War of attrition (game)
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 evolutionarily 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).
The war of attrition cannot be properly solved using the payoff matrix. 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 α, there is a value β that is greater. Attempting to put all possible bids onto the matrix, however, will result in an ∞×∞ 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 lower bid. If the player who bids the lesser value bids b, then that player loses b and the other player will benefit by an amount of V-b. If both players bid the same amount b, they split the value of V, each gaining V/2-b.
The premise that the players may bid any number is important to analysis of the game. The bid may even exceed the value of the resource that is contested over. This at first appears to be irrational, being seemingly foolish to pay more for a resource than its value; however, remember 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. The player who bid the lesser value b loses b and the one who bid more loses b -V. This situation is commonly referred to as a Pyrrhic victory. For a tie such that b>V/2, they both lose b-V/2. Luce and Raiffa () 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 no dominant strategy. However, this fact and the above argument do not preclude the existence of Nash Equilibria. Any pair of strategies with the following characteristics is a Nash Equilibrium:
- One player bids zero
- The other player bids any value equal to V or higher, or mixes among any values V or higher.
With these strategies, one player wins and pays zero, and the other player loses and pays zero. It is easy to verify that neither player can strictly gain by unilaterally deviating.
Another popular formulation of the war of attrition is as follows: Two players are involved in a dispute. The value of the object to each player is . Time is modeled as a continuous variable which starts at zero and runs indefinitely. Each player chooses when to concede the object to the other player. In the case of a tie, each player receives utility. Time is valuable, each player uses one unit of utility per period of time. This formulation is slightly more complex since it allows each player to assign a different value to the object. Its equilibria are not as obvious as the other formulation.The evolutionarily stable strategy is a mixed ESS, in which the probability of persisting for a length of time t is:
The evolutionarily 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.
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+.
It has also been shown that even if the individuals can only play pure strategies, the time average of the strategy value of all individuals converges precisely to the calculated ESS. In such a setting, one can observe a cyclic behavior of the competing individuals.3
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 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, D.T. & Cannings, C. (1978) A generalized war of attrition. Journal of Theoretical Biology 70: 85-124.
- K. Chatterjee, J.G. Reiter, M.A. Nowak: "Evolutionary dynamics of biological auctions". Theoretical Population Biology 81 (2012), 69 - 80
- 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