Counterfactual Prediction in Complete Information Games: Point Prediction Under Partial Identification

We study the problem of counterfactual prediction in discrete decision games with complete information, pure strategies, and Nash equilibria. We show that the presence of multiple equilibria poses unique challenges for the problem of counterfactual prediction even if the payoff structure is known in its entirety. We show that multiple types of counterfactuals can be defined and that the prediction probabilities are not generally point–identified. We establish the sharp identified bounds of the prediction probabilities. We further propose, compare, and contrast various decision methods for the purpose of producing a  point prediction, namely midpoint prediction, a decision–theoretic possibility using a Dirichlet–based prior, and a maximum–entropy approach. On balance, we conclude that the maximum–entropy approach is the least of several evils. Our results have implications for counterfactual prediction in other models with partial identification.