Mixed Strategies in Simultaneous and Sequential Play of a 2 Player Game

We take a class of games with two players and two actions which only have mixed strategy Nash equilibria. We show that such games can only have hybrid equilibria if played sequentially with one player moving first. The hybrid equilibrium has the leader p laying a mixed strategy but the follower playing a pure strategy. We apply this result to a game between a debtor and a lender following a loan contract. The debtor can have high or low revenues and has to report his state to the lender. The lender can choose whether or not to undertake a costly audit. With simultaneou s play there is only a mixed strategy Nash equilibrium with random cheating in reports and auditing. With sequential play if the debtor moves first, there is zero auditing and the debtor cheats as much as possible without giving the incentive to audit. We argue that the setting of the game and the valuable first mover advantage of the debtor mean that we should expect the game to be played sequentially with this hybrid outcome. This is important in the context of the loan contract since the hybrid outcome makes the contract renegotiation proof. Alternatively if the timing allowed the lender to move first, then the equilibrium would have the debtor reporting truthfully and the monitor auditing just sufficiently to ensure truthtelling by the debtor. This ha s strong links to the optimal debt contract with no commitment (Mookherjee-Png, 1989 and Jost, 1996). However we argue that the natural timing of events makes the debtor the leader. We then consider other examples and show that the same outcome emerges in matching pennies and in a generic inspection game involving adverse selection in labour markets.