Title: A Game of Martingales,
Abstract: We consider a two player dynamic game played over $T \leq \infty$ periods. In
each period each player chooses any probability distribution with support on
$[0,1]$ with a given mean, where the mean is the realized value of the draw
from the previous period. The player with the highest realization in the period
achieves a payoff of $1$, and the other player, $0$; and each player seeks to
maximize the discounted sum of his per-period payoffs over the whole time
horizon. We solve for the unique subgame perfect equilibrium of this game, and
establish properties of the equilibrium strategies and payoffs in the limit.
The solution and comparative statics thereof provide insight about
intertemporal choice with status concerns. In particular we find that patient
players take fewer risks.