Going for Broke and Playing it Safe: Dynamic Competition with Choice of Variance

We consider a differential game, in which the joint choices of the two players influences the variance, but not the mean, of the one-dimensional state variable. We interpret this state variable as a summary of how far ‘ahead’ player 1 is in the game. The flow payoffs that the players enjoy are a continuous, monotonic and bounded function of this state variable. In any Markov Perfect Equilibrium, patient players chose to play it safe when ahead and to take risks when behind. Thus, in some sense, risk choice in this model follows from the first derivative properties of payoffs. We also give a simple condition that implies both players choose risky strategies when neither one is too far ahead, a situation that ensures a dominant player emerge “quickly”.