Game Theory of Mind

Abstract

This paper introduces a model of ‘theory of mind’, namely, how we represent the intentions and goals of others to optimise our mutual interactions. We draw on ideas from optimum control and game theory to provide a ‘game theory of mind’. First, we consider the representations of goals in terms of value functions that are prescribed by utility or rewards. Critically, the joint value functions and ensuing behaviour are optimised recursively, under the assumption that I represent your value function, your representation of mine, your representation of my representation of yours, and so on ad infinitum. However, if we assume that the degree of recursion is bounded, then players need to estimate the opponent's degree of recursion (i.e., sophistication) to respond optimally. This induces a problem of inferring the opponent's sophistication, given behavioural exchanges. We show it is possible to deduce whether players make inferences about each other and quantify their sophistication on the basis of choices in sequential games. This rests on comparing generative models of choices with, and without, inference. Model comparison is demonstrated using simulated and real data from a ‘stag-hunt’. Finally, we note that exactly the same sophisticated behaviour can be achieved by optimising the utility function itself (through prosocial utility), producing unsophisticated but apparently altruistic agents. This may be relevant ethologically in hierarchal game theory and coevolution.

Author Summary

The ability to work out what other people are thinking is essential for effective social interactions, be they cooperative or competitive. A widely used example is cooperative hunting: large prey is difficult to catch alone, but we can circumvent this by cooperating with others. However, hunting can pit private goals to catch smaller prey that can be caught alone against mutually beneficial goals that require cooperation. Understanding how we work out optimal strategies that balance cooperation and competition has remained a central puzzle in game theory. Exploiting insights from computer science and behavioural economics, we suggest a model of ‘theory of mind’ using ‘recursive sophistication’ in which my model of your goals includes a model of your model of my goals, and so on ad infinitum. By studying experimental data in which people played a computer-based group hunting game, we show that the model offers a good account of individual decisions in this context, suggesting that such a formal ‘theory of mind’ model can cast light on how people build internal representations of other people in social interactions.