Behavioral Ecology P3

Foundations of Conflict and Evolutionary Game Theory

• Ubiquity of Conflict vs. Rarity of Physical Battles:

  • Animals routinely compete for essential resources including food, mates, and territories.

  • Despite intense competition, escalation to physical battles is relatively rare in nature. Conflicts are frequently settled through low-cost displays or signaling behaviors.

  • Pikas demonstrate territorial defense by guarding food piles (haypiles) against neighboring pikas inclined to steal rather than forage independently.

• Limitations of Optimality Models:

  • Simple cost-benefit optimality models fail in conflict scenarios because they evaluate decisions in a static environment.

  • Conflicts inherently involve other individuals whose dynamic and unpredictable behaviors alter the costs and benefits of a chosen action.

  • An individual's optimal decision depends directly on the dynamic strategy selected by its opponent.

• Introduction of Game Theory to Behavioral Ecology:

  • In 1982, John Maynard Smith, a professor at the University of Sussex, applied game theory—a branch of mathematics previously used in economics—to model biological conflict.

  • Contests over resources are framed as "games" where participants (players) execute specific strategies.

  • Natural selection favors players whose choices maximize evolutionary fitness (survival and reproductive success).

  • An animal should initiate or escalate a fight whenever doing so increases its overall fitness relative to alternative actions.

• Historical Foundations of Game Theory:

  • 1944: Mathematician John von Neumann and economist Oskar Morgenstern published Theory of Games and Economic Behavior, applying strategic modeling to business and economic decisions.

  • 1950: Mathematician John Nash formulated the Nash equilibrium to identify optimal outcomes in multi-player decision-making games.

  • Modern applications extend to car pricing negotiations, national debt restructuring (such as Greece's debt negotiations), corporate profit strategies, psychology, and behavioral biology.

The Hawk-Dove Game Model

• Core Assumptions of the Model:

  • Formulated by John Maynard Smith to evaluate strategic choices during resource contests.

  • Models a symmetric contest between two equally matched contestants competing for a discrete resource (e.g., a vacant territory or haypile).

  • Contestants do not know their opponent's strategy beforehand.

• Strategies Defined:

  • Hawk: An aggressive strategy. The player escalates fights for the contested resource and risks grievous injury when facing another Hawk. Hawks always defeat Doves, but win only $50\%$ of contests against other Hawks.

  • Dove: A passive or display-oriented strategy. The player does not engage in physical fighting and never suffers injury. Doves flee immediately when confronted by Hawks. When facing another Dove, they engage in low-cost display behaviors ("showing off"), winning $50\%$ of contests.

• Nomenclature Conventions:

  • The terms hawk and dove derive from political descriptions of war versus diplomacy.

  • Individual players are capitalized (e.g., Challenger, Opponent, Hawk, Dove), whereas the underlying strategies are lowercase (e.g., hawk strategy, dove strategy).

• Possible Contest Outcomes:

  1. Challenger plays hawk; Opponent plays hawk.

  2. Challenger plays hawk; Opponent plays dove.

  3. Challenger plays dove; Opponent plays hawk.

  4. Challenger plays dove; Opponent plays dove.

Payoff Matrix Structure and Mathematical Formulation

• General Average Payoff Formula:   \n  E = (P_{\text{win}} \times V) - (P_{\text{lose}} \times I) - D\n  

  • $E$ = Expected average fitness payoff.

  • $V$ = Benefit of victory (value of the disputed resource, such as a haypile).

  • $I$ = Cost of injury ($I = 0$ if risk of injury is $0\%$).

  • $D$ = Cost of display ($D = 0$ if no display occurs).

  • $P_{\text{win}}$ = Probability of winning.

  • $P_{\text{lose}}$ = Probability of losing ($P_{\text{win}} + P_{\text{lose}} = 1$).

