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Oct 2, 2013 (4 years and 12 days ago)

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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.