Go for broke or play it safe? Dynamic competition with choice of variance

We consider a differential game in which the joint choices of the two players influence 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. At each moment in time, players receive a flow payoff which is continuous, bounded, and satisfies a simple single crossing property. We show that a pure strategy perfect equilibrium in stationary Markov strategies (ME) exists and has the property that patient players choose to play it safe when sufficiently ahead and to take risks when sufficiently behind. We also provide a simple condition that implies both players choose risky strategies when neither one is too far ahead, a situation that ensures a dominant player emerges “quickly.” Additionally, we compare the ME outcome with the control that maximizes joint discounted profits (the efficient outcome). We find that the ME outcome is efficient when players are close together and inefficient when players are far apart