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game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applic ...
, 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 John Maynard Smith (6 January 1920 – 19 April 2004) was a British theoretical and mathematical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics u ...
; a mixed
evolutionarily stable strategy An evolutionarily stable strategy (ESS) is a strategy (or set of strategies) that is ''impermeable'' when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy (or set o ...
(ESS) was determined by Bishop & Cannings. An example is a second price ''all-pay''
auction An auction is usually a process of buying and selling goods or services by offering them up for bids, taking bids, and then selling the item to the highest bidder or buying the item from the lowest bidder. Some exceptions to this definition ex ...
, 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).


Examining the game

To see how a war of attrition works, consider the all pay auction: Assume that each player makes a bid on an item, and the one who bids the highest wins a resource of value ''V''. Each player pays his bid. 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''. Finally, think of the bid ''b'' as time, and this becomes the war of attrition, since a higher bid is costly, but the higher bid wins the prize. The premise that the players may bid any number is important to analysis of the all-pay, sealed-bid, second-price auction. 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'' (where, in this scenario, b>V). This situation is commonly referred to as a
Pyrrhic victory A Pyrrhic victory ( ) is a victory that inflicts such a devastating toll on the victor that it is tantamount to defeat. Such a victory negates any true sense of achievement or damages long-term progress. The phrase originates from a quote from P ...
. For a tie such that ''b''>''V''/2, they both lose ''b''-''V''/2.
Luce Luce may refer to: People * Luce (name), as a given name and a surname * Luce (singer) Places * Luče, a town in Slovenia * Luce, Minnesota, an unincorporated community * Luce Bay, a large Bay in Wigtownshire in southern Scotland * Luce ...
and Raiffa referred to the latter situation as a "ruinous situation"; 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 In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The o ...
. However, there are multiple asymmetric weak
Nash Equilibria In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equi ...
in pure strategies. For example, either player could commit to any bid ''b''≥''V''. The other player's best response is to bid zero as there is no bid with which they can win the prize and receive a positive payoff. The player with the positive bid pays nothing in equilibrium. So, they have no incentive to bid less. This equilibrium is subgame perfect. There is also a symmetric equilibrium in mixed strategies.


Symmetric Nash equilibrium

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 v_i > 0. 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 v_i / 2 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. We assume that both players know the valuation of the other player. Thus, the game is a complete information game. The unique symmetric Nash equilibrium is defined by the following
survival function The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time. The survival function is also known as the survivor function or reliability function. The te ...
for ''t'': S_i(x)=e^ The value S_i(x), for a player ''i'' whose opponent values the resource at V_j over time ''t'', is the probability that ''t ≥ x''. This strategy does not guarantee the win for either player. Rather, it is the optimal strategy given that your opponent also plays a strategy of the same form. Note that if V_1 > V_2, S_1(x) = e^ < e^ = S_2(x) So, the player with the lower value persists longer than the player with the higher value. This means that the player with the lower value has a higher probability of winning the war. Note that there does not exist any ''x'' such that the survival function equals zero. So, the distribution of bids has full support. Moreover, both players receive an expected payoff of zero because their payoff is zero at ''t''=''0'' and their expected payoff must be equal at every value of ''t''.


Dynamic formulation and evolutionarily stable strategy

The unique
evolutionarily stable strategy An evolutionarily stable strategy (ESS) is a strategy (or set of strategies) that is ''impermeable'' when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy (or set o ...
coincides with the symmetric Nash equilibrium. This follows from the fact that any ESS must be a Nash equilibrium and the fact that no pure persistence time can be an ESS. 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+\delta''. 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.


See also

*
Evolutionary game theory Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John M ...
* Bishop-Cannings theorem * Rubinstein bargaining model *
Hawk-dove game The game of chicken, also known as the hawk–dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while the ideal outcome is for one player to yield (to avoid the worst outcome if ...
*
Dollar auction The dollar auction is a non- zero sum sequential game explored by economist Martin Shubik to illustrate a paradox brought about by traditional rational choice theory in which players are compelled to make an ultimately irrational decision based co ...
*
Attrition warfare Attrition warfare is a military strategy consisting of belligerent attempts to win a war by wearing down the enemy to the point of collapse through continuous losses in personnel and materiel. The word ''attrition'' comes from the Latin root , ...


References

{{Game theory Non-cooperative games