Playing Anonymous Games Using Simple Strategies Yu Cheng Ilias Diakonikolas Alistair Stewart University of Southern California
Anonymous Games β’ π players, π = π(1) strategies β’ Payoff of each player depends on β’ Her identity and strategy β’ The number of other players who play each of the strategy β’ NOT the identity of other players Jan 16, 2017 Yu Cheng (USC)
Anonymous Games Jan 16, 2017 Yu Cheng (USC)
Nash Equilibrium β’ Players have no incentive to deviate β’ π -Approximate Nash Equilibrium ( π -ANE): Players can gain at most π by deviation Jan 16, 2017 Yu Cheng (USC)
Previous Work π -ANE of π -player π -strategy anonymous games: π )/+ , -. β’ [DPβ08]: First PTAS β’ [CDOβ14]: PPAD-Complete when π = 2 01 2 and π = 5 Jan 16, 2017 Yu Cheng (USC)
How small can π be so that an π -ANE can be computed in polynomial time?
π Running time # of strategies π )/+ , -. π > 2 [DPβ08a] π = 2 01 2 π = 5 [CDOβ14] PPAD Complete poly π β 1/π :(;<= > (?/+)) π = 2 [DPβ08b] π = π 0?/@ poly π π = 2 [GTβ15] poly π β 1/π :(;<= (?/+)) π = 2 [DKSβ16a] [DKSβ16b] [DDKTβ16] π B<;C()) β 1/π ) ;<= ?/+ ,(-) π > 2 Jan 16, 2017 Yu Cheng (USC)
Our Results Fix any π > 2, π > 0 ? β’ First poly-time algorithm when π = 1 GHI ? β’ A poly-time algorithm for π = 1 GJI βΉ FPTAS π = 1/2 1L π = 1/π ?/@ π = 1 π = 0.01 π = 1/2 1L π = 1/π L π = 1 π = 1/π Jan 16, 2017 Yu Cheng (USC)
Our Results Fix any π > 2, π > 0 ? β’ First poly-time algorithm when π = 1 GHI ? β’ A poly-time algorithm for π = 1 GJI βΉ FPTAS ? β’ A faster algorithm that computes an π β 1 G/. equilibrium Jan 16, 2017 Yu Cheng (USC)
οΏ½ Anonymous Games β’ Player π βs payoff when she plays strategy π S = 0.3 S = 0.7 π£ R π£ R S : Ξ 10? ) β’ π£ R β 0, 1 ) = {(π¦ ? , β¦ , π¦ ) ) | β π¦ S β’ Ξ 10? = π β 1} S Jan 16, 2017 Yu Cheng (USC)
οΏ½οΏ½ Poisson Multinomial Distributions β’ π -Categorical Random Variable ( π -CRV) π S is a vector random variable β {π -dimensional basis vectors } β’ An (π, π) -Poisson Multinomial Distribution (PMD) is the sum of π independent π -CRVs π = β π S Jan 16, 2017 Yu Cheng (USC)
Player 1 plays strategy 1 Player 2 plays strategy 1 or 2 Player 3 plays strategy 2 or 3 Jan 16, 2017 Yu Cheng (USC)
Poisson Multinomial Distributions Jan 16, 2017 Yu Cheng (USC)
Poisson Multinomial Distributions (PMDs) = Sum of independent random (basis) vectors = Mixed strategy profiles of anonymous games Jan 16, 2017 Yu Cheng (USC)
Better understanding of PMDs Faster algorithms for π -ANE of anonymous games Jan 16, 2017 Yu Cheng (USC)
Our Results Fix any π > 2, π > 0 ? β’ First poly-time algorithm when π = 1 GHI ? β’ A poly-time algorithm for π = 1 GJI βΉ FPTAS ? β’ A faster algorithm that computes an π β 1 G/. equilibrium Jan 16, 2017 Yu Cheng (USC)
Pure Nash Equilibrium Player 1 Player 2 Strategy 1 Strategy 2 β¦ Strategy 3 Player n Jan 16, 2017 Yu Cheng (USC)
Lipschitz Games β’ An anonymous game is π -Lipschitz if S β π£ R S ( ) β€ π β ? π£ R β’ [DPβ15, ASβ13] Every π -Lipschitz π -strategy anonymous game admits a 2ππ -approximate pure equilibrium Jan 16, 2017 Yu Cheng (USC)
