Just & Sun: Evolution of Contest Escalation
2/16/16
1
2
3
4
5
Simulating the Evolution of Contest Escalation
6
WINFRIED JUST1 AND XIAOLU SUN2
7
8
1
Department of Mathematics and 2Edison Biotechnology Institute
9
Ohio University, Athens, OH 45701, U.S.A.
10
11
12
13
14
Running Headline: Evolution of Contest Escalation
15
16
Correspondence: W. Just, Department of Mathematics, Ohio University, Athens
17
OH 45701 (e-mail: just@math.ohiou.edu; Tel.: +1-740-593-1260; fax.: +1-740-
18
593-9805 )
19
20
Word count: 2,698
1
Just & Sun: Evolution of Contest Escalation
1
Empirical studies of animal contests have shown that escalation to costly fights is
2
usually initiated by their likely winners; in some species however, it is the likely
3
losers who tend to initiate escalation; see (Morris et al., 1995) for a particularly
4
striking example. Game-theoretic models developed in (Just and Morris, 2003)
5
and (Just et al., in review) show a possible explanation for the latter phenomenon.
6
Here we test an extension of these models with simulated evolution.
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
2
Just & Sun: Evolution of Contest Escalation
1
2
3
1. Introduction
Animals often resolve conflicts over limited resources by ritualized
3
displays, but such contests can escalate to costly fights. The decisions animals
4
make in these contests should be influenced by the difference in their fighting
5
ability, or resource holding power (abbreviated RHP; Parker, 1974). It would be
6
expected that, in general, individuals with higher RHP should benefit from
7
escalation since lower RHP individuals are not likely to gain resources by fighting
8
but will often pay a cost of fighting (Parker, 1974). However, it is less clear which
9
of the contestants should escalate first if both contestants would prefer escalation
10
to giving up. Individuals with the relatively higher RHP (likely winners) initiate
11
escalated contests in several different organisms (e.g., fishes, Figler and Einhorn,
12
1983; Keeley and Grant, 1993; Turner and Huntingford, 1986; Barlow et al., 1986;
13
hermit crabs, Dowds and Elwood, 1983; sea anemones, Brace and Pavey, 1978;
14
and mollusks, Zack, 1975). Recently, however, empirical studies have detected
15
cases where likely losers are more aggressive than expected (Morris et al., 1995;
16
Enquist and Jackobsson, 1986; Dow et al., 1976). In particular, (Morris et al.,
17
1995) studied contests between males in two species of swordtail fishes
18
(Xiphophorus nigrensis and X. multilineatus) in the laboratory. When the
19
difference in size between fish was very large, contests were settled without fights
20
with the smaller animal retreating. However, 78% of the observed fights were
21
initiated by the smaller animal, and in 70% of the fights, the fish that delivered the
22
first bite lost the contest.
23
Just & Sun: Evolution of Contest Escalation
4
1
Models for explaining this counterintuitive behavior have been developed
2
by Just and Morris (2003) and Just et al. (in review). These models predict that in
3
a large region of the relevant parameter space, escalation of most contests should
4
be initiated by a player precisely when this player perceives himself as slightly
5
(but not overwhelmingly) weaker than the opponent. These models have been
6
studied analytically. However, in order to make an analytical approach feasible,
7
the process of acquisition of partial information during the display stage had to be
8
either ignored (Just et al., in review) or restricted to one particularly important
9
scenario (Just and Morris, 2003). Here we consider an extended model that allows
10
for a variety of ways in which partial information about the difference in RHP may
11
be acquired during the display stage. This makes the model too complicated for an
12
analytical treatment; we explore it with computer simulations instead.
