Learning models do not in general imply that weakly dominated strategies are irrelevant or justify the related concept of “forward induction,” because rational agents may use dominated strategies as experiments to learn how opponents play, and may not have enough data to rule out a strategy that opponents never use. Learning models also do not support the idea that the selected equilibria should only depend on a game's reduced normal form. However, playing the extensive form of a game is equivalent to playing the normal form augmented with the appropriate terminal node partitions so that two games are information equivalent, i.e., the players receive the same feedback about others' strategies.

Fix such a symmetric static equilibrium α∗, and let σ recommend α∗ at every record pair (r, r′). Then (σ, μ) is an equilibrium for any steady state μ. Moreover, note that ûr,r′(a, a ′) = (1− γ)u(a, a′) + γu(α∗, α∗), for any r, r′, a, a′. Thus, (α, α′) is a (possibly mixed) augmented-game Nash equilibrium if and only if it is a Nash equilibrium of the stage game. Since (α∗, α∗) is not Pareto-dominated by another static equilibrium, there is no augmented-game Nash equilibrium (α, α′) satisfying (u(α, α′), u(α′, α)) > (u(α∗, α∗), u(α∗, α∗)), and hence there is no augmented-game Nash equilibrium (α, α′) satisfying (ûr,r′(α, α ′), ûr′,r(α ′, α)) > (ûr,r′(α ∗, α∗), ûr′,r(α ∗, α∗)) for any r, r′. That is, (σ, μ) is coordination-proof.

We study adverse selection problems in which the agent’s type is twodimensional and there is a contractible ex-post signal of one component of the agent’s type. We analyze the extent to which the principal can use the ex-post signal to mitigate the effects of the asymmetric information concerning this component, and we characterize when the principal can completely eliminate such effects and achieve the same payoff as in the benchmark where this component is public information. Using this characterization, we show that, under broad conditions, the principal can never achieve the same payoff as the public information benchmark with any ex-post signal that has only two realizations. However, the principal can achieve this payoff with ex-post signals that satisfy a “positive curvature criterion.” Throughout, we focus on the example of a firm employing an informed worker on a project, where the worker has superior information both about the difficulty of the project and the project’s potential quality.

Justified communication equilibrium (JCE) is an equilibrium refinement for signaling games with cheap-talk communication. A strategy profile must be a JCE to be a stable outcome of nonequilibrium learning when receivers are initially trusting and senders play many more times than receivers. In the learning model, the counterfactual "speeches" that have been informally used to motivate past refinements are messages that are actually sent. Stable profiles need not be perfect Bayesian equilibria, so JCE sometimes preserves equilibria that existing refinements eliminate. Despite this, it resembles the earlier refinements D1 and NWBR, and it coincides with them in co-monotonic signaling games.

We introduce a new model of repeated games in large populations with random matching, overlapping generations, and limited records of past play. We prove that steady-state equilibria exist under general conditions on records. When the updating of a player’s record can depend on the actions of both players in a match, any strictly individually rational action can be supported in a steady-state equilibrium. When record updates can depend only on a player’s own actions, fewer actions can be supported. Here, we focus on the prisoner’s dilemma and restrict attention to strict equilibria that are coordination-proof, meaning that matched partners never play a Pareto-dominated Nash equilibrium in the one-shot game induced by their records and expected continuation payoffs. Such equilibria can support full cooperation if the stage game is either “strictly supermodular and mild” or “strongly supermodular,” and otherwise permit no cooperation at all. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting any cooperation when the stage game is “severe.”

We introduce a new model of repeated games in large populations with random matching, overlapping generations, and limited records of past play. We prove that steady-state equilibria exist under general conditions on records. When the updating of a player’s record can depend on the actions of both players in a match, any strictly individually rational action can be supported in steady-state equilibrium. When record updates can depend only on a player’s own actions, fewer actions can be supported. Here we focus on the prisoner’s dilemma, and restrict attention to strict equilibria that are coordination-proof, meaning that matched partners never play a Pareto-dominated Nash equilibrium in the one-shot game induced by their records and expected continuation payoffs. Such equilibria can support full cooperation if the stage game is either “strictly supermodular and mild” or “strongly supermodular,” but permit no cooperation at all for a near-complementary parameter set. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting maximal cooperation when the stage game is “severe.”

Indirect reciprocity is a foundational mechanism of human cooperation. Existing models of indirect reciprocity fail to robustly support social cooperation: Image-scoring models fail to provide robust incentives, while social-standing models are not informationally robust. Here we provide a model of indirect reciprocity based on simple, decentralized records: Each individual's record depends on the individual's own past behavior alone, and not on the individual's partners' past behavior or their partners' partners' past behavior. When social dilemmas exhibit a coordination motive (or strategic complementarity), tolerant trigger strategies based on simple records can robustly support positive social cooperation and exhibit strong stability properties. In the opposite case of strategic substitutability, positive social cooperation cannot be robustly supported. Thus, the strength of short-run coordination motives in social dilemmas determines the prospects for robust long-run cooperation.

