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We derive a necessary and sufficient condition for the existence of equilibria with only two active players in the all-pay auction with complete information and identity-dependent externalities. This condition shows that the generic equilibrium of the standard all-pay auction is robust to the introduction of small identity-dependent externalities. In general, however, the presence of identity-dependent externalities invalidates well-established qualitative results concerning the set of equilibria of the first-price all-pay auction with complete information. With identity-dependent externalities equilibria are generally not payoff equivalent, and identical players may earn different payoffs in equilibrium. These observations show that Siegel's (2009) results characterizing the set of equilibrium payoffs in all-pay contests, including the all-pay auction as a special case, do not extend to environments with identity-dependent externalities. We further compare the all-pay auction with identity-dependent externalities to the first-price winner-pay auction with identity-dependent externalities. We demonstrate that the equilibrium payoffs of the all-pay auction and winner-pay auction cannot be ranked unambiguously in the presence of identity-dependent externalities by providing examples of environments where equilibrium payoffs in the all-pay auction dominate those in the winner-pay auction and vice versa.

This article investigates the impact of the distribution of preferences on equilibrium behavior in conflicts that are modeled as all-pay auctions with identity-dependent externalities. In this context, we define centrists and radicals using a willingness-to-pay criterion that admits preferences more general than a simple ordering on the line. Through a series of examples, we show that substituting the auction contest success function for the lottery contest success function in a conflict may alter the relative expenditures of centrists and radicals in equilibrium. Extremism, characterized by a higher per capita expenditure by radicals than centrists, may persist and lead to a higher aggregate expenditure by radicals, even when they are relatively small in number. Moreover, we show that centrists may in the aggregate expend zero, even if they vastly outnumber radicals. Our results demonstrate the importance of the choice of the institutions of conflict, as modeled by the contest success function, in determining the role of extremism and moderation in economic, political, and social environments.

This article investigates the impact of societal structure on behavior in competitions that can be modeled as all-pay auctions with identity-dependent externalities. The consideration of identity-dependent externalities, which naturally arise in the most common applications of all-pay auctions, enables us to define players' individual characteristics in society (in particular radicalism and centrism) not only for arrangements on the line but more generally. We ?nd that even with a high ratio of centrists in comparison to radicals extremism, characterised by higher expenditure by radicals in comparison to centrists, persists. Moreover, for environments with two radical players we show that there exists a symmetric equilibrium in which all moderates bid zero with certainty. This equilibrium is the unique symmetric equilibrium if there is only a single centrist player. Our results suggest that the active participation of centrists in equilibrium remains characteristic for lottery success functions in the presence of identity-dependent externalities, and therefore that the chosen contest success function was crucial for predictions about extremism and moderation that can be found in the political economy literature.