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No Regret Algorithms in Games Georgios Piliouras Georgia Institute of Technology John Hopkins University

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No Regret Algorithms in Games Georgios Piliouras Georgia Institute of Technology John Hopkins University

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No Regret Algorithms in Games Georgios Piliouras Georgia Institute of Technology John Hopkins University

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No Regret Algorithms in Social Interactions Georgios Piliouras Georgia Institute of Technology John Hopkins University

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No Regret Algorithms in Social Interactions Georgios Piliouras Georgia Institute of Technology John Hopkins University

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No Regret Behavior in Social Interactions Georgios Piliouras Georgia Institute of Technology John Hopkins University

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No Regret Behavior in Social Interactions Georgios Piliouras Georgia Institute of Technology John Hopkins University

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“Reasonable” Behavior in Social Interactions Georgios Piliouras Georgia Institute of Technology John Hopkins University

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No Regret Learning Regret(T) in a history of T periods: total profit of algorithm total profit of best fixed action in hindsight - An algorithm is characterized as “no regret” if for every input sequence the regret grows sublinearly in T. [Blackwell 56], [Hannan 57], [Fundberg, Levine 94],… 10 01 (review) No single action significantly outperforms the dynamic.

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10 01 Weather Profit Algorithm 3 Umbrella 3 Sunscreen 1 No Regret Learning (review)

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Games (i.e. Social Interactions) Interacting entities Pursuing their own goals Lack of centralized control

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Games n players Set of strategies S i for each player i Possible states (strategy profiles) S= × S i Utility u i :S→ R Social Welfare Q:S→ R Extend to allow probabilities Δ(S i ), Δ(S) u i (Δ(S))=E(u i (S)) Q(Δ(S))=E(Q(S)) (review)

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Games & Equilibria 0, 0-1, 11, -1 0, 0-1, 1 1, -10, 0 Rock Paper Scissors Nash: A product of mixed strategies s.t. no player has a profitable deviating strategy. 1/3

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0, 0-1, 11, -1 0, 0-1, 1 1, -10, 0 Nash: A probability distribution over outcomes, that is a product of mixed strategies s.t. no player has a profitable deviating strategy. Choose any of the green outcomes uniformly (prob. 1/9) Rock PaperScissors Rock Paper Scissors 1/3 Games & Equilibria

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0, 0-1, 11, -1 0, 0-1, 1 1, -10, 0 Nash: A probability distribution over outcomes, s.t. no player has a profitable deviating strategy. Rock PaperScissors Rock Paper Scissors 1/3 Games & Equilibria Coarse Correlated Equilibria (CCE):

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A probability distribution over outcomes, s.t. no player has a profitable deviating strategy. Rock PaperScissors Rock Paper Scissors Games & Equilibria Coarse Correlated Equilibria (CCE): 0, 0-1, 11, -1 0, 0-1, 1 1, -10, 0

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A probability distribution over outcomes, s.t. no player has a profitable deviating strategy. Rock PaperScissors Rock Paper Scissors Games & Equilibria Coarse Correlated Equilibria (CCE): 0, 0-1, 11, -1 0, 0-1, 1 1, -10, 0 Choose any of the green outcomes uniformly (prob. 1/6)

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. Rock PaperScissors Rock Paper Scissors Algorithms Playing Games Online algorithm: Takes as input the past history of play until day t, and chooses a randomized action for day t+1. 0, 0-1, 11, -1 0, 0-1, 1 1, -10, 0 Alg 2 Alg 1

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. Rock PaperScissors Rock Paper Scissors Today’s class Online algorithm: Takes as input the past history of play until day t, and chooses a randomized action for day t+1. 0, 0-1, 11, -1 0, 0-1, 1 1, -10, 0 Alg 2 Alg 1 No-Regret

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No Regret Algorithms & CCE A history of no-regret algorithms is a sequence of outcomes s.t. no agent has a single deviating action that can increase her average payoff. A Coarse Correlated Equilibrium is a probability distribution over outcomes s.t. no agent has a single deviating action that can increase her expected payoff.

