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Theory and Decision

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01-01-2015

Cooperation and signaling with uncertain social preferences

Authors: John Duffy, Félix Muñoz-García

Published in: Theory and Decision | Issue 1/2015

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Abstract

This paper investigates behavior in finitely repeated simultaneous and sequential-move prisoner’s dilemma games when there is one-sided incomplete information and signaling about players’ concerns for fairness, specifically, their preferences regarding “inequity aversion.” In this environment, we show that only a pooling equilibrium can be sustained, in which a player type who is unconcerned about fairness initially cooperates in order to disguise himself as a player type who is concerned about fairness. This disguising strategy induces the uninformed player to cooperate in all periods of the repeated game, including the final period, at which point the player type who is unconcerned about fairness takes the opportunity to defect, i.e., he “backstabs” the uninformed player. Despite such last-minute defection, our results show that the introduction of incomplete information can actually result in a Pareto improvement under certain conditions. We connect the predictions of this “backstabbing” equilibrium with the frequently observed decline in cooperative behavior in the final period of finitely repeated experimental games.

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Players in PD games face similar strategic incentives as those in public good games and, more generally, games where players’ actions are strategic substitutes. Sequential-move versions are also used to characterize firm-worker wage-effort decisions and the notion of “gift-exchange.”

For instance, for the finitely repeated PD game, Andreoni and Miller’s (1993) experiment shows that cooperation peaks in the first round of interaction (86% of subjects cooperate), stays above 50% until round 6, and then falls to about zero in the last (tenth) round.

In particular, the worker’s type is assumed to be either reciprocator or selfish, since Healy’s results do not derive from any preference specification.

In particular, we examine a twice-repeated sequential-move PD game where the first mover is uninformed about the second mover’s social preferences. In this game, the first mover cooperates only when he believes that the second mover will reciprocate afterward (which occurs when the second mover is highly concerned about fairness). These strategic incentives coincide with those in the gift-exchange game analyzed by Healy (2007) whereby the firm manager only offers high wages when he believes that the worker is a reciprocating type.

In a study of the hold-up problem under incomplete contracts, Siemens (2009) considers a signaling game whereby the seller of a good initially invests in the good’s quality, and then the seller and a buyer interact in an ultimatum bargaining game where the buyer makes a take-it-or-leave-it offer to the seller. Similar to our results, his paper also shows that the seller’s initial investment can serve to conceal his privately observed fairness concerns to the buyer in order to condition the buyer’s offers in the ensuing bargaining.

In such a context, the uninformed player defects at every stage of the game and, therefore, the informed player cannot affect the uninformed player’s actions. This eliminates the possibility of information transmission; see Kreps et al. (1982), p. 251. If, in contrast, the model in Kreps et al. (1982) is modified to allow for one-sided uncertainty where it is common knowledge that reciprocation is a dominant strategy for the uninformed player, then both our model and theirs would yield a similar cooperative outcome. Such cooperation, nonetheless, originates from inequity aversion in our model (which has regularly been observed in experiments), while the irrational reciprocation in Kreps et al. (1982), i.e., which arises when their parameter \(a\) is lower than 1, would be more difficult to support experimentally.

We also consider the usual second condition on the parameters of PD games, \( 2a>b+c\), to guarantee that, in the iterative version of the game, mutual cooperation provides a larger payoff than that arising from alternating cooperation and defection.

Intuitively, \(\alpha _{i}\ge \beta _{i}\) implies that players (weakly) suffer more from inequality directed at them than inequality directed at others. Empirically, estimates of \(\alpha _i\) have been found to be 2–3 times higher than estimates of \(\beta _i\). On the other hand, \(\beta _{i}\ge 0\) means that players dislike being better off than others (this assumption rules out cases in which individuals are status seekers but serves to simplify the analysis). Finally, \(\beta _{i}<1\) suggests that when player \(i\)’s payoff is higher than that of player \(j\)’s by one unit (e.g., a dollar), player \(i\) is never willing to give up more than one unit in order to reduce this inequality. For more details, see Fehr and Schmidt (1999).

Note that this best response function is similar to what Cooper et al. (1996) call “best response altruists,” namely players for whom cooperate (defect) is their best response to cooperation (defection, respectively). This result also relates with that of Rabin (1993) for psychological games, where he assumes that players are motivated by the kindness they infer from other players’ actions. Rabin (1993) assumes, however, that individuals’ kindness parameters are common knowledge among the players. In contrast, we extend our study by allowing for incomplete information.

This best response function for the second mover resembles that of Falk and Fischbacher (2006). In particular, assuming that individuals are perfectly informed about each others’ reciprocal motivations, they show that the second mover might respond by “matching” the first mover’s choice if the second mover is sufficiently reciprocally motivated. When the second mover is insufficiently motivate to reciprocate, he responds to any action of the first mover with defection.

Clark and Sefton (2001) provide an experimental test of this best response function. Specifically, they modify the payoff structure in the sequential PD game so that the second mover can obtain a “temptation payoff” if he is the only player defecting. Note that this payoff structure resembles ours, since payoffs associated to the (C,C) and (D,D) outcomes are unmodified, relative to the standard PD game (with selfish players), but those in which only the second mover defects vary. In particular, they find that the second mover is more likely to respond to cooperation with cooperation as the “temptation payoff” from defecting decreases. This experimental observation is in line with our result, since the second mover has greater incentives to respond to cooperation with cooperation if his concerns for fairness are relatively high (when the “temptation payoff” from defecting decreases), but responds by defecting against any choice by the first mover when he (the second mover) is unconcerned about fairness (when the “temptation payoff” increases).

