War of Attrition (game) - Examining The Game

Examining The Game

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.