Finitely repeated games with social preferences

Abstract

A well-known result from the theory of finitely repeated games states that if the stage game has a unique equilibrium, then there is a unique subgame perfect equilibrium in the finitely repeated game in which the equilibrium of the stage game is being played in every period. Here I show that this result does in general not hold anymore if players have social preferences of the form frequently assumed in the recent literature, for example in the inequity aversion models of Fehr and Schmidt (Quartely Journal of Economics 114:817–868, 1999) or Bolton and Ockenfels (American Economic Review 100:166–193, 2000). In fact, repeating the unique stage game equilibrium may not be a subgame perfect equilibrium at all. This finding should have relevance for all experiments with repeated interaction, whether with fixed, random or perfect stranger matching.

Appendix

Proof of the Claim

We can analyze the game as a game between P1 and P2 since these are the only active players (although the payoffs of R1 and R2 do matter when P1 and P2 have social preferences).

After the first period, the following 9 subgames can originate, $(a_{1}^{1},a_{2}^{1})\in \{(100,0),(100,100), (0,100),(0,0),(100,40),(40,100),(0,40),(40,0),(40,40)\}$, where $a_{i}^{t}$ denote the amounts that proposer i keeps for himself in period t.

To prove that a SPE exists in which P1 and P2 keep 100 in the first period and 0 in the second, we consider first the subgame (100,100). The (overall) Fehr–Schmidt utilities for the actions $a_{i}^{2}\in \{0,40,100\}$ in this subgame are given by the following matrix (since the payoff matrix is symmetric, only the utilities of P1 are shown).

		P2
		0	40	100
P1	0	100	$100-20\beta -\frac{40}{3}\alpha$	$100-\frac{100}{3}\beta -\frac{100}{3}\alpha$
	40	140−60β	$140-\frac{200}{3}\beta$	140−80β−20α
	100	$200-\frac{400}{3}\beta$	200−140β	$200-\frac{400}{3}\beta$

Thus, (0, 0) is a Nash equilibrium in this subgame for β≥3/4. It can be checked that P1 can obtain no higher utility in any other subgame. Thus for P1 and P2 to keep 100 in the first period and 0 in the second is a SPE.

To show that the repetition of (40,40) is not SPE for some parameters, consider the subgame (40,40) which results in the following Fehr-Schmidt utility matrix.

		P2
		0	40	100
P1	0	$40-\frac{200}{3}\alpha$	40−60α	$40-\frac{200}{3}\alpha$
	40	$80-\frac{40}{3}\beta -20\alpha$	80	$80-\frac{40}{3}\beta -20\alpha$
	100	$140-\frac{200}{3}\beta$	$140-\frac{220}{3}\beta$	$140-\frac{200}{3}\beta$

While (100, 100) is always a Nash equilibrium of this subgame, (40, 40) is a Nash equilibrium only for $80\geq 140-\frac{220}{3}\beta$, or β≥9/11. Hence, for β<9/11, the repetition of the stage game Nash equilibrium is not a SPE.

Together these facts imply that the repetition of (40,40) is never the unique SPE. □