13
14
15
2. The model
16
17
We model animal contests that have up to two stages: a display stage,
18
during which no physical contact occurs, followed in some cases by a fight stage
19
during which physical contact occurs. Note that this structure implies that passage
20
from the display stage to the fight stage requires escalation by only one of the
21
contestants. The objective of a contest is to gain access to a resource that has a
22
value V for each of the contestants. The payoff for each individual is determined
23
by the outcomes of contests, and an individual may be involved in one or more
Just & Sun: Evolution of Contest Escalation
5
1
contests during its lifetime. Displaying carries a small cost to each of the
2
contestants that is assumed proportional to the duration of the display stage. The
3
cost of a fight has two components: a cost born only by the loser (denoted by K)
4
and a cost that is incurred by both the winner and the loser (denoted by L). It is
5
easy to see that a player will derive an expected positive payoff from a fight if and
6
only if his probability of winning the fight exceeds (K+L)/(V+K). The number
7
(K+L)/(V+K) will be referred to as the threshold for escalation.
8
9
We assume that during the display stage contestants try to assess the
10
probability of winning a fight. It is assumed that from the point of view of a given
11
contestant, this probability is partitioned into four classes; it is either very low (less
12
than (K+L)/(V+K); escalation to fighting would be disadvantageous), low (less
13
than 0.5 so that the opponent is more likely to win, but bigger than (K+L)/(V+K);
14
so that escalation to the fighting stage would be advantageous), high (the opponent
15
is more likely to lose, but his winning probability is still above the threshold for
16
escalation), or very high (opponent’s winning probability is below the threshold
17
for escalation). We are only interested in parameter settings where the classes low
18
and high can actually occur. A contestant may somewhat misperceive the class of
19
the probability of winning; for example, if this probability is low, then a contestant
20
may misperceive it as high (with probability q[0]) or very low (with probability
21
q[1]), but not as very high. The probability q[0] of misperception of role and q[1]
22
of misperception of magnitude of difference in RHP are crucial parameters of the
23
model.
Just & Sun: Evolution of Contest Escalation
6
1
2
At the beginning of the display stage, the players have no information
3
about the probability of winning, and during the process of displaying they may
4
gain partial and eventually full information about the winning probability. We
5
assume that partial information is obtained in two bits: one bit designates the role
6
of a given player (Pl if the given player has the lower RHP, PH if the player has the
7
higher RHP); the other bit tells whether the probability of winning for the weaker
8
player is below or above the threshold for escalation. We say that the absolute
9
value of the difference in fighting ability is large in the former case and small in
10
the latter. While the model of Just and Morris (2003) and the simulations of Just et
11
al. (2000) assume that information about the role is known at the beginning of an
12
encounter, and while the model of Just et al. (in review) does not allow for partial
13
information, we assume here that the two bits of information can be obtained in
14
any order. Thus at any given time of the display stage, a player will be in one of
15
eight perception states: (no info, no info), (Pl, no info), (PH, no info),
16
(no info, small), (Pl, small), (PH, small), (Pl, large), (PH, large). For example, a
17
player in perception state (no info, small) will assume that the probability of
18
winning is either low or high; a player in perception state (Pl, large) will have
19
determined that the probability of winning is very low. Note that a perception state
20
(no info, large) is implausible and should never be reached.
21
22
In each of the perception states, a player has a choice between three
23
actions: R (retreat), D (continue displaying), and F (escalate to the fighting stage).
Just & Sun: Evolution of Contest Escalation
1
Thus a strategy can be conceptualized as a function that specifies one of the three
2
actions for each of the eight possible perception states, and we can code strategies
3
as strings of eight letters from the alphabet {D, F, R}. Note that there are
4
altogether 38 = 6,561 different strategies in this game. This number makes it
5
difficult to study the game analytically and calls for an approach using simulated
6
evolution.
7
7
8
9
The display stage of an encounter lasts until one of the players takes action
R or F. Displaying for time t carries a cost of d·t, for each contestant. In an
10
encounter where neither contestant ever performs action R or F, the display stage
11
is arbitrarily terminated after c units of time. The numbers c and d are user-
12
definable parameters.
