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

Yu Cheng (USC)

• 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

Anonymous Games

Yu Cheng (USC) Jan 16, 2017

Nash Equilibrium

Yu Cheng (USC)

• Players have no incentive to deviate
• 𝜗-Approximate Nash Equilibrium (𝜗-ANE):

Players can gain at most 𝜗 by deviation

Jan 16, 2017

Previous Work

𝜗-ANE of 𝑜-player 𝑙-strategy anonymous games:

• [DP’08]:

First PTAS 𝑜 )/+ , -.

• [CDO’14]: PPAD-Complete when 𝜗 = 2012 and 𝑙 = 5

Yu Cheng (USC) Jan 16, 2017

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

Yu Cheng (USC)

Running time 𝜗 # of strategies [DP’08a] 𝑜 )/+ , -. 𝑙 > 2 [CDO’14] PPAD Complete 𝜗 = 2012 𝑙 = 5 [DP’08b] poly 𝑜 ⋅ 1/𝜗 :(;<=>(?/+)) 𝑙 = 2 [GT’15] poly 𝑜 𝜗 = 𝑜0?/@ 𝑙 = 2 [DKS’16a] poly 𝑜 ⋅ 1/𝜗 :(;<= (?/+)) 𝑙 = 2 [DKS’16b] [DDKT’16] 𝑜B<;C()) ⋅ 1/𝜗 ) ;<= ?/+ ,(-) 𝑙 > 2

Jan 16, 2017

Our Results

Fix any 𝑙 > 2, 𝜀 > 0

• First poly-time algorithm when 𝜗 =

? 1GHI

• A poly-time algorithm for 𝜗 =

? 1GJI ⟹ FPTAS

Yu Cheng (USC)

𝜗 = 1 𝜗 = 1/21L 𝜗 = 1/𝑜L 𝜗 = 1/𝑜 𝜗 = 1 𝜗 = 1/𝑜?/@ 𝜗 = 1/21L 𝜗 = 0.01

Jan 16, 2017

Our Results

Fix any 𝑙 > 2, 𝜀 > 0

• First poly-time algorithm when 𝜗 =

? 1GHI

• A poly-time algorithm for 𝜗 =

? 1GJI ⟹ FPTAS

• A faster algorithm that computes an 𝜗 ≈

? 1G/. equilibrium

Yu Cheng (USC) Jan 16, 2017

Anonymous Games

Yu Cheng (USC)

• Player 𝑗’s payoff when she plays strategy 𝑏
• 𝑣R

S : Π10? )

→ 0, 1

• Π10?

)

= {(𝑦?, … , 𝑦)) | ∑ 𝑦S

• S

= 𝑜 − 1}

Jan 16, 2017

𝑣R

S = 0.3

𝑣R

S = 0.7

Poisson Multinomial Distributions

• 𝑙-Categorical Random Variable (𝑙-CRV) 𝑌S 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 = ∑ 𝑌S

• Yu Cheng (USC)

Jan 16, 2017

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

Poisson Multinomial Distributions (PMDs) = Sum of independent random (basis) vectors = Mixed strategy profiles of anonymous games

Yu Cheng (USC) Jan 16, 2017

Yu Cheng (USC) Jan 16, 2017

Better understanding of PMDs

Faster algorithms for 𝜗-ANE

• f anonymous games

Our Results

Fix any 𝑙 > 2, 𝜀 > 0

• First poly-time algorithm when 𝜗 =

? 1GHI

• A poly-time algorithm for 𝜗 =

? 1GJI ⟹ FPTAS

• A faster algorithm that computes an 𝜗 ≈

? 1G/. equilibrium

Yu Cheng (USC) Jan 16, 2017

Yu Cheng (USC) Jan 16, 2017

Player 1 Player 2 Player n …

Pure Nash Equilibrium

Strategy 1 Strategy 2 Strategy 3

• An anonymous game is 𝜇-Lipschitz if
• [DP’15, AS’13] Every 𝜇-Lipschitz 𝑙-strategy anonymous

game admits a 2𝑙𝜇 -approximate pure equilibrium

Lipschitz Games

Yu Cheng (USC) Jan 16, 2017

𝑣R

S − 𝑣R S ( ) ≤ 𝜇 − ?

