Conditional Probability Improves Game Theory Models

Conditional Belief Types, a recent research article by Alfredo Di Tillio (Department of Economics) joint with Joseph Y. Halpern (Cornell University) and Dov Samet (Tel Aviv University), published in Games and Economic Behavior (Volume 87, September 2014, 253-268, doi: 10.1016/j.geb.2014.05.012) introduces a new model of beliefs for strategic settings. Unlike in the standard models of asymmetric information commonly used in game theory and economics, in this model the players' beliefs are modeled by conditional probabilities, not just probabilities. Thus, for each event in a given family, a state of the world in the model specifies a player's belief conditional on that event, even if that event is in fact given probability zero by the player.

Allowing conditional beliefs is essential to a better understanding of strategic reasoning and behavior, especially (though not only) in dynamic problems. Consider a worker who agrees to perform a certain task, believing that refusing to do so would make her worse off (perhaps because the firm would then fire her). What happens if she refuses is counterfactual, as she actually agrees. If we assume, as is natural, that she knows she is indeed agreeing, then we must also assume that she assigns probability zero to the event that she refuses. As a result, in the standard models where only probabilistic beliefs are specified, the sentence "believing that refusing to do so would make her worse off" is not formalized, because conditional probabilities are not defined when the conditioning event has probability zero.

This paper investigates the natural solution to the above problem, which is modeling the players' beliefs as conditional probabilities rather than probabilities. In this richer model, a player's belief is a family of probability distributions, one for each conditioning event in a given set. The authors analyze the role of various natural restrictions on the players' conditional beliefs. The main one, which they call the "echo axiom," requires that a player's conditional belief must reflect the unconditional belief the player has when the condition is actually true. In the example of the worker, for instance, if whenever the worker refuses, she is sure she will be fired, then (even if she in fact agrees) conditionally on refusing, she is also sure she will be fired. The authors also show that the echo axiom provides a formalization of the familiar interpretation of conditional probability as "belief when the conditioning event is known to be true."