What is Evolutionary Game Theory?


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1 What is Evolutionary Game Theory? Origins: Genetics and Biology Explain strategic aspects in evolution of species due to the possibility that individual fitness may depend on population frequency Basic Entities: Genes Basic Example Genes induce characteristics (behaviors) into organisms Fitness of a gene: # of offsprings (organisms) carrying some characteristics induced by that gene Selection in presence of frequencydependent fitness: The number of actual offsprings (organisms) embodying a given gene depends on: frequency of organisms embodying that gene; frequencies of organisms embodying all other genes. Evolutionary Model as a dynamic model whose law of motion reflects three basic forces: 1. Selection: Objects with higher fitness tend to spread 2. Mutation: Introduces new objects (variety) in the system 3. Inheritance: Transfers (successful) behavior across time
2 Why biology metaphor can be applied to social sciences?  Models as As if metaphors  Economic agents as simple, myopic, entities living in complicated environments (trial & error, learning)  Agents carry a set of simple behavioral rules ( genes ) allowing them to cope with the environment  Inheritance of BR through imitation and other learning mechanisms  Social and economic institutions act as selection devices (e.g. firms in markets)  Fitness of a gene depends on: realized outcomes induced by BR (i.e. realized profits) not only on frequency of same BR in the population but also on frequencies of other BR  Selection acts directly on realized outcomes and agents (e.g. firms) and indirectly on genes (only firms carrying genes that generate successful behaviors are reproduced)  Mutations as discoveries which introduce variety and novelty in genetic pool (BR)  Biological evolution understood as cultural evolution
3 Historical Development of EGT Models 1. Original Approach: Evolutionary Stable Strategies No explicit dynamics, no gametheoretic foundations Suppose a strategy is played by all agents in the population We ask: Is it stable to invasion of any mutant strategy? If yes: It is an ESS and can withstand to the evolutionary pressure of mutation, inheritance and selection 2. Gametheoretic approach: Det. Replicator Dynamics Population of agents repeatedly playing bilateral games Each agent is hardwired to play a certain strategy In each time period there are many random bilateral meetings (and payoffs) (Alternatively: Each agent plays against the field ) The number of offsprings each agent (strategy) will have will be higher the larger his payoff (fitness) (Alternatively: Each strategy will be adopted the more its fitness is higher) No mutations and novelty: We assume that mutations are independent and occur relatively infrequently; population is sufficiently large that one can reason in terms of expected values and deterministic processes
4 3. Gametheoretic approach: Stochastic Evolution As before but now mutations are relevant and occur very often Deterministic Force: Inheritance and Selection Stochastic Force: Mutations Variety  Do deterministic predictions change? EGT, Economics and Game Theory EGT vs. Evolutionary Models (EM)  Evolutionary Models as a general framework to describe economies where rationality, selection, heterogeneity, dynamics and endogenous novelty are central  EGT as a particular instance of EM  Why EGT has been so successful in economics and why is it so strongly related to games Models are simple and analytically solvable (cf. EM) Tool for addressing deficiencies in game theory o Hyperrationality o Dynamics o Equilibrium Selection
5 The HyperRationality Problem Standard NonCooperative Game Theory Assumptions  Fully Rational Players with Unbounded Computational Skills  Perfect knowledge of their own and opponent s strategies  Well defined preferences over uncountable sets of lotteries  Perfect knowledge of other preferences  Rationality is common knowledge Example: Prisoner dilemma (1,2) C D C (3,3) (0,5) D (5,0) (1,1) Agents do not play (C,C), (C,D) or (D,C) because everyone knows that the other will deviate as they know each other to be rational. Experimental results tell us that people are not rational If EGT is able to explain social behavior, then maybe rationality is not necessary!
