[Paper Review] Sequential Sampling Equilibrium

Duarte V. Gonçãlves|arXiv (Cornell University)|Dec 15, 2022

Experimental Behavioral Economics Studies4 citations

TL;DR

This paper introduces Sequential Sampling Equilibrium (SSE), a game-theoretic framework in which players rationally acquire costly, informative signals about opponents' actions to resolve strategic uncertainty. By modeling decision-making as an optimal stopping problem over sequential sampling, SSE endogenously generates stochastic choices, belief dynamics, and response times—rationalizing deviations from Nash equilibrium and predicting novel empirical patterns, while recovering Nash equilibrium as a limit when sampling costs vanish.

ABSTRACT

This paper introduces an equilibrium framework based on sequential sampling in which players face strategic uncertainty over their opponents' behavior and acquire informative signals to resolve it. Sequential sampling equilibrium delivers a disciplined model featuring an endogenous distribution of choices, beliefs, and decision times, that not only rationalizes well-known deviations from Nash equilibrium, but also makes novel predictions supported by existing data. It grounds a relationship between empirical learning and strategic sophistication, and generates stochastic choice through randomness inherent to sampling, without relying on indifference or choice mistakes. Further, it provides a rationale for Nash equilibrium when sampling costs vanish.

Motivation & Objective

• To develop a disciplined equilibrium concept that endogenously generates stochastic choices, beliefs, and decision times in strategic settings.
• To rationalize well-known deviations from Nash equilibrium, such as slower decisions being associated with greater choice randomness and lower strategic sophistication.
• To provide a microfoundation for stochastic choice based on inherent randomness in sequential sampling, avoiding reliance on indifference or choice mistakes.
• To show that Nash equilibrium emerges as a limiting case when sampling costs vanish.
• To ground a relationship between strategic sophistication and response time through a model of costly information acquisition.

Proposed method

• Players face strategic uncertainty and hold a prior belief over opponents’ action distributions.
• Each player sequentially samples noisy signals about opponents’ behavior, with a constant additive cost per observation.
• Players solve an optimal stopping problem, balancing the expected value of further sampling against sampling costs.
• Upon stopping, players choose an action that maximizes expected payoff given their posterior beliefs.
• The equilibrium is defined as a fixed-point in which each player’s optimal sampling and action choice are consistent with the empirical distribution of opponents’ actions.
• The model uses conjugate Dirichlet priors to enable tractable empirical estimation and finite-horizon approximation of the infinite-horizon problem.

Experimental results

Research questions

• RQ1 How can stochastic choice and response time be endogenously generated in equilibrium without relying on indifference or choice errors?
• RQ2 What is the relationship between decision time, strategic sophistication, and choice randomness in strategic environments?
• RQ3 How does costly information acquisition rationalize deviations from Nash equilibrium in experimental settings?
• RQ4 Under what conditions does sequential sampling equilibrium converge to Nash equilibrium?
• RQ5 Can the model predict novel empirical patterns supported by existing data on response times and action distributions?

Key findings

• Sequential Sampling Equilibrium endogenously generates stochastic choice through the randomness inherent in sequential sampling, without requiring indifference or choice mistakes.
• The model rationalizes the empirical regularity that slower decisions are associated with weaker preference intensity and greater choice randomness.
• Faster decisions are linked to less strategic sophistication, while longer decision times correlate with higher strategic reasoning, consistent with experimental evidence.
• When sampling costs vanish, the model converges to Nash equilibrium, providing a microeconomic foundation for it as a limiting case.
• The model predicts that in games with incomplete information, players may fail to converge to Nash equilibrium if priors are misspecified, leading to non-termination of sampling.
• Empirical estimation is feasible using Dirichlet priors, which allow for a compact parameter space and finite-horizon approximation, enabling maximum likelihood estimation.