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THE PRICE OF STOCHASTIC ANARCHY Christine Chung University of Pittsburgh Katrina Ligett Carnegie Mellon University Kirk Pruhs University of Pittsburgh Aaron Roth Carnegie Mellon University Load Balancing on Unrelated Machines 2 n players, each with a job to run, chooses one of m machines to run it on Time Machine Machine Machine Machine Needed 1 2 1 2 Job 1 Job 2 Job 3 Each player’s goal is to minimize her job’s finish time. NOTE: finish time of a job is equal to load on the machine where the job is run. Load Balancing on Unrelated Machines 3 n players, each with a job to run, chooses one of m machines to run it on Time Machine Machine Machine Machine Needed 1 2 1 2 Job 1 Job 2 Job 3 Each player’s goal is to minimize her job’s finish time. NOTE: finish time of a job is equal to load on the machine where the job is run. Load Balancing on Unrelated Machines 4 n players, each with a job to run, chooses one of m machines to run it on Time Machine Machine Machine Machine Needed 1 2 1 2 Job 1 Job 2 Job 3 Each player’s goal is to minimize her job’s finish time. NOTE: finish time of a job is equal to load on the machine where the job is run. Load Balancing on Unrelated Machines 5 n players, each with a job to run, chooses one of m machines to run it on Time Machine Machine Machine Machine Needed 1 2 1 2 Job 1 Job 2 Job 3 Each player’s goal is to minimize her job’s finish time. NOTE: finish time of a job is equal to load on the machine where the job is run. Unbounded Price of Anarchy in the Load Balancing Game on Unrelated Machines 6 Price of Anarchy (POA) measures the cost of having no central authority. Let an optimal assignment under centralized authority be one in which makespan is minimized. POA = (makespan at worst Nash)/(makespan at OPT) Bad POA instance: 2 players and 2 machines (L and R). OPT here costs δ. 1 Worst Nash costs 1. L R δ Price of Anarchy: job 1 δ 1 cost of worst Nash 1 job 2 1 δ cost at OPT Drawbacks of Price of Anarchy 7 A solution characterization with no road map. If there is more than one Nash, don’t know which one will be reached. Strong assumptions must be made about the players: e.g., fully informed and fully convinced of one anothers’ “rationality.” Nash are sometimes very brittle, making POA results feel overly pessimistic. Evolutionary Game Theory 8 Young (1993) specified a model of adaptive play. Evolutionary Game Theory 9 dispense (1993) specified a model understand “I Young with the notion that people fully of structure of the games allows us to they have the adaptive play that they play, that predict a which solutions will be chosen they can make coherent model of others’ behavior, thatin the long run calculations of infinite complexity, and that rational by self-interested decision-making all this is common knowledge. and resources. ofagents with limited info Instead I postulate a world in which people base their decisions on limited data, use simple predictive models, and sometimes do unexplained or even foolish things.” – P. Young, Individual Strategy and Social Structure, 1998 Evolutionary Game Theory 10 Young (1993) specified a model of adaptive play. Adaptive play allows us to predict which solutions will be chosen in the long run by self-interested decision-making agents with limited info and resources. L R Adaptive Play Example job 1 δ 1 job 2 1 δ 11 In each round of play, each player uses some simple, reasonable dynamics to decide which strategy to play. E.g., imitation dynamics Sample s of the last mem strategies I played Play the strategy whose average payoff was highest (breaking ties uniformly at random) best response dynamics Sample the other player’s realized strategy in s of the last mem rounds. Assume this sample represents the probability distribution of what the other player will play the next round, and play a strategy that is a best response (minimizes my expected cost). L R Adaptive Play Example job 1 δ 1 job 2 1 δ 12 In each round of play, each player uses some simple, reasonable dynamics to decide which strategy to play. E.g., imitation dynamics Sample s of the last mem strategies I played Play the strategy whose average payoff was highest (breaking ties uniformly at random) best response dynamics Sample the other player’s realized strategy in s of the last mem rounds. Assume this sample represents the probability distribution of what the other player will play the next round, and play a strategy that is a best response (minimizes my expected cost). L R Adaptive Play Example: job 1 δ 1 a Markov process job 2 1 δ 13 Let mem = 4. (Then there are 2^8 = 256 total states in the state space.) player 1 LLLL LLLR LLLL RRRR RRRR player 2 LLLL LLLL LLLR ... LRRR RRRR 3/4 1/4 1 1 LLRR LLRL LLLL LLLL If s = 3, each player randomly samples three past plays from the memory, and picks the strategy among them that worked best (yielded the highest payoff). L R Absorbing Sets of the job 