1. Simulating the Evolution of Contest Escalation Winfried Just and Xiaolu Sun Department of Mathematics and Edison Biotechnology Institute Ohio University

2. Background Most published studies of escalated animal contests show that it is usually the likely winner of a contest who initiates escalation to the more costly stage. However, in some species the situation is reversed and escalation is much more often initiated by the eventual loser. For example, such a situation was reported for swordtail fishes Xiphophorus multilineatus and X. nigrensis by Morris et al. (1995). We developed a game-theoretic model that shows a possible reason for such counterintuitive behavior. Here we report on the results of testing the model under simulated evolution.

3. Game-theoretic models of animal contests In game-theoretic models of animal behavior, animals are treated as players that try to maximize their payoffs (Darwinian fitness) in a game. They are supposed to follow genetically coded strategies (prescriptions for behavior). A strategy is evolutionarily stable (an ESS) if a population of players who all follow this strategy cannot be invaded by a mutant strategy.

4. The model We model animal contests that have up to two stages: a display stage, during which no physical contact occurs, followed in some cases by a fight stage during which physical contact occurs. Note that this structure implies that passage from the display stage to the fight stage requires escalation by only one of the contestants. Payoffs: V for obtaining the contested resource, -L for engaging in a fight - (L + K) for losing a fight Note that engaging in a fight is advantageous if and only if the Probability of winning a fight is above (K+L)/(V+K).

5. The probability classes of winning a fight We are interested only in parameter settings where 0 < (K+L)/(V+K) < 0.5, so that sometimes both players will prefer escalation to unilateral retreat. We assume that during the display stage contestants try to assess the probability of winning a fight. It is assumed that from the point of view of a given contestant, this probability is partitioned into four classes: zvery low: escalation to fighting would be disadvantageous; zlow: opponent is more likely to win, but escalation to the fighting stage would be still be advantageous; zhigh: the opponent is more likely to lose, but still should prefer escalation to the fighting stage over unilateral retreat; zvery high: the opponent should retreat.

6. Perception of probability classes We assume that a player may misperceive his probability class of winning a fight as each neighboring one with probability q. At each time during the display stage, a player will have partial or full information about his probability class (possibly incorrect information). Such partial information is modeled as a perception state of a player. For example, a player may perceive that his winning probability is either very low or low, but may not have reached a decision yet as to which one it is. Encounters start with none of the players having any information about their winning probability, and the estimates of the winning probability become more refined as the encounter progresses.

7. Strategies A strategy prescribes one of the three actions D (continue displaying), R (retreat), or E (escalate) to each one of the eight perception states we consider in our model. Thus there is a total of 3 8 = 6,561 possible strategies. In the simulations, strategies are coded as strings of letters. They are fixed throughout the lifetime of each player, and inherited from the parents with crossover and mutations. Encounters are modeled by letting the contestants carry out the prescribed actions as the perception states become more refined. The outcomes of fights are randomly generated according to given parameter settings of winning probabilities and the actual (not necessary perceived) probability classes.

8. Predictions of the model With a total of 6,561 strategies, the model is not analytically tractable. However, simplified versions of the model have been analyzed by Just and Morris (in review) and Just, Morris, and Sun (in review). These models ignore or greatly simplify the process of refinement of partial information and suggest that for typical parameter settings with probability of misperception q > 0, a player should retreat if he perceives his winning probability as very low, should escalate if he perceives his winning probability as low, and should continue displaying if he perceives his winning probability as high or very high. This would lead to a population of players where most fights are Initiated by their eventual losers.

9. Our simulations For two parameter settings suggested by the results of Just, Morris, and Sun (in review) we run 120 simulations each with q > 0 and 30 simulations each with q = 0. Some of these simulations started from random initial populations; other simulations started from initial populations where all players followed a fixed strategy that was different from the predicted ESS. We simulated the evolution of strategies in populations of 3,000 players over 100,000 mating seasons. Each player was characterized for life by its innate fighting ability and its strategy. In each mating season, each player had on average 6 encounters per mating season, and lived for 10 mating seasons.

10. Results The results of these simulations confirm that for the particular parameter settings studied, the results of the simplified model of Just, Morris, and Sun (in review) carry over to our model: zIn the simulations with q > 0, over 75% of all fights were initiated by their likely loser, and most of the time, a mix of strategies in which the ESS predicted by the simpler model dominated was observed. zIn the simulations with q = 0, the percentage of fights initiated by the weaker contestant was not significantly different from 50%, and no (mixed or pure) ESS appeared to evolve.

11. Open problems However, exploratory runs for several other parameter settings did show patterns that differed from the predictions of Just, Morris, and Sun (in review). Characterizing the region of the parameter space where the results of the latter model remain valid if the process of Information acquisition is explicitly modeled remains an open problem. Further directions or research include investigating how robust our findings are if more probability classes are considered or if escalation can proceed in more than just two stages.

12. References 1.W. Just and M. R. Morris (in review). The Napoleon Complex: Why Smaller Males Pick Fights. 2.W. Just, M. R. Morris, and X. Sun (in review). The evolution of aggressive losers. 3.M. R. Morris, L. Gass, and M. J. Ryan (1995). Assessment and individual recognition of opponents in the swordtails Xiphophorus nigrensis and X. multilineatus. Behavioral Ecology and Sociobiology 37:303--310.

13. Acknowledgement This work was partially supported by NSF grant DBI-9904799 to W.J.