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Web Topic 11.1
A Detailed Description of Three Conflict Resolution Models

Introduction

War of attrition games provided the earliest models for conflict resolution that involved strings of successive displays or other actions per contest (see Web Topic 10.5). Although these contests involve repeated bouts, and are thus technically sequential games, the war of attrition models got around the sequential aspects by having each player select a persistence time at the outset, and then letting them display at each other until the shorter time bid had run out. Whichever individual picked the shorter time was then the loser. To prevent cheating that would undermine any ESS, players were not allowed to vary either the intensity or the rate of repetition of their display during the contest. The symmetric war of attrition assumed all players suffered the same rate of cost accrual during the contest and valued the contested commodity equally. Under these restrictive conditions, a mixed ESS was possible favoring settling of conflicts by repeated displays. Since player equality is unlikely to be the case in reality, the asymmetric war of attrition was proposed. Here, players were allowed to differ in cost accrual rates and/or contested commodity valuation. To achieve an ESS in this situation, the asymmetric war of attrition assumed that players could glean sufficient information about their own and their opponent’s fighting abilities before the contest to assign themselves to either “loser” or “winner” roles. Each player then drew a persistence time from a random distribution appropriate to that role. As with the symmetric war of attrition, players were not allowed to vary display intensity or repetition rates. This combination of assumptions and strategies did lead to a stable ESS.

One problem with the asymmetric war of attrition is that it assumes that players can assess cues in each other before the contest that are correlated with likely fighting abilities. While body size is an obvious factor that might both affect fight outcomes and be assessable by players before a contest, there are many other likely determinants of fight outcome that are not so assessable. Examples include physical condition, energy reserves, prior experience, and motivation. In addition, fight outcomes might also depend on relative stress or damage accumulated by players during an escalation; this might be very difficult to estimate before a contest but play a major role in outcomes. Clearly, the asymmetric war of attrition cannot handle any of these cases.

In this Web Topic, we review models in which each of three different types of initially non-assessable factors are assumed to play significant roles in the outcomes of contests with repeated and sequential displays. In the first example, the sequential assessment model, all of the same assumptions are made as for the asymmetric war of attrition except that no useful assessments are possible before the contest begins. Instead, players gradually extract information about their own and their opponent’s relative fighting abilities as the contest proceeds. At some point, these estimates become sufficiently accurate that the players can confidently identify who would be a “winner” and who a “loser” if they escalated; the contest then ends with the presumed loser quitting. In a second example, the critical factor that affects contests is the amount of stored energy or some equivalent resource prior to the contest. This resource gets “used up” as the contest proceeds, and eventually, one player reaches some ceiling or threshold beyond which it cannot afford to use up the remaining resource. It quits and the contest ends with it being the loser. The third type of model focuses on the role of damage accrual and/or other externally inflicted costs that might accumulate as a contest progresses. Again, each player will have its own threshold level of such costs that it can afford to suffer before it quits. The three models are similar in that each focuses on some state variable (information, energy reserves, or damage, respectively) that changes progressively as the contest proceeds. This raises a second issue that is normally associated with sequential games: what is the optimal policy for scheduling the rate of change in that state variable at different points in the contest? We take up this question for each of the three types of models (see Web Topic 10.5 for a review of terms and logic of game theory).

The sequential assessment model (SAM)

The sequential assessment model, like the asymmetric war of attrition, assumes that each player uses the same display repeatedly and does not vary either display intensity or instantaneous repetition rates (Enquist and Leimar 1983). Also like the asymmetric war of attrition, the goal of the interaction is to obtain an accurate estimate of the relative fighting abilities of the two parties. The state variable that changes during the course of the contest is the amount of error associated with those estimates. The sought ESS is a policy that identifies how accurate the estimates need be before one party assigns itself a “loser” role and quits.

The model assumes that the true fighting abilities of two contesting players (A and B) depend on the rate at which each accrues costs during an escalated contest, both from their own actions and from those of their opponent. If cA is the true rate at which A accrues costs when contesting with B, and cB is the equivalent rate for B, relative fighting ability from A’s point of view can be measured as

When θA > 0, B suffers higher cost accrual than A, and thus A can be considered to have higher relative fighting ability; when θA < 0, then A suffers more than B and has the lesser relative fighting ability. It should be obvious that θA = – θB.

