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In this paper we propose a “behavioral equilibrium” definition for a class of dynamic games of perfect information. We document various experimental studies of the Centipede Game in the literature that demonstrate that players rarely follow the subgame perfect equilibrium strategies. Although some theoretical modifications have been proposed to explain the outcomes of the experiments, we offer another: players can choose whether or not to believe that their opponents use subgame perfect equilibrium strategies. We define a “behavioral equilibrium” for this game; using this equilibrium concept, we can reproduce the outcomes of those experiments.

In dynamic games of perfect information, the concept of subgame perfect equilibrium is most commonly used in the prediction of players’ behavior. Consider a generic game of finitely many moves, the subgame perfect equilibrium always uniquely exists. While the equilibrium concept is easily understood and the equilibrium characterization is usually straightforward, challenges to its ability to predict players’ behavior grow in the literature, both on theoretical front and experimental front.

Rosenthal [

Various centipede game experiments have been conducted to test the predictive power of the concept of SPE. McKelvey and Palfrey [

In an attempt to reconcile the differences between the theory and the experimental outcomes, various modifications to the assumptions of the games used in the experiments have been proposed. McKelvey and Palfrey [^{1}. A few years later, McKelvey and Palfrey [

In the theoretical literature, game theorists have proposed alternatives to some key assumptions that lead to SPE, including the common knowledge of rationality and backward induction. Aumann [^{2}. Caplan [

Advances in psychology also help explain why players in experiments may behave differently than SPE predicts. Epstein et al. [

In this paper, we argue along the lines of the above psychological findings and propose another theoretical explanation of the failure of the SPE as a predictor of behavior. We emphasize on the observation that even if all players understand fully the concept of subgame perfect equilibrium and even if no players believe that other players are altruists, they still do not follow the SPE strategies when playing the centipede game. We assume that a player can choose to play SPE, i.e. be “rational”, or else may choose to be “behavioral”. If being “behavioral” yields a better expected outcome than being “rational”, then a player would choose to be “behavioral” (or, in terms of standard game theory terminology, “irrational”). Our intuition is as follows. SPE strategies are optimal for a player only when other players follow them. If players do not believe that other players will follow SPE strategies, then their own SPE strategies are not, in general, optimal. In the model, we specify an alternative belief for each player regarding the behavior of other players. Each player then has a choice of selecting his belief (between the SPE strategy and the alternative one) at the beginning of the game and then optimizing given the selected belief. A “behavioral equilibrium” is formed if each player is better off in the actual outcomes by selecting the alternative belief. These outcomes of the game are determined by the strategies the players actually used in the game.

The basic idea behind the “behavioral equilibrium” concept is that players can choose to believe that their counterparts can be either fully rational (such that SPE strategies are the best response) or somewhat irrational (so that SPE strategies are not best response any more). Given any belief, the players still optimize by choosing the best strategy. This is the same as in a subgame perfect equilibrium. However, the difference between a behavioral equilibrium and a subgame perfect equilibrium is that those alternative beliefs in a behavioral equilibrium do not usually coincide with those players’ actual strategies. If the two are the same, a subgame perfect equilibrium is formed. Therefore, these alternative beliefs are somewhat irrational. Still, these irrational beliefs generate better payoffs than those SPE beliefs. Thus, players will choose these irrational beliefs rationally.

The origin of irrational beliefs is an interesting and open question. Epstein et al. [

One real life example related to the centipede games that we examine in this paper is the rotating-savings and credit associations (Roscas), commonly found in many developing countries. (See Besley et al. [

The rest of this paper is organized as follows. In Section 2, we analyze a few centipede games using the concept of “behavioral equilibria”. In Section 3, we analyze some of the experiments in centipede games in the literature. In Section 4, we conclude.

We begin with a general description of the centipede games.

There are two players, 1 and 2, playing the centipede game of n moves in

In this game,

Now suppose that before the start of the game, the two players choose a belief secretly and simultaneously. Player 1 chooses a belief from

player 2 chooses a belief from

The subgame perfect equilibrium belief

In summary, the game we are examining is as follows. Both players simultaneously select their beliefs before the start of the game. Once the belief is selected, it remains the same throughout the game. Given these beliefs regarding an opponent’s behavior, players play the above centipede game. Each player’s goal is to maximize his expected payoff given his chosen belief.

To simplify our analysis, we assume that the beliefs are not updated during the game. (Even if we allow for belief updating, we will not get back the SPE beliefs as long as the initial belief is somewhat incorrect.)

