War of Attrition: When There Is No End in Sight

A couple weeks ago, I went over the concept of the Hawk-Dove evolutionary model. This time, I plan to look at another Maynard-Smith derived model, namely the War of Attrition. Most people know the phrase by the actual model of warfare, where one side continues to hammer against the opponent regardless of its own losses until the other side has been severely weakened and can fight no more. The game theoretic model has a similar idea but is different in its thought process. Let’s take an example to demonstrate how the model works. Imagine there are two  people, Sam and Jaclyn, bidding on cash from an auctioneer, Judy. Not an item of any kind but pure cash. 100,000 dollars. Like the normal English auction we are familiar with, the person with the highest bid gets the cash, only discrete amounts of money are accepted (Pi dollars cannot be a bid), and there is no jumping of the bid (if a person bids $10,000 and the intervals are $1000, then the next person cannot bid higher than $11,000). Unlike the English auction though, this auction has special rules, namely one must pay their highest bid whether they won the cash or not. If the second highest bid is $85,000, then that person must pay the $85,000.

So the bidding starts. Sam bids $1000, then Jaclyn bids $2000, then Sam bids $3000 and so on and so forth. Eventually, it gets until $50,000 and $51,000. Judy has won at this point as she will receive $101,000 in payment but only forfeits $100,000. But the bidding does not stop there. Sam and Jaclyn still gain money by bidding higher. And so it continues until we get to the $99,000 bid from Sam and the $100,000 from Jaclyn. Surely it should stop there. Sam does not gain by bidding $101,000; he will lose $1000. Logically though, it should not stop there. Remember, Sam still has to pay the $99,000 he has bid. The question now is whether he should bid $101,000 but gain $100,000 to come up $1000 short, or pay the $99,000 and gain nothing. The answer is obvious. Bid. So he does. Now what’s Jaclyn’s best option? Sam just bid the $101,000. Should she bid to only lose $2000 ($102,000-$100,000) or should she just pay the $100,000? The definite answer is bid. Now Sam’s best choice? Bid again. And Jaclyn’s best choice? Continue bidding. And what should Judy do? Just smile and take it all in. At this point, the bidders best choice are to continue bidding no matter how high. Afterall, a $900,000,000 payment is better than a $900,098,000. So this continues ad infinitum. Judy is literally laughing all the way to the bank from the pure ecstasy while Sam and Jaclyn plot their revenge in secret.

The war of attrition can be thought of a being tangentially related to the Hawk-Dove game. Instead of simple Hawk, we can add degrees of Hawkishness, where some actions are more hawkish than others. When two creatures meet, the one with the more hawkish action wins. Also unlike Hawk-Dove, let’s assume that each creature can choose how hawkish to be (allowing for multiple choices in each meeting) and a cost proportional to how hawkish the action is is payed no matter the outcome. This is exactly like our dollar auction scenario. And just like the dollar auction scenario, when two animals meet over a resource, the response is to keep escalating no matter how much the cost that has already been payed. That’s why animals (of a similar size and strength) fighting over a mate can quickly escalate the battle until one or more of the individuals dies. The cost of escalating the fight is less than the reward of being able to mate and have ones genes persist into the next population.

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