John Nash's Equilibrium

John Nash's Equilibrium 

by : Aria Ratmandanu 

Figure.1 Professor John Nash (1928-2015)

        Classical game theory (Luce & Raiffa 1957, Fudenberg & Tirole 1991) has two different branches. In cooperative game theory, the phenomenon of cooperation is to a certain extent assumed and thus not subject to a complete analysis. Noncooperative game theory, on the other hand, seeks to fully explain cooperation as well as noncooperation. This branch is the one that matters in evolutionary biology. It was established in classical game theory by Nash (1951). He suggested that one should study a game by looking for a combination of strategies (one for each player) with the following property. If all players act according to this combination, then "everybody achieves his maximum payoff against the strategies of all other players." This means that nobody would have an incentive to unilaterally deviate from such a combination. The idea can be rephrased in technical terms.

A Nash equilibrium is a combination of strategies for the players of a game, such that each player's strategy is a best response to the other players' strategies. A best response is a strategy which maximizes a player's expected payoff against a fixed combination of strategies played by the others.

        In order to illustrate this, let us consider a game taking place in real life. In a psychological experiment, envelopes are distributed to three subjects. They are each asked to put any amount of money between 0 and 100 units in their envelope. Nobody is given a chance to observe what the others contribute. The experimenter then collects the envelopes and proceeds according to the following rule which is known to everybody. All contributions are thrown into the same box. If this box contains 30 money units or more, the experimenter himself will throw 15 additional units into the box and all the accumulated money will be split equally among the three subjects. However, if the box contains less than 30 units, all money goes to the experimenter.

        Let us treat this as a three-person game played by the subjects. What would be a Nash equilibrium for them ? We only want to ask here for symmetric equilibria, where everybody plays the same strategy and thus gives the same amount. Obviously, (10, 10, 10) is a symmetric Nash equilibrium. The reason is that no player has an incentive to unilaterally deviate from this solution: If the other two players each play 10, the third player achieves his maximum by also playing 10. In this Nash equilibrium, the players cooperatively exploit their resource—the experimenter—and achieve a net equilibrium payoff of 5 units. We now have to ask whether this is the only solution to the game. There is indeed another symmetric Nash equilibrium, (0, 0, 0), in which all three subjects hand over empty envelopes to the experimenter. Obviously, nobody has an incentive to deviate from this solution. Furthermore, no cooperation takes place, the resource remains unexploited, and the equilibrium payoff is zero.

           This type of game is well known and has been played in numerous experiments. It is presented here to give the reader a feel for the Nash equilibrium. We observe a phenomenon which is typical for games, namely that there is more than one such equilibrium. One would think that the players should play the cooperative solution. However, imagine a player who worries about the risk involved if other players come to a different conclusion about the choice of equilibrium. If this player plays zero, he is relatively safe and can only miss the small cooperative payoff consisting of 5 units. Otherwise, he might lose the larger amount of 10 units if he cooperates and another player fails to do so. This would be an argument in favor of the noncooperative solution. Obviously, it can be difficult to decide between alternative Nash equilibria.

         Turning back now to biology, the important solution concept developed by Maynard Smith and Price (1973), and foreshadowed by Hamilton (1967), is that of an evolutionarily stable strategy (ESS). Intuitively speaking, ESS theory draws the modeler's attention to population states which are resistant against the forces of selection and mutation. An ESS sensu Maynard Smith (1982) is a strategy with the following property: If all members of a population are genetically coded to play this strategy, any initially rare mutant strategy would receive negative selection pressure in this population.

           Let us see what this means in the formal context of equation (1). Suppose that the population plays strategy s{x} and that a mutant s2 arises. Let the strategy frequen- cies be jt, = 1—e and x2 = e, so that e is the mutant frequency. The evolutionary stability of 5] means that for sufficiently small e the difference w](x) —w2(x) must be positive. If we are dealing with random pairwise interactions and if fitness is defined as in (2), this difference can be written as follows:

                                                                                                                                   (1)

                       (2)

(3)

            We are now able to see how the Nash equilibriumemerges in evolutionary biology. If for some mutant strategy the first square bracket in (3) is negative, then wt(x) —w2(x) becomes negative for sufficiently small values of e. In order to exclude this possibility, one has to require for an ESS SL that it should be a best response to itself in the phenotypic game. The latter requirement can be rephrased by saying that the symmetric pair of strategies (S|,S]) should be a Nash equilibrium.

        This is a necessary, but not sufficient, condition for (3) to be positive. Suppose there is another strategy s2 which is different from s,, but also a best response to s^. The first square bracket in (3) then is zero and the second square bracket needs to be positive in order for selection to act against the mutant under consideration. Clearly, this is only the case if E(sl,s2)>E(s2,s2). In order to ensure evolutionary stability, this second condition needs to hold for all strategies s2 that are alternative best responses to 5,.

         We have now recapitulated Maynard Smith's original thoughts using the language of game theory. Using this language again, his technical definition of an ESS can be described as follows. It relates to a symmetric game in strategic form and to the dynamic context of (1) with fitness function (2). A strategy st is called evolutionarily stable if it satisfies the following two conditions:

1. Property of a symmetric Nash equilibrium: .v, is a best response to {s}. In other words, if an opponent plays this strategy, one receives the highest possible payoff by also playing this strategy.

2. Stability against alternative best responses: If a strategy s2 is different from s, and E(s2,sl) =E(sl,s]), then the inequality £(s,,.s2) > E(s2,s2) holds. In other words, if another strategy also achieves the highest payoff against s^ then it is better to play against this other strategy than to play the latter strategy s2 against itself.

          From an economist's point of view, it is interesting to note that we only have to study the phenotypic game model in order to see whether these two ESS conditions hold. The selection equation (1) is completely hidden in the background theory that gave rise to these conditions. Nash himself foresaw the possibility of dynamic interpretations of the Nash equilibrium when he mentioned in his work the "mass action interpretation" of his solution concept as an alternative to decision-theoretic interpretations. However, it surely was Maynard Smith (1982) who first created such a theory. Furthermore, the now famous statement of Nash was buried in his Ph.D. thesis and cannot be found in his publications. It was only rediscovered when Nash received the Nobel Prize in 1994.