The Structure of Networks

1. The Structure of Networks with emphasis on information and social networks T-214-SINE Summer 2011 Chapter 7 Ýmir Vigfússon

2. Game theory • Regular game theory • Individual players make decisions • Payoffs depend on decisions made by all • The reasoning about what other players might do happens simultaneously • Evolutionary game theory • Game theory continues to apply even if no individual is overtly reasoning or making explicit decisions • Decisions may thus not be conscious • What behavior will persist in a population?

3. Background • Evolutionary biology • The idea that an organism‘s genes largely determine its observable characteristics (fitness) in a given environment • More fit organisms will produce more offspring • This causes genes that provide greater fitness to increase their representation in the population • Natural selection

4. Evolutionary game theory • Key insight • Many behaviors involve the interaction of multiple organisms in a population • The success of an organism depends on how its behavior interacts with that of others • Can‘t measure fitness of an individual organism • So fitness must be evaluated in the context of the full population in which it lives • Analogous to game theory! • Organisms‘s genetically determined characteristics and behavior = Strategy • Fitness = Payoff • Payoff depends on strategies of organisms with which it interacts = Game matrix

5. Motivating example • Let‘s look at a species of a beetle • Each beetle‘s fitness depends on finding and processing food effectively • Mutation introduced • Beetles with mutation have larger body size • Large beetles need more food • What would we expect to happen? • Large beetles need more food • This makes them less fit for the environment • The mutation will thus die out over time • But there is more to the story...

6. Motivating example • Beetles compete with each other for food • Large beetles more effective at claiming above-average share of the food • Assume food competition is among pairs • Same sized beetles get equal shares of food • A large beetle gets the majority of food from a smaller beetle • Large beetles always experience less fitness benefit from given quantity of food • Need to maintain their expensive metabolism

7. Motivating example • The body-size game between two beetles • Something funny about this • No beetle is asking itself: “Do I want to be small or large?“ • Need to think about strategy changes that operate over longer time scales • Taking place as shifts in population under evolutionary forces

8. Evolutionary stable strategies • The concept of a Nash equilibrium doesn‘t work in this setting • Nobody is changing their personal strategy • Instead, we want an evolutionary stable strategy • A genetically determined strategy that tends to persist once it is prevalent in a population • Need to make this precise...

9. Evolutionarily stable strategies • Suppose each beatle is repeatedly paired off with other beetles at random • Population large enough so that there are no repeated interactions between two beetles • A beetle‘s fitness = average fitness from food interactions = reproductive success • More food thus means more offspring to carry genes (strategy) to the next generation • Def: • A strategy is evolutionarily stable if everyone uses it, and any small group of invaders with a different strategy will die off over multiple generations

10. Evolutionarily stable strategies • Def: More formally • Fitness of an organism in a population = expected payoff from interaction with another member of population • Strategy T invades a strategy S at level x (for small x) if: • x fraction of population uses T • 1-x fractionof population uses S • Strategy S is evolutionarily stable if there is some number y such that: • When any other strategy T invades S at any level x < y, the fitness of an organism playing S is strictly greater than the fitness of an organism playing T

11. Motivating example • Is Smallan evolutionarily stable strategy? • Let‘s use the definition • Suppose for some small number x, a 1-xfraction of population use Small and x use Large • In other words, a small invader population of Large beetles • What is the expected payoff to a Small beetle in a random interaction? • With prob. 1-x, meet another Smallbeetle for a payoff of 5 • With prob. x, meet Large beetle for a payoff of 1 • Expected payoff: 5(1-x) + 1x = 5-4x

12. Motivating example • Is Smallan evolutionarily stable strategy? • Let‘s use the definition • Suppose for some small number x, a 1-xfraction of population use Small and x use Large • In other words, a small invader population of Large beetles • What is the expected payoff to a Large beetle in a random interaction? • With prob. 1-x, meet a Smallbeetle for payoff of 8 • With prob. x, meet another Large beetle for a payoff of 3 • Expected payoff: 8(1-x) + 3x = 8-5x

13. Motivating example • Expected fitness of Large beetles is 8-5x • Expected fitness of Small beetles is 5-4x • For small enough x (and even big x), the fitness of Large beetles exceeds the fitness for Small • Thus Smallis not evolutionarily stable • What about the Largestrategy? • Assume x fraction are Small, rest Large. • Expected payoff to Large: 3(1-x) + 8x = 3+5x • Expected payoff to Small: 1(1-x) + 5x = 1+4x • Large is evolutionarily stable

14. Motivating example • Summary • A few large beetles introduced into a population consisting of small beetles • Large beetles will do really well: • They rarely meet each other • They get most of the food in most competitions • Population of small beetles cannot drive out the large ones • So Small is not evolutionarily stable

15. Motivating example • Summary • Conversely, a few small beetles will do very badly • They will lose almost every competition for food • A population of large beetles resists the invasion of small beetles • Large is thus evolutionarily stable • The structure is like prisoner‘s dilemma • Competition for food = arms race • Beetles can‘t change body sizes, but evolutionarily forces over multiple generations are achieving analogous effect

16. Motivating example • Even more striking feature! • Start from a population of small beetles • Evolution by natural selection is causing the fitness of the organisms to decrease over time • Does this contradict Darwin‘s theory? • Natural selection increases fitness in a fixed environment • Each beetle‘s environment includes all other beetles • The environment is thus changing • It is becoming increasingly more hostile for everyone • This naturally decreases the fitness of the population

17. Evolutionary arms races • Lots of examples • Height of trees follows prisoner‘s dilemma • Only applies to a particular height range • More sunlight offset by fitness downside of height • Roots of soybean plants to claim resources • Conserve vs. Explore • Hard to truly determine payoffs in real-world settings

18. Evolutionary arms races • One recent example with known payoffs • Virus populations can play an evolutionary version of prisoner‘s dilemma • Virus A • Infects bacteria • Manifactures products required for replication • Virus B • Mutated version of A • Can replicate inside bacteria, but less efficiently • Benefits from presence of A • Is B evolutionarily stable?

