News and Notes 4/13 HW 3 due now HW 4 distributed today, due Thu 4/22 Final exam is Mon May 3 11 AM Levine 101 Today: –intro to evolutionary game theory –demonstration of EGT applet –intro to strategy language Thursday: class exercise/competition

Evolutionary Game Theory Networked Life CSE 112 Spring 2004 Prof. Michael Kearns

Game Theory/Econ Review A mathematical theory designed to model: –how rational individuals should behave –when individual outcomes are determined by collective behavior –strategic behavior –rational usually means selfish What we have examined: –basic theory and equilibrium concepts (Nash, correlated, market) –repeated games, learning in games –behavioral modifications (e.g. inequality aversion) –game and economics on networks interdependent security on the air carrier transfer network our in-class market economy experiment Notions of rationality and measures of performance were at the level of the individuals –individual wealth, individual adaptation,… –forces of change act on individual players

Evolution Without Biology Darwinian dynamics: –large population of individuals –interacting with each other and their environment –given population and environment, an abstract measure of “fitness” fitness is a property of individuals measures how well-suited the individual is to current conditions e.g. it’s good to be furry in cold weather it’s good to be a hawk if there are lots of doves around –assume individuals replicate according to their fitness –can also incorporate processes of mutation, crossover, etc. –so in cold climates “next generation” of population will be furrier So “unfit” properties or strategies can die out over time Note: can also apply to many non-biological settings –e.g. to the survival/propagation of ideas (mimetics) –it’s good to be a simple, comforting idea in troubled times –to the development of technology, companies,… –thefacebook = Friendster + mutation? How can we marry this broad framework with game theory?

Hawks and Doves Two parties confront over a resource of value V May simply display aggression, or actually have a fight Cost of losing a fight: C > V Assume parties are equally likely to win or lose There are three Nash equilibria: –(hawk, dove), (dove, hawk) and (V/C hawk, V/C hawk) Alternative interpretation for C >> V: –the Kansas Cornfield Intersection game (a.k.a. Chicken) –hawk = speed through intersection, dove = yield hawkdove hawk(V-C)/2, (V-C)/2V, 0 dove0, VV/2, V/2

Evolutionary Hawks and Doves Now imagine an infinite population of H&D players Each individual playing some (mixed or pure) strategy –think of an individual’s strategy as their “genetic material” Assume random but pairwise conflicts between individuals –choose individuals x and y uniformly at random from population –x is playing Pr[hawk] = p_x, y is playing Pr[hawk] = p_y –each will have some (expected) payoff in this conflict Natural notion of an individual’s fitness: –view population average as the opponent –i.e. x’s fitness f_x is the expected payoff against a randomly chosen y –x’s fitness dependent on p_x and current population Evolutionary dynamics: –for each x, create a number of copies n_x of x in next generation –n_x proportional to fitness of x –e.g. let fraction of x in next generation be ~ f_x/(  f_x) Key questions: –will population converge to stable profile of strategies? (e.g 1/3 H, 2/3D) –if so, to which one?

Evolutionary Stable Strategies An evolutionary notion of equilibrium Want to capture idea that population profile of strategies is stable Let p be the probability that a randomly chosen player plays hawk: –draw individual x at random  Pr[hawk] = p_x –flip coin with bias p_x; –over draw of both x and coin flip, what’s the probability of hawk? Now suppose a small fraction e of “mutants” is injected –mutants all play hawk with probability q <> p New population probability of playing hawk: –p’ = (1-e )p + eq We call p an ESS if for any q, the population Pr[hawk] will return to p, as long as the invasion fraction e is small enough –so the mutants will die out under evolutionary dynamics Hawks and Doves: –fraction V/C of pure hawks, 1 – V/C of pure doves is ESS

Formal ESS Conditions Let E(p,q) be expected payoff of playing strategy p against opponent q (<> E(q,p)) If p is an ESS, then for any q, must have: –(1-e)E(p,p) + eE(p,q) > (1-e)E(q,p) + eE(q,q) as e  0 –think of the q’s as a very small population of invading mutants There are two ways this condition can be satisfied: –if E(p,p) > E(q,p), then we can make e small enough to satisfy > above –else, if E(p,p) = E(q,p), we need E(p,q) > E(q,q) –if E(p,p) < E(q,p) then p cannot be an ESS

Remarks on ESS and EGT Individuals are not “rational” per se If their fitness is better than others they will propagate But at ESS, they will be playing optimally against the current population Special case of Nash equilibrium for 2-player games –demands symmetry --- both players playing the same strategy at equilibrium –may not exist! EGT a serious branch of biology –many field studies –has been applied to explain many behavioral phenomena in animals male fly waiting times at cowpats gender ratios

The EGT Applet Many thanks to Ben Packer and Nick Montfort –applet installation, NW version, strategy language, cool examples,… Applet simulates EGT dynamics for repeated games Repeated game strategies are history-dependent Can (and will) program our own strategies (Thursday!) Basic usage: –choose game matrix and collection of strategies in initial population –choose number of rounds in each repeated-game meeting –applet shows “tournament score” for each strategy overall fitness against an population evenly divided among strategies of course, highest tournament score is no guarantee of survival! because the population fractions will change with time –choose initial population fractions for each strategy –choose network structure which strategies compete with which others –applet simulates evolution dynamics shows fraction of each strategy in population over time Now let’s go to the appletapplet

Thursday’s Class We will divide into a few groups Participation required, will have lists for each group Each group will discuss and develop a strategy –game will be Prisoner’s Dilemma –strategies should be programmable in STRANGE keep them simple and short –can be history-dependent –Nick, Kilian, MK will go around to help with coding We will then run the EGT applet –equal numbers of each strategy in initial population Will repeat several times –including with network structure Come armed with ideas –drive your fellow classmates to extinction!