"Third down with a yard to go": recursive expected utility and the Dixit¯Skeath conundrum

In two-outcome strictly competitive games, equilibrium mixed strategies do not depend on ultimate prizes. Dixit and Skeath [Games of Strategy (1999) Norton, New York] find this `counter-intuitiveʹ. We show this invariance comes from reduction, not independence; and provide conditions for `intuitiveʹ comparative statics under recursive expected utility.