Chapter 21: Conflict
Resolution
Single threat display
Why so many threat displays?
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Hawk-Dove Game: a first step
Hawk
Hawk ½(V – D)
Dove
zero
½V
2
Hawk-Dove-Assessor Game
Dove
V
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• New strategies:
– Bully
If V > D, then it always
pays to be a hawk: pure
ESS
• Escalate on first move, continue if opponent
displays, retreat if they escalate
– Retaliator
But if V < D, then no
single best strategy:
mixed ESS
• Display first, escalate if opponent does
– Prober-retaliator
• Like retaliator, but occasionally throws in an
escalation
A bit too simple, but does predict that high costs favour
ritualization and escalation more likely with low costs.
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Hawk-Dove-Assessor Game
Assessor Strategy
• New strategies:
• Assessors display first, evaluate whether
they would win or lose from information
provided by the display, and then retreat if
they assess themselves to be the loser.
– Bully
• Bluffing – doesn’t do well
– Retaliator
• Successful
– Prober-retaliator
• Successful
• Responsiveness to opponent is better than
all-or-none strategies.
– Better still, if the responsiveness includes an
estimate of chances of winning: Assessment
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Correlated asymmetry
– Assessor strategy: if larger play hawk, if smaller play
dove
– If owner and intruder are equally frequent and get equal
payoffs:
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Dove
V
V/2
3V/4
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Assessor Threat Displays
• Opponents differ in RHP
• Example: hawk - dove - assessor
Opponent:
Hawk
Assessor
(V- D)/2
Actor:
Hawk
Dove
0
Assessor V/2
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• Costless and accurate don’t usually go
together.
• What is effect of inaccuracy
– Or: how much accuracy is necessary?
• Not that much.
(V- D)/2
V/4
V/2
When there is a cost to fighting, Assessor is pure ESS
assuming that assessment is costless and accurate
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– Enough to be just better than random
• If there is a cheap unbluffable display that
provides reasonably accurate information
on, for e.g., body size, this is all you need.
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Assessor
• Over a wide range of reasonable values
for fighting costs and asymmetries,
Assessor is a pure ESS over Hawk and
Dove. D can be > or < V.
• A single threat behaviour will only be
stable when the display is constrained to
reveal the truth about an asymmetry that
determines the probability of winning.
Usually index signals.
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Calling in toads
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Variable Length Contests
• One display repeated a variable number of
times.
• Symmetric and asymmetric War of Attrition
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War of attrition - the waiting
game
War of attrition - Payoff matrix
• Assumptions
– Resource cannot be shared
– Cost of display increases with length of contest
– No information is received during contest and
opponents are symmetrical
– Winner is the contestant willing to accept the higher
cost
– The cost to both contestants equals the cost
acceptable to the loser
– The range of actions of each contestant is
continuous
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Probabilistic strategy
Play all values of x
with varying
probability – lower
values more
probable.
P(x)
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Payoff to:
Actor :
xA > xB
Opponent: x A < x B
Player A
V - kx B
- kx A
Player B
- kx B
V - kx A
No pure ESS is possible, since an opponent that displayed a little
bit longer would have higher fitness
Solution is a mixed ESS where the probability of leaving at any
time is a constant. The times an individual stays should be
distributed as a negative exponential.
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Asymmetric war-of-attrition
P(x) = k/V*exp( -kx/ V )
k/V
xi = amount of time individual i displays
k = rate at which costs are expended
V = value of resource
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• If animals experience different costs of
display or the resource differs in value to
them, the game is asymmetric
• Which player has the largest V/k will win,
but this may not be known
• This may lead to two different giving-uptime strategies
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War-of-attrition solutions
War of Attrition with Assessment
• Initial display is how contestants decide their
role, then decide on their “bid” accordingly
• Contest costly for both, so point at which V=k
will determine maximum investment
• Small asymmetry, greater uncertainty of
assessment, increased average and variance of
contest duration
• Greater resource value, greater contest duration
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Fight duration and resource value in
newts
Signals and assessment of roles
Males fight longer over
larger females. Larger
females carry more eggs
and are, therefore, more
valuable.