When the contest begins, neither party has a good estimate of the true θi values. They then try to get more accurate estimates by observing each other’s displays during the contest. During each sample, an estimated value of θi will depend both on the true value and on some random error. Suppose the random error in any one sample i is ziA and that by B is ziB. The model assumes that these random errors are drawn from a normal distribution with mean zero and standard deviation σ. The best estimates at any step n in the contest would then be the cumulative average of the current and all prior samples. If at contest step n, A’s current estimate of θA is xnA, and B’s estimate of θB is xnB, then

At the beginning of the contest, each individual has only a crude estimate of the θi values. As the contest progresses, the effects of the random errors begin to average out to zero (e.g., the sampling error for the cumulative estimate after n steps is SE = σ/√n), and the current estimate approaches the true relative fighting ability. The model assumes that once one animal estimates that its θi is negative, it will end the contest by retreating or giving up. Since the standard error of the estimates should decrease as n increases, a smaller |θI| is necessary to trigger quitting as n increases. The evolutionarily stable policy can thus be described as a giving-up line on a graph of current estimates versus the number of steps so far, n. When one contestant’s current estimate crosses this line, it is sufficiently certain of its lower fighting ability that it makes the decision to quit (Figure 1).

Figure 1: Graphical solution for the sequential assessment game. Current estimates by each party of its relative fighting ability, xA (red line) and xB (blue line), are plotted on the vertical axis for successive numbers of steps n (horizontal axis). The solid horizontal line indicates equal estimated fighting abilities (e.g., xi = 0). The dashed line shows the ESS policy for giving-up. Early in the fight, a player’s assessment of itself, x, must be quite low for it to give up because of the high level of uncertainty (e.g., large standard error of estimate). As the contest progresses and the estimates of true relative fighting ability become more accurate with repeated sampling, the giving-up line rises towards the 0 line. When an individual’s x crosses the line, it quits the fight. In this example, that was the blue player. (After Enquist and Leimar 1983.)

The smaller the difference in true fighting ability, and the higher the error of assessment, the longer the fight will be on average (Figure 2). Contests between near equals are also more variable in duration than those between unequal competitors and the (slightly) poorer contestant may sometimes win (Figure 3).

Figure 2: Average duration of a fight (in number of steps) as function of the absolute value of relative fighting ability. The three curves represent increasing (black to red to blue) standard deviations in the error levels of opponent assessment: higher levels of errors result in a slower decrease in fight durations. (After Enquist and Leimar 1983.)

Figure 3: The probability of victory (vertical axis) as a function of relative fighting ability. Losing or winning becomes more certain when contestants are more disparate in ability.

As the cost of fighting increases, the giving up line moves higher towards the x=0 line, meaning fight duration is shorter (Figure 4).

Figure 4: Giving-up line as a function of the cost of the interaction. The red curve is the giving-up line for a high level of interaction cost. Each line below it represents half the cost of the one above. As cost increases, the giving-up line moves up and contests are shorter. A similar series of curves is generated when the value of the resource decreases. As the value increases, the giving-up line is shifted down and contests are longer. (After Enquist and Leimar 1983.)

This game is a somewhat more realistic model than the asymmetric war of attrition, because it allows contestants to control the cost (duration) of the fight as it proceeds and to decide whether to continue or quit based on information gained during the fight. On the other hand, it yields some similar predictions to those of the asymmetric war of attrition, namely that contest length is short if the individuals are very different in size or fighting ability, and highly variable but generally longer when the contestants are similar.

In the original sequential assessment model, players were allowed to shift between multiple behaviors as long as these gave the same kind and amount of information. However, real animals often seem to have a sequence of escalating behaviors that are seen in conflicts. Enquist et al. (1990) later developed a different version of the sequential assessment game with several behavioral options that could be adopted in stages. Thus when the additional information provided by any given display stage became asymptotic, players could then switch to another display that provided more or different information. Usually, better information requires riskier or more expensive displays. In fish for example, lateral display may provide only partial information on size, tail beating leads to better but riskier size estimation, and mouth wrestling or head butting provides even better information. The model predicts that the alternative stages should be ordered so as to be maximally efficient in assessing relative fighting ability. This model thus provides an optimal policy for adjusting display type at different stages in the contest. In this extended model: (1) all contests should be organized into phases consisting of one or several behavior patterns with constant intensities and rates of repetition within a phase; (2) the contest should begin with the least costly behaviors that provide some information about fighting ability asymmetry, and after repetition with diminishing returns, switch to new, more costly but more effective behaviors in subsequent stages; (3) the division into phases should be independent of relative fighting ability; and (4) contests with great asymmetry in relative fighting ability should end in an early phase, whereas matched individuals may proceed through a series of escalations reaching a final phase of more dangerous fighting. Figure 5 shows a sample trajectory with three behaviors.

