Self-referential thinking and equilibrium as states of mind in games: fMRI evidence

Sixteen subjects' brain activity were scanned using previous termfMRInext term as they made choices, expressed beliefs, and expressed iterated 2nd-order beliefs (what they think others believe they will do) in eight games. Cingulate cortex and prefrontal areas (active in “theory of mind” and social reasoning) are differentially activated in making choices versus expressing beliefs. Forming self-referential 2nd-order beliefs about what others think you will do seems to be a mixture of processes used to make choices and form beliefs. In equilibrium, there is little difference in neural activity across choice and belief tasks; there is a purely neural definition of equilibrium as a “state of mind.” “Strategic IQ,” actual earnings from choices and accurate beliefs, is negatively correlated with activity in the insula, suggesting poor strategic thinkers are too self-focused, and is positively correlated with ventral striatal activity (suggesting that high IQ subjects are spending more mental energy predicting rewards)

A Formal Separation Between Strategic and Nonstrategic Behavior

It is common in multiagent systems to make a distinction between "strategic" behavior and other forms of intentional but "nonstrategic" behavior: typically, that strategic agents model other agents while nonstrategic agents do not. However, a crisp boundary between these concepts has proven elusive. This problem is pervasive throughout the game theoretic literature on bounded rationality and particularly critical in parts of the behavioral game theory literature that make an explicit distinction between the behavior of "nonstrategic" level-0 agents and "strategic" higher-level agents (e.g., the level-k and cognitive hierarchy models). Overall, work discussing bounded rationality rarely gives clear guidance on how the rationality of nonstrategic agents must be bounded, instead typically just singling out specific decision rules and informally asserting them to be nonstrategic (e.g., truthfully revealing private information; randomizing uniformly). In this work, we propose a new, formal characterization of nonstrategic behavior. Our main contribution is to show that it satisfies two properties: (1) it is general enough to capture all purportedly "nonstrategic" decision rules of which we are aware in the behavioral game theory literature; (2) behavior that obeys our characterization is distinct from strategic behavior in a precise sense

Strategic Learning in Teams

This paper analyzes a two-player game of strategic experimentation with three-armed exponential bandits in continuous time. Players face replica bandits, with one arm that is safe in that it generates a known payoff, whereas the likelihood of the risky arms’ yielding a positive payoff is initially unknown. It is common knowledge that the types of the two risky arms are perfectly negatively correlated. I show that the efficient policy is incentive-compatible if, and only if, the stakes are high enough. Moreover, learning will be complete in any Markov perfect equilibrium with continuous value functions if, and only if, the stakes exceed a certain threshold

Regret in Dynamic Decision Problems

The paper proposes a framework to extend regret theory to dynamic contexts. The key idea is to conceive of a dynamic decision problem with regret as an intra-personal game in which the agent forms conjectures about the behaviour of the various counterfactual selves that he could have been. We derive behavioural implications in situations in which payoffs are correlated across either time or contingencies. In the first case, regret might lead to excess conservatism or a tendency to make up for missed opportunities. In the second case, behaviour is shaped by the agent’s self-conception. We relate our results to empirical evidence

Negatively Correlated Bandits

We analyze a two-player game of strategic experimentation with two-armed bandits. Each player has to decide in continuous time whether to use a safe arm with a known payoff or a risky arm whose likelihood of delivering payoffs is initially unknown. The quality of the risky arms is perfectly negatively correlated between players. In marked contrast to the case where both risky arms are of the same type, we find that learning will be complete in any Markov perfect equilibrium if the stakes exceed a certain threshold, and that all equilibria are in cutoff strategies. For low stakes, the equilibrium is unique, symmetric, and coincides with the planner's solution. For high stakes, the equilibrium is unique, symmetric, and tantamount to myopic behavior. For intermediate stakes, there is a continuum of equilibria

Behavioral game theory: Plausible formal models that predict accurately

Many weaknesses of game theory are cured by new models that embody simple cognitive principles, while maintaining the formalism and generality that makes game theory useful. Social preference models can generate team reasoning by combining reciprocation and correlated equilibrium. Models of limited iterated thinking explain data better than equilibrium models do; and they self-repair problems of implausibility and multiplicity of equilibria

Learning and payoff externalities in an investment game

This paper examines the interplay of informational and payoff externalities in a two-player irreversible investment game. Each player learns about the quality of his project by observing a private signal and the action of his opponent. I characterize the unique symmetric equilibrium in a timing game that features a second-mover advantage, allowing for arbitrary correlation in project qualities. Despite private learning, the game reduces to a stochastic war of attrition. In contrast to the case of purely informational externalities, all investments happen at the same real time instant—irrespective of the sign of the correlation—and beliefs never get trapped in a no-learning region, provided that the second-mover advantage is sufficiently high.Accepted manuscrip

Umbrella Branding and the Provision of Quality

Consider a two-product firm that decides on the quality of each product. Product quality is unknown to consumers. If the firm sells both products under the same brand name, consumers adjust their beliefs about quality subject to the performance of both products. We show that if the probability that low quality will be detected is in an intermediate range, the firm produces high quality under umbrella branding whereas it would sell low quality in the absence of umbrella branding. Hence, umbrella branding mitigates the moral hazard problem. We also find that umbrella branding survives in asymmetric markets and that even unprofitable products may be used to stabilize the umbrella brand. However, umbrella branding does not necessarily imply high quality; the firm may choose low-quality products with positive probability