In game theory, the war of attrition is a dynamic timing game in which players choose a time to stop, and fundamentally trade off the strategic gains from outlasting other players and the real costs expended with the passage of time.
Its precise opposite is the pre-emption game, in which players elect a time to stop, and fundamentally trade off the strategic costs from outlasting other players and the real gains occasioned by the passage of time.
The model was originally formulated by John Maynard Smith;[1] a mixed evolutionarily stable strategy (ESS) was determined by Bishop & Cannings.
[2] An example is a second price all-pay 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).
In other words, if a player bids b, then his payoff is -b if he loses, and V-b if he wins.
Finally, assume that if both players bid the same amount b, then 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 all-pay, sealed-bid, second-price auction.
There is a catch, however; if both players bid higher than V, the high bidder does not so much win, but loses less.
This situation is commonly referred to as a Pyrrhic victory.
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, there are multiple asymmetric weak Nash Equilibria in pure strategies.
[4] The player with the positive bid pays nothing in equilibrium.
Another popular formulation of the war of attrition is as follows: two players are involved in a dispute.
Time is modeled as a continuous variable which starts at zero and runs indefinitely.
This formulation is slightly more complex since it allows each player to assign a different value to the object.
The unique symmetric Nash equilibrium is defined by the following survival function for t:[6]
This means that the player with the lower value has a higher probability of winning the war.
[4] Note that there does not exist any x such that the survival function equals zero.
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.