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How good are the CCE? It depends… Which class of games are we interested in? Which notion of social welfare? Today Class of games: potential games Social welfare: makespan [Kleinberg, P., Tardos STOC 09]

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Congestion Games n players and m resources (“edges”) Each strategy corresponds to a set of resources (“paths”) Each edge has a cost function c e (x) that determines the cost as a function on the # of players using it. Cost experienced by a player = sum of edge costs xxxx 2x xx Cost(red)=6 Cost(green)=8

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Equilibria and Social Welfare Load Balancing Makespan: Expected maximum latency over all links c(x)=x … …

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Equilibria and Social Welfare Pure Nash Makespan = 1 c(x)=x 111 … …

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Equilibria and Social Welfare (Mixed) Nash Makespan = Ω(logn/loglogn) > 1 c(x)=x 1/n [Koutsoupias, Mavronicolas, Spirakis ’02], [Czumaj, Vöcking ’02] Makespan = Θ(logn/loglogn) > 1 … …

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Equilibria and Social Welfare Coarse Correlated Equilibria Makespan = Ω(√n) >> Θ(logn/loglogn) c(x)=x … [Blum, Hajiaghayi, Ligett, Roth ’08] …

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Linear Load Balancing Δ(S) Pure Nash OPT Q(worst Nash)= Θ(logn/loglogn) Q(worst CCE) = Θ(√n) >> Q(OPT) CCE Q(worst pure Nash) Nash >> Q=makespan

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Linear Load Balancing Δ(S) Pure Nash OPT Price of Anarchy = Θ(logn/loglogn) Price of Total Anarchy = Θ(√n) Q=makespan >> 1 CCE Price of Pure Anarchy Nash >>

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Natural no-regret algorithms should be able to steer away from worst case equilibria. Our Hope

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The Multiplicative Weights Algorithm (MWA ) [Littlestone, Warmuth ’94], [Freund, Schapire ‘99] Pick s with probability proportional to (1-ε) total(s), where total(s) denotes combined cost in all past periods. Provable performance guarantees against arbitrary opponents No Regret: Against any sequence of opponents’ play, avg. payoff converges to that of the best fixed option (or better).

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(t) is the current state of the system (this is a tuple of randomized strategies, one for each player). Each player tosses their coins and a specific outcome is realized. Depending on the outcome of these random events, we transition to the next state. (Multiplicative Weights) Algorithm in (Potential) Games Δ(S) (t) (t+1) Infinite Markov Chains with Infinite States O(ε)

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Problem 1: Hard to get intuition about the problem, let alone analyze. Let’s try to come up with a “discounted” version of the problem. Ideas?? (Multiplicative Weights) Algorithm in (Potential) Games Δ(S) (t) (t+1) Infinite Markov Chains with Infinite States O(ε)

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Idea 1: Analyze expected motion. (Multiplicative Weights) Algorithm in (Potential) Games Δ(S) (t) (t+1) Infinite Markov Chains with Infinite States O(ε)

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The system evolution is now deterministic. (i.e. there exists a function f, s.t. I wish to analyze this function (e.g. find fixed points). (Multiplicative Weights) Algorithm in (Potential) Games Δ(S) (t) E[ (t+1)] O(ε) E[ (t+1)]= f ( (t), ε ) Idea 1: Analyze expected motion.

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Idea 2: I wish to analyze the MWA dynamics for small ε. Use Taylor expansion to find a first order approximation to f. (Multiplicative Weights) Algorithm in (Potential) Games Δ(S) (t) E[ (t+1)] O(ε) f ( (t), ε) = f ( (t), 0) + ε ×f ´ ( (t), 0) + O(ε 2 ) Problem 2: The function f is still rather complicated.

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Idea 2: I wish to analyze the MWA dynamics for small ε. Use Taylor expansion to find a first order approximation to f. (Multiplicative Weights) Algorithm in (Potential) Games Δ(S) (t) E[ (t+1)] O(ε) f ( (t), ε) ≈ f ( (t), 0) + ε ×f ´ ( (t), 0) Problem 2: The function f is still rather complicated.