• Assessment must be imperfect
• Displays should not reveal intention or
motivation (only potential strength)
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Bowl & doily spiders
Bowl & doily spider
Resource value equal
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Sequential assessment
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Sequential Assessment
• Fixed “bid” aspect of WoA not very realistic
xna = 1/n Σ ?AB + zia
– What about updating estimate of chances as contest
continues?
Random error
• Assumptions
– Animals display in order to acquire information about each
other’s fighting ability and resource value
– Fights only occur when animals are closely matched
?AB = - ? BA relative fighting ability
• Predictions
Each repeated display gives one “sample” of
asymmetry. Accuracy increases with number of
samples: SDx = s /sqrt(n )
– Fight duration increases as the asymmetry in fighting ability
decreases and/or as resource value increases
– The cost of a fight increases as the asymmetry in fighting
ability decreases
– Probability of winning increases with asymmetry
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6
0.004
?±z
0.000
0.002
dnorm(1:500, 250, 50)
0.006
0.008
Sequential assessment ESS
0
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200
300
400
500
Index
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Ownership effects in spider fights
Sequential assessment
• Predictions similar to WoA
– Longer, more variable contest with smaller
asymmetry
– Short contest with large asymmetry
• But more plausible than WoA
– No “sealed bid”
– Outcome determined by relative fighting abilities
– Contestant control their investment as contest
proceeds
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Ownership effects in spider fights
Fixed sequence contests
• Sequential assessment game with several
different behaviours
– Behaviours occur in a consistent sequence
– Each is an escalation over previous
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Assessment in red deer
Jumping spider contest stages
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Sequential assessment in
phases
A brief digression
Low frequency: honest, but still maximizing their potential
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Variable sequence contests
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Why give multiple signals?
• Many different displays, no consistent order
• Each display serves a different function
• Examples: the web-building spider Agelenopsis
with 33 different behaviors used during contests,
penguins with 22 behaviours, and crickets with
12 different behaviours.
• Most behaviours are unilateral
• Behaviours not typically repeated (in a row) –
rather a chain of actions/responses
• Rapid escalation and de- escalation
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– Maybe in crickets
• Signal erosion: threat displays lose effectiveness
as the frequency of bluff increases
– Maybe some theoretical support
• Displays transmit graded information about
display intentions
– Multiple signals may lower potential risk associated
with escalation
– Maybe little blue penguins
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Little blue penguin
Cricket display costs
Different functions
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Little blue penguin display repertoire
Little blue penguin fights
• Burrow dwellers
– Similar to crickets (in signalling system):
intermediate repetoire size; little redundancy
within functional categories
• Cave dwellers
– Larger repetoire ; multiple signals within each
functional category
Some populations nest in caves, some in burrows.
• Why?
Cave dwellers use more displays, but have fewer escalated fights.
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More or fewer displays?
More or fewer displays?
• Remember asymmetric WoA
• More displays can be seen as dividing the
signal axis more and more finely
– It pays to use a conventional settlement
• If there is a cheap unbluffable display that
provides reasonably accurate information on, for
e.g., body size, this is all you need.
– No need for complex repetoire with many displays
• But also remember that not all contests are so
simply decided.
– More accurate representation of your relative
position
• The more signals, the smaller the
probability of escalated fight.
– e.g. ownership, resource value differences, etc.
contribute to more contingency in outcome
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Penguins again
What’s the point
• Cave dwellers are closer together, with
fewer intervening barriers. Burrow dwellers
more isolated.
• Frequency of conflict three times higher in
cave dwellers, but frequency of escalated
fights is lower.
– Due to more subtle and elaborate signalling
repetoire.
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• Diversity of complex behaviours
associated with aggressive contests
– What are the structures of stable resolution
mechanisms?
– How do they arise
• Must find appropriate model to understand
behavioural dynamics.
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