Figure 5: The sequential assessment game with three actions assigned to successive phases. In phase 1, only Action A is used until repetition provides no more useful information. In phase 2, Action B is added. In this example, the giving-up line is crossed during the second phase, so the contest ends before escalating to dangerous fighting, Action D. (After Enquist et al. 1990.)

Limited energy models

The sequential assessment game assumes that each time a combatant performs the same display, it provides additional information about its ability to fight should the contest escalate. This mechanism works best for assessing instantaneous attributes such as coordination or motivation. It may not be a good way to assess an opponent’s stamina and endurance, which could also play an important role in escalated contests. Like the asymmetric war of attrition and the sequential assessment game, the two models below focus on populations in which players do not share the same cost accrual rates or disputed commodity valuation. Both assume that each player has its own reserves (e.g., energy or some other limited resource) that get used up during the contest. No initial assessment of these reserves is possible, and players just display until one hits a threshold in cumulative costs beyond which it is not prepared to continue. It then quits and becomes the loser. Note that the outcomes of these contests are predetermined before the displays even begin; it is just that players cannot assess what these are. It is only after a contest has finished that this information is revealed. The first model below asks whether there can be an ESS in such contests if initial assessments are ignored and opponents just play out the endurance competition. The second model assumes that such an ESS exists and examines the optimal policy schedule for cost accumulation during such an endurance contest.

Limited energy models/War of attrition without assessment (WOAWA): The WOAWA model (Mesterton-Gibbons et al. 1996) assumes that individuals differ in the amount of energy or other limiting resource that is available for a protracted contest; distributions of maximal resources among the population’s individuals are assumed to be unimodal with a long tail at higher values. The longer a given player continues a contest, the less resource is available for other fitness-enhancing functions. Key parameters for this model are β, the rate at which a contest uses up this resource, and α, the efficiency with which residual resource after a contest can be turned into fitness. The cost/benefit ratio, β/α, is denoted by θ. A second key parameter is κ, the coefficient of variation in the amount of total resource held by different contestants in the population. Analysis of the model shows that an ESS can exist when players do not assess each other prior to initiating a contest as long as the relative cost/benefit ratio θ is small enough and/or the variation among contestants, κ, large enough (Figure 6A). In addition, the ESS identifies the maximal fraction, υ, of the total available resource at the start of the contest that an animal should commit before quitting (Figure 6B).

Figure 6: ESS outcomes of WOA model. (A) Combinations of coefficients of variation in initial resources (κ) and cost/benefit ratios of contests (θ) that preclude (blue) or favor (tan) a stable ESS in which contestants do not assess each other before beginning a contest, but just play out their resource until it hits a critical fraction, υ, of their total resource available. First opponent to hit this ceiling quits and is therefore loser. High cost/benefit ratios and/or low variation among combatants in initial resource level do not support this ESS. Combinations that favor the ESS may leave losers with less variation in residual resources after the contest than winners (light tan) or winners may show less variation than losers (dark tan). (B) ESS fraction of total available resource that should be assigned to a contest υ as a function of population variation in initial resources available (horizontal axis) and cost/benefit ratio (different colored lines). As cost/benefit ratio increases, average fraction of resources that should be allocated to a contest decreases. (After Mesterton-Gibbons et al. 1996.)

These authors also considered which statistical model best fit the distributions of resource identified in wild populations; the Weibull distribution (http://en.wikipedia.org/wiki/Weibull_distribution) appeared to give a better fit than a lognormal or gamma distribution. They also pointed out that while the asymmetric war of attrition predicts an inverse correlation between actual contest durations and the asymmetry in player fighting abilities, the WOAWA model predicts a positive correlation between contest duration and the residual resource remaining in losers of contests. This provides some interesting tests for comparing which of these two models, if either, fits a real system.

Limited energy models/Energetic war of attrition model (EWOA): The EWOA analysis looks for the optimal allocation of display effort during an endurance contest (Payne & Pagel 1996). Since the relative frequencies of players with different maximal endurance times are assumed to be fixed, and adoption of particular effort schedules by various players has no effect on those frequencies, this model is more of a simple optimization problem than a game. However, it provides some interesting predictions about when an endurance display system can or cannot pay for its costs.

