Belief-based Learning

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1 Belief-based Learning Algorithmic Game Theory Marcello Restelli

2 Lecture Outline Introdutcion to multi-agent learning Belief-based learning Cournot adjustment Fictitious play Bayesian learning Equilibrium learners Targeted learning Learning to cooperate Evolutionary game theory

3 Cournot Adjustment The simplest approach to model other agents Assumes that other agents will repeat the same action this stage as they did the previous time BR 1 For every θ The agent generates a best response to 2 the line states this model Each agent has a prior model of the other agent θ 2 θ t = (θ t 1, θ t2 ) the BR of player 1 against it. The value for player 1 is the height at point θ 2 θ t 2 θ t+1 Can you convince yourself this point is a Nash? BR 2 θ t 1 New BR if 2 plays θ t 2 θ 1

4 Cournot Dynamics A movement between profiles such that θ t+1 = f(θ t ), f i (θ t ) = BR i (θ t -i ) A steady state is θ s so that θ s = f(θ s ) Once θ t =θ s the system remains there Claim: θ s is a Nash equilibrium Proof: by definition for every player θ s =BR(θ -i ), so players don't want to move So: Every steady state is a Nash equilibrium

5 Best Response and Adjustment Fighters and Bombers problem Suppose that the fighter believes that the bomber will look up every time We can compute the expected values E(Sun) = 0.95*1 + 1*0 = 0.95 E(Bottom) = 1*1 + 0*0 = 1 best response What happens if the bomber looks down? Fighter/Bomber Look Up Look Down Sun Attack Bottom Attack 1 0 The best response is to attack from sun Sometimes Cournot dynamics find the Nash equilibrium (PD) Other times Cournot dynamics cycles...

6 Cycles Rock-Paper-Scissors paper Prior beliefs P row (Rock)=1, P row (Paper)=0, P row (Scissors)=0 P row (Rock)=1, P row (Paper)=0, P row (Scissors)=0 Next round Row/Column Rock Paper Scissors Rock (0,0) (-1,1) (1,-1) Paper (1,-1) (0,0) (-1,1) Scissors (-1,1) (1,-1) (0,0) P row (Rock)=0, P row (Paper)=1, P row (Scissors)=0 P row (Rock)=0, P row (Paper)=1, P row (Scissors)=0 The cycle continues indefinitely

7 Fictitious Play Agents behave assuming that they are facing stationary opponents The learner explicitly mantains a belief about the opponent's strategy At each iteration Play a best response to the assessed opponent's strategy Observe the opponent's action and update beliefs The opponents' payoffs are unknown The agent knows his own payoff matrix in the stage game

8 Two-Player Fictitious Play Player i starts with a weight function Each time an opponent strategy is played, its weight is update by adding 1 The belief about player i playing s -i at time t is

9 Two-Player Fictitious Play A FP is any rule that is a best response to the estimated opponent strategy Many best responses are possible Several tie-breaking rules can be used Example: prefer pure strategies over mixed strategies Which rule is used is not much important FP is very sensitive to players' initial beliefs The prior beliefs cannot be (0,...,0) If the initial weights are positive, there is no finite sample to which the beliefs assign probability zero

10 Example: Matching Pennies As the number of rounds tends to infinity, the empirical distribution of each player will converge to (0.5,0.5)

11 Convergence: Pure Strategies Does FP converge? Steady state: a strategy profile is a steady state of FP whenever it is played in every period after some finite time T If a pure-strategy profile is a strict Nash equilibrium of a stage game, then it is a steady state of FP in the repeated game If a pure-strategy profile is a steady state of FP in the repeated game, then it is a (possibly weak) Nash equilibrium in the stage game

12 Convergence: Mixed Strategies What happens if there are no pure Nash? If all the equilibria of the stage game are mixed, FP cannot converge to a pure profile In matching pennies FP cycles and never converges to Nash, but the empirical distribution over player i's strategy converges to (1/2,1/2) So the product of the two empirical marginal distributions is the mixed-strategy equilibrium of the game If the empirical distribution of each player's strategies converges in fictitious play, then it converges to a Nash equilibrium

13 Convergence: More Results Under FP the empirical distributions converge if the stage game is: Generic payoff and 2x2 Zero sum Identical interest Solvable by iterated strict dominance When empirical distribution converge, what about the outcome? Empirical distributions need not converge

14 Example: Anti-Coordination Two pure Nash equilibria Each player earns 1 One mixed Nash equilibrium Each player earns 0.5 Using FP each player converge to the mixed strategy, but... each player earns 0

15 Example: Shapley's RPS The empirical play of this game never converges to any fixed distribution

16 Generalizations of FP Mainly generalize the update rule for the belief Adaptive forecasting E.g., exponentially weigthed If opponent follows a fixed mixed-strategy, then the assessments do not converge If β goes to 1 then the rule is asymptotically empirical, and the standard FP results hold

17 Rational Learning Also known as Bayesian learning Allows to consider a richer set of beliefs about opponents' strategies than FP Repeated-game strategies, e.g. Tit-for-Tat Each player begins with some prior beliefs Beliefs are updated using Bayesian updating

18 Example: Iterated Prisoner's Dilemma Suppose that the support of the prior belief is g 1,g 2,...,g g : trigger strategy g T : trigger strategy before T, defect later Suppose that the best response is selected from among g 0,g 1,g 2,...,g After each round, each player performs a Bayesian update after history h t Assuming that the other player has always cooperate up to t

19 Convergence Analysis Formal analysis focuses on self-play Under some conditions, self-play rational learning results in correct beliefs about opponent's strategy Under some conditions, self-play rational learning converges toward a Nash equilibrium with high probability The main condition is an absumption of absolute continuity Let X be a set and let μ, μ' ϵ Π(X) be probability distributions over X. Then the distribution μ is said to be absolutely continuous with respect to the distribution μ' iff for x ⅽ X that is measurable it is the case that if μ(x) > 0 then μ'(x) > 0

20 Belief Accuracy Player's beliefs will eventually converge if Use Bayesian updating Play best respone The play predicted by the other players' real strategies is absoutely continuous with respect to that predicted by his belief If the above conditions are met, then, for every ε and for every history, there is a time T such that for all t > T the distribution induced by the player's beliefs is ε-close to the actual strategy profile Prediction is accurate only for the on-path portion

21 Nash Equilibrium Player's strategies will eventually converge to ε-nash if Use Bayesian updating Play best respone The play predicted by the other players' real strategies is absoutely continuous with respect to that predicted by his belief The space of repeated-games equilibria is huge: which equilibrium? Optimism can lead to high rewards (Tit-for-Tat) Pessimism can lead to low rewards (defect, grim-trigger)

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