6 The Dynamics Problem Standard NonCooperative Game Theory Assumptions  Games are typically played once  If games are multistage or repeated, rationality assumptions prevents one from appreciate dynamics entirely (all is compressed in a unique decision) EGT is inherently dynamic  Agents are myopic and repeatedly adapt to the environment in a trial & error fashion  The state of the system today (e.g. frequency with which a certain strategy is played in the population) depends on the past states of the system
7 The Equilibrium Selection Problem What if Multiple Equilibria Arise?  Example: Coordination Game  No inbuilt explanation: Everything can happen  Preplay commitment  Some equilibrium may be more appealing than others Perfect NE: A NE which does not involve playing weakly dominated strategies ChainStore Game Subgame Perfect Equilibrium many others  There are so many refinements that in any multiple equilibrium game, each equilibrium may be justified under some refinement!!  From equilibrium selection to refinement selection!!  EGT can be a useful tool to address (and solve) equilibrium selection problems Some questions that we will ask  Are strictly/weakly dominated strategies wiped away by selection (i.e. does EG select equilibria where agents play rationally)?  Can evolutionary processes select a NE?  Do evolutionary processes select among NE? And if yes, how? Do they select efficient equilibria?  What if no equilibria exist?
8 The ChainStore Game (1,2) Acquiesces Retaliates Stays Out (0,4) (0,4) Enters (2,2) ( 4, 4) 1=Potential Entrant; 2=Incumbent Nash Equilibria: (Stays Out, Retaliates), (Enters, Acquiesces) Which one will be played? (Stays Out, Retaliates) seems implausible i. If 2 plays R then it must take into account that if 1 enters he gets 4 ii. If 2 chooses R, it must be because he knows that the threat of retaliation will induce 1 to stay out iii. However, 1 may think that the threat is not credible, because 2 would choose A if he enters If fact, (Stays Out, Retaliates) is not a Perfect NE because involves playing weakly dominated strategies: for 2 strategy A does strictly better than R if 1 enters (2 4) and does at least as well as R if 1 stays out (4 4). Therefore one should eliminate R and concentrate on the game (1,2) Acquiesces Stays Out (0,4) Enters (2,2)
9 Two Important Issues 1. Selection, Rationality and Equilibrium Perfect rationality is not a widespread property of realworld agents (and often cannot be!) As if hypothesis: Equilibrium does not appear because agents are rational, but agents appear rational because only rationality can survive in equilibrium Agents can be assumed to behave as if they were rational But a behavior can be selected in equilibrium only if: (i) (ii) it is feasible and it is present from the start in the head of the agents the environment does not change during selection process. Both conditions are hardly met in reality! 2. Is EGT a Theory of Learning? Players are either hardwired or adaptive EGT is not a model of individual learning Agents do not start from some model of reality and they refine it across time EGT is a populationlevel learning theory: It is the population that learns over time
10 Important Concepts in GameTheory Focus: Noncooperative, finite, games in normal form Set of players: I = {1, 2 } Set of pure strategies: S i = { s 1, s 2,, s k }, S = i S i = S 1 S 2 Payoff Matrices (G is k x k) Mixed Strategies 1. Symmetric Games: G (i) = {g (i) hl}, i=1,2, G (2) = [G (1) ] T 2. Asymmetric Games: G (i) = {g (i) hl},  Probability Distribution over S i : x i = { x i1, x i2,, x ik } k  Subjective vs. frequency interpretation  Pure strategies = vertices of k  Population Profile: x =(x 1,, x n )  Expected payoffs u i when the profile is x Example: Symmetric Game with k=2 strategies Player 1: x 1 = (x 11, x 12 )=(x 11, 1 x 11 ) Player 2: x 2 = (x 21, x 22 )=(x 21, 1 x 21 ) u i (x) = x i G (i) x j, i j  Expected payoffs u i when i plays a pure strategy h and j i plays some other strategy z j u i ( e h i, z j ) = e h i G (i) z j = G (h) (i) z j
11 Weak and strict dominance (for some agent i)  x i WD y i u i ( x i, z ) u i ( y i, z ) all z, > for some z  x i SD y i u i ( x i, z ) > u i ( y i, z ) all z Example for pure strategies (symmetric PD game) (1,2) C D C (3,3) (0,5) D (5,0) (1,1) Iteratively Strictly Dominated Strategies and Strictly Dominance Solvable games  SD pure strategies are never played by rational players (and thus may be eliminated)  Requirement: Everybody knows about each other payoffs (so that they can eliminate each other s SD strategies)  A pure strategy is not iteratively strictly dominated if it is not SD in any reduced game (1) A B C A B C NB: Symmetry!