In this model, players are competing for some commodity of value V, and each player has its own cost ceiling for a particular contest. This cost ceiling could be reached by performing high intensity or rapidly repeated displays throughout a short duration contest, or alternatively low intensity/infrequently repeated displays over a longer period. A focal animal’s instantaneous level of display (intensity, repetition rate, or both) at any time t in the contest is denoted by a(t) and the cumulative “signal” generated by this and all prior display in this contest is denoted by s(t). The cumulative cost of displays at point t in the contest is

C(t) = F(t) + T(t)

where F(t) is the cumulative energy cost and T(t) is the cumulative time lost (or fatigue acquired), both scaled in the same currency. At frequent intervals during the contest, each player compares its cumulative signal to its threshold value X. As long as its cumulative display is less than its X and its opponent is still displaying, it also continues to display. Once s(t) ≥ X, the player either flees (ending the contest), or escalates it into a more violent stage. Assuming no player escalates after s(t) ≥ X, denoting the average duration of a contest against an opponent B that has a threshold XB by τ(XB), and denoting the distribution of players with different values of XB by N(XB), the average payoff of endurance contests for a focal player A is:

The first term reflects cases where A won, and the second cases where B won.

Holding V and N(XB) constant, the authors then examined how the costs C might vary with different emphases on display intensity versus contest duration for a given X. Replacing C with expressions defining its dependence on a(t), and setting the first derivative of that equation to zero and the second derivative to negative values identified the values of a(t) during a contest that maximized the payoff. Three types of outcomes were identified (Figure 7).

Figure 7: Possible outcomes for the EWOA analysis. A Type I outcome arises when cumulative time costs of continued display increase rapidly with contest duration whereas cumulative energy costs rise only minimally. Then a single high-intensity display is favored (maximal a(t) for the short duration of the contest). Type II contests arise when both cumulative energy and time costs increase during the contest. Type III contests arise when energetic costs rise rapidly with contest duration but time costs remain low. A minimally energetic display (low a(t)) is then given repeatedly for long periods as in the classical war of attrition. Whether a constant or changing display level is optimal during a contest depends on whether the cumulative time costs increase in accelerating (top dashed red line), linear (solid red line), or decelerating (lower dashed red line) manner. (Modified from Payne & Pagel 1996.)

In Type I contests, the optimal strategy is to produce a single maximum intensity signal; the duration of the contest then provides no information on player endurance. This situation can arise if the T(t) costs increase non-linearly with t but energetic costs, F(t), increase only slowly if at all during the contest. In Type III contests, the opposite occurs: it is best for each player to produce as costless a display as possible for as long a time as possible. In some respects, this is the classical war of attrition model. This is likely if energetic costs increase nonlinearly with contest duration, whereas time costs are minimal. Of greatest interest in this analysis are Type II contests. Here, a stable endurance game, in which each player’s stamina is honestly displayed by its maximal display duration, is only compatible with the relevant ESS if the cumulative energetic and cumulative time costs both increase significantly during the contest. In contrast with the sequential assessment game, in which each opponent should produce successive displays identically so that the average converges on their true fighting ability, the EWOA model can favor players increasing or decreasing the intensity or repetition rate of their displays as the contest proceeds: if time costs increase in an accelerating way, players should increase a(t) as t increases; if cumulative time costs increase in a decelerating way, players should decrease a(t) as t increases. This could explain changes in display intensity in natural examples for reasons other than escalating to obtain more information (as in staged sequential assessment games). Note however that for stable outcomes, all players have to adopt the same escalated or de-escalated display at the same time. How this “matching” might be achieved was not discussed in this paper, but is taken up in the next model.

Cumulative assessment model (CAM)

Whereas the sequential assessment model focused on cumulative acquisition of information, and the two prior models focused on the cumulative energy costs of protracted contests, the cumulative assessment model combines the accumulated effects of energy consumption and acquired damage during a contest. It focuses on a contestant’s successive decisions about whether to continue or quit given the cumulative sum of these combined costs (Payne 1998). It is most relevant to species that employ ritualized fighting in which only a certain amount of direct physical damage or stress can be tolerated. It can also be applied to non-contact interactions as long as the contestants are subject to external time costs not under their control, such as predation risk or lost foraging time. At no point are rivals assessed, and instead players only self-assess their own energetic expenditures and accumulated externally caused effects like damage.

