Simulating the Evolution of Contest
Escalation
Winfried Just and Xiaolu Sun
Department of Mathematics and
Edison Biotechnology Institute
Ohio University
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.
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.
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).
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:
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.
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.
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.
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.
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.
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 > 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.
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.
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.
Acknowledgement
This work was partially supported by NSF grant DBI

9904799 to W.J.
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