Since liquidity competition may generate beneficial round trips, an effect that can be interpreted as a form of arbitrage and should therefore be ruled out, we introduce two related trading constraints into the model: a market drop-out constraint and a short-sale constraint. These constraints lead to new liquidation games, both in the mean-field setting and in the finite-player framework.
In each case, we show that the equilibria (for both the mean-field and the finite-player game) are characterized as solutions to a nonlinear higher-order integral equation with an endogenous terminal condition. We establish the existence and uniqueness of a solution to this integral equation, which in turn yields the existence and uniqueness of equilibrium of the game.