• Baseline Arbitrary Fitness Unit Values:

  • Benefit of Loss ($L$) = $0$

  • Benefit of Victory ($V$) = $50$

  • Cost of Injury ($I$) = $100$

  • Cost of Display ($D$) = $10$

• Derivation of Payoffs for Baseline Scenarios:

  • Hawk vs. Hawk ($E_{\text{H,H}}$):     \n    E_{\text{H,H}} = \left(\frac{1}{2} \times V\right) - \left(\frac{1}{2} \times I\right) - D\n         \n    E_{\text{H,H}} = \left(\frac{1}{2} \times 50\right) - \left(\frac{1}{2} \times 100\right) - 0 = 25 - 50 = -25.0\n    

  • Hawk vs. Dove ($E_{\text{H,D}}$):     \n    E_{\text{H,D}} = (1.0 \times V) - (0 \times I) - 0\n         \n    E_{\text{H,D}} = (1.0 \times 50) - 0 - 0 = 50.0\n    

  • Dove vs. Hawk ($E_{\text{D,H}}$):     \n    E_{\text{D,H}} = (0 \times V) - (1.0 \times 0) - 0 = 0.0\n    

  • Dove vs. Dove ($E_{\text{D,D}}$):     \n    E_{\text{D,D}} = \left(\frac{1}{2} \times V\right) - (0 \times I) - D\n         \n    E_{\text{D,D}} = \left(\frac{1}{2} \times 50\right) - 0 - 10 = 25 - 10 = 15.0\n    

• Summary Payoff Matrix (Challenger Payoffs):

  • Challenger Hawk vs Opponent Hawk: $E = -25.0$

  • Challenger Hawk vs Opponent Dove: $E = 50.0$

  • Challenger Dove vs Opponent Hawk: $E = 0.0$

  • Challenger Dove vs Opponent Dove: $E = 15.0$

• Payoff Matrix Variable Dependencies:

  • Benefit of Victory ($V$) affects expected payoffs in all contests where the Challenger can win ($E_{\text{H,H}}$, $E_{\text{H,D}}$, and $E_{\text{D,D}}$).

  • Cost of Injury ($I$) affects only contests involving fights ($E_{\text{H,H}}$).

  • Cost of Display ($D$) affects only contests settled by displays ($E_{\text{D,D}}$).

Evolutionarily Stable Strategies (ESS) and Frequency-Dependent Selection

• Definition of Evolutionarily Stable Strategy (ESS):

  • An ESS is a strategy that, if adopted by all members of a population, cannot be invaded by any alternative mutant strategy through natural selection.

• Frequency-Dependent Selection Mechanics:

  • An evolutionary process where the fitness of a phenotype depends on its relative frequency within the population.

  • Under standard baseline parameters ($V=50, I=100, D=10$):

  • Hawk invading pure Dove population: A single Hawk in a population of $999$ Doves encounters only Doves. The Hawk receives a payoff of $50.0$, compared to Doves receiving $15.0$. Hawk fitness is higher, causing Hawks to rapidly increase in frequency.

  • Dove invading pure Hawk population: A single Dove in a population of $999$ Hawks encounters only Hawks. The Dove receives a payoff of $0.0$, while Hawks facing each other receive $-25.0$. Dove fitness is higher ($0.0 > -25.0$), allowing the Dove to successfully invade.

  • Rare Strategy Advantage: Under these parameters, a strategy experiences a fitness advantage whenever it is rare in the population, driving the system toward a stable equilibrium co-existence of both phenotypes.

  • Initial population counts do not alter the final equilibrium ratio of Hawks and Doves.

• Pure ESS vs. Mixed ESS:

  • Pure ESS: A single, non-invadable strategy adopted by all individuals.

  • Example: If Benefit of Victory ($V=100$), Cost of Injury ($I=50$), and Cost of Display ($D=10$):

    • $E_{\text{H,H}} = \left(\frac{1}{2} \times 100\right) - \left(\frac{1}{2} \times 50\right) = 25.0$

    • $E_{\text{D,H}} = 0.0$

    • $E_{\text{H,D}} = 100.0$

    • $E_{\text{D,D}} = 40.0$

    • Hawk strictly yields a higher payoff regardless of whether the opponent plays hawk ($25.0 > 0.0$) or dove ($100.0 > 40.0$). Thus, hawk is a pure ESS.

  • Rule: Dove can never be a pure ESS because a Hawk strategy can always invade a population composed entirely of Doves.

  • Mixed ESS: A stable equilibrium composed of two or more strategies where the average fitness of all strategies at equilibrium is equal.

