221: BioNB Lecture 3
Sept. 3, 2008
Lecture 3: Game Theory and Hamilton's Rule
Professor H. Kern Reeve
A.
Introduction
The goal of every scientific discipline is to describe, explain and predict phenomena with the aid of theoretical models that picture the essential relationships between these phenomena and their causes. These models can be verbal or mathematical. The virtue of mathematical models is that they (i) force one to be explicit about the assumptions behind the model, (ii) allow one to explore the consequences of models that are too complex or subtle for verbal intuition to be a reliable guide, and (iii) generate precise, quantitative predictions (and thus especially sensitive tests of the model). Thus it is no surprise that mathematical models have played an important role in understanding the evolution of animal behavior. Today, you will be introduced to two classes of mathematical models that have been especially useful in understanding the evolution of social interactions in animals and plants: i.e., GAME THEORY and KIN SELECTION THEORY. In game theory, organisms are pictured as players in games, with the payoffs of the game being units of individual fitness. In kin selection theory, we include the possibility that socially interacting organisms are genetic relatives, i.e., that they share gene copies through common descent.
B.
Evolutionary game theory is the mathematical apparatus for predicting
the combinations of social behaviors that should be exhibited by interacting organisms as the result of natural selection. It deals with cases in which an organism's fitness depends on the behaviors exhibited by other members of the population, i.e., cases in which fitness is frequency dependent (dependent on the frequencies of alternative behavioral strategies in the population). CLASSROOM DEMONSTRATION. Game theory generates predictions about which combinations of social behaviors will be stable over evolutionary time, i.e., which strategies will be ESS's (evolutionary stable strategies). An ESS is a strategy that cannot be invaded by an alternative mutant strategy (Note: the set of allowed alternatives must be exactly specified). Let's analyze a simple game:
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BioNB 221: Lecture 3
Sept. 3, 2008
Imagine a game in which Producers search for food, whereas Scroungers lazily wait until Producers find food and then take a piece of it. A Scrounger paired with a Producer gets a fitness payoff b > 0, where b is the fitness value of the piece that it acquires from a Producer, but a Scrounger paired with another Scrounger gets nothing, since no one looks for food! Assume that paired Producers rarely discover the same food item and that such items have fitness value a. Let the fitness cost of searching for food be equal to c. Here is the payoff matrix: Producer Producer Scrounger ac b Scrounger a- b -c 0
When is Producer an ESS? a c > b or a b > c Can Scrounger be an ESS? a - b - c < 0 or a b < c The above is a pair-wise game in which either pure Producer or pure Scrounger is always the ESS (positively frequency-dependent selection); this is not true of all pairwise games. Now, let's consider a multi-player producer/scrounger-like game: Suppose that if you raise your hand, you get ONE bonus point on the next exam. Suppose that if you DON'T raise your hand, you get a number of bonus points on the next exam equal to the number of hands that are raised! What happens?!
C.
Kin Selection
1. W.D. Hamilton's hypothesis (J. Theoret. Biol. 7: 1, 1964) states that: "The social behaviour of a species evolves so that in distinct each behaviourevoking situation the individual will seem to value his neighbour's fitness against his own according to the coefficients of relationship appropriate to that situation."
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BioNB 221: Lecture 3 2.
Sept. 3, 2008
The coefficient of relatedness -- "r" -- is the probability that the alleles in one individual are identical copies, due to common ancestry, of alleles in another individual (see your readings). There is a genetic equivalence between personally reproducing and rearing descendants ("direct" fitness) versus assisting in rearing nondescendant kin ("indirect" fitness). Hamilton's rule: (r X b) - c > 0 [ (r X benefit to the recipient) - cost to the actor > 0 } Under Hamilton's rule, would you lay down your life for your sister? In this case, c=1 and r=1/2 so b > 2 to favor such self-sacrifice (i.e., you'd have to save at least two siblings to make up for the loss)
3.
4.
Using Hamilton's Rule Suppose an actor is related by 1/2 to a potential recipient (e.g., a full sister). Will the actor be favored to help the recipient if the reproductive outputs are as below? Help occurs 4 offspring 0 offspring Help doesn't occur 1 offspring 1 offspring
Recipient Actor b = 4 1 = 3 offspring c = 1 0 = 1 offspring rb = (1/2)*3 = 1.5 > c = 1. Extensions to Hamilton's Rule
1. Altruism between generations (e.g., between parents and offspring) Offspring of your parents are just as related to you as are your own offspring (even though you are more related to yourself than to your parents), so the simple version of Hamilton's rule doesn't work in this case. Here is a formula that works in every case: rb = relatedness to recipient's offspring rs = relatedness to own offspring rbb rsc > 0 This more general form reduces to simple form rb c > 0 when altruist and beneficiary are of the same generation because rb = r/2 and rs = 1/2 to yield (r/2)b-(1/2)c > 0 which is the same as rb c > 0.
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BioNB 221: Lecture 3 2. Altruism involving multiple recipients r1b1 + r2b2 +... rsc > 0 or just
Sept. 3, 2008
" ri bi > 0
where there are n partners (including the altruist itself) and the ith recipient experiences a fitness change bi as the result of altruism. Note that this compact formula includes the -rsc term. To see this, let the altruist itself be the jth relative (itself). Since the fitness change for the altruist as the result of the altruism is -c and its relatedness to its offspring is rs, then rj = rs and bj = c. 3. Inclusive fitness and Hamilton's rule. Hamilton's rule is satisfied when the inclusive fitness for the altruist exceeds that of a non-altruist. If an altruist produces x offspring and it's partner produces y offspring, then it's inclusive fitness is x + ry. If a non-altruist produces z offspring and it's partner produces w offspring, then the nonaltruist's fitness is z + rw. Altruism spreads in a population when the inclusive fitness of the altruist is higher than that of the non-altruist, i.e., when x + ry > z + rw which is the same as r(y - w) > z - x This is the same as Hamilton's rule because y - w = b and z - x = c. Inclusive fitness can be expanded to encompass multiple recipients in the same way that we expanded Hamilton's rule to include multiple recipients. In sum, Hamilton's rule being satisfied for the evolution of one strategy over another occurs when the inclusive fitness of the favored strategy is higher than that of the alternative strategy. This applies to the evolution of any behavior, not just altruism.
n
i =1
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