“ Third down with a yard to go ” : the Dixit - Skeath conundrum on equilibria in competitive

In strictly competitive games, equilibrium mixed strategies are invariant to changes in the ultimate prizes. Dixit & Skeath (1999) argue that this seems counter-intuitive. We show that this invariance is robust to dropping the independence axiom, but is removed if we drop the reduction axiom. The conditions on the resulting recursive expected-utility model to get the desired outcome are analogous to conditions used in the standard model of comparative statics under risk.