Lipschitz Games β’ 2ππ -approximate pure equilibrium Bad case: π = 1 S = 0 S = 1 π£ R π£ R Jan 16, 2017 Yu Cheng (USC)
S = 0 π£ R S = 1 π£ R Nov 11, 2016 Yu Cheng (USC)
S = 0 π£ R S = 1 π£ R π fg = 1 π fg βͺ 1 Jan 16, 2017 Yu Cheng (USC)
οΏ½ Smoothed Game [GTβ15] β’ Given a game π» , construct a new game π» j π£β = π½ π£( ) ? n β’ G j is π -Lipschitz 1j Jan 16, 2017 Yu Cheng (USC)
οΏ½ οΏ½ p (1/ π ?/q )-ANE in Polynomial Time O β’ A 2ππ -ANE of π» j is a 2ππ + π -equilibrium of π» j j β’ Gain at most 2ππ by switching to 1 β π, )0? , β¦ , )0? β’ Gain at most 2ππ + π by switching to 1, 0 β¦ 0 ? ? ? n n n β’ π = π βΉ π = π + π = π 1 G/. 1j 1j Jan 16, 2017 Yu Cheng (USC)
οΏ½ n( 1 π fg , β€ π ) β ? ππ β πͺ(π ? , Ξ£ ? ) πͺ(π w , Ξ£ w ) β’ Size-free multivariate Central Limit Theorem [DKSβ16]: an (π, π) -PMD is poly(π/π) close to discrete Gaussians β’ Two Gaussians with similar mean and variance are close Jan 16, 2017 Yu Cheng (USC)
Our Results Fix any π > 2, π > 0 ? β’ First poly-time algorithm when π = 1 GHI ? β’ A poly-time algorithm for π = 1 GJI βΉ FPTAS ? β’ A faster algorithm that computes an π β 1 G/. equilibrium Jan 16, 2017 Yu Cheng (USC)
O(1/π x.yy ) -ANE Jan 16, 2017 Yu Cheng (USC)
Player 1 Player 2 β¦ Player n Jan 16, 2017 Yu Cheng (USC)
Quasi-PTAS when π = O(1/π L ) β’ Small d TV βΉ Similar payoffs β’ Limitation: β’ Cover-size lower bound [DKSβ16]: even when π = 2 Any proper π -cover π must have π β₯ π (1/π) |(;<= ?/+ ) Jan 16, 2017 Yu Cheng (USC)
π = 1/π x.yy π = 1/π ?/q log (1/π) moments π 1 moments Two moments Jan 16, 2017 Yu Cheng (USC)
Moment Matching Lemma β’ For two PMDs to be π -close in d TV [DPβ08, DKSβ16] need first log (1/π) moments to match β’ We provide quantitative tradeoff between β’ The number of moments we need to match β’ The size of the variance Jan 16, 2017 Yu Cheng (USC)
Moment Matching Lemma β’ Multidimensional Fourier transform β’ Exploit the sparsity of the Fourier transform β’ Taylor approximations of the log Fourier transform β’ Large variance βΉ Truncate with fewer terms Jan 16, 2017 Yu Cheng (USC)
O(1/π x.yy ) -ANE in Polynomial Time β’ There always exists an equilibrium with variance ππ = π 0x.yy β π = π x.x? β’ Construct a poly-size π -cover of large variance PMDs β’ Polynomial-size: Match only degree π(1) moments Jan 16, 2017 Yu Cheng (USC)
NE All (π, π) -PMDs π β₯ π (1/π) |(;<= ?/+ ) Jan 16, 2017 Yu Cheng (USC)
? 1 ~.β’β’ -ANE π S = 1 β π π π + ππ 0π PMDs with ? π = βπ S 1 G/. -ANE large variance π = π ) , G/(GH~.β’β’ ) π β€ π )0? Jan 16, 2017 Yu Cheng (USC)
Conclusion Computing π -ANE of π -player anonymous games ? β’ First poly-time algorithm when π = 1 GHI β’ New moment-matching lemma for PMDs ? β’ A poly-time algorithm for π = 1 GJI βΉ FPTAS π = 1/2 1L π = 1/π L π = 1 π = 1/π Jan 16, 2017 Yu Cheng (USC)
Open Problems FPTAS ? π = 1/2 1L π = 1/π L π = 1 π = 1/π Jan 16, 2017 Yu Cheng (USC)
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