex °li.r+t, and its altitude is the vertical distance between the vertex ~r÷l and the face Yi(r)of the simplex Yi(r + I).Therefore, the altitudes of all pyramids r/k(r + 1) will be equal in length.

Yet, by assumption, the Lebesgue measures k[z/k(r)] of the sets r/k(r) are proportional to the payoffs u/k for k = 1 . . . . . K. In view of Proposition t0, this implies that the Lebesgue measures h['r~(r + 1)] of the pyramids r~(r + 1) will be likewise proportional to the payoffs u~ for k = 1 . . . . . K - which is what Proposition 12 asserts.

PROPOSITION 13. The Lebesgue measures h(p~) o f the image sets p~ are proportional to the payoffs u~ for k = 1 . . . . . K.

Proof. By Proposition I1, the Lebesgue measures h[r~(K)] are proportional to the payoffs u k in the case r = K. Proposition 12 enables us to extend this statement inductively to higher and higher values of r, including r = M. Yet, by (A. 19), z/k(M) = OF for k = 1 . . . . . K. This establishes the truth of Proposition 13.


Finally, Proposition 13 implies Lemma II of Section 3.4.

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HARSANYI, J. C. (1977). Rational Behavior and Bargaining Equilibrium in Games and Social Situations. Cambridge: Cambridge Univ. Press.

HARSANYI, J. C., AND SF,LTF,N, R. (1988). A General Theory o f Equilibrium Selection in Games. Cambridge, MA: MIT Press.

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