Go for broke or play it safe? Dynamic competition with choice of variance
研究一个微分博弈，其中两个玩家的共同选择影响一维状态变量的方差但不影响均值，发现耐心玩家在领先时选择安全策略、落后时选择冒险策略，并给出双方都选择冒险策略的条件。
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 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.”