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As ε→0, the equation specifies a vector on each point of our state space (i.e. a vector field). This vector field defines a system of ODEs which we are going to analyze. (Multiplicative Weights) Algorithm in (Potential) Games Δ(S) (t) f ( (t), ε)-f ( (t), 0) = f ´ ( (t), 0) ε f ´ ( (t), 0)

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As ε→0, the equation specifies a vector on each point of our state space (i.e. a vector field). This vector field defines a system of ODEs which we are going to analyze. (Multiplicative Weights) Algorithm in (Potential) Games Δ(S) (t) f ( (t), ε)-f ( (t), 0) = f ´ ( (t), 0) ε f ´ ( (t), 0)

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Deriving the ODE Taking expectations: Differentiate w.r.t. ε, take expected value: This is the replicator dynamic studied in evolutionary game theory.

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Motivating Example c(x)=x

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Motivating Example Each player’s mixed strategy is summarized by a single number. (Probability of picking machine 1.) Plot mixed strategy profile in R 2. Pure Nash Mixed Nash

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Motivating Example Each player’s mixed strategy is summarized by a single number. (Probability of picking machine 1.) Plot mixed strategy profile in R 2.

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Motivating Example Even in the simplest case of two balls, two bins with linear utility the replicator equation has a nonlinear form.

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The potential function The congestion game has a potential function Let Ψ=E[Φ]. A calculation yields Hence Ψ decreases except when every player randomizes over paths of equal expected cost (i.e. is a Lyapunov function of the dynamics). [Monderer-Shapley ’96].

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Unstable vs. stable fixed points The derivative of ξ is a matrix J (the Jacobian) whose spectrum distinguishes stable from unstable fixed points. Unstable if some eigenvalue has positive real part, else neutrally stable. Non-Nash fixed points are unstable. (Easy) Which Nash are unstable?

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Unstable Nash.5 c(x)=x.5

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Motivating Example.51 c(x)=x.5.49

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Motivating Example.51 c(x)=x.6.4.49

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Motivating Example.65 c(x)=x.6.4.35

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Unstable vs. stable fixed points The derivative of ξ is a matrix J (the Jacobian) whose spectrum distinguishes stable from unstable fixed points. Unstable if some eigenvalue has positive real part, else neutrally stable. Non-Nash fixed points are unstable. (Easy) Which Nash are unstable?

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Unstable vs. stable Nash J p submatrix of J of strategies with positive prob. Every eigenvalue of J p is an eigenvalue of J. Computing the entries of J p reveals that at Nash, If Tr(J p )=0 and no eigenvalue has positive real part, then they all are pure imaginary, so Tr(J p 2 ) ≤ 0. But clearly Tr(J p 2 ) ≥ 0. Hence E[cost i (R,Q,s -i,j )] = E[cost i (R’, Q,s -i,j )] whenever p iR,p iR’,p jQ’ >0. ∴ A new refinement of Nash equilibria!

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Weakly stable Nash equilibrium Definition: A weakly stable Nash equilibrium is a mixed Nash equilibrium (σ 1,...,σ n ) such that: for all players i,j... if i switches to using any pure strategy in support(σ i )... then j remains indifferent between all the strategies in support(σ j ).

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Weakly stable Nash are Nash Δ(S) Nash Weakly stable Nash

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Weakly stable Nash equilibrium Definition: A weakly stable Nash equilibrium is a mixed Nash equilibrium (σ 1,...,σ n ) such that: for all players i,j... if i switches to using any pure strategy in support(σ i )... then j remains indifferent between all the strategies in support(σ j ).