  • Occurs when Cost of Injury $I$ is higher relative to Victory $V$ (e.g., $V=40, I=120, D=10, L=0$):

    • $E_{\text{H,H}} = \left(\frac{1}{2} \times 40\right) - \left(\frac{1}{2} \times 120\right) = -40.0$

    • Against Hawks, playing Dove is better ($0.0 > -40.0$).

    • Against Doves, playing Hawk is better ($40.0 > 10.0$).

  • Forms of Mixed ESS in Nature:

    1. Each individual permanently plays a distinct strategy, maintaining fixed phenotype ratios in the population at equilibrium.

    2. Every individual in the population plays both strategies conditionally or probabilistically with specific frequencies.

• Ratio of Benefit of Victory to Cost of Injury ($V/I$):

  • Hawks are most prevalent at equilibrium when the ratio of victory benefit to injury cost ($V/I$) is high.

  • Setting $I = 40$, $V = 50$, $D = 10$ results in $E_{\text{H,H}} = 5.0$, enabling Hawk to completely displace Dove.

  • Maximizing the equilibrium proportion of Doves requires a low $V/I$ ratio and high display cost $D$ (e.g., $V=20, I=120, D=40$).

Case Studies in Biological Contests

• High-Stakes Escalation: Male Elephant Seals (Mirounga angustirostris):

  • Males engage in violent physical battles to secure breeding rights over a harem of females.

  • Only dominant males breed; non-dominant males reproduce at a rate near zero.

  • Because the benefit of victory $V$ is massive, the $V/I$ ratio remains high despite significant injury risks, favoring high physical escalation (hawk strategy).

• Low-Cost Escalation: Male Toads (Bufo bufo):

  • Male toads fight frequently over female mates.

  • Toads lack deadly natural weaponry, making the cost of injury $I$ near zero.

  • Low injury costs yield a high $V/I$ ratio, leading to frequent fights because there is minimal fitness penalty for escalating.

• Kleptoparasitism in Bald Eagles (Haliaeetus leucocephalus):

  • Thousands of bald eagles congregate annually along the Chilkat River in the Alaskan panhandle to feed on chum salmon (Oncorhynchus keta) carcasses.

  • Eagles execute two distinct strategies:

  • Hunter: Searches for unclaimed salmon carcasses.

  • Pirate: Acts as a kleptoparasite, stealing salmon from other eagles.

  • Doctoral research by Andrew Hansen (University of Virginia) demonstrated:

  • Eagles hunted $42\%$ of the time and pirated $58\%$ of the time.

  • Injury rates were identical and negligible ($0/25$ for Hunters, $0/36$ for Pirates).

  • Feeding rates were nearly identical ($4.5\text{ bites/min}$ for Hunters, $4.2\text{ bites/min}$ for Pirates).

  • Payoffs for both strategies are equal at these frequencies, confirming a mixed ESS maintained by frequency-dependent selection.

• Genetic Polymorphism in Gouldian Finches (Erythrura gouldiae):

  • Displays genetically controlled head color morphs associated with behavioral strategies.

  • Black-faced morphs are relatively passive (dove-like), whereas red-faced morphs are aggressive (hawk-like).

  • Maintained in natural populations via Hawk-Dove frequency-dependent dynamics (Kokko et al., 2014).

Asymmetrical Contests and the Bourgeois Strategy

• Asymmetry in Contests:

  • Real-world animal contests are rarely symmetrical; competitors differ in resource holding power, body size, or resource valuation.

  • A territory owner typically values a territory more than an intruder due to established knowledge of escape routes, hiding spots, and cached food.

• The Hawk-Dove-Bourgeois Game:

  • Incorporates ownership asymmetry into the Hawk-Dove model through a conditional strategy known as bourgeois (named after the French term for land-owning middle class).

  • Bourgeois Rule: Play hawk if owning the territory; play dove if intruding on another's territory.

  • Prevents physical fights and injury by using territory ownership as a convention to settle disputes peacefully.

• Empirical Demonstration: Speckled Wood Butterfly (Pararge aegeria):

  • Research conducted by Nick Davies established bourgeois dynamics in male speckled wood butterflies.

  • Males defend sunlit patches on the forest floor to attract females; non-territorial males reside in the less effective tree canopy.

  • Davies' Experiments:

  • When a blue butterfly owns a sunlit territory and a yellow butterfly intrudes, the blue butterfly wins.