As with the EWOA model, this analysis examines how contestants who differ in some fighting ability or intrinsic quality related to fighting, q, should alter the intensity, here denoted by R, of their actions over the course of a contest to maximize their expected payoffs. R can be a measure of the magnitude of a display and/or its instantaneous rate of repetition. All opponents are able to vary R during the course of a contest. It is assumed that each contestant persists in the interaction until the total costs including damage inflicted by the opponent surpass some threshold, and ignores the effects of its own attack upon the opponent, other than to note whether the opponent is still fighting or has fled. Like the EWOA model described earlier, this is a model of fighting tactics (how to perform optimally against a given opponent for a certain threshold) and not a strategy of how to choose the best threshold.

In this model, the overall cost suffered by each contestant over time is a combination of energetic costs F(t) and damage costs D(t):

C(t) = F(t) + D(t)

Consider a focal player with quality q that enters a contest with another player of quality . The rate at which energy costs accrue to the focal player at any time t in the contest depends on the intensity of the actions R(q,t) that it chooses to adopt, and the rate that it accrues damage costs depends on the intensity of the actions R(, t) adopted by its opponent over which the focal player has no control. The overall rate of cost accrual is then

where a and d are scaling parameters, and n is an exponent that allows for the possibility that energetic and damage costs do not accumulate with the same power of R over time.

The contest continues until one contestant flees at time T(q,) which occurs when its tolerance threshold X(q) has been reached. The threshold reflects the costs an animal is willing to suffer in the contest, which in turn depend on contextual factors such as the value of the commodity being contested (V) and its own quality q. The overall expected payoff is given by

where p() is the probability density function of possible opponent qualities. As with the EWOA model, the first term accounts for contests that are won (opponent flees first), and the second term accounts for contests that are lost. The goal is to choose the optimal policy for each contestant as expressed by R(q,t) that maximizes E(q).

The author then examines a likely general case in which at least one party linearly escalates its R at which point the other should also increase its R and again is limited to linear increases. The question is then what the optimal intercept (initial R) and slope (rate of increase in R over time) should be for players with different quality (q) values. Stable policies are only present if the exponent n is greater than 1: energetic costs must rise in an accelerating manner as R is increased or no policy can be stable. The predicted ESS policy intercepts and slopes for contestants with linearly increasing R(t) are shown for high and low q individuals in Figure 8.

Figure 8: Optimal policies for increasing display intensity (R(t)) during a contest according to CAM model. In all cases, red line shows optimal policy (particularly intercept and slope) for high quality individuals and blue line indicates optimal policy for low quality individuals. High quality individuals should always adopt a higher initial value (intercept) than low quality ones. (A) When contests are likely to be brief, high quality individuals should adopt a high initial R and lower quality individuals a lower intensity. There is no time for the latter to catch up and damage the former enough to change outcome. They should just quit. (B) When contests are likely to be long, both parties should pick low initial intensities. Since high quality individuals can always afford to increase intensity at a steeper rate than low quality, the latter can never catch up. (C) For intermediate duration contests, high quality individuals again choose a high initial R. However, there is a chance low quality players can inflict sufficient damage on high quality opponents if they increase intensities faster and thus have a chance to win. Intermediate duration ensures that this strategy does not trap lower quality players into a long period of high costs. (After Payne 1998.)

Optimal policies differ depending on the likely duration of a contest. In all cases, higher q individuals should begin display at a higher R than lower q individuals. Thus the relevant linear trajectories always have a higher intercept for high q than for low q individuals. There are three general cases:

A critical assumption of this model is that there needs to be some component of the cumulative costs that is beyond the control of each individual. Individuals can control their energy expenditures, but they cannot control the damage or stress imposed by the rival’s actions. Other possible external sources of cost include attraction of predators, increased vulnerability to parasites, lost foraging time, impaired ability to mate guard, and lost mating opportunities. The CAM model thus differs from the EWOA model primarily in the addition of cumulative costs due to external factors out of the focal animal’s control. Note that in addition to the escalation patterns illustrated in Figure 8, the cumulative assessment model predicts that contest duration will be positively correlated with the quality (energy reserves and defensive skill) of the loser. However, controlling for loser quality, contest duration should be negatively correlated with quality of the winner, since a higher quality winner will inflict costs on the loser at a higher rate. Depending on the relative importance of energetic and damage costs, these two predictive curves could vary in their strength.

