Playing Anonymous Games Using Simple Strategies Yu Cheng Ilias - PowerPoint PPT Presentation

Playing Anonymous Games Using Simple Strategies Yu Cheng Ilias Diakonikolas Alistair Stewart University of Southern California

Anonymous Games • 𝑜 players, 𝑙 = 𝑃(1) strategies • Payoff of each player depends on • Her identity and strategy • The number of other players who play each of the strategy • NOT the identity of other players Jan 16, 2017 Yu Cheng (USC)

Anonymous Games Jan 16, 2017 Yu Cheng (USC)

Nash Equilibrium • Players have no incentive to deviate • 𝜗 -Approximate Nash Equilibrium ( 𝜗 -ANE): Players can gain at most 𝜗 by deviation Jan 16, 2017 Yu Cheng (USC)

Previous Work 𝜗 -ANE of 𝑜 -player 𝑙 -strategy anonymous games: 𝑜 )/+ , -. • [DP’08]: First PTAS • [CDO’14]: PPAD-Complete when 𝜗 = 2 01 2 and 𝑙 = 5 Jan 16, 2017 Yu Cheng (USC)

How small can 𝜗 be so that an 𝜗 -ANE can be computed in polynomial time?

𝜗 Running time # of strategies 𝑜 )/+ , -. 𝑙 > 2 [DP’08a] 𝜗 = 2 01 2 𝑙 = 5 [CDO’14] PPAD Complete poly 𝑜 ⋅ 1/𝜗 :(;<= > (?/+)) 𝑙 = 2 [DP’08b] 𝜗 = 𝑜 0?/@ poly 𝑜 𝑙 = 2 [GT’15] poly 𝑜 ⋅ 1/𝜗 :(;<= (?/+)) 𝑙 = 2 [DKS’16a] [DKS’16b] [DDKT’16] 𝑜 B<;C()) ⋅ 1/𝜗 ) ;<= ?/+ ,(-) 𝑙 > 2 Jan 16, 2017 Yu Cheng (USC)

Our Results Fix any 𝑙 > 2, 𝜀 > 0 ? • First poly-time algorithm when 𝜗 = 1 GHI ? • A poly-time algorithm for 𝜗 = 1 GJI ⟹ FPTAS 𝜗 = 1/2 1L 𝜗 = 1/𝑜 ?/@ 𝜗 = 1 𝜗 = 0.01 𝜗 = 1/2 1L 𝜗 = 1/𝑜 L 𝜗 = 1 𝜗 = 1/𝑜 Jan 16, 2017 Yu Cheng (USC)

Our Results Fix any 𝑙 > 2, 𝜀 > 0 ? • First poly-time algorithm when 𝜗 = 1 GHI ? • A poly-time algorithm for 𝜗 = 1 GJI ⟹ FPTAS ? • A faster algorithm that computes an 𝜗 ≈ 1 G/. equilibrium Jan 16, 2017 Yu Cheng (USC)

� Anonymous Games • Player 𝑗 ’s payoff when she plays strategy 𝑏 S = 0.3 S = 0.7 𝑣 R 𝑣 R S : Π 10? ) • 𝑣 R → 0, 1 ) = {(𝑦 ? , … , 𝑦 ) ) | ∑ 𝑦 S • Π 10? = 𝑜 − 1} S Jan 16, 2017 Yu Cheng (USC)

�� Poisson Multinomial Distributions • 𝑙 -Categorical Random Variable ( 𝑙 -CRV) 𝑌 S is a vector random variable ∈ {𝑙 -dimensional basis vectors } • An (𝑜, 𝑙) -Poisson Multinomial Distribution (PMD) is the sum of 𝑜 independent 𝑙 -CRVs 𝑌 = ∑ 𝑌 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 𝜗 -ANE of anonymous games Jan 16, 2017 Yu Cheng (USC)

Our Results Fix any 𝑙 > 2, 𝜀 > 0 ? • First poly-time algorithm when 𝜗 = 1 GHI ? • A poly-time algorithm for 𝜗 = 1 GJI ⟹ FPTAS ? • A faster algorithm that computes an 𝜗 ≈ 1 G/. equilibrium Jan 16, 2017 Yu Cheng (USC)

Pure Nash Equilibrium Player 1 Player 2 Strategy 1 Strategy 2 … Strategy 3 Player n Jan 16, 2017 Yu Cheng (USC)

Lipschitz Games • An anonymous game is 𝜇 -Lipschitz if S − 𝑣 R S ( ) ≤ 𝜇 − ? 𝑣 R • [DP’15, AS’13] Every 𝜇 -Lipschitz 𝑙 -strategy anonymous game admits a 2𝑙𝜇 -approximate pure equilibrium Jan 16, 2017 Yu Cheng (USC)

Lipschitz Games • 2𝑙𝜇 -approximate pure equilibrium Bad case: 𝜇 = 1 S = 0 S = 1 𝑣 R 𝑣 R Jan 16, 2017 Yu Cheng (USC)