13
14
15
3. Our simulations
16
17
We simulated the evolution of strategies in populations of 3,000 players
18
over 100,000 mating seasons. Each player was characterized for life by its innate
19
fighting ability and its strategy. In each mating season, each player had on average
20
6 encounters per mating season, and lived for 10 mating seasons. The outcomes of
21
individual encounters were simulated taking into account the participant's innate
22
fighting abilities and strategies. The times at which the player's perception states
23
changed to more informative ones as well as the actual outcomes of the fights were
Just & Sun: Evolution of Contest Escalation
1
randomized (Pl won with probability a if the difference in fighting ability was
2
large and with probability b if this difference was small, where 0 < a < b < 0.5
3
were user-definable parameters). The fitness of each individual was set to an
4
initial fitness W at the beginning of each mating season and was then decreased or
5
increased according to the outcomes of this player's encounters. At the end of a
6
mating season, the fitness was used to determine the probability that males will
7
mate (all individuals had an equal probability of being treated as male or female).
8
After each mating season the 300 oldest players were removed and replaced by
9
new players. The new players inherited the actions in their strategy from their
10
parents with mutations and either uniform crossover or a crossover operator that
11
favored inheritance of certain actions in blocks. A detailed description of the
12
program as well as the source code can be found at the following URL:
13
http://www.math.ohiou.edu/~just/Escalate/.
8
14
15
16
3. Results
17
18
The model of Just et al. (in review) suggests that if the probabilities of
19
misperception q[0] and q[1] are positive, then for a wide range of parameter
20
settings a mix of the strategies DDDDFDRD and DDDFFFRD should emerge as
21
an evolutionarily stable strategy (abbreviated ESS). The first of these strategies
22
calls for a player to escalate to fighting precisely when the probability of winning
23
is perceived as low, to retreat when the probability of winning is perceived as very
Just & Sun: Evolution of Contest Escalation
9
1
low, and to continue displaying in all other perception states. The second of these
2
strategies calls for a player to escalate to fighting when the probability of winning
3
is perceived as low or high, to retreat when the probability of winning is perceived
4
as very low, and to continue displaying in all other perception states. Any
5
nontrivial mix of these two strategies will lead to a situation where most fights are
6
initiated by their eventual losers. In contrast, the same model suggests that if the
7
probabilities of misperception q[0] and q[1] are zero, then no ESS should emerge
8
and about half of all fights should be initiated by their eventual losers.
9
10
For two of the parameter settings where the model of Just et al. (in review)
11
makes the above predictions we run 120 simulations each with q[0], q[1] > 0, and
12
30 simulations each with q[0] = q[1] = 0. Some of these simulations started from
13
random initial populations; other simulations started from initial populations where
14
all players followed a fixed strategy that was different from the predicted ESS.
15
The program monitored the proportion of selected strategies between mating
16
seasons 10,000 and 100,000, as well as the proportion of fights that were initiated
17
by the likely loser. Output files of all our simulations as well as more detailed
18
summaries of our results than can be given here can be found at the following
19
URL: http://www.math.ohiou.edu/~just/Escalate/.
20
21
The results of these simulations confirm that for the particular parameter
22
settings studied, the results of Just et al. (in review) that were obtained without
23
modeling the process of information acquisition carry over to our model where the
Just & Sun: Evolution of Contest Escalation
10
1
process of information acquisition is considered. In the simulations with positive
2
probability of perception error, we observed the following outcomes: For the first
3
parameter setting that we studied (a = 0.25, b = 0.45, c = 25, d = 1, q[0] = q[1] =
4
0.12, V = 99, K = 33, L = 18, W = 25), the strategy DDDDFDRD that calls for
5
escalation only in perception state (Pl, small) was used on average by 48.1% of all
6
players throughout all monitored seasons. A mix of DDDDFDRD/DDDFFFRD
7
was used on average by 84.2% of all players. Of all the observed fights, 72.7%
8
were initiated by Pl, and 54.6% of all fights were initiated by their eventual losers.
9
For the second parameter setting that we studied (a = 0.29, b = 0.48, c = 16, d =
10
1, q[0] = q[1] = 0.033, V = 81, K = 67, L = 2, W = 25), the strategy
11
DDDDFDRD that calls for escalation only in perception state (Pl, small) was used
12
on average by 64.5% of all players throughout all monitored seasons. A mix of
13
DDDDFDRD/DDDFFFRD was used on average by 81% of all players. Of all the
14
observed fights, 86% were initiated by Pl, and 52.1% of all fights were initiated by
15
their eventual losers. These outcomes indicate a pattern similar to the ones
16
observed in the experiments of (Morris et al., 1995), and we conclude that, at least
17
for some parameter settings, our model could be a possible explanation for the
18
behavior observed in these animals. In the simulations where we set q[0] = q[1]
19
= 0 and kept the remaining parameters as in the previous experiments, the
20
percentage of fights initiated by the weaker contestant was not significantly
21
different from 50%, and no (mixed or pure) ESS appeared to evolve. This
22
confirms that the possibility of perception error is crucial to our findings.
