uch of the existing literature on 'wars of attrition' focuses on games with only two players, or on the straightforward generalization to N+1 players competing for N prizes. Multiplayer wars of attrition, however, have numerous practical applications. Examples include winner-takes-all markets, originally contested by more than two firms, but in which only one firm is likely to survive; negotiations among firms in industry standard-setting committees; and building voting majorities in narrowly balanced situations. In such games, except in the final stage, a player’s departure will not end the game, and players may continue to incur at least some costs even after they have conceded.

In Discussion Paper No. 1564, Jeremy Bulow and Paul Klemperer generalize the model to allow for N+K firms competing for N prizes, so K players must exit for the game to end. Two special cases are of particular interest. First, if firms continue to pay their full costs after dropping out (as in a standard-setting context), each firm’s exit time is independent both of K and of the actions of other players. Second, in the limit in which firms pay no costs after dropping out (as in a natural oligopoly problem), the field is immediately reduced to N+1 firms. Furthermore, there is perfect sorting, so it is always the K–1 lowest-value players who drop out in zero time, even though each player’s value is private information to the player. The authors apply their model to politics, using the example of the 1993 US Congressional budget battle. Their model explains why rounding up most of the necessary votes for a bill might take very little time, but gathering the last few votes may be time consuming and costly.