Unilaterally Competitive Multi-Player Stopping Games

A multi-player competitive Dynkin stopping game is constructed. Each player can either exit the game for a fixed payoff, determined a priori, or stay and receive an adjusted payoff depending on the decision of other players. The single period case is shown to be “weakly unilaterally competitive”. We present an explicit construction of the unique value at which Nash and optimal equilibria are attained. Multiple period generalisations are explored. The game has interpretations in economic and financial contexts, for example, as a consumption model with bounded resources. It also serves as a starting point to the construction of multi-person financial game options. In particular, the concept of optimal equilibria becomes pivotal in the pricing of the game options via super-replication.

💡 Research Summary

The paper develops a multi‑player competitive Dynkin stopping game in which each participant faces a binary choice at any decision epoch: either exit the game and receive a predetermined fixed payoff, or stay in the game and obtain a payoff that depends on the stopping decisions of the other players. The authors first formalise the single‑period version of the game, showing that it satisfies a novel property they call “weakly unilaterally competitive” (WUC). WUC means that when a single player unilaterally changes his strategy, the payoffs of all other players do not increase (or do not decrease), a condition that lies between the extremes of zero‑sum competition and full cooperation.

Exploiting the WUC structure, the authors prove that the game possesses a unique value vector (V = (V_1,\dots,V_N)) and that the strategy profile attaining this value is simultaneously a Nash equilibrium and an “optimal equilibrium”. The optimal equilibrium notion is stronger than the usual Nash concept: each player’s strategy maximises his own payoff while guaranteeing that no unilateral deviation by any other player can improve the deviator’s payoff without hurting the original player. The value for player (i) is given by the saddle‑point expression
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