12 BestReplies  Pure: β i (y) = {h S i : u i ( e h i, y) u i ( e l i, y), all l S i }  Mixed: α i (y) = {x i : u i (x i, y ) u i (z i, y ), all z i } Nash Equilibrium  A mixed strategy profile x is a NE iff x α(x) = i α i (x) 2 players, symm: x is a NE x G x y G x for all y A pure strategy profile (s 1,, s n ) is a NE iff any player will not deviate unilaterally  A NE is Strict iff α(x) = { x }, i.e. there are no alternative BR. A strict NE cannot involve mixed strategies (by convexity)!  Let Θ NE the set of all NE of the game; for any finite game there is at least one NE in mixed strategies, but not necessarily a pure strategy one!  Example. No pure NE, unique mixed NE: (½,½), (½,½) (1,2) A B A (1,1) (1,1) B (1,1) (1,1)  NE is invariant to positive affine transformations of all payoffs and to additions of some constant number to all payoffs to a given player associated to a given pure strategy played by the opponent (i.e. adding a number to a given column)  A NE cannot be SD but it can be WD (see EntryDeterrence)!!
13 NE Refinements: TremblingHand Perfection  A (TremblingHand) Perfect NE is a NE that is robust to trembles or mistakes in strategy plays  Consider for each player i some numbers µ i = (µ i1,, µ ik ) where µ ih (0,1) is the probability with which i plays h by mistake  Consider the perturbed game associated to the original game and compute the new set of NE: Θ NE (µ)  Then x is a PNE iff, for some sequence µ 0, there exists a sequence of x(µ) Θ NE (µ) such that x(µ) x  Important If x is a PNE then it is not WD In a 2player game, any NE which is not WD is a PNE Hence in a 2player game, x is a PNE iff it is not WD
14 Two Player Symmetric Games  I = {1,2}, S 1 = S 2  Payoff matrices are s.t. G (1) = (G (2) ) T  Example: Two Strategies, S 1 = S 2 = {1,2} G (1) 1 2 G (2) g' 11 g' 12 1 g" 11 g" 12 2 g' 21 g' 22 2 g" 21 g" 22 g' 11 = g" 11, g' 22 = g" 22, g' 12 = g" 21 and g' 21 = g" 12 PD 1 2 MP 1 2 ED (4,4) (0,5) 1 (1,1) (1,1) 1 (2,2) (0,0) 2 (5,0) (3,3) 2 (1,1) (1,1) 2 (1,4) (1,4) Two Player DoublySymmetric Games  TwoPlayer Symmetric Game where G (i) = [G (i) ] T, i =1,2  Example: Two Strategies g' 12 = g' 21, g 12 = g (2,2) (0,0) 2 (0,0) (1,1) 2 player Symmetric NE: Is a NE (x,y) where x=y i.e. where both players use the same (mixed or pure) strategy
15 Exercises Find NE in mixed and pure strategies for the games: 1. Prisoner Dilemma Game (Weibull, Ex 1.1, p.2) 2. Coordination Game (Weibull, Ex 1.10, p.26) 3. HawkDove Game (Weibull, Ex 1.11, p.27) 4. RockScissorPaper Game (Weibull, Ex 1.12, p.28) Recalling the invariance results about NE, reelaborate in all details the classification in Weibull, par , p Find for any of the 4 categories the NE of the game. 2. Find why games 1,2 and 3 of the exercise above belong to one of the 4 categories 3. Which is the difference between Pareto efficient and risk dominant equilibria in a coordination game (p.31)?
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