Lecture #20 (Chapter 19: Game theory) Chapter 20: Honest signaling Chapter 21: Conflict resolution 4/19/11

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Combinations of game criteria Strategies Discreet Continuous Game Symmetric Asymmetric # opponents Contest Scramble Interactions Independent Dependent

Hawk vs Dove Four possible combinations Hawk Dove (V-D) / 2 V V/2 Opposing individual Hawk Dove (V-D) / 2 V V/2 V is what victor wins; D is cost of defeat (injury) Focal individual

So compare payoffs and put dot where greatest Two choices if V > D then acting like a hawk is ESS V = what victor gets D = costs of defeat So as what you win gets small compared to the cost of fighting, you shouldn’t risk a fight. So often beneficial not to fight and using signals to avoid this fight might be favored

So compare payoffs and put dot where greatest if V < D then ESS is stable mixture with fraction f=V/D hawks and 1-V/D doves As V gets smaller, more animals should settle things peacefully So as what you win gets small compared to the cost of fighting, you shouldn’t risk a fight. So often beneficial not to fight and using signals to avoid this fight might be favored

Let’s try this : V=10 D = 2 Hawk Dove (V-D)/2 = 4 V = 10 V/2 = 5 Your opponent Hawk Dove (V-D)/2 = 4 V = 10 V/2 = 5 You

Average payoffs V = 10 D = 2 # hawks Win +10 Lose -2 Average hawk # doves Average dove 2 10 24 12 5 9 1 8.8 14 4 2.9

Payoff varies with % hawks V = 10 D = 2

Average payoffs V = 10 D = 20 # hawks Win +10 Lose -20 Average hawk # doves Average dove 2 10 24 12 5 9 1 7 14 4 2.9 -5

Payoff varies V = 10 D = 20 Note: nothing has changed for the doves. The only difference is now the hawks are paying more penalties for fighting. So it becomes advantageous to be a dove if there are too many hawks. Balance point when dove and hawks get same payoff at 50% hawks 50% doves in this case.

Discreet symmetric contests: Take games Demonstrates impact of fact that most animals are selfish Two strategies: passive vs cheat Eg. Gulls vs terns fishing Payoffs If passive, minds own business and get own fish P If cheat, steal B fish from others but gives up some of fishing time so only gets P -C fish itself

Take game Note this lowers fitness of entire population which seems counterintuitive. Passive vs cheat will only get P-B fish each day. Cheat vs passive will get P-C fish plus B that it steals. Cheat vs cheat: each gets P-C fish 1/2 time get B from other cheat but 1/2 time lose B to other so P-C in end.

Take game If B < C it is best to be passive Note this lowers fitness of entire population which seems counterintuitive. Passive vs cheat will only get P-B fish each day. Cheat vs passive will get P-C fish plus B that it steals. Cheat vs cheat: each gets P-C fish 1/2 time get B from other cheat but 1/2 time lose B to other so P-C in end.

Take game If B > C, then Cheat is an ESS Note this lowers fitness of entire population which seems counterintuitive. Passive vs cheat will only get P-B fish each day. Cheat vs passive will get P-C fish plus B that it steals. Cheat vs cheat: each gets P-C fish 1/2 time get B from other cheat but 1/2 time lose B to other so P-C in end. If B > C, then Cheat is an ESS

Selfish herd of gazelles Cheetah will capture closest gazelle. Gazelles on edge of herd most at risk. Passive animals will just graze. If all passive, will graze and get P. Cheats will move to interior of herd. This movement costs C so get P-C. All animals will take on cheat strategy so fitness of herd goes from P to P-C.

Give game Passives - mind own business, payoff P Donors - give B to others at cost C

Give game - payoffs 1/2 time donor gives b to passives 1/2 time donor give B to donors At a cost C or k

Give game One ESS 2nd ESS Always be passive If B - C >b then be donor : might gain more than lose

Vampire bats Bat go out and drink blood from cows. If prevent one bat from feeding, that bad will beg and be fed by other returning bats. Not determined just by relatedness. Also determined by if bats were “friends”. If bat A feeds bat B then later on, bat B would feed bat A.

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Honesty in signaling If cheaters often win, why doesn’t everyone cheat How can receivers be more discriminating? Require honest signals - those difficult to generate Handicap principle Honest signals are a handicap, difficult to make and maintain so must be high fitness to make them (can’t be faked)

Carotenoids as honest signals So bright plumage is correlated with better provisioning of young in nest. It is also heritable from father to son. So females use it as a signal of which birds are most fit and therefore best to mate with. Fig 20.5

Badges of status - house sparrow If badge labels dominant status, can prevent fights by just attending to badge. Bad to cheat as will get in more fights which can’t win. What is missing is how this badge is made? Does higher status correlate with more testosterone which results in more melanin?? Fig 20.6

Ch. 21: Conflict resolution Two individuals both want a resource Need to assess each other’s ability Avoid fight - especially if will lose Both are senders AND receivers - need to assess other’s ability relative to their own Contests btn 2 individuals

Types of contests Simple threat display contests The Hawk - Dove - Assessor Variable length contests War of attrition Fixed sequence contests Variable sequence contests