  • When the yellow butterfly is established as the owner and the blue butterfly intrudes, the yellow butterfly wins.

  • Alternative/Complementary Hypothesis:

  • Resident butterflies are positioned in sunny spots, making them physically warmer. Warmer flight muscles allow them to fly longer and more vigorously, giving them a physical advantage in territory defense.

Non-Confrontational Contests and the Prisoner's Dilemma

• Marauding in Satin Bowerbirds:

  • Male bowerbirds build elaborate stick structures (bowers) to attract females.

  • Males engage in indirect competition by destroying competitors' bowers (marauding).

  • Cost of Marauding: Time spent traveling to and destroying a neighbor's bower leaves a male's own bower unguarded and open to destruction or female visits while absent.

• Pruett-Jones Game Theory Model:

  • Stephen and Melinda Pruett-Jones measured male bowerbird time budgets (time at own bower, time repairing bower, travel time to neighbors).

  • Formulated a game with two strategies: Guard (G) and Maraud (M).

  • Empirical Payoff Matrix (Challenger row vs Opponent column):

  • Guard vs Guard: $2.50$

  • Guard vs Maraud: $1.24$

  • Maraud vs Guard: $2.71$

  • Maraud vs Maraud: $1.30$

• Payoff Matrix Analysis:

  • If Opponent plays Guard: Maraud yields $2.71$ vs. Guard's $2.50$ ($2.71 > 2.50$).

  • If Opponent plays Maraud: Maraud yields $1.30$ vs. Guard's $1.24$ ($1.30 > 1.24$).

  • Marauding yields a strictly higher fitness payoff regardless of the opponent's choice, making Pure Maraud the pure ESS of the game.

• The Prisoner's Dilemma Paradox:

  • If all male bowerbirds collectively agreed to guard their own bowers without marauding, every male would receive an average payoff of $2.50$.

  • In a population of all Marauders, every male receives an average payoff of only 1.30$.\n - Despite collective cooperation yielding a higher overall payoff (2.50 > 1.30), individual selection favors the selfish choice (marauding) in every contest, driving the population to a sub-optimal equilibrium.\n\n# Practice Problems and Analytical Scenarios\n\n- **Evaluating Game Theory Assumptions**:\n - *Incorrect Assumption*: "Game theory assumes that a strategy's payoff is not dependent on strategies adopted by other players."\n - *Correct Principle*: Game theory explicitly evaluates payoffs as dynamic variables dependent on the actions and relative frequencies of other players.\n\n- **Loon Territory Dispute Analysis**:\n - Loons are territorial, fish-eating birds nesting on lakes. Fights over nesting territories mirror a Hawk-Dove game.\n - **Payoff Matrix**:\n - Challenger Hawk vs Opponent Hawk: -40\n - Challenger Hawk vs Opponent Dove: 20\n - Challenger Dove vs Opponent Hawk: 0\n - Challenger Dove vs Opponent Dove: 10\n - **Strategic Decisions**:\n - When Opponent plays Hawk: Challenger should play Dove (0 > -40).\n - When Opponent plays Dove: Challenger should play Hawk (20 > 10).\n - **Environmental Shift (Increased Rainfall / Expanded Lakes)**:\n - As lake size and territory abundance increase, the fitness value V of a single territory declines.\n - *Outcome*: Doves increase in proportion because the benefit of victory V is reduced relative to injury costs.\n - **Morphological Shift (Blunted Bills)**:\n - Blunting loon bills reduces the physical danger and injury cost I of fighting.\n - *Outcome*: Loons fight more frequently (Hawks increase in proportion) because the cost of injury decreases.\n\n- **Cat-Dog Game ESS Analysis**:\n - **Payoff Matrix** (Challenger row vs Opponent column):\n - Cat vs Cat: 10\n - Cat vs Dog: 35\n - Dog vs Cat: 5\n - Dog vs Dog: 22\n - **Strategic Analysis**:\n - Against Cat opponent: Playing Cat yields 10$; playing Dog yields$5$($10 > 5).\n - Against Dog opponent: Playing Cat yields 35$; playing Dog yields$22$($35 > 22$$).

  • Conclusion: Playing Cat yields a strictly higher payoff against both strategies. The expected ESS for this game is Pure Cat.