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Pure Nash Weakly stable Δ(S) Pure Nash Weakly stable Nash p1-p c(x)=x

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How bad can a weakly stable NE be? Price of Anarchy Price of Total Anarchy >> 1 Price of Pure Anarchy >> Δ(S) Pure Nash Weakly stable Nash

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How bad can a weakly stable NE be? Price of Anarchy >> 1 Price of Pure Anarchy Δ(S) Pure Nash Weakly stable Nash

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How bad can a weakly stable NE be? Price of Anarchy 1 Price of Pure Anarchy Δ(S) Pure Nash Weakly stable Nash = Price of weakly stable Anarchy

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Mixed weakly stable NE are due to rare coincidences Relies on a coincidence: two edges having equal cost functions. If we perturb the cost functions — e.g. scale each one by an independent random factor between 1-δ and 1+δ — then the game has no mixed equilibrium with the same support sets. p1-p c(x)=x Example:

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Games as vectors If one fixes the set of players, facilities and strategy sets (combinatorial structure), the game is determined by the vector of edge costs R {c}. Game + Strategy profile: R {p} ×R {c}. C e (1) C e (2) C e’ (1) C e’ (2) … …

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Weakly Stable Nash Restrictions The assertion {p iR } iεN,RεP(i) is a fully mixed weakly stable eq. of the game with edge costs {c e (x)} eεE, 1≤x≤n entails many equations among {p i },{c e (x)}: These are all polynomial equations. (In fact, multilinear.) Do they imply a nonzero polynomial equation among {c e (x)}?

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Sard’s Theorem The solution set of these polynomials is an algebraic variety X in R {p} ×R {c}. Project it onto R {c}. Is the image dense? Sard’s Theorem: If f:X→Y is smooth, the image of the set of critical points of f has measure zero.

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Sard’s Theorem The solution set of these polynomials is an algebraic variety X in R {p} ×R {c}. Project it onto R {c}. Is the image dense? Sard’s Theorem: If f:X→Y is smooth, the image of the set of critical points of f has measure zero. Use an alg. geom. version of Sard’s Theorem and prove the derivative of f is rank-deficient everywhere.

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Summary 1 Price of Pure Anarchy ???? Multiplicative Updates in Potential Games Price of Anarchy Price of Total Anarchy

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Summary 1 Price of Pure Anarchy Multiplicative Updates in Potential Games Price of Anarchy Price of Total Anarchy Expectations & ε→0

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Summary 1 Price of Pure Anarchy Multiplicative Updates in Potential Games Price of Anarchy Price of Total Anarchy Expectations & ε→0

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Summary 1 Price of Pure Anarchy Multiplicative Updates in Potential Games Price of Anarchy Expectations & ε→0 Jacobian weakly stable Nash

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Summary Expectations & ε→0 1 Price of Pure Anarchy Jacobian weakly stable Nash Algebraic Geometry Multiplicative Updates in Potential Games

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Summary Expectations & ε→0 1 Price of Pure Anarchy Jacobian weakly stable Nash Algebraic Geometry Multiplicative Updates in Potential Games Taylor Series Manipulations [Kleinberg, P., Tardos STOC 09]

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Learning Algs + Social Welfare Price of Pure Anarchy Price of Anarchy How low can we go? [Kleinberg, P., Tardos STOC 09] [Roughgarden STOC 09] Any no-regret alg in potential games, when social welfare is the sum of utilities.

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Learning Algs + Social Welfare Price of Pure Anarchy Price of Anarchy Price of Stability How low can we go? [Kleinberg, P., Tardos STOC 09] [Roughgarden STOC 09] [Balcan, Blum, Mansour ICS 2010] In specific classes of potential games via public advertising. >>

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Learning Algs + Social Welfare Price of Pure Anarchy Price of Anarchy >> 1 (=OPT) Price of Stability How low can we go? [Kleinberg, P., Tardos STOC 09] [Roughgarden STOC 09] [Balcan, Blum, Mansour ICS 2010] [Kleinberg, Ligett, P.,Tardos ICS 2011] Cycles of the replicator dynamics in specific games. >>

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Learning Algs + Social Welfare Price of Pure Anarchy Price of Anarchy >> 1 (=OPT) Price of Stability How low can we go? [Kleinberg, P., Tardos STOC 09] [Roughgarden STOC 09] [Balcan, Blum, Mansour ICS 2010] [Kleinberg, Ligett, P.,Tardos ICS 2011] … >>

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