War of Attrition or the “ Waiting Game ”

1. War of Attrition or the “Waiting Game” Question: “How much time do you spend waiting for females to arrive? Answer: It depends on how long other males are waiting... Too short ... longer waiting males claims all late arriving females But if all males wait a long time, it pays to leave early and claim females arriving at new cowpats Male Dung flies guard fresh cowpats

2. Winners are decided simply by a contest involving waiting (or displaying) for different amounts of time – War of Attrition Imagine two males, A and B, that wait time XA and XB, respectively, and assume XA > XB, that is male A wins the contest Payoff to A = V – cB Payoff to B = - cB where V = value of the resource and c is the cost associated with waiting

3. There is no single optimum value when contests are settled by persistence But there is an ESS Payoff to A = V – cB Payoff to B = - cB Obviously B would have done better if XB > XA, so.... if XA = 10 sec, then XB should be 10.1 sec, but if XB = 10.1 sec, then XA should be 10.2........ Choose a random waiting time – in other words, be unpredictable

4. The Data: (1) Flies waiting times appear negatively exponential (2) Male mating success appears equal regardless of waiting time ESS: Distribution of waiting times as a function of resource value

6. Define two strategies: HAWK and DOVE HAWK – Always fight to injure or kill their opponents at a potential risk of injury to themselves DOVE – Display as if to fight, but never engage in a fight Payoffs: Winner = 50 Injury = -100 Loser = 0 Display = 10

7. Payoffs: Winner = 50 Injury = -100 Loser = 0 Display = 10 Payoff matrix: • Hawk meets Hawk • Half the encounters an individual wins • and half it loses and sustains injury Payoff = (1/2)50 + (1/2)(-100) Hawk Dove -25 Hawk Dove

8. Payoffs: Winner = 50 Injury = -100 Loser = 0 Display = 10 Payoff matrix: • Hawk meets Dove • Hawk beats Dove – Always • Dove always immediately retreat when encountering Hawk Payoff = (1)50 Hawk Dove -25 50 Hawk Dove

9. Payoffs: Winner = 50 Injury = -100 Loser = 0 Display = 10 Payoff matrix: • Dove meets Hawk • Dove loses to Hawk - Always Payoff = (1)(0) Hawk Dove -25 50 Hawk Dove 0

10. Payoffs: Winner = 50 Injury = -100 Loser = 0 Display = 10 Payoff matrix: • Dove meets Dove • There is always a display and half the • encounters an individual wins Payoff = (1/2)(50-10) + (1/2)(0-10) Hawk Dove -25 50 Hawk Dove 0 15

11. Hawk Dove What is the ESS? -25 50 Hawk h = frequency of Hawks (1-h) = frequency of Doves Mean Hawk Payoff = -25(h) + 50(1-h) Mean Dove payoff = 0(h) + 15(1-h) Dove 0 15

12. Hawk favored when rare Dove favored when rare DOVE HAWK

13. In nature, contests are much more complex than simple Hawk, Dove, Bourgeois strategies The value of conceptualizing these games is to make clear the best fighting strategy of an individual depends on the strategies of others – a Game The ESS depends on the relative pay-offs in the game (how individuals value resources and pay costs), and thus it highlights what kinds of data we need to gather • Other points theoretical treatment highlights: • Avoidance of serious fights • Respect for ownership

14. The Speckled Wood Butterfly treats patches of sunlight as territories and immediately set up shop when they come upon one If an intruder arrives it immediately retreats

16. In lions too, “ownership” of females in estrus appears to be decided on who gets there first. Fights are very rare, and a bourgeois-like strategy holds, probably b/c fights are potentially very costly and the benefits low

17. We want to continue these themes today, Specifically, (1) Individuals often differ in their fighting ability (2) Assessment of fighting ability often follows ritualized behaviors/displays (3) Differences in Resource Value (inherently or thru individual differences)

18. Red Deer (Cervus elephus) Male’s reproductive success depends on fighting ability – the strongest stags maintain harems and huge rep. success But at a cost: almost all males suffer small injuries, 20-30% become permanently injured, e.g., broken leg, blinded by antler These risks are minimized by opponents assessing each other’s fighting potential so as to attempt to fights the fights that can be won.

