Backward induction and repeated games with strategy constraints: An inspiration from the surprise exam paradox

Backward induction has led to some controversy in specific games, the surprise exam paradox and iterated prisoner's dilemma for example, despite its wide use in solving finitely repeated games with complete information. In this paper, a typical misuse of backward induction is revealed by analyzing the surprise exam paradox, and the reason why backward induction may fail is investigated. The surprise exam paradox represents a set of repeated games with strategy constraints and has not been fully investigated in game theory. The agents in real-world activities always face constraints in decision making, for example, a budget limitation. In a repeated game with strategy constraints, the players' choices in different stages are not independent and later choices depend on previous choices because of the strategy constraints. Backward induction cannot be applied in its normal use and it needs to be combined with Bayes' theorem in solving these kinds of problems. We also investigate how the strategy constraints influence the equilibrium and show how to solve repeated games with strategy constraints by analyzing a repeated battle of the sexes game with a budget constraint.