Testing the models

The three models discussed above that predict contest trajectories (SAM, EWOA, and CAM), are sufficiently different that they generate divergent predictions that can be tested empirically. In the following table, different measures of relative fighting ability are all collected under a general term: resource holding potential (RHP) (see Chapter 11 in the text for more detailed definitions of this term). Below, we contrast a variety of predictions of these three trajectory models (an abbreviated version of this table also appears in the text):

Table 1. Summary of the predictions of the SAM, EWOA, and CAM models. (Sources: Briffa & Elwood 2009; Payne 1998.)

Sequential Assessment (SAM) Energetic War of Attrition (EWOA) Cumulative Assessment (CAM)
Decision based on: Difference between average opponent RHP Sum of own actions Sum of opponent’s actions
Assumes display level matching in population: Depends on version Yes No
Assessment of opponent: Yes No No
Escalation: Not within a phase, but in sequential phases Escalation and de-escalation possible Escalation and de-escalation possible
Contest duration most strongly correlated with: RHP asymmetry between opponents (–) Loser RHP (+) Loser RHP (+) and winner RHP (–)
Contest duration increases with increasing mean opponent RHP? No Yes Possible
Display characteristics: Non-dangerous index signals or ritualized fighting tactics Energetically costly chasing or handicap signals with enforced intensity matching Dangerous displays

It is easy to stage contests between known sized opponents and measure contest duration. This sort of data has been collected for a large number of species and used to test the contest duration predictions as a way to distinguish among the models. Taylor and Elwood (2003) importantly pointed out that these patterns can be misleading. They showed with simple simulations that if a pure self-assessment process such as the EWOA was in operation, and there was a perfect positive correlation between loser RHP and contest duration, a spurious negative correlation between RHP asymmetry and contest duration could arise. This occurs because in a population with a normal spread of body size or RHP, smaller individuals would usually have large opponents. Body size is thus negatively correlated with contestant size asymmetry, generating the spurious correlation between asymmetry and duration. Moreover, if true assessment is occurring, such that RHP asymmetry is negatively correlated with contest duration, a spurious positive correlation between loser RHP and duration, and a spurious negative correlation between winner RHP and duration, would be generated.

To make matters even more complex, if the CAM model is in operation, the same positive correlation with loser RHP, negative correlation with winner RHP, and negative correlation with RHP asymmetry will be generated. Although the CAM predicts that the loser and winner correlations should be stronger than the RHP asymmetry correlation, while the SAM predicts a stronger RHP asymmetry correlation, noisy data can obscure such subtle differences. Below, we show some of these primary versus spurious predictions of the three models:

Figure 9: Predicted relationships of contest duration as a function of winner and loser characteristics for the three fighting trajectory models. Contest duration is on the y-axis in each of these graphs. (1) indicates a primary prediction; (2) indicates a spurious correlation. (After Gammell & Hardy 2003; Taylor & Elwood 2003.)

As a consequence of these complications, it is essential in a study testing the fit to alternative models to examine the other differences among the models, including the dynamics of escalation, the presence of matching intensities, and the type of displays or fighting tactics employed (Briffa & Elwood 2009).

Literature cited

Briffa, M. & R.W. Elwood. 2009. Difficulties remain in distinguishing between mutual and self-assessment in animal contests. Animal Behaviour 77: 759–762.

Enquist, M. & O. Leimar. 1983. Evolution of fighting behavior — decision rules and assessment of relative strength. Journal of Theoretical Biology 102: 387–410.

Enquist, M., O. Leimar, T. Ljungberg, Y. Mallner & N. Segerdahl. 1990. A test of the sequential assessment game — fighting in the cichlid fish Nannacara anomala. Animal Behaviour 40: 1–14.

Gammell, M.P. & I.C.W. Hardy. 2003. Contest duration: sizing up the opposition? Trends in Ecology & Evolution 18: 491–493.

Mesterton-Gibbons, M., J.H. Marden, and L.A. Dugatkin. 1996. On wars of attrition without assessment. Journal of Theoretical Biology 181: 65–83.

Payne, R.J.H. and M. Pagel. 1996. Escalation and time costs in displays of endurance. Journal of Theoretical Biology 183: 185–193.

Payne, R.J.H. 1998. Gradually escalating fights and displays: the cumulative assessment model. Animal Behaviour 56: 651–662.

Taylor, P.W. & R.W. Elwood. 2003. The mismeasure of animal contests. Animal Behaviour 65: 1195–1202.

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