19. Ritualized Red Deer assessment of opponent’s strength

20. Ritualized contests of strength in Copperheads

21. Dall sheep, Buffalo, and other ungulates assess strength thru head-on clashes

22. What we can conclude from these examples and others: • Displays used in assessment appear to be reliable signals of size and strength, • e.g., vocal displays, pushing and shoving • Display also must involve some degree of cooperation – after all it should pay for • both individuals to avoid a fight if at all possible and they have a common interest in • obtaining info about each other • (3) Contests often proceed through a set pattern of assessment – i.e., ritualized

23. The Sequential Assessment Model (Enquist and Leimar) Assessment is analogous to statistical sampling...One bout contains random error and so to increase the accuracy of assessment requires repeated estimations involving contests that become increasingly dangerous When you realize your opponent’s strength exceeds yours – “Give up”

24. a - Lateral Orientation, fins erect b – Tail-beating c – Frontal Orientation d – Biting e – Mouth Wrestling f – Loser gives up and signals by changing color, fins down

25. Individuals may give-up after any stage,but as size differences decreases individual aggression escalates through the series 20 min contests between evenly matched rivals Brief matches between asymmetric rivals

26. We want to continue these themes today, Specifically, (1) Individuals often differ in their fighting ability (2) Assessment of fighting ability often follows ritualized behaviors/displays (3) Differences in Resource Value (inherently or thru individual differences)

27. When do we expect to see contests settled by arbitrary convention of ownership (?) as opposed to real differences in fighting ability? Asymmetric war of Attrition Bourgeois strategy – No specified relationship between ownership and fighting ability, its an arbitrary convention However, what if animals could control their risk along a continuum, adjusting for differences in strength and how they value the resource? If that’s the case, Bourgeois is no longer an ESS

28. Solution to an Asymmetric War of Attrition: VA VB A Gives-up and retreats when: < KA KB V = resource value (benefit) K = rate of accruing costs – correlated to fighting ability

29. Steven Austad’s work on Bowl-and-Doily Spiders (Frontinella pyramitela) Once mature, males feed very little and spend most of their time looking for females. They only live 3 days as mature males.

30. Male copulation Rapid diminishing returns Leads to Asymmetry in the value of the female to “resident” and intruder males

32. - Contests settled purely by persistence - contestants engage in conflicts at 2 levels of risk: display (low) and fight (high) - Uncorrelated asymmetry to settle contests (respect ownership) - Continuum of engagement based on asymmetry in strength • Continuum of engagement based on asymmetry in strength • and value of the resource - ESS distribution of waiting times (be unpredictable) - Mixed strategy ESS - Single strategy ESS: Bourgeois - Did not examine as a specific Game - ESS giving-up point based on cost/benefit ratio of contestants War of Attrition Hawk-Dove Hawk-Dove-Bourgeois Sequential Assessment Asymmetric War of Attrition War of Attrition Hawk-Dove Hawk-Dove-Bourgeois Sequential Assessment Asymmetric War of Attrition

34. Summary of results from Rohwer and Rohwer (1978) manipulation experiments on subordinate Harris’ Sparrows Look dominant Behave dominant Change in status Y N N Paint black Inject testosterone Both N Y N Y Y Y Cheating is prevented b/c status is ultimately checked by escalation Badges of Status The size of the black bib in male house sparrows signals dominance rank and eliminates undo fighting between males “Lieutenant” “Captain” “Private” “Sergeant”

35. Summary: • Its a Game…actually several types of games • Simple “everything else equal” Games become more complicated • No one wants to fight, but everyone wants to win…ESS will be a balance between these two