[Paper Review] Cursed Sequential Equilibrium

Meng-Jhang Fong, Po-Hsuan Lin+1 more|arXiv (Cornell University)|Jan 27, 2023

Game Theory and ApplicationsDecision Sciences3 citations

TL;DR

This paper introduces Cursed Sequential Equilibrium (CSE), a dynamic extension of cursed equilibrium that incorporates sequential rationality and belief updating in multi-stage games with incomplete information. Unlike standard cursed equilibrium, CSE captures systematic deviations from Bayesian Nash equilibrium in dynamic settings—particularly in games with private values—by modeling players who partially neglect the dependence of others' strategies on their private types, leading to distinct predictions in signaling, voting, and reputation games.

ABSTRACT

This paper develops a framework to extend the strategic form analysis of cursed equilibrium (CE) developed by Eyster and Rabin (2005) to multi-stage games. The approach uses behavioral strategies rather than normal form mixed strategies, and imposes sequential rationality. We define cursed sequential equilibrium (CSE) and compare it to sequential equilibrium and standard normal-form CE. We provide a general characterization of CSE and establish its properties. We apply CSE to five applications in economics and political science. These applications illustrate a wide range of differences between CSE and Bayesian Nash equilibrium or CE: in signaling games; games with preplay communication; reputation building; sequential voting; and the dirty faces game where higher order beliefs play a key role. A common theme in several of these applications is showing how and why CSE implies systematically different behavior than Bayesian Nash equilibrium in dynamic games of incomplete information with private values, while CE coincides with Bayesian Nash equilibrium for such games.

Motivation & Objective

• To extend the static cursed equilibrium (CE) framework of Eyster and Rabin (2005) to dynamic, multi-stage games with incomplete information.
• To address the limitation of standard CE in failing to distinguish between dynamic games with identical strategic forms but different timing of moves.
• To model how players with cursed beliefs—partially ignoring the link between others’ types and actions—update beliefs sequentially and deviate systematically from Bayesian Nash equilibrium in private-value games.
• To demonstrate that CSE yields qualitatively different predictions than both CE and sequential equilibrium in key applications, especially where higher-order beliefs and belief updating matter.

Proposed method

• Adapts the cursed equilibrium concept to multi-stage games with observed actions and private types, using behavioral strategies instead of normal-form mixed strategies.
• Imposes sequential rationality by requiring players to best-respond to cursed beliefs at each public history, where beliefs are updated via Bayes’ rule but incorporate a cursed belief about others’ average action distributions.
• Introduces a cursedness parameter $\chi \in [0,1]$, where $\chi = 1$ denotes full cursedness (ignoring type dependence in others’ strategies), and $\chi = 0$ reduces to sequential equilibrium.
• Applies the framework to five dynamic games: signaling games, public goods with preplay communication, centipede games with altruists, sequential voting, and the dirty faces game.
• Characterizes CSE generally and compares it to both sequential equilibrium and standard CE, showing that CSE can refine pooling equilibria in signaling games.
• Uses a belief-updating mechanism where players form incorrect beliefs about others’ average action distributions based on prior type beliefs, leading to persistent misinference in later stages.

Experimental results

Research questions

• RQ1 How can cursed equilibrium be generalized to dynamic games with sequential moves and incomplete information?
• RQ2 In what ways does CSE differ from both sequential equilibrium and standard cursed equilibrium in predicting behavior in dynamic games?
• RQ3 Does CSE predict systematic deviations from Bayesian Nash equilibrium in private-value dynamic games, where standard CE coincides with BNE?
• RQ4 How does cursedness affect information aggregation in sequential voting and preplay communication games?
• RQ5 To what extent does CSE predict earlier or later stopping in coordination games like the dirty faces game, compared to sequential equilibrium?

Key findings

• CSE predicts that in private-value games, such as signaling games and sequential voting, behavior systematically differs from both Bayesian Nash equilibrium and standard CE, even though CE and BNE coincide in such settings.
• In signaling games, $\chi$-CSE refines pooling equilibria in ways not captured by belief-based refinements or $\chi$-CE, aligning with experimental evidence of non-credible signaling behavior.
• In the public goods game with preplay communication, CSE predicts that communication becomes less effective as the number of players ($N$) or contribution cost ($K$) increases—contrary to CE and sequential equilibrium, which are independent of $N$ and $K$.
• In the dirty faces game, $\chi$-CSE predicts stopping at intermediate stages due to incorrect belief propagation, expanding the set of equilibria beyond sequential equilibrium.
• In the centipede game with altruists, CSE predicts earlier termination than sequential equilibrium, reflecting the impact of cursedness on reputation and cooperation incentives.
• CSE provides qualitatively testable predictions in sequential voting and dirty faces games, suggesting experimental avenues to distinguish it from CE and sequential equilibrium.