All-Pay Hex: A Multibattle Contest With Complementarities∗

In this paper, we examine a modified 2 × 2 game of Hex in which control of each cell is determined by a Tullock contest. The player establishing a path of cells within his control between his two sides wins a fixed prize. Examining the polar cases of all cells being contested simultaneously versus all four cells being contested sequentially, we show that there is an increase in the total expected value for the players in the sequential case compared to the simultaneous case. Furthermore, due to the players having different, albeit symmetric winning combinations, in the sequential case one player may have a greater expected value than their opponent, which depends on the order of the cell contests. We thus provide a canonical model of a multibattle contest in which complementarities between battlefields are heterogeneous across both battlefields and players.