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In game theory the war of attrition is a model of aggression in which two contestants compete for a resource of value V by persisting while constantly accumulating costs over the time t that the contest lasts. The model was originally formulated by John Maynard Smith^[1], a mixed evolutionary stable strategy (ESS) was determined by Bishop & Cannings^[2]. Strategically, the game is an auction, in which the prize goes to the player with the highest bid, and each player pays the loser's low bid (making it an all-pay sealed-bid second-price auction).

Contents 1 Examining the game 2 Evolutionary stable strategy 3 The ESS in popular culture 4 Conclusions 5 See also 6 References 7 Sources 8 External link

	a	B
a	V/2-a,V/2-a	V-a, -a
B	-a, V-a	V/2-B, V/2-B
War of attrition: a and B are any amount such that 0

The war of attrition cannot be properly solved using the payoff matrix. For reasons detailed in this section, there is no Nash equilibrium, nor indeed any set value other than V. The players available resources are the only limit to the maximum value of bids; bids can be any number if available resources are ignored, meaning that for any value of a, there is a value B that is greater. Attempting to put all possible bids onto the matrix, however, will result in an ∞x∞ matrix. One can, however, use a pseudo-matrix form of war of attrition to understand the basic workings of the game, and analyze some of the problems in representing the game in this manner.

The game works as follows: Each player makes a bid; the one who bids the highest wins a resource of value V. Each player pays the lowest bid, a

There are four possible outcomes to the game:

• If both players bid amounts greater than V, but one bids more, he or she will have a Pyrrhic victory
• If both players bid different amounts less than V, the higher bidder will gain and the lower bidder will lose.
• If both players bid the same amount less than V/2, they both gain a small amount.
• If both players bid the same amount greater than V/2, they both suffer equal losses, and there is no winner.

The premise that the players may bid any number is important to analysis of the game. They may even exceed the value of the resource that is contested over. This at first appears to be a non sequitur, as it would be foolish to pay more than the value for a resource. Remember, however that each bidder only pays the low bid. Therefore, it would seem to be in each player's best interest to bid the maximum possible amount rather than an amount equal to or less than the value of the resource.

There is a catch, however; if both players bid higher than V, the high bidder does not so much win as lose less. This situation is commonly referred to as a Pyrrhic victory. In contrast, if each player bids less than V, the player bidding a will lose, and the other player will benefit by an amount of V-a. If each player bids the same amount for a less than V/2, they split the value of V, each gaining V/2-a. For a tie such that a>V/2, they both lose the difference of V/2 and a. Luce and Raffia referred to the latter situation as a "ruinous situation"; the point at which both players suffer, and there is no winner.

The conclusion one can draw from this pseudo-matrix is that there is no value to bid which is beneficial in all cases, so there is not dominant strategy. Further, bidding a larger amount increases one's chances of catastrophic loss, and bidding a lower amount increases the chances of an opponent's victory and also the opponents benefit, so there is no advantage to either extreme. The unrestricted nature of the bids prevents a Nash equilibrium. There is always a higher possible bid.

The evolutionary stable strategy below represents the most probable value of a. The value p(t) for a contest with a resource of value V over time t, is the probability that t = a. This strategy does not guarantee the win; rather it is the optimal balance of risk and reward. The outcome of any particular game cannot be predicted as the random factor of the opponent's bid is too unpredictable.

The evolutionary stable strategy is a mixed ESS, in which the probability of persisting for a length of time t is:

That no pure persistence time is an ESS can be demonstrated simply by considering a putative ESS bid of x, which will be beaten by a bid of x+δ.

The evolutionarily stable strategy when playing this game is a probability density of random persistence times which cannot be predicted by the opponent in any particular contest. This result has led to the prediction that threat displays ought not to evolve, and to the conclusion in The Illuminatus! Trilogy that optimal military strategy is to behave in a completely unpredictable, and therefore insane, manner. Neither of these conclusions appear to be truly quantifiably reasonable applications of the model to realistic conditions.

By examining the unusual results of this game, it serves to mathematically prove another piece of old wisdom: "Expect the unexpected". By making the assumption that an opponent will act irrationally, one can paradoxically better predict their actions, as they are limited in this game. They will either act rationally, and take the optimal decision, or they will be irrational, and take the non-optimal solution. If one considers the irrational as a bluff and the rational as backing down from a bluff, it transforms the game into another game theory game, Hawk and Dove.

• Bishop-Cannings theorem
• Hawk-dove game
• Dollar auction

1. ^ Maynard Smith, J. (1974) Theory of games and the evolution of animal contests. Journal of Theoretical Biology 47: 209-221.
2. ^ Bishop, D.T. & Cannings, C. (1978) A generalized war of attrition. Journal of Theoretical Biology 70: 85-124.

• Bishop, D.T., Cannings, C. & Maynard Smith, J. (1978) The war of attrition with random rewards. Journal of Theoretical Biology 74:377-389.
• Maynard Smith, J. & Parker, G. A. (1976). The logic of asymmetric contests. Animal Behaviour. 24:159-175.
• Luce,R.D. & Raiffa, H. (1957) "Games and Decisions: Introduction and Critical Survey"(originally published as "A Study of the Behavioral Models Project, Bureau of Applied Social Research") John Wiley & Sons Inc., New York
• Rapaport,Anatol (1966) "Two Person Game Theory" University of Michigan Press, Ann Arbor

• Exposition of the derivation of the ESS - From Ken Prestwich's Game Theory website at College of the Holy Cross

view	Topics in game theory
Definitions	Normal form game · Extensive form game · Cooperative game · Information set · Preference
Equilibrium concepts	Nash equilibrium · Subgame perfection · Bayesian-Nash · Perfect Bayesian · Trembling hand · Proper equilibrium · Epsilon-equilibrium · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy · Risk dominance
Strategies	Dominant strategies · Mixed strategy · Tit for tat · Grim trigger · Collusion
Classes of games	Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design · Stochastic game · Nontransitive game
Games	Prisoner's dilemma · Traveler's dilemma · Coordination game · Chicken · Volunteer's dilemma · Dollar auction · Battle of the sexes · Stag hunt · Matching pennies · Ultimatum game · Minority game · Rock, Paper, Scissors · Pirate game · Dictator game · Public goods game · Nash bargaining game · Blotto games  · War of attrition
Theorems	Minimax theorem · Purification theorems · Folk theorem · Revelation principle · Arrow's theorem