Complex Games
Efforts to resolve questions about animal behavior, especially contest
behavior, invite collaboration between mathematicians and biologists using
analytical tools called games [1]. A game in this context is a mathematical model of
strategic interaction, which arises whenever the outcome of one
individual's actions depends on actions to be taken by others. In my
research, usually that one individual is a non-human animal, and the others
are the rest of its population. Constructing such games is the focus of my
work.
Once a game has been constructed, the goal is to find an evolutionary
stable strategy or ESSbroadly speaking, a population strategy
yielding a higher reward than any feasible alternative. Much
game-theoretic literature deals with games that are simple in the sense
that interaction is assumed to be dyadic, role asymmetry (if any) is
assumed to be static, and intrinsic variation among individuals in, e.g.,
fighting abilityaka resource holding potential or RHPis assumed
either to be absent or to have no affect on payoffs (thus, for example,
weak individuals and strong individuals would both be aggressive half the
time if the ESS called for aggressive behavior with probability 0.5). If
any of these three assumptions is relaxed, thenlargely for want of a
better phraseI say that the game is complex; and this is the sense in
which complex games are the primary focus of my research. Of course, the
phrase is most apt when all three assumptions are relaxed. Indeed
polyadicity, role-change dynamics and intrinsic variation are merely three
of many factors that make strategic interactions complex; and insofar as
these are the only complexifying factors that I tend to consider, perhaps
what I do is best described as simple complex
games!
Typically, a well designed model has an ESS that, albeit unique, varies
across ecological parameter space. Through such variation, game-theoretic
models capture differences across real populations in behavior that has
evolved by natural selection. In this regard, recent and ongoing
collaborative work
explores the following overlapping themes.
Triadic games: As noted above, much
game-theoretic literature deals only with dyadic interactions, which
preclude network effects (but are perfectly adequate for studying many
other aspects of contest behavior). Because triads are both the smallest
groups in which network effects arise and the groups beyond dyads for which
analytical models are most likely to be tractableespecially when
allowing for intrinsic variation in RHPa major thrust of recent work
has been developing triadic games, in collaboration with Tom
Sherratt. Insights gleaned from these models are summarized in [2] and [3]. For example, animals are known to gather information
through eavesdropping. We were motivated to model such behavior by an
earlier game-theoretic model, which counterintuitively predicted that
eavesdropping increases the frequency of mutually aggressive contests.
However, this conclusion was predicated on zero variance of strength.
Allowing for such variation, we showed instead that eavesdropping reduces
aggression [4]. Again, it is
known that often two animals form a pact against a third. Somewhat
counterintuitively, we predicted that the probability of such a pact is
higher when there exists a degree of antergy in combining fighting
strengthsthe effective strength of a coalition is less than the sum
of the individual strengths of the pairand is especially high when
variance in strength and reliability of strength difference as a predictor
of fight outcome are both also high [5]. Yet these conditions do appear to characterize
coalitions found in primate
societies.
Two male fiddler crabs grappling. The male on the right is
a territory owner, the male on the left a floating intruder
without a territory. |
Coalition or alliance formation: When
should an animal step in to help a neighbor having a territorial dispute
with an intruder? To address this question, we constructed a model whose
ESS depends on three size thresholds, as follows: A territory owner who is
sufficiently strong (above the first size threshold) should help a neighbor
who is sufficiently weak (below the second threshold), but otherwise not
help; and a ''floater'' seeking a territory should challenge for ownership
only if sufficiently strong (above the third threshold). Our model
[6] characterizes the dependence of these thresholds on a host of ecological
factors and identifies a range of ecological parameters where the second
threshold is always lower than the first, matching observations of fiddler
crabs, in which helpers have always been larger and hence stronger. But
the model also identifies other ranges where the second threshold is higher
than the first, so that an animal should sometimes come to the aid of a
stronger neighbor. This less intuitive prediction yields a test of the
model for future field studies.
Coalition or alliance formation theory is discussed more generally in
[7].
Inter-individual variation: Although in
nature a contested food resource is most often monopolized by a clear
winner, contests between roller beetles have been observed to result in
sharingmost commonly when opponents were approximately equal in size.
So mutual assessment has been suggested as instrumental to the outcome.
The idea is intuitively appealing. Indeed it has been widely assumed for
over 30 years that animals compare one another's fighting abilities before
contests. Yet there appears to be virtually no evidence from animal
studies to support such mutual assessment. It is also not essential. To
make this point, we constructed a game where individuals vary in size but
are unable to assess their opponent's size
[8]. Yet the model still predicts
sharing; moreover, individuals observed sharing are unlikely to be closely
matched, even though closely matched individuals are likely to share. This
result is hard to intuit without a game-theoretic
model.
