Beyond Nash Equilibrium - Correlated Equilibrium and Evolutionary Equilibrium
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Beyond Nash Equilibrium - Correlated Equilibrium and Evolutionary Equilibrium

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Beyond Nash Equilibrium - Correlated Equilibrium and Evolutionary Equilibrium Beyond Nash Equilibrium - Correlated Equilibrium and Evolutionary Equilibrium Presentation Transcript

  • Beyond Nash Equilibrium Correlated Equilibrium and Evolutionary Equilibrium Jie Bao 2003-05-05 Iowa State University
  • Equilibrium in Games
    • Pure strategy Nash equilibrium
    • Mixed strategy Nash equilibrium
    • Correlated equilibrium
    • Evolutionary equilibrium
    • Bayesian Nash equilibrium
    • NE is too strict on what is “ration”…
  • Correlated Equilibrium
    • Aumann 1974
    • A generalization of “rational” solution
    • In a CE
      • the action played by any player is a best response (in the expected payoff sense) to the conditional distribution over the other players given that action,
      • and thus no player has a unilateral incentive to deviate from playing their role in the CE.
    • Example: Traffic Signal- a single bit of shares information allows a fair split of waiting times.
      • “ running a light” can’t bring greater expected payoff
      • The actions of players are “correlated”
    View slide
  • Example : BoS
    • Bach or Stravinsky Game (or Battle of Sex)
    • NE: strategy profile -> payoff profile
      • (Bach, Stravinsky) ->(2 ,1)
      • (Stravinsky, Bach) ->(1 ,2)
      • (1/3 Bach, 1/3 Bach) -> (2/3, 2/3)
    • Another equilibrium: the player observe the outcome of a public coin toss, which determines which of the two pure strategy Nash equilibria they play.->(3/2,3/2)
    ? 1,2 1,0 Stravinsky 0,1 2,1 Bach Stravinsky Bach View slide
  • BoS: Mixed Strategy   (b) Agent 2’s expected utility, blue line is it’s best response to given p (agent 1’s strategy) (d) Look from the top, (1/3,1/3) is the only mixed NE. pure NEs include (0,1) and (1,0) (a) Agent 1’s expected utility, red line is it’s best response to given q (agent 2’s strategy) (c) Overlap of a and b
  • BoS: CE - Toss Coin Equ.
    • <N,(A i ),(u i )>= <{1,2} ,{Bach , Stravinsky}, payoff matrix U>
    • Probability space ( Ω,π )
      • Ω = {x,y}
      • π(x) = π(y) = ½
    • information partition of each agent
      • P1 = P2 = {{x},{y}}
    • For each player {1,2} action function σ i : Ω->A i ,
      • σ 1 (x) = σ 2 (x) = Bach
      • σ 1 (y) = σ 2 (y) = Stravinsky
    • Payoff profile: ½(B,B), ½(S,S) -> (3/2, 3/2)
      • Compared with (for agent 1):
      • ½(S,B), ½(S,S) ->1/2
      • ½(B,B), ½(B,S) ->1
      • ½(S,B), ½(B,S) ->0
  • CE: formal definition
    • A strategic game <N,(A i ),(u i )>
    • A finite probability space ( Ω,π )
      • Ωis a set of states and π is a probability measure on π
    • For each player i∈N, a partition P i of Ω(player i’s information partition)
    • For each player i∈N, a function σ i : Ω->A i , with σ i (w)= σ i (w’) whenever w∈ P i , and w’∈ P i for some P∈ P i , (σ i is player i’s strategy)
      • such that for every I∈N and every function τ i : Ω->A i for which τ i (w)=τ i (w’) whenever w∈ P i , and w’∈ P i for some P∈ P i (i.e. for every strategy of player i) we have
    • Note that we assume the players share a common belief about the probabilities with which the states occur.
  • CE contains Mixed NE
    • For every mixed strategy Nash equilibriumαof a finite game <N,(A i ),(u i )>, there is a correlated equilibrium <( Ω,π ), P i ,σ i > in which for each player i∈N, the distribution on A i induced by σ i is α i .
    • Construct CE from Mixed NE
      • Ω= A=(X j ∈N A j ) – strategy profiles
      • π(α)=Π j ∈N α j (a j ) – prob. Of the strategy profiles
      • P i (b i )={a ∈A: a i = b i }, P i consist of the |A i | sets P i (b i )
      • σ i (a)=a i
    • CE is a more general concept than Mixed NE and Pure NE
    NE CE Mixed NE
  • Convex Combination of CE->CE
    • Let G=<N,(A i ),(u i )> be a strategic game. Any convex combination of correlated equilibrium payoff of G is a correlated equilibrium payoff of G
    • Interpret: first a public random device determines which of the K correlated equilibria is to played, and then the random variable corresponding to the k th correlated equilibrium is realized.
