Simulating the Evolution of Contest Escalation

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.

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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.

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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
  (KL)/(VK).

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The probability classes of winning a fight

• We are interested only in parameter settings
  where
• 0 lt (KL)/(VK) lt 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
• very low escalation to fighting would be
  disadvantageous
• low opponent is more likely to win, but
  escalation to the fighting stage would be still
  be advantageous
• high the opponent is more likely to lose, but
  still should prefer escalation to the fighting
  stage over unilateral retreat
• very high the opponent should retreat.

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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.

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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 38 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.

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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 gt 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.

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Our simulations

• For two parameter settings suggested by the
  results of Just, Morris,
• and Sun (in review) we run 120 simulations each
  with q gt 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.

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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
• In the simulations with q gt 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.
• In 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.

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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.

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References

• W. Just and M. R. Morris (in review). The
  Napoleon Complex Why Smaller Males Pick Fights.
• W. Just, M. R. Morris, and X. Sun (in review).
  The evolution of aggressive losers.
• 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 37303--310.

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Acknowledgement

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