Last Man Standing

Game theory has all sorts of interesting applications. In their classic textbook from 1991, Drew Fudenberg and Jean Tirole (this year’s Nobel prize winner in economics, by the way) present an example for a War of Attrition from biology. Two animals are fighting for a prize, let’s say food or a mating partner. Fighting, as you might expect, is costly in terms of energy and because of the possible injuries you might suffer. Both animals do not know the strength of the other. Which means, they do not know how long exactly the opponent is able to withstand* (they might have had some sparring sessions though). As the fight goes on, both animals become more and more pessimistic about the strength of the other. And because the costs of fighting are accumulating, eventually one of them will drop out, leaving the prize up to the other.

Male impalas fighting
Male impalas fighting during the breeding season. Source:

Although we occasionally observe fights to the death, these are very rare and most of the times they end with exhaustion and only minor injuries. The goal is winning a fight with the least possible amount of effort. Anything beyond that, risking serious injuries, would be a huge waste of resources, even for the winner. When your enemy appears to be too strong, you rather opt out and try it next year than going all-in.

But how do hoofed animals in the African savanna turn out to be so wise? The answer is, due to evolution. Animals might not always have had rational strategies. But those who didn’t simply had a lower rate of survival. Thus, by showing a lower evolutionary fitness, their types were slowly driven out of the population.

Reading the introduction to this article, you might wonder why economists care about evolutionary biology? They don’t. Biology just provides an instructive example here**. The analog in economics are two firms who are fighting for a prize. And this prize are the customers in a market.

Yuya Takahashi has an interesting paper where he applies the concept of war of attrition to the single-screen movie theater industry in the U.S. between 1950 and 1960. With the advent of affordable TV sets, this industry was doomed to decline. Whereas before people were filling up the theater seats even to watch the news, a lot of them now could comfortably do so at home. Single-screen movie theaters were not able to adjust to this decline in demand by down-sizing their capacity, as they only had one screen and were often heavily mortgaged. A lot of them simply went bankrupt. Sure, movie theaters did not stop to exist. We still see them nowadays, even small ones, but the industry was forced to drastically change their business model.

So, customers are the prize. But there are less and less of them during the 1950s. Every movie theater in a town faces this problem. But what if a theater owner could wait until a rival drops out of the market first? The remaining customers might still be enough to secure a decent profit, at least for a while. The market might be too small for two rivals, but being the only theater in town could still pay the bills. If both owners have the same idea, they will try to outlast each other. The result is a war of attrition. You see that the notion of “fighting” is more symbolic here. However, firms are willing to incur costs by staying in a market longer than if they wouldn’t take the behavior of rivals into account. They are willing to take even negative profits into account. This is what makes the competition so fierce.

Since a war of attrition game is relatively simple compared to other games with many more strategic dimensions, Takahashi is able to estimate a model of the movie theatre industry. His results suggest that, due to strategic behavior, firms on average delay their exit from the market by 2.7 years. This delay results in reduced profits of 4.9% across the whole industry.

These results are relevant because, as Takahashi points out in his paper, many declining industries are still important in terms of total output. Imagine 4.9% loss of total profits in the global steel industry just because firms are playing a waiting game of “who moves first”. A war of attrition also makes the life of antitrust authorities harder. It’s much more difficult to decide about how healthy market competition should look like when dynamic strategic incentives are present. As economists, we only start to understand these incentives, and more work needs to be done in this direction.

* Here, I’m talking about the asymmetric information version with pure strategies according to Fudenberg and Tirole (1986).

** Another example is the attrition warfare during World War I, when armies of the two blocs were facing each other in battle for years without any conquest of substantial territory. Regarding this example, however, I’m  less convinced about the rationality of the employed strategies.


Fudenberg, D. and J. Tirole (1986): “A Theory of Exit in Duopoly,” Econometrica, 54(4), 943-960.

Fudenberg, D. and J. Tirole (1991): “Game Theory,” The MIT Press.

Takahashi, Y. (2014): “Estimating a War of Attrition: The Case of the US Movie Theatre Industry,” American Economic Review, forthcoming.

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