Appendix: Games with Mixed Actions

deviation (a payoff of 3 for one period and 1 thereafter). This discussion leads to the following corollary:

Corollary 2.1

In an infinitely repeated game cooperation is easier to sustain when players have a higher time discount factor.

2.3.3 A discussion of repeated games and cooperation

In this section we have shown that a one-shot game with a unique non-cooperative Nash equilibrium can have a cooperative SPE when it is repeated infinitely. However, note that in the repeated game, this SPE is not unique. For example, it is easy to show that the noncooperative outcome where each country plays WAR in every period constitutes a SPE also. Moreover, the Folk Theorem (Folk, because it was well known to game theorists long before it was formalized) states that for a sufficiently high time discount factor, a large number of outcomes in the repeated game can be supported as a SPE. Thus, the fact that we merely show that cooperation is a SPE is insufficient to conclude that a game of this type will always end up with cooperation. All that we managed to show is that cooperation is a possible SPE in an infinitely repeated game.

Finally, let us look at an experiment Robert Axelrod conducted in which he invited people to write computer programs that play the Prisoners' Dilemma game against other computer programs a large number of times. The winner was the programmer who managed to score the largest sum over all the games played against all other programs. The important result of this tournament was that the program that used a strategy called Tit-for-Tat won the highest score. The Tit-for-Tat strategy is different from the trigger strategy defined in Definition 2.12 because it contains a less severe

punishment in case of deviation. In the Tit-for-Tat strategy, a player would play in period t what the opponent played in period t - 1. Thus, even if deviation occurred, once the opponent resumes

cooperation, the players would switch to cooperation in a subsequent period. Under the trigger strategy, once one of the players deviates, the game enters a noncooperative phase forever.

2.4 Appendix: Games with Mixed Actions

The tools developed in this appendix are not implemented in this book, and are brought up here only for the sake of completeness. Thus, this appendix is not necessary to study this book successfully, and the beginning readers are urged to skip this appendix.

Games with mixed actions are those in which the players randomize over the actions available in their action sets. Often, it is hard to

motivate games with mixed actions in economics modeling. This is not because we think that players do not choose actions randomly in real life. On the contrary, the reader can probably recall many instances in which he or she decided to randomize actions. The major reason why games with mixed actions are hard to interpret is that it is not always clear why the players benefit from

randomizing among their pure actions.

The attractive feature of games with mixed actions is that a Nash equilibrium (in mixed actions) always exists. Recall that Proposition 2.2 demonstrates that a Nash equilibrium in pure actions need not always exist.

In what follows, our analysis will concentrate on the Top-Bottom-Left-Right given in Table 2.5. The reason for focusing on the game in

Ms. β

Table 2.5 is that we show that a Nash equilibrium in mixed actions exists despite the fact that a Nash equilibrium in pure actions does not (the reader is urged to verify that indeed a Nash equilibrium in pure actions does not exist).

We now wish to modify a game with pure strategies to a game where the players choose probabilities of taking actions from their action sets. Recall that by Definition 2.1, we need to specify three elements: (a) the list of players (already defined), (b) the action set available to each player, and (c) the payoff to each player at each possible outcome (the payoff function for each player). probability λ and plays R with probability 1 -λ

3. An action profile of a mixed actions game is a list (τ, λ) (i.e., the list of the mixed action chosen by each player).

4. An outcome of a game with mixed actions is the list of the realization of the actions played by each player.

Definition 2.13 implies that the mixed-action set of each player is the interval [0,1] where player α picks a and player β picks a . The reader has probably noticed that Definition 2.13 introduces a new term, action profile, which replaces the term outcome used in normal form games, Definition 2.1. The reason for introducing this term is that in a game with mixed actions, the players choose only probabilities for playing their strategies, so the outcome itself is random. In games with pure actions, the term action profile and the term outcome mean the same thing since there is no uncertainty. However, in games with mixed actions, the term action profile is used to describe the list of probability distributions over actions chosen by each player, whereas the term outcome specifies the list of actions played by each player after the uncertainty is resolved.

Our definition of the "mixed extension" of the game is incomplete unless we specify the payoff to each player under all possible action profiles.

Definition 2.14

A payoff function of a player in the mixed-action game is the expected value of the payoffs of the player in the game with the pure actions. Formally, for any given action profile (λ, τ), the expected payoff to player i, i = α, β, is given by

According to Definition 2.1 our game is now well defined, since we specified the action sets and the payoff functions defined over all possible action profiles of the mixed actions game.

Applying the NE concept, defined in Definition 2.4, to our mixed-actions game, we can state the following definition:

Definition 2.15

An action profile (where , ), is said to be a Nash equilibrium in mixed actions if no player would find it beneficial to deviate from her or his mixed action, given that the other player does not deviate from her or his mixed action. Formally,

We now turn to solving for the Nash equilibrium of the mixed-actions extension game of the game described in Table 2.5. Substituting the payoffs associated with the "pure" outcomes of the game in Table 2.5 into the "mixed" payoff functions given in Definition 2.14 yields

and

Restating Definition 2.15, we look for a pair of probabilities that satisfy two conditions: (a) for a given , maximizes given in (2.6), and (b) for a given , maximizes given in (2.7).

It is easy to check that the players' payoffs (2.6) and (2.7) yield best-response functions (see Definition 2.5) given by

That is, when player β plays R with a high probability (1 - λ> 1/2), player α's best response is to play T with probability 1 (τ = 1) in order to minimize the probability of getting a payoff of -1.

However, when player β plays L with a high probability (λ > 1/2), player α's best response is to play B with probability 1 (τ = 0) in order to maximize the probability of getting a payoff of +1. Similar explanation applies to the best-response function of player β.

The best-response functions of each player are drawn in Figure 2.3. Equations (2.8) and Figure 2.3 show that when β plays λ = 1/2, player α is indifferent to the choice among all her actions. That is, when λ = 1/2, the payoff of player α is the same (zero) for every mixed action . In particular, player α is indifferent to the choice between playing a pure strategy (meaning that τ = 0 or τ = 1) and playing any other mixed actions (0 < τ < 1). Similarly, player β is indifferent to the choice among all her mixed actions , when player α plays τ= 3/4.

Although a NE in pure actions does not exist for the game described in Table 2.5, the following proposition shows:

Proposition 2.6

There exists a unique NE in mixed actions for the game described in Table 2.5. In this equilibrium, τ

= 3/4 and λ = 1/2.

Figure 2.3:

Best-response functions for the mixed-action extended game

The proposition follows directly from the right-hand side of Figure 2.3 that shows that the two best-response functions given in (2.8) have a unique intersection.

Finally, the best-response functions given in (2.8) have a property of being composed of horizontal or vertical line segments. Since the equilibrium occurs when the two curves intersect in their

"middle" sections, we have it that under the NE mixed outcome, each player is indifferent to the choice among all other probabilities that can be played, assuming that the other player does not deviate from the mixed action. This result makes the intuitive interpretation of a mixed-action game rather difficult, because there is no particular reason why each player would stick to the mixed action played under the NE.