To analyze the modified centipede game, first note the following. If

If player 1 chooses belief

Let

Let

The proposed pure strategy for player 1 is to select

Definition 1

where

In this behavioral equilibrium, players are better off selecting these non-SPE beliefs than selecting the SPE beliefs. These beliefs are reinforced if the players play these games again later.

Now consider mixed strategy “behavioral equilibria”. Suppose that there are more than one j’s that maximize (2), or there are more than one i’s that maximize (1), mixed strategies could be used by the players. Let

Definition 2

Again, in this behavioral equilibrium, players are better off selecting these non-SPE beliefs than selecting those SPE beliefs. We can generalize the concept of behavioral equilibria to any general game G with n players and normal-form payoff

Definition 3 Suppose that

Note that in the above definition, a player’s belief may not be correct; that is,

Note also that the subgame perfect equilibrium strategy profile

Below, we focus on centipede games to illustrate our equilibrium concept.

Example 1 Consider the eight-move centipede game in

Suppose that

It is easy to see that

Example 2 Consider the six-move centipede game in

In this game, we can construct pure-strategy behavioral equilibria similarly to the last example. Let

Now consider a mixed-strategy behavioral equilibrium. Suppose that

that is,

Similarly, for player 2,

Suppose that

To construct a behavioral equilibrium, player 1’s mixed strategy

To summarize,

^{3}Payoffs

McKelvey and Palfrey [^{3} Session 4 is a high-payoff four-move centipede game where the payoffs are quadrupled. Sessions 5 to 7 are six-move centipede games with the following payoffs:

We cannot infer a player’s belief in playing these games from the data since many different beliefs could lead to the same observed strategy. Therefore, in each session, we assume that a player’s belief corresponds exactly to his rival’s revealed strategy and calculate the player’s optimal action according to that belief. In the calculations, we assign the players a utility function with a constant degree of absolute risk aversion of 0.5 so

Session | Optimal Action | |||||||
---|---|---|---|---|---|---|---|---|

1 | player 1 | 0.06 | 0.61 | Take at Node 3 (61%) | ||||

player 2 | 0.28 | 0.61 | Take at Node 4 (61%) | |||||

2 | player 1 | 0.10 | 0.69 | Take at Node 3 (69%) | ||||

player 2 | 0.42 | 0.52 | Take at Node 4 (52%) | |||||

3 | player 1 | 0.06 | 0.42 | Pass at Node 3 (42%) | ||||

player 2 | 0.46 | 0.33 | Take at Node 4 (33%) | |||||

4 | player 1 | 0.15 | 0.57 | Take at Node 3 (57%) | ||||

player 2 | 0.44 | 0.39 | Take at Node 2 (44%) | |||||

5 | player 1 | 0.02 | 0.43 | 0.50 | Take at Node 5 (50%) | |||

player 2 | 0.09 | 0.51 | 0.20 | Take at Node 4 (51%) | ||||

6 | player 1 | 0.00 | 0.04 | 0.70 | Take at Node 5 (70%) | |||

player 2 | 0.02 | 0.48 | 0.42 | Take at Node 4 (48%) | ||||

7 | player 1 | 0.00 | 0.15 | 0.55 | Take at Node 5 (55%) | |||

player 2 | 0.07 | 0.51 | 0.40 | Take at Node 4 (51%) |

that the players are modestly risk averse. That is,

In this paper, we propose a concept of behavioral equilibrium to explain the observed behavior of players in centipede games. Experimental evidence suggests that players’ behavior is inconsistent with game theoretic predictions. We allow players to abandon the “logic” of subgame perfect equilibrium and to choose an alternate belief of opponents’ expected behavior formed from previous experience in similar situations. We show that, under certain conditions, players are better off abandoning the “logic” of subgame perfect equilibrium and choosing the alternative belief instead. We argue this reinforces the players’ subjective belief that subgame perfect equilibrium may not work well in these games and, by extension, that the alternative belief becomes the belief of choice. We support our theory by re-examining the results of centipede game experiments conducted by other researchers.

We thank the referees, Jim Bergin, Lester Kwong and Jasmina Arifovic for helpful com- ments. Ruqu Wang’s research is supported by the Social Sciences and Humanities Research Council of Canada. Xiaoting Wang acknowledges support from the National Natural Science Foundation of China (#71571038).

Dunbar, G., Wang, R.Q. and Wang, X.T. (2016) Rationalizing Irrational Beliefs. Theoretical Economics Letters, 6, 1219-1229. http://dx.doi.org/10.4236/tel.2016.66115