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Ranking Games that have Competitiveness-based Strategies Leslie Goldberg, Paul Goldberg, Piotr Krysta and Carmine Ventre University of Liverpool

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Ἐ ν ἀ ρχ ῇ ἦ ν ὁ ἀ ρρεψία Nash, κα ὶ ὁ ἀ ρρεψία Nash ἦ ν πρ ὸ ς τ ὸ ν ο ἰ κονόμος... In the beginning was the Nash equilibrium, and the Nash equilibrium was with Economists…

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... but then in the summer of Daskalakis, Goldberg & Papadimitriou show that computing NEs is “hard” (in terms of PPAD) for graphical games Later, [DP, Chen & Deng] show “hardness” for 3-player games and CD show “hardness” for 2-player games! Q: Is NE a “meaningful” concept? “If your laptop can't find it, neither can the market.” Kamal Jain. A1: Define interesting classes of games (ie, describing the world) for which it is A2: Compute efficiently approximate NEs

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Ἐ ν ἀ ρχ ῇ ἦ ν ὁ ἀ ρρεψία Nash, κα ὶ ὁ ἀ ρρεψία Nash ἦ ν πρ ὸ ς τ ὸ ν ο ἰ κονόμος, κα ὶ μπορεί ἀ κμήν ἐ γγίων ἀ ρρεψία κα ὶ / ή ἀ ξιόλογος ἀ στροθετέω των ἄ εθλος In the beginning was the Nash equilibrium, and the Nash equilibrium was with Economists, and it may still be for approximate equilibria and/or an interesting class of games

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Every morning in Africa a Gazelle wakes up. It knows it must run faster than the fastest lion or it will be killed. Every morning a Lion wakes up. It knows it must outrun the slowest Gazelle or it will starve to death. It doesn't matter whether you are a Lion or a Gazelle... when the sun comes up, you'd better be running 0 mph 25 mph50 mph 25 mph 50 mph

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Ranking games 1st 2nd 1st 2nd A1: Define interesting classes of games (describing the world) for which NE is “meaningful”

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Ranking games describe the world but NE is not “meaningful” for them (ie, these games are “hard”). [Brandt, Fischer, Harrenstein & Shoham, 2009] Ranking games describe the world but NE is not “meaningful” for them (ie, these games are “hard”). [Brandt, Fischer, Harrenstein & Shoham, 2009]

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Competitiveness-based ranking games 0 mph 25 mph 50 mph 25 mph 50 mph Increasing effort cost (effort) return (speed) Aside note: Returns allow compact representation of these games

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A1A2 Our algorithmic results # players# prizes# actionsResult return-symmetric games O(1) anyPTAS any O(1)PTAS O(1)1anyFPTAS any 2Exact pure tie-free games22anyExact linear-prize gamesany# playersanyExact A (F)PTAS computes an Ɛ -NE in time polynomial in the input (and 1/ Ɛ )

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Games without ties Return values are all different E.g., no two players ranked first, Google page rank Algorithm to find NEs of any 2-player such game 1 wins 2 wins 1.The support of a NE is a prefix of the strategies available to a player 2.There is a polynomial number of possible supports 3.It is well known that once having the support we can efficiently solve a 2-player game (essentially LP)

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Games without ties (further results) Characterization of NEs for games with a single prize: “One player has expected payoff positive, all the others have expected payoff 0.” Games without ties and single prize can be solved in polytime given the knowledge of the support Reduction to polymatrix games [DP09] when prizes are linear (rank j has a prize a-jb) Polymatrix games and thus linear-prize ranking games are solvable in polytime [DP09]

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Return-symmetric games (RSGs) All players have n actions, all with the same return while cost-per-action is player specific E.g., lion-gazelle game Actions’ returns: all speeds in [0,50] mph Effort for speed s is animal/player-dependant NEs of these games can be studied wlog* for our class of ranking games cost 2 (r) = cost 2 (r’’) r’r’’ r r’ r’’ r’ < r < r’’ r r’ r’’ r’r’’r * A game with O(1) actions can be reduced to a game with a polynomial number of actions

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PTAS for RSGs with O(1) players 1. Round down each cost (normalized to [0,1]) to the nearest integer multiple of Ɛ 2. Eliminate dominated strategies 3. Brute force search for an Ɛ -NE of the reduced game using discretized probability vectors (prob’s are integer multiple of δ) (in time (k+1) (#players/δ) ) 1 n 1 n 10 Ɛ =1/k, δ= Ɛ /(k+1) for k in N Regret of 3 Ɛ regret of Ɛ regret of 2 Ɛ After step 2 each player has only k+1 strategies polytime

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FPTAS for RSGs, O(1) players and single prize worth 1 j 1j-1jj+1n win share lose Definition of Ɛ -NE: 1.x’s are probability distribution 2.

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FPTAS: left-to-right is a collection of vectors of admissibile values that are multiple of Ɛ,e.g., … … Discard : sequences whose first 8 values are different than last 8 values of previous sequences

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FPTAS: right-to-left Output the x’s in … … Overall regret of Ɛ = O(1/ Ɛ 9 ) FPTAS

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Conclusions Introduction of ranking games with competitiveness-based strategies Interesting games (describing real life) Encouraging initial positive results (wrt both A1, A2) Work in progress: FPTAS works for many prizes Open problems What is the hardness of these games? Related to the unknown hardness of anonymous games Polytime algorithms for 2-player RSGs?

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