    • CE: <(Ω k ,π k ), P i k ,σ i k >
    • CE payoff profiles: U 1 ,.. U k
    • c 1 ,…c k , all c i >=0, and Σ c i =1
    • Construct a new CE
      • Ω = union of all Ω k
      • π(w) = c k π k (w) , if w in Ω k
      • P i =union (on k) of P i k
      • σ i (w) = σ i k (w)
    • Payoff profiles Σ c k U k
  • Example
    • Pure NE payoff profile: (7,2) (2,7)
    • Mixed NE payoff profile: (4 2 / 3 ,4 2 / 3 )
    • CE: Ω={x,y,z}, π(x)= π(y)= π(z)=1/3, P1={{x},{y,z}}, P2={{x,y},{z}}, σ 1 (x)=B, σ 1 (y)= σ 1 (z)=T, σ 2 (x)= σ 2 (y)=L, σ 2 (z)=R, -> (5,5)
    • The CE is outside the convex hull of Pure / Mixed NE payoff profiles
    0,0 7,2 B 2,7 6,6 T R L 1 2 - x B z y T R L 1 2
  • States and outcomes in CE
    • Let G=<N,(A i ),(u i )> be a strategic game. Every probability distribution over outcomes that can be obtained in a CE of G can be obtained in a CE’ in which
      • the set of state Ω’ is A and
      • for each i∈N, player i’s information partition P i ’ consists of all sets of the form {a∈A i : a i =b i } for some action b i ∈A i
      • [π’(a) = π k ({w ∈ Ω: σ(w)=a}) σ i ’ (a) = σ i k (a i )]
    • This theorem allows us to confine attention to equilibria in which the set states is the set of outcomes
  • Notes about CE
    • If players hold different beliefs, additional equilibrium payoff profiles are possible.
    • Nash equilibrium is a special case of CE in which we demand that πbe a product distribution for some distribution π i , so every player acts independently of all others
    • It’s possible to compute CE via linear programming in polynomial time, while NE is exponentially complex!
  • Evolutionary Equilibrium
    • ESS: Maynard Smith & Price, 1972
    • A steady state in which all organism take this action and not mutant can invade the population .
    • Example: the sex ratio in bee population is 1(male):3(female)
    Selfish Selfish Selfish Selfless Selfless Selfless Selfless Selfless Selfless Selfish
  • ESS: Definition:
    • Let G=<{1,2},(B,B),(u i )>be a symmetric strategic game, where u 1 (a,b)=u 2 (b,a)=u(a,b) for some function u.
    • An evolutionarily stable strategy(ESS) of G is an action
      • b*∈B for which (b*,b*) is a NE of G
      • and u(b,b)<u(b*,b) for every best response b∈B to b* with b≠b*.
  • Example: Hawk--Dove
    • Choose to be Hawk or dove?
    • Pure NE: (D,H) & (H,D)
    • Mixed NE: (0.5D/0.5H, 0.5D/05H)
    • If the players have the freedom to choose to be hawk or dove in a repeated game, and utility will be used to reproduce their offspring, what’ll be the optimal strategy?
    (1-c/)2, (1-c/)2 1,0 H 0,1 ½, ½ D H D
  • HD Game – mixed strategy (a) Agent 1’s expected utility, red line is it’s best response to given q (agent 2’s strategy) (b) Agent 2’s expected utility, blue line is it’s best response to given p (agent 1’s strategy) (c) Overlap of a and b (d) Look from the top, (0.5,0.5) is the only mixed NE. pure NEs include (0,1) and (1,0) With c = 2, Action = {'Hawk' , 'Dove'} ;
  • HD Game – ESS invading
    • In all those games, ESS (half-dove-half-hawk mixed strategy (0.5,0.5)) starts from a percentage of 0.1 in the population
    • Population: 200, Game Round = 1000
    • Reproduce: proportional to total utility of each type
    • Note that Dove is not completely eliminated
    ESS vs. Dove ESS vs. Hawk ESS vs. 1/4 Hawk ESS vs. 1/4 Dove
  • HD Game – ESS being invaded
    • In all those games, ESS (half-dove-half-hawk mixed strategy (0.5,0.5)) starts from a percentage of 0.9 in the population
    • Setting is same to the last slide
    • ESS can successfully defend the invasion of mutants, although it may not be a completely expelling
    • The experiment shows that ESS can be taken to be the set of mixed strategy over some finite set of actions
    ESS vs. Dove ESS vs. Hawk ESS vs. 1/4 Hawk ESS vs. 1/4 Dove
  • Not all NE are ESS
    • A strict equilibrium b* is an ESS
      • (b*,b*) if a symmetric NE
      • and no strategy other than b* is a best response to b*
    • A nonstrict equilibrium may not be an ESS
    • Mixed NE (1/3, 1/3, 1/3) expected payoff t/3 Can be invaded by any pure strategy
      • Receives expected utility t/3 when it encounters MixedNE
      • Receives expected t when it encounters another pure strategy
    Example: ESS Mixed NE t,t -1,1 1,-1 1,-1 t,t -1,1 -1,1 1,-1 t,t
  • More about ESS
    • Widely used in sociobiology
      • See Dawkins <selfish gene>, chapter 6
      • Wilson , < sociobiology – New Synthesis>, chapter 5
    • And in politic science and sociology
      • See <the evolution of cooperation>, where tit-for-tat is a ESS in Evolutionary Pensioner Dilemma Game