Dynamic stability of the Nash equilibrium for a bidding game

A one-sided limit order book is modeled as a noncooperative game for several players. An external buyer asks for an amount X > 0 of a given asset. This amount will be bought at the lowest available price, as long as the price does not exceed an upper bound P. One or more sellers offer various quantities of the asset at different prices, competing to fulfill the incoming order. The size X of the order and the maximum acceptable price P are not a priori known, and thus regarded as random variables. In this setting, we prove that a unique Nash equilibrium exists, where each seller optimally prices his assets in order to maximize his own expected profit. Furthermore, a dynamics is introduced, assuming that each player gradually adjusts his pricing strategy in reply to the strategies adopted by all other players. In the case of (i) infinitely many small players or (ii) two large players with one dominating the other, we show that the pricing strategies asymptotically converge to the Nash equilibrium.