A Cognitive Hierarchy Model Applied to the Lemonade Game

One of the challenges of multiagent decision making is that the behavior needed to maximize utility can depend on what other agents choose to do: sometimes there is no "right" answer in the absence of knowledge of how opponents will act. The Nash equilibrium is a sensible choice of behavior because it represents a mutual best response. But, even when there is a unique equilibrium, other players are under no obligation to take part in it. This observation has been forcefully illustrated in the behavioral economics community where repeated experiments have shown individuals playing Nash equilibria and performing badly as a result. In this paper, we show how to apply a tool from behavioral economics called the Cognitive Hierarchy (CH) to the design of agents in general sum games. We attack the recently introduced "Lemonade Game" and show how the results of an open competition are well explained by CH. We believe this game, and perhaps many other similar games, boils down to predicting how deeply other agents in the game will be reasoning. An agent that does not reason enough risks being exploited by its opponents, while an agent that reasons too much may not be able to interact productively with its opponents. We demonstrate these ideas by presenting empirical results using agents from the competition and idealizations arising from a CH analysis.