Otherwise, player \(j\) would find defection to be a strictly dominant strategy in the second-period simultaneous-move game, and the first-period player \(i\)’s actions would not affect player \(j\)’s future play.

In order to guarantee that the observation of envy parameter \(\alpha _{i}\) does not allow the uninformed player \(j\) to infer the guilt parameter \(\beta _{i}\), consider that \(\alpha _{i}\) is distributed according to a continuous distribution function \(G(\alpha _{i})\), which assigns a positive probability to all \(\alpha _{i}\ge \beta _{i}^\mathrm{H}\). In such setting, observing the precise realization of \(\alpha _{i}\) does not provide player \(j\) with any additional information about \(\beta _{i}\), other than that \(\beta _{i}\) must satisfy \(\alpha _{i}\ge \beta _{i}\), which holds by assumption. Note that if, instead, \(\alpha _{i}\) was distributed according to a discrete probability distribution by which \(\alpha _{i}\) could only take two possible values, the mere observation of the realization of \(\alpha _{i}\) would allow the uninformed player \(j\) to infer the value of \(\beta _{i}\), thus nullifying the signaling role of player \(i\)’s actions.

Therefore, the uninformed player \(j\) (being highly concerned about fairness) does not know whether the game he plays with player \(i\) is: (1) a Pareto coordination game, which arises when player \(i\) is also highly concerned about fairness; or (2) a game where defection is a strictly dominant strategy for player \(i\), while player \(j\) still prefers to mimic the action selected by his opponent, which ensues when player \(i\)’s concerns for fairness are low. Hence, the latter game can neither be interpreted as a standard PD game or as a Pareto coordination game.

For simplicity, we ignore discounting. However, for completeness at the end of this section we demonstrates that the equilibrium predictions of Proposition 1 are unaffected by allowing for discounting. In addition, note that we use “pooling” equilibrium to refer to strategy profiles in which both types of informed player \(i\) cooperate during the first-period game. For robustness, we show that this pooling equilibrium survives Cho and Kreps’ (1987) Intuitive Criterion; see Appendix 2. That appendix also provides conditions under which a “non-cooperative” pooling equilibrium—where both types of player \(i\) defect in the first period—can be sustained, and under which parameter values it survives Cho and Kreps’ (1987) Intuitive Criterion.

If players’ guilt parameters satisfied \(\beta _{i}^\mathrm{H}>\beta _{i}^\mathrm{L}\), but envy concerns were lower for those concerned about fairness than for unconcerned players, i.e., if \(\alpha _{i}^\mathrm{H}\le \alpha _{i}^\mathrm{L}\), then such a separating strategy profile could be sustained in equilibrium. However, following the experimental evidence by Bellemare et al. (2008) as described at the beginning of this section, we think it is more plausible to assume that individuals with high concerns for inequity aversion exhibit both larger guilt and envy concerns than players with low concerns, thus implying that this case will not arise.

If, instead, priors are sufficiently low, i.e., \(q<q^\mathrm{Sim}(\alpha _{j},\beta _{j})\), the uninformed player \(j\) defects in the first period of the game and, as a consequence, an alternative pooling equilibrium emerges in which the informed player \(i\) defects in the first period of interaction, both when he is concerned about fairness and when he is not. Player \(j\)’s action in the second-period of interaction, however, depends on his off-the-equilibrium beliefs: when they are relatively high, the uninformed player \(j\) chooses to cooperate after observing that player \(i\) cooperated in the first period, but defects otherwise. In such a setting, player \(i\) does not find it profitable to cooperate in the first-period game, thus leading both types of player \(i\) to defect. For more details about this pooling equilibrium, see Appendix 1.

If the simultaneous-move PD game is, instead, repeated for \(T>2\) periods, the informed player \(i\) becomes more attracted to cooperation, since his cooperation triggers a longer stream of cooperative outcomes, yielding an overall payoff that exceeds that from defecting under larger parameter conditions. Therefore, the pooling equilibrium can be sustained for less restrictive conditions as the number of repetitions increases.

For a detailed analysis of these conditions, see Appendix 3.

In order to facilitate our utility comparisons, we hereafter assume that, in settings where both outcomes (C,C) and (D,D) can be sustained in equilibrium as in the shaded area of Fig. 1 for the simultaneous-move PD game under complete information, players can resort to some coordination mechanism, such as social norms or a stochastic randomization, that enables them to coordinate on the efficient cooperative outcome (C,C). Alternatively, if a pre-play communication stage exists, Demicheli and Weibull (2008) show that, in the context of Pareto coordination games such as that arising when \(\beta _{i},\beta _{j}\ge \frac{b-a}{b-c}\) in our model, every evolutionary stable equilibrium induces players to asymptotically coordinate on the cooperative outcome (C,C).

In particular, this result corresponds to proposition 1 in Healy (2007) where the past actions of all players are observable, but workers’ types are not.

Note that, in the context of the simultaneous-move PD game, our equilibrium results in Sect. 5.1 would not be affected if we modified which player holds private information about his social preferences, either player 1 (the row player) or 2 (the column player), since our results in those propositions are valid for any player \(i=\{1,2\}\) and \(j\ne i\).

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Title: Cooperation and signaling with uncertain social preferences
Authors: John Duffy
Félix Muñoz-García
Publication date: 01-01-2015
Publisher: Springer US
Published in: Theory and Decision / Issue 1/2015
Print ISSN: 0040-5833
Electronic ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-013-9400-5

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