The war of attrition

Setup

One indivisible resource worth V. Two contestants. Each privately commits to a persistence time. Cost accrues continuously to both for as long as the contest runs. Whoever quits first loses, and both pay the cost of the contest up to that moment. There is no way to pay less by losing gracefully.

That is the all-pay structure, and it is what separates this from an auction where only the winner pays. It is also what makes the contest expensive: the loser has spent everything and gained nothing.

The model

The exponential is not an arbitrary choice. It is the only distribution with no memory, which is exactly what is required: the chance an opponent quits in the next instant must not depend on how long they have already persisted. If it did, persisting longer would carry information and could be exploited.

Results

The equilibrium dissipates the entire value of the resource. Expected payoff is zero, so the population as a whole gains nothing from having the resource contested at all. This is a general property of all-pay contests, which is why the same structure is used to model lobbying, patent races and advertising wars.

Bishop and Cannings (1978) generalised the result to a broad class of contests with the same shape. Hammerstein and Parker (1982) showed what happens when the symmetry breaks. If contestants differ in fighting ability or in what the resource is worth to them, and can perceive the difference, the randomised solution collapses into a conditional rule: the one with less to gain withdraws immediately. Contests become short and settled by assessment rather than long and settled by exhaustion. That asymmetric version, not the symmetric one, describes most observed animal contests.

Evidence

Parker and Thompson (1980) tested the symmetric model against struggles between male yellow dung flies (Scatophaga stercoraria) over females at cowpats. They compared the distribution of struggle durations against the exponential prediction. The fit was reasonable through the bulk of the distribution and departed in the tail, roughly what the asymmetric extensions would lead you to expect.

The cleaner test is Marden and Waage (1990) on black-winged damselflies (Calopteryx maculata). Territorial contests between males take the form of prolonged aerial chases. They measured fat reserves before and after and found that contest duration was predicted by the loser’s energy stores, and that the loser was the individual who ran out first. The contest was an energetic war of attrition in the literal sense, with a measurable currency and an outcome determined by who could pay longest rather than by any decisive act.

That is what field game theory can do that a payoff matrix cannot. It identified the currency being spent and then went and measured it.

A third case cuts against the theory in a useful way. Dawkins and Brockmann (1980) examined fights between female digger wasps over jointly occupied nests, measuring each wasp’s prior investment as the number of prey items she had already carried in. Persistence should depend on what the nest is worth from here, not on what has already been spent. What they found was the opposite: the wasp with the smaller prior investment was the one more likely to give up, which is behaviour consistent with the sunk-cost reasoning the theory says should be irrelevant. They titled the paper after the Concorde fallacy and were candid that the result resisted a tidy optimality account.

Limitations

The symmetric model is the wrong model for most real contests, and the people who built it said so first. Animals assess each other. They use size, displays, prior experience and residency, all of which break the symmetry the exponential solution depends on. Where assessment is possible the observed distribution of contest durations is not exponential and should not be.

So the zero-payoff result is best read as a boundary case: it tells you what a contest costs when nothing can be inferred about the opponent, which is the worst case rather than the usual one. The value of the model is that it explains why animals invest so heavily in signals that make assessment possible. Every reliable signal moves the contest away from this equilibrium and towards something cheaper.

The simulator plays the symmetric game only. Adding perceivable asymmetry would remove the flat payoff line entirely.

Related This model shares its mechanism with Cyclic dominance (Paper 02) and Colonel Blotto (Paper 06) and Professionals play minimax (Paper 08). In each of them, no fixed choice is stable, so the solution is a distribution or a cycle rather than a move.

References

1. Bishop, D. T., & Cannings, C. (1978). A generalized war of attrition. Journal of Theoretical Biology, 70(1), 85–124. https://doi.org/10.1016/0022-5193(78)90304-1
2. Dawkins, R., & Brockmann, H. J. (1980). Do digger wasps commit the Concorde fallacy? Animal Behaviour, 28(3), 892–896. https://doi.org/10.1016/S0003-3472(80)80149-7
3. Hammerstein, P., & Parker, G. A. (1982). The asymmetric war of attrition. Journal of Theoretical Biology, 96(4), 647–682. https://doi.org/10.1016/0022-5193(82)90235-1
4. Marden, J. H., & Waage, J. K. (1990). Escalated damselfly territorial contests are energetic wars of attrition. Animal Behaviour, 39(5), 954–959. https://doi.org/10.1016/S0003-3472(05)80960-1
5. Maynard Smith, J. (1974). The theory of games and the evolution of animal conflicts. Journal of Theoretical Biology, 47(1), 209–221. https://doi.org/10.1016/0022-5193(74)90110-6
6. Maynard Smith, J., & Price, G. R. (1973). The logic of animal conflict. Nature, 246(5427), 15–18. https://doi.org/10.1038/246015a0
7. Parker, G. A., & Thompson, E. A. (1980). Dung fly struggles: A test of the war of attrition. Behavioral Ecology and Sociobiology, 7(1), 37–44. https://doi.org/10.1007/BF00302516

Verification Every reference on this page has been checked against Crossref, so the works cited are real and correctly identified. The claims this page makes about what those sources report have not yet been independently checked. Of the nine papers audited so far, eight required a correction, so treat the specifics here as unconfirmed.