23
Intuitively, our model predicts that the player with the larger RHP can sufficiently
Just & Sun: Evolution of Contest Escalation
11
1
often avoid a costly fight by leaving the initiative to the adversary in the hope that
2
the adversary will retreat by mistake.
3
4
However, exploratory runs for several other parameter settings did show
5
patterns that differed from the predictions in Just et al. (in review). Characterizing
6
the region of the parameter space where the results of the latter model remain valid
7
if the process of information acquisition is explicitly modeled remains an open
8
problem.
9
10
11
Acknowledgment
12
13
This work was partially supported by NSF grant DBI-9904799 to W.J.
14
15
16
References
17
18
Barlow, G. W., Rogers, W. and Fraley, N., 1986. Do Midas Cichlids win through
19
prowess or daring? It depends. Behav. Ecol. Sociobiol. 19, 1-8.
20
21
Brace, R. C., Pavey, J., 1978. Size-dependent dominance hierarchy in the
22
anemone Actinia equina. Nature 273, 752-753.
23
Just & Sun: Evolution of Contest Escalation
1
Dow, M., Ewing, A. W., Sutherland, I., 1976. Studies on the behaviour of
2
cyprinodont fish III. The temporal patterning of aggresssion in Aphyosemion
3
striatum (Boulenger). Behaviour 59, 252-268.
12
4
5
Dowds, B. M., Elwood, R. W., 1983. Shell wars: assessment strategies and
6
the timing of decisions in hermit crab shell fights. Behaviour 85, 1-24.
7
8
Enquist, M., Jackobsson, S., 1986. Decision making and assessment in the
9
fighting behaviour of Nannacara anomala (Cichlidae, Pisces). Ethology 72,
10
143-153.
11
12
Figler, M. H., Einhorn, D. M., 1983. The territorial prior residence effect in
13
convict cichlids (Chichlasoma nigrofasciatum Gunther): Temporal aspects
14
of establishment and retention and proximate mechanisms. Behaviour 85,
15
157-181.
16
17
Just, W., Morris, M. R., 2003. The Napoleon Complex: Why Smaller Males Pick
18
Fights. Evol. Ecol. 17(5), 509-522
19
20
Just, W., Morris, M. R., Sun, X., in review. The evolution of aggressive losers.
21
22
Just, W., Wu, M., Holt, J. P., 2000. How to Evolve a Napoleon Complex.
Just & Sun: Evolution of Contest Escalation
13
1
In A. Zalzala (ed.), Proceedings of 2000 Congress on Evolutionary Computation,
2
851-856. IEEE Press.
3
4
Keeley, E. R., Grant, J. W. A., 1993. Asymmetries in the expected value of
5
food do not predict the outcome of contests between convict cichlids. Anim.
6
Behav. 45, 1035-1037.
7
8
Morris, M. R., Gass, L., Ryan, M. J., 1995. Assessment and individual recognition
9
of opponents in the swordtails Xiphophorus nigrensis and X. multilineatus.
10
Behav. Ecol. Sociobiol. 37, 303-310.
11
12
Parker, G. A., 1974. Assessment strategy and evolution of fighting
13
behaviour. J. Theor. Biol. 47, 223-243.
14
15
Turner, G. F., Huntingford, F., 1986. A problem for game theory analysis:
16
assessment and intention in male mouthbrooder contests. Anim. Behav. 34, 961-
17
970.
18
19
Zack, S., 1975. A description and analysis of agonistic behaviour patterns in an
20
opisthobranch mollusk, Hermissenda crassicornis. Behaviour 5, 238-267.
21