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

Nash Equilibria (revising Hollywood) 0,02,1 1,20,0

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…

... 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

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

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

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

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]

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

Ranking games with competitiveness- based strategies: setting Finite game in which outcomes are ranking of players Players are awarded prizes according to the rank (“And the 1 st prizes goes to...”)  Prizes normalized in [0,1] For each strategy we have a player-specific:  Return: used to rank the players  Cost: cost to play such an action Costs normalized in [0,1]

Ranking games: (sharing) prizes Fast bee (who ranks first) gets all the pollen of the daisy (ie, the first prize) and the four-eyed bee (ranked second) gets the pollen of the withered narcissus (ie, the second prize) The bees (who both rank first) share the pollen of the daisy and the pollen of the narcissus (ie, they both get half the first prize plus half the second)

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 )

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)

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]

Polymatrix games [DP09] 21 3 Constant- sum game L payoff 2 ( )+payoff 2 ( ) LR Polymatrix games are solvable in polytime [DP09] payoff 1 ( ) L payoff 3 ( ) R R

Reduction to polymatrix games 3 players linear-prize ranking games (prize for rank j is a-jb, payoff is a-jb-cost) Payoff 0 for who ranks better and –b for who ranks worst N Payoff of a–b–cost for the player and cost+b–a for N

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

L L H H 1 wins1,2 win 2 wins 1,2 win Computing Pure NEs for RSGs with 2 actions per player There is a pure NE A PNE consists of a (potentially empty) set of players playing H, all of whom have a cost for playing H lower than the ones playing L Algorithm: Order players non-decreasingly in cost of H and try all such prefixes P 2 – cost 1 (L) < P 1 /2+P 2 /2 – cost 1 (H) P 1 – cost 1 (H) < P 1 /2+P 2 /2 – cost 1 (L)

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

PTAS for RSGs with O(1) actions For O(1) actions and poly #players, naive brute force over discretized probabilities is not efficient (time about #actions #players ) Each player is identified by: In a RSG, players only differentiate by their cost vectors (type of a player) If we round down the costs to nearest integer multiple of Ɛ the number of possible types (rounded cost vectors) is #types=#actions 1/ Ɛ +1 Can do brute force over #actions #types, poly for #actions=O(1) Less efficient and simple PTAS known for the more general class of anonymous games [DP09] 1 actions return cost Ɛ 2Ɛ2Ɛ 3Ɛ3Ɛ 4Ɛ4Ɛ 5Ɛ5Ɛ 6Ɛ6Ɛ 7Ɛ7Ɛ

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.

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

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

Conclusions Introduction of ranking games with competitiveness-based strategies  Interesting games (describing real life)  Encouraging initial positive results (wrt both A1, A2)  Class of games resorting Nash’s solution concept 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?