19. Virus game • Look at interactions between two viruses • Viruses in a pure A population do better than viruses in pure B population • But regardless of what other viruses do, higher payoff to be B • Thus B is evolutionarily stable • Even though A would have been better • Similar to the exam-presentation game

20. What happens in general? • Under what conditions is a strategy evolutionarily stable? • Need to figure out the right form of the payoff matrix • How do we write the condition of evolutionary stability in terms of these 4 variables, a,b,c,d? Organism 2 Organism 1

21. What happens in general? • Look at the definition again • Suppose again that for some small number x: • A 1-x fraction of the population uses S • An x fraction of the population uses T • What is the payoff for playing S in a random interaction in the population? • Meet another S with prob. 1-x. Payoff = a • Meet T with prob. x. Payoff = b • Expected payoff = a(1-x)+bx • Analogous for playing T • Expected payoff = c(1-x)+dx

22. What happens in general? • Therefore, S is evolutionarily stable if for all small values of x: • a(1-x)+bx > c(1-x)+dx • When x is really small (goes to 0), this is • a > c • When a=c, the left hand side is larger when • b > d • In other words • In a two-player, two-strategy symmetric game, S is evolutionarily stable pricely when either • a > c, or • a = c, and b > d

23. What happens in general? • Intuition • In order for S to be evolutionarily stable, then: • Using S against S must be at least as good as using T against S • Otherwise, an invader using T would have higher fitness than the rest of the population • If S and T are equally good responses to S • S can only be evolutionarily stable if those who play S do better against T than what those who play T do with each another • Otherwise, T players would do as well against the S part of the population as the S players

24. Relationship with Nash equilibria • Let‘s look at Nash in the symmetric game • When is (S,S) a Nash equilibrium? • S is a best response to S: a ≥ c • Compare with evolutionarily stable strategies: • (i)a > c or (ii)a = c and b > d • Very similar!

25. Relationship with Nash equilibria • We get the following conclusion • Thm: If strategy S is evolutionary stable, then (S,S) is a Nash equilibrium • Does the other direction hold? • What if a = c, and b < d? • Can we construct such an example?

26. From yesterday • Stag Hunt • If hunters work together, they can catch a stag • On their own they can each catch a hare • If one hunter tries for a stag, he gets nothing • Two equilibria, but “riskier“ to hunt stag • What if other player hunts hare? Get nothing • Similar to prisoner‘s dilemma • Must trust other person to get best outcome!

27. Counterexample • Modify the game a bit • Want: a = c, and b < d

28. Counterexample • Modify the game a bit • Want: a = c, and b < d • We‘re done!

29. Relationship with Nash equilibria • We get the following conclusion: • Thm: If strategy S is evolutionarily stable, then (S,S) is a Nash equilibrium • Does the other direction hold? • What if a = c, and b < d? • Can we construct such an example? Yes! • However! Look at a strict Nash equilibrium • The condition gives a > c • Thm: If (S,S) is a strict Nash equilibrium, then strategy S is evolutionarily stable • The equilibrium concepts refine one another

30. Summary • Nash equilibrium • Rational players choosing mutual best responses to each other‘s strategy • Great demands on the ability to choose optimally and coordinate on strategies that are best responses to each other • Evolutionarily stable strategies • No intelligence or coordination • Strategies hard-wired into players (genes) • Successful strategies produce more offspring • Yet somehow they are almost the same!

31. Mixed strategies • It may be the case that no strategy is evolutionarily stable • The Hawk-Dove game is an example • Hawk does well in all-Dove population • Dove does well in population of all Hawks • The game even has two Nash equilibria! • Yesterday we introduced mixed strategies to study this • How should we define this in our setting?

32. Mixed strategies • Suppose: • Organism 1 plays S with probability p • Plays T with probability 1-p • Organism 2 plays S with probability q • Plays T with probability 1-q • Expected payoff for organism 1 • Probability pq of (S,S) pairing, giving a • Probability p(1-q) of (S,T) pairing, giving b • Probability (1-p)qof (T,S) pairing, giving c • Probability p(1-q) of (T,T) pairing, giving d • In total: • V(p,q) = pqa+p(1-q)b+(1-p)qc+(1-p)(1-q)d

33. Mixed strategies • Fitness of an organism = expected payoff in a random interaction • More precisely: • Def: p is an evolutionary stable mixed strategy if there is a small positive number y s.t. • when any other mixed strategy q invades p at any level x<y, then • the fitness of an organism playing p is strictly greater than the fitness of an organism playing q

34. Mixed strategies • Let‘s dig into this condition • p is an evolutionarily stable mixed strategy if: • For some y and any x < y, the following holds for all mixed strategies q ≠ p: • (1-x)V(p,p) + xV(p,q) > (1-x)V(q,p) + xV(q,q) • This parallels what we saw earlier for mixed Nash equilibria • If p is an evolutionarily stable mixed strategy then V(p,p) ≥ V(q,p), • Thus p is a best response to q • So (p,p) is a mixed Nash equilibrium

35. Example: Hawk-Dove • See book

36. Interpretation • Can interpret this in two ways • All participants in the population are mixing over two possible pure strategies with given probability • Members genetically the same • Population level: 1/3 of animals hard-wired to play D and 2/3 are hard-wired to always play H