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

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

5. 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

6. 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

7. 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

8. 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

9. 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

10. Payoff varies with % hawks V = 10 D = 2

11. 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

12. 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.

13. 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

14. 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.

15. 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.

16. 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

17. 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.

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

19. 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

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

21. 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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23. 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)

24. 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

25. 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

26. 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

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

28. 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

29. 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

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

31. 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

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

33. 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

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

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

36. 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

37. 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

38. 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

39. 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.

40. 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

41. 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

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

43. 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

44. 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

45. 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.

46. 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.

47. 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

48. 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

49. Using multiple behaviors in sequential assessment

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