The aforementioned model was one of a suite of twelve (arising from two
interaction patterns, three payoff and two information structures)
developed in
[9] to assess how
inter-individual variation in RHP affects the propensity to share or
otherwise cooperate. These models have yielded numerous insights. For
example, increasing variance of RHP can promote cooperation, inhibit it, or
have no effect (though only if costs are constant, so that the probability
of cooperation is independent of variance); somewhat counterintuitively,
precluding fights between hawks and doves can decrease the level of
cooperation; and ignorance can be more conducive to
cooperation than partial knowledge.
Victory displays: Victory
displaysperformed by the winner of a contest but not the
loserhave been reported in a variety of species, not merely in
humans. For example, the video on the right shows post-victory
stridulation by the winner of a fight between crickets (towards the
left-hand corner of the picture after the fight).
John Bower proposed two explanations for their
function. The "advertising" rationale is that victory displays are
attempts to communicate victory to other members of a social group that do
not pay attention to contests or cannot otherwise identify winners. The
"browbeating" rationale is that victory displays are attempts to decrease
the probability that the loser of a contest will initiate a future contest
with the same individual. Bower's reasoning was purely verbal. To check
it out, we constructed a pair of game-theoretic modelsdistinguished
as Model A for advertising and Model B for browbeatingto explore the
logic of each rationale. Our models showed that both rationales are
logically sound
[10]; and that
all other things being equal, victory displays are most intense through
advertising if reproductive advantage of dominance is low, but most intense
through browbeating if reproductive advantage of dominance is high.
Interestingly, two of Bower's favorite candidates for the advertising
rationale, tropical boubous and song sparrows, are socially monogamous;
whereas many species of crickets and wetasfor which Bower favored the
browbeating rationaleexhibit resource defense polygyny, with clear
potential for high reproductive skew. Thus our predictions appear to
accord quite well with current empirical
knowledge.
A more recent model
[11]
relaxes two assumptions of our earlier Model B.
Two male Jackson's chameleons (Chamaeleo jacksonii) push each other in a contest of strength. |
Self- versus mutual
assessment: Knowing one’s enemy, to paraphrase Sun-Tzu,
allows humans never to face peril in battle. Knowing one’s enemy was
once thought to be valuable in the animal kingdom as well. However, recent
studies on animal contests suggest that individuals commonly do not gather
information on their opponents. Why? A game developed in collaboration
with
Steve Heap suggests a possible rationale. This
model
[12] addresses two ways in
which individuals might make inappropriate decisions in an encounter:
Weaker fighters might persist in fights that they cannot win, and stronger
fighters might flee even if victory is assured. Individuals can, however,
reduce the likelihood of these errors by gathering information about their
opponent. This prospect leads to two trade-offs that can affect whether
individuals benefit from assessment. First, individuals that spend more
effort in assessment make better decisions in any single battle, but fight
less frequently; however, fortune can favor the bold, because individuals
that fight blindly stand to gain more contested resources over time by
fighting more often, triumphing in the long run over more prudent
strategists. Second, weaker individuals risk revealing their inferiority
by allowing reciprocal assessment to occur, and may thus prefer contests to
be settled without assessment. Analysis of these two trade-offs offers a
resolution to the paradox of animals ignoring information on their
opponents, showing in particular that strategic error is necessary for
assessment strategy to vary. In essence, there is not only physical
conflict between animals, but also an evolutionary dilemma over the value
of gathering opponent information.
A shell fight between two hermit crabs (Pagarus bernhardus) in which the defender has been pulled from its shell. |
Respect for Ownership: Respect for
property is widespread in the animal kingdom even without any third-party
enforcement [13]. Owners are
frequently left unchallenged by potential competitors, and tend to win
contests when disputes arise. Game theory shows that respect for
ownershipBourgeois behaviorcan arise as an arbitrary convention
to avoid costly disputes. However, the same theory predicts that a
paradoxical respect for lack of ownershipanti-Bourgeois
behaviorcan evolve under the same conditions, and in some cases is
the only stable outcome. Despite these predictions, anti-Bourgeois
behavior is rare in nature, whereas respect for ownership is common. A
frequently invoked resolution of this paradox is that two individuals
employing anti-Bourgeois behaviour over repeated rounds would be swapping
roles continually, a potentially inefficient outcome known as infinite
regress. Another potential resolution is that there is often confusion
over ownership in the natural world, mediated for example by the temporary
absence of a true owner. Further games, again in collaboration with Tom Sherratt, explore the validity of these verbal
rationales by explicitly modelling role-change dynamics with infinite
regress [14][15] or confusion over ownership [16][17]. What the models show is that although either infinite
regress or mistakes over ownership can facilitate the evolution of
Bourgeois-like conventions, neither seems to fully explain the extreme
rarity of anti-Bourgeois behavior in nature.