We introduce a new model of repeated games in large populations with random matching, overlapping generations, and limited records of past play. We prove that steady-state equilibria exist under general conditions on records. We then focus on “trigger-strategy” equilibria. When the updating of a player’s record can depend on the actions of both players in a match, steady-state equilibria in trigger strategies can support the play of a wide range of actions, including any action that Pareto-dominates a static Nash equilibrium. When updates can depend only on a player’s own actions, fewer actions can be supported by steady-state equilibria. We provide sufficient conditions for trigger equilibria to support a given action, along with somewhat more permissive necessary conditions. When players have access to a form of decentralized public randomization, the sufficient conditions expand to match the necessary conditions. ∗We thank Nageeb Ali, V Bhaskar, Glenn Ellison, Sander Heinsalu, Yuval Heller, Takuo Sugaya, Satoru Takahashi, and Caroline Thomas for helpful comments and conversations, and NSF grants SES 1643517 and 1555071 and Sloan Foundation grant 2017-9633 for finanical support.

We study when and how cooperation can be supported in the repeated prisoner’s dilemma in a large population with random matching and overlapping generations, when players have only first-order information about their current partners: a player’s record tracks information about their past actions only, and not their partners’ past actions (or their partners’ partners’ actions). We restrict attention to strict equilibria that are coordination-proof, meaning that two matched players never play a Pareto-dominated Nash equilibrium in the stage game induced by their records and expected continuation payoffs. We find that simple strategies can support limit efficiency if the stage game is either “mild” or “strongly supermodular,” and that no cooperation can occur in equilibrium for a near-complementary parameter set. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting maximal cooperation when the stage game is “severe.” ∗We thank Nageeb Ali, V Bhaskar, Glenn Ellison, Sander Heinsalu, Yuval Heller, Takuo Sugaya, Satoru Takahashi, and Caroline Thomas for helpful comments and conversations, and NSF grants SES 1643517 and 1555071 and Sloan Foundation grant 2017-9633 for financial support.

Indirect reciprocity is a foundational mechanism of human cooperation.1,2,3,4,5,6,7,8 Existing models of indirect reciprocity fail to robustly support social cooperation: image scoring models5 fail to provide robust incentives, while social standing models2,9,10,11 are not informationally robust. Here we provide a new model of indirect reciprocity based on decentralized, individualistic records: each individual’s record depends on its own past behavior alone, and not on its partners’ or partners’ partners’ behavior. When social dilemmas exhibit a coordination motive (or strategic complementarity), tolerant trigger strategies based on individualistic records can robustly support positive social cooperation and exhibit strong stability properties. In the opposite case of strategic substitutability, positive social cooperation cannot be robustly supported. Thus, the strength of short-run coordination motives in social dilemmas determines the prospects for robust long-run cooperation. People (and perhaps also other animals) often trust each other to cooperate even when they know they will never meet again. Such indirect reciprocity relies on individuals having some information about how their partners have behaved in the past. Existing models of indirect reciprocity fall into two paradigms. In the image scoring paradigm, each individual carries an image that improves when it helps others, and (at least some) individuals help only those with good images. In the standing paradigm, each individual carries a standing that (typically) improves when it helps others with good standing, but not when it helps those with bad standing, and individuals with good standing help only other good-standing individuals.

We study when and how cooperation can be supported in the repeated prisoner’s dilemma in a large population with random matching and overlapping generation, when players have only first-order information about their current partners: a player’s record tracks information about her past actions only, and not her partners’ past actions (or her partners’ partners’ actions). We also restrict attention to strict equilibria that are coordination-proof, meaning that two matched players never play a Pareto-dominated Nash equilibrium in the stage game induced by their records and expected continuation payoffs. We find that simple strategies can support limit efficiency if the stage game is either “mild” or “strongly supermodular,” and that no cooperation can occur in equilibrium for a near-complementary parameter set. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting maximal cooperation when the stage game is “severe.” ∗We thank V Bhaskar, Glenn Ellison, Sander Heinsalu, Yuval Heller, Takuo Sugaya, Satoru Takahashi, and Caroline Thomas for helpful comments and conversations, and NSF grants SES 1643517 and 1555071 and Sloan Foundation grant 2017-9633 for financial support.

Proof of Claim 1. Since M̄ is a metric space under the sup norm topology, it suffices to show that it is sequentially compact. Consider a sequence {μ}n∈N of μ ∈ M̄ . By a standard diagonalization argument, there exists some μ̃ ∈ [0, 1]Z and some subsequence {μ}m∈N such that limm→∞ μm k = μ̃k for all k ∈ Z. This, along with the fact that ∑ t′<t ∑ k∈R(t′) μm k ≥ 1− γ 1− γ for all t ∈ N and m ∈ N implies that ∑ t′<t ∑ k∈R(t′) μ̃k ≥ 1− γ 1− γ

Indirect reciprocity is a foundational mechanism of human cooperation. Existing models of indirect reciprocity fail to robustly support social cooperation: image scoring models fail to provide robust incentives, while social standing models are not informationally robust. Here we provide a new model of indirect reciprocity based on simple, decentralized records: each individual's record depends on their own past behavior alone, and not on their partners' past behavior or their partners' partners' past behavior. When social dilemmas exhibit a coordination motive (or strategic complementarity), tolerant trigger strategies based on simple records can robustly support positive social cooperation and exhibit strong stability properties. In the opposite case of strategic substitutability, positive social cooperation cannot be robustly supported. Thus, the strength of short-run coordination motives in social dilemmas determines the prospects for robust long-run cooperation.