Discrete bidding strategies for a random incoming order

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 Ω_ν = {(i/ν)P¯ i = 1, . . . , ν}, 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 ν → ∞, 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, P¯].