Discrete Bidding Strategies for a Random Incoming Order

Abstract

This paper is concerned with a model of a one-sided limit order book, viewed as a noncooperative game for $n$ players. Agents offer various quantities of an asset at different prices, ranging over a finite set $\Omega_\nu=\{(i/\nu)\overline{P};\ i=1,\ldots,\nu\}$, competing to fulfill an incoming order, whose size $X$ is not known a priori. Players can have different payoff functions, reflecting different beliefs about the fundamental value of the asset and probability distribution of the random variable $X$. For a wide class of random variables, we prove that the optimal pricing strategies for each seller form a compact and convex set. By a fixed point argument, this yields the existence of a Nash equilibrium for the bidding game. As $\nu\to\infty$, we show that the discrete Nash equilibria converge to an equilibrium solution for a bidding game where prices range continuously over the whole interval $[0,\overline{P}]$.