Hawk - Dove - Assessor Hawk vs dove uses all or none response Behavior could be dependent on other player’s actions, status, or size Assessor - displays like a dove while assesses the other contestant If other is smaller - attacks (hawk) If other is bigger - retreats (dove) 3rd possible role to add to game

Assessor’s display is used as an opportunity to assess opponent Assess size, strength, fighting ability Display is inexpensive Far from opponent so no risk of injury Vocal and visual Not huge performance energy Not take too much time Display must be honest and accurate

Payoff matrix Hawk Dove Assessor (V-D)/2 V V/2 V/4 3V/4

So being an assessor is advantageous Hawk Dove Assessor (V-D)/2 V V/2 V/4 3V/4 Against dove it is better to be a hawk as get more. But against hawk or assessor it is better to be assessor as don’t pay costs of fighting

What traits can be used to represent fighting ability Body size - tough to cheat so honest Vocal calls Amplifiers Color patches / badges

Auditory vs visual indications of body size Allow different size frogs to amplex with female. But rubber band mouth shut so can’t call. Then play back either high or low pitch call. Other males attack more if high pitch to try to take female. If male is bigger, don’t attack. Fig 21.2

Amplifiers of body size Fish often use broadside display to assess each other so amplifiers that accentuate size from side are important

2. Amplifiers - traits that make it easier to assess size Colors outline and accentuate size

Color patch - great tit Fig 21.3a Patch is larger in males than females. Width of stripe is correlated with male dominance. Displayed in upright threat posture. Size hue and chroma are important in assessing threat. Why not cheat? If badge is too big, bird will have to fight more and often lose. These color patches correlated with long term fighting ability. Short term motivation is best displayed by movement, posture and coverable patches. Fig 21.3a

Variable length contests - war of attrition game Continuous contest ( not discreet) Each opponent decides ahead how much to invest in winning Time, energy spent displaying kx One with biggest investment wins V Symmetric war All players same benefit, V, and same costs kx If win 1/2 time, benefit is V/2 - kx

Symmetric war of attrition: Pick an investment at random based on probability Must decide ahead of time how much to invest and then make a sealed bid which abide by but which other opponent doesn’t know about Lower investments picked more often

Probability of investment (time), P(x) k/V K is cost per time. This is a bit counter intuitive. The higher the curve, the larger the probability that a particular time will be selected. So green curve - it is more likely will pick a short time than a long time. If k is large, then higher curve and more likely pick a shorter time.

Assymetric war of attrition Each contestant again decides duration of its fight Differs in accumulated costs with time, k And in perceived value of winning, V Animal with higher V/k should persist longer

Assymetric war of attrition Decide if winner or loser: Loser picks a time btn 0 and S Winner picks a time btn S and ∞ So winner is more likely to pick a longer time and so to win! S depends on V/k If V is bigger, longer battles If k is smaller, longer battles V = what winner wins; k = cost per time to battle

Prediction As reward, V, gets larger, duration of contests gets longer Holds true in newts, Notophthalmus viridescens

Prediction: Assymetric war of attrition As difference in body size gets smaller, roles will be more uncertain. Fights will be longer and more variable in length Bowl and doily spider seems to support this Female lives on underside of bowl. Bowl is for catching prey which she can then eat. Males come to bowl to mate with female. If both arrive, will spar with forelegs and grapple with each other till one retreats by falling off the web

Sequential assessment Probably unlikely that animals choose persistence time at very beginning and then don’t change it More likely to gather information and reassess as battle goes on Initially uncertain if could win Gather signals from opponent As signal is repeated, get more and more sure about probability of winning

Sequential assessments - probability that contestants A or B will win Initially estimate are close to 0. Then as aggressive acts are repeated n times, the two individuals each re-evaluate their likelihood of winning. When one individuals estimate crosses the giving up line, it is pretty sure it will lose and so gives up. If different in fighting ability, one will give up fairly quickly. If more similar, will take longer for one to give up. Note: fighting ability is ln(cA/cB) where c is cost that can inflict on other individual.

Compare sequential assessment with data Bowl and doily spiders: If 2 males introduced at same time, size asymmetry predicts who will win If one is present first (owner) then this precedence can compensate for some difference in size Dashed lines are probability of winning predictions of sequential assessment game theory. Solid lines show data. If introduced at same time, then size asymmetry is symmetric about 0 (equal size). If one is present first (owner) it can be 5% smaller and still win.

Fixed sequence contests Fighting behavior may pass through stages with each stage escalating Roaring - frequency tells them about age. Loudness and rate of roaring is indication of males fitness Parallel walk so can assess relative size Pushing Red deer 1) roaring 2) Parallel walk 3) Pushing contest

Jumping spider five stages Hunch display - arch abdomen and raise leg III Stave display - raise leg III and flick it at the leg of opponent Head butt Cheliceral lock (try lift other with mandibles) Grappling and biting

Using multiple behaviors in sequential assessment

Sequential display in cichlid fish Broadside display Tail beating Frontal orientation Biting Mouthwrestling Termination