S = 0 𝑣 R S = 1 𝑣 R Nov 11, 2016 Yu Cheng (USC)

S = 0 𝑣 R S = 1 𝑣 R 𝑒 fg = 1 𝑒 fg ≪ 1 Jan 16, 2017 Yu Cheng (USC)

� Smoothed Game [GT’15] • Given a game 𝐻 , construct a new game 𝐻 j 𝑣’ = 𝔽 𝑣( ) ? n • G j is 𝑃 -Lipschitz 1j Jan 16, 2017 Yu Cheng (USC)

� � p (1/ 𝑜 ?/q )-ANE in Polynomial Time O • A 2𝑙𝜇 -ANE of 𝐻 j is a 2𝑙𝜇 + 𝜀 -equilibrium of 𝐻 j j • Gain at most 2𝑙𝜇 by switching to 1 − 𝜀, )0? , … , )0? • Gain at most 2𝑙𝜇 + 𝜀 by switching to 1, 0 … 0 ? ? ? n n n • 𝜇 = 𝑃 ⟹ 𝜗 = 𝑃 + 𝜀 = 𝑃 1 G/. 1j 1j Jan 16, 2017 Yu Cheng (USC)

� n( 1 𝑒 fg , ≤ 𝑃 ) − ? 𝑜𝜀 ≈ 𝒪(𝜈 ? , Σ ? ) 𝒪(𝜈 w , Σ w ) • Size-free multivariate Central Limit Theorem [DKS’16]: an (𝑜, 𝑙) -PMD is poly(𝑙/𝜏) close to discrete Gaussians • Two Gaussians with similar mean and variance are close Jan 16, 2017 Yu Cheng (USC)

Our Results Fix any 𝑙 > 2, 𝜀 > 0 ? • First poly-time algorithm when 𝜗 = 1 GHI ? • A poly-time algorithm for 𝜗 = 1 GJI ⟹ FPTAS ? • A faster algorithm that computes an 𝜗 ≈ 1 G/. equilibrium Jan 16, 2017 Yu Cheng (USC)

O(1/𝑜 x.yy ) -ANE Jan 16, 2017 Yu Cheng (USC)

Player 1 Player 2 … Player n Jan 16, 2017 Yu Cheng (USC)

Quasi-PTAS when 𝜗 = O(1/𝑜 L ) • Small d TV ⟹ Similar payoffs • Limitation: • Cover-size lower bound [DKS’16]: even when 𝑙 = 2 Any proper 𝜗 -cover 𝑇 must have 𝑇 ≥ 𝑜 (1/𝜗) |(;<= ?/+ ) Jan 16, 2017 Yu Cheng (USC)

𝜗 = 1/𝑜 x.yy 𝜗 = 1/𝑜 ?/q log (1/𝜗) moments 𝑃 1 moments Two moments Jan 16, 2017 Yu Cheng (USC)

Moment Matching Lemma • For two PMDs to be 𝜗 -close in d TV [DP’08, DKS’16] need first log (1/𝜗) moments to match • We provide quantitative tradeoff between • The number of moments we need to match • The size of the variance Jan 16, 2017 Yu Cheng (USC)

Moment Matching Lemma • Multidimensional Fourier transform • Exploit the sparsity of the Fourier transform • Taylor approximations of the log Fourier transform • Large variance ⟹ Truncate with fewer terms Jan 16, 2017 Yu Cheng (USC)

O(1/𝑜 x.yy ) -ANE in Polynomial Time • There always exists an equilibrium with variance 𝜗𝑜 = 𝑜 0x.yy ⋅ 𝑜 = 𝑜 x.x? • Construct a poly-size 𝜗 -cover of large variance PMDs • Polynomial-size: Match only degree 𝑃(1) moments Jan 16, 2017 Yu Cheng (USC)

NE All (𝑜, 𝑙) -PMDs 𝑇 ≥ 𝑜 (1/𝜗) |(;<= ?/+ ) Jan 16, 2017 Yu Cheng (USC)

? 1 ~.•• -ANE 𝑌 S = 1 − 𝑞 𝒇 𝒌 + 𝑞𝒇 0𝒌 PMDs with ? 𝑌 = ∑𝑌 S 1 G/. -ANE large variance 𝑇 = 𝑜 ) , G/(GH~.•• ) 𝑇 ≤ 𝑜 )0? Jan 16, 2017 Yu Cheng (USC)

Conclusion Computing 𝜗 -ANE of 𝑜 -player anonymous games ? • First poly-time algorithm when 𝜗 = 1 GHI • New moment-matching lemma for PMDs ? • A poly-time algorithm for 𝜗 = 1 GJI ⟹ FPTAS 𝜗 = 1/2 1L 𝜗 = 1/𝑜 L 𝜗 = 1 𝜗 = 1/𝑜 Jan 16, 2017 Yu Cheng (USC)

Open Problems FPTAS ? 𝜗 = 1/2 1L 𝜗 = 1/𝑜 L 𝜗 = 1 𝜗 = 1/𝑜 Jan 16, 2017 Yu Cheng (USC)