Work in progress: Ongoing collaborative
projects are a mix of revisiting previously studied themes and branching
out in new directions. For example, I have been using game-theoretic
modelling to explore territory size and shape in convict cichlids, in
collaboration with Yao Dai [18];
musth in male elephants, in collaboration with Max Wyse, Ian Hardy and Lisa Yon [19]; and volatile chemical emissions as
weapons of rearguard action, in collaboration with Yao Dai, Ian Hardy and Marlène Goubault [20].
(full text downloadable from
here in most cases)
- M. Mesterton-Gibbons and E.S. Adams. Animal contests as
evolutionary games. American
Scientist 86, 334-341. BACK
- M. Mesterton-Gibbons and T. N.
Sherratt. Animal network phenomena: insights from triadic
games. Complexity 14, 44-50. BACK
- T. N.
Sherratt and M. Mesterton-Gibbons. Models of Group or
Multi-Party Contests. In Animal Contests (I.C.W. Hardy and M. Briffa eds, Cambridge
University Press), in press. BACK
- M. Mesterton-Gibbons and T. N. Sherratt. Social eavesdropping:
a game-theoretic analysis. Bulletin of Mathematical Biology
69,
1255-1276. BACK
- M. Mesterton-Gibbons and T. N.
Sherratt. Coalition formation: a game-theoretic analysis.
Behavioral Ecology 18, 277-286. BACK
- M. Mesterton-Gibbons and T. N. Sherratt. Neighbor intervention: a game-theoretic
model. Journal of Theoretical Biology 256,
263-275. BACK
- M. Mesterton-Gibbons, S. Gavrilets, J. Gravner and
E. Akçay.
Models of coalition or alliance formation. Journal of Theoretical
Biology 274, 187-204. BACK
- M. Mesterton-Gibbons and T. N. Sherratt. Divide
and conquer: when and how should competitors share? Evolutionary
Ecology 26, 943-954. BACK
- M. Mesterton-Gibbons and T. N. Sherratt.
Information, variance and cooperation: minimal models.
Dynamic Games and Applications 1, 419-439. BACK
- M. Mesterton-Gibbons and T. N. Sherratt. Victory displays: a game-theoretic analysis. Behavioral Ecology
17, 597-605. BACK
- M. Mesterton-Gibbons and T. N. Sherratt.
Signalling victory to ensure dominance: a continuous model. Annals
of the International Society of Dynamic Games 12,
25-38. BACK
- M. Mesterton-Gibbons and S. M. Heap. Variation between self-
and mutual assessment in animal contests. American Naturalist
183, 199-213. BACK
- T. N. Sherratt and M. Mesterton-Gibbons.
The evolution of respect for property.
Journal of Evolutionary Biology 28, 1185-1202. BACK
- M. Mesterton-Gibbons and T. N. Sherratt.
Bourgeois versus anti-Bourgeois: a model of infinite regress.
Animal Behaviour 89, 171-183. BACK
- M. Mesterton-Gibbons, T. Karabiyik and T. N.
Sherratt. On the evolution of partial respect for ownership. Dynamic Games
and Applications 6, 359-395. BACK
- M. Mesterton-Gibbons, T. Karabiyik and T. N.
Sherratt. The iterated Hawk-Dove game revisited: the effect of
ownership uncertainty on Bourgeois as a pure convention. Dynamic Games
and Applications 4, 407-431. BACK
- M. Mesterton-Gibbons and T. N. Sherratt. How residency duration affects the outcome of a territorial contest:
Complementary game-theoretic models. Journal of Theoretical
Biology 394,
137-148. BACK
- M. Mesterton-Gibbons and Y. Dai.
An effect of landmarks on territory shape in a convict cichlid.
Bulletin of Mathematical Biology 77, 2366-2378. BACK
- J. M. Wyse, I.C.W. Hardy, L. Yon and M.
Mesterton-Gibbons. The impact of competition on elephant musth
strategies: a game theoretic model. Journal of Theoretical
Biology 417, 109-130. BACK
- M. Mesterton-Gibbons, Y. Dai, I.C.W. Hardy and M. Goubault. Volatile chemical emission
as a weapon of rearguard action: A game-theoretic model of contest
behavior. Click here for unpublished manuscript. BACK
Last updated March 16, 2017.