1 δ 1 14 Markov Process job 2 1 δ An absorbing set is a set of states that are all reachable from one another, but cannot reach any states outside of the set. In our example, we have 4 absorbing sets: RRRR RRRR LLLL LLLL RRRR LLLL RRRR LLLL 1 1 NASH OPT1 1 But which state we end up in depends on our initial state. Hence we perturb our Markov process as follows: During each round, each player, with probability ε, does not use imitation dynamics, but instead chooses a machine at random. L R Stochastic Stability job 1 δ 1 job 2 1 δ 15 The perturbed process has only one big absorbing set (any state is reachable from any other state). Hence we have a unique stationary distribution με (where μεP = με). The probability distribution με is the time-average asymptotic frequency distribution of Pε. A state z is stochastically stable if lim ( z ) 0 0 L R Finding Stochastically job 1 δ 1 16 Stable States job 2 1 δ Theorem (Young, 1993): The stochastically stable states are those states contained in the absorbing sets of the unperturbed process that have minimum stochastic potential. RRRR LLLL RRRR LLLL 1 RRRR LLLL 1 1 LLLL RRRR 1 L R Finding Stochastically job 1 δ 1 17 Stable States job 2 1 δ Theorem (Young, 1993): The stochastically stable states are those states contained in the absorbing sets of the unperturbed process that have minimum stochastic potential. RRRR LLLL RRRR LLLL RRRR LLLL 1 LLLL 3 RRRR L L L L RRRR LLLL L L L L RRRR R R R R RRRR L L L L L R Finding Stochastically job 1 δ 1 18 Stable States job 2 1 δ Theorem (Young, 1993): The stochastically stable states are those states contained in the absorbing sets of the unperturbed process that have minimum stochastic potential. = cost of min spanning tree RRRR rooted there 2 6 LLLL RRRR LLLL RRRR LLLL 1 LLLL 3 RRRR L R Finding Stochastically job 1 δ 1 19 Stable States job 2 1 δ Theorem (Young, 1993): The stochastically stable states are those states contained in the absorbing sets of the unperturbed process that have minimum stochastic potential. = cost of min spanning tree 6 RRRR rooted there 1 6 2 LLLL RRRR LLLL RRRR LLLL 3 LLLL RRRR L R Finding Stochastically job 1 δ 1 20 Stable States job 2 1 δ Theorem (Young, 1993): The stochastically stable states are those states contained in the absorbing sets of the unperturbed process that have minimum stochastic potential. = cost of min spanning tree 6 RRRR rooted there 1 6 1 LLLL RRRR LLLL RRRR 5 LLLL 3 LLLL RRRR L R Finding Stochastically job 1 δ 1 21 Stable States job 2 1 δ Theorem (Young, 1993): The stochastically stable states are those states contained in the absorbing sets of the unperturbed process that have minimum stochastic potential. = cost of min spanning tree 6 RRRR rooted there 1 6 1 LLLL RRRR LLLL RRRR 2 5 LLLL Stochastically LLLL RRRR 4 Stable! Recap: Adaptive Play Model 22 Assume the game is played repeatedly by players with limited information and resources. Use a decision rule (aka “learning behavior” or “selection dynamics”) to model how each player picks her strategy for each round. This yields a Markov Process where the states represent fixed-sized histories of game play. Add noise (players make “mistakes” with some small positive probability and don’t always behave according to the prescribed dynamics) Stochastic Stability 23 The states in the perturbed Markov process with positive probability in the long-run are the stochastically stable states (SSS). In our paper, we define the Price of Stochastic Anarchy (PSA) to be cost of SSS max SSS cost at OPT 23 L R PSA for Load Balancing job 1 δ 1 job 2 1 δ 24 Recall bad instance: POA = 1/δ (unbounded) But the bad Nash in this case is not a SSS. In fact, OPT is the only SSS here. So PSA = 1 in this instance. Our main result: Ω(m) ≤ PSA ≤ m∙Fib(n)(mn+1) For the game of load balancing on unrelated machines, while POA is unbounded, PSA is bounded. Specifically, we show PSA ≤ m∙(Fib(n)(mn+1)), which is m times the (mn+1)th n-step Fibonacci number. We also exhibit instances of the game where PSA > m. (m is the number of machines, n is the number of jobs/players) Closing Thoughts 25 In the game of load balancing on unrelated machines, we found that while POA is unbounded, PSA is bounded. Indeed, in the bad POA instances for many games, the worst Nash are not stochastically stable. Finding PSA in these games are interesting open questions that may yield very illuminating results. PSA allows us to determine relative stability of equilibria, distinguishing those that are brittle from those that are more robust, giving us a more informative measure of the cost of having no central authority. L R Conjecture job 1 δ 1 job 2 1 δ 26 You might notice in this game that if players could coordinate or form a team, they would play OPT. Instead of being unbounded, [AFM2007] have shown the strong price of anarchy is O(m). We conjecture that PSA is also O(m), i.e., that a linear price of anarchy can be achieved without player coordination.