Lipschitz Games

Yu Cheng (USC)

• 2𝑙𝜇 -approximate pure equilibrium

Bad case: 𝜇 = 1

𝑣R

S = 0

𝑣R

S = 1

Jan 16, 2017

Nov 11, 2016 Yu Cheng (USC)

𝑣R

S = 0

𝑣R

S = 1

Yu Cheng (USC)

𝑣R

S = 0

𝑣R

S = 1

𝑒fg = 1 𝑒fg ≪ 1

Jan 16, 2017

Yu Cheng (USC)

• Given a game 𝐻, construct a new game 𝐻j

𝑣’ = 𝔽 𝑣( )

• Gj is 𝑃

n

? 1j

• Lipschitz

Smoothed Game [GT’15]

Jan 16, 2017

O p(1/𝑜?/q)-ANE in Polynomial Time

Yu Cheng (USC)

• A 2𝑙𝜇 -ANE of 𝐻j is a 2𝑙𝜇 + 𝜀 -equilibrium of 𝐻
• Gain at most 2𝑙𝜇 by switching to 1 − 𝜀,

j )0? , … , j )0?

• Gain at most 2𝑙𝜇 + 𝜀 by switching to 1, 0 … 0
• 𝜇 = 𝑃

n

? 1j

• ⟹ 𝜗 = 𝑃

n

? 1j

• + 𝜀 = 𝑃

n

? 1G/.

Jan 16, 2017

Yu Cheng (USC)

𝑒fg , ≤ 𝑃 n( 1 𝑜𝜀

• ) − ?
• 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

𝒪(𝜈?, Σ?) 𝒪(𝜈w, Σw)

≈

Our Results

Fix any 𝑙 > 2, 𝜀 > 0

• First poly-time algorithm when 𝜗 =

? 1GHI

• A poly-time algorithm for 𝜗 =

? 1GJI ⟹ FPTAS

• A faster algorithm that computes an 𝜗 ≈

? 1G/. equilibrium

Yu Cheng (USC) Jan 16, 2017

O(1/𝑜x.yy)-ANE

Yu Cheng (USC) Jan 16, 2017

Yu Cheng (USC) Jan 16, 2017

Player 1 Player 2 Player n …

Quasi-PTAS when 𝜗 = O(1/𝑜L)

• Small dTV ⟹ Similar payoffs
• Limitation:
• Cover-size lower bound [DKS’16]: even when 𝑙 = 2

Any proper 𝜗-cover 𝑇 must have 𝑇 ≥ 𝑜 (1/𝜗)|(;<= ?/+ )

Yu Cheng (USC) Jan 16, 2017

Yu Cheng (USC) Jan 16, 2017

𝜗 = 1/𝑜?/q Two moments log (1/𝜗) moments 𝜗 = 1/𝑜x.yy 𝑃 1 moments

Moment Matching Lemma

Yu Cheng (USC)

• For two PMDs to be 𝜗-close in dTV

[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

Moment Matching Lemma

Yu Cheng (USC)

• 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

O(1/𝑜x.yy)-ANE in Polynomial Time

• There always exists an equilibrium with variance

𝜗𝑜 = 𝑜0x.yy ⋅ 𝑜 = 𝑜x.x?

• Construct a poly-size 𝜗-cover of large variance PMDs
• Polynomial-size: Match only degree 𝑃(1) moments

Yu Cheng (USC) Jan 16, 2017

Yu Cheng (USC)

All (𝑜, 𝑙)-PMDs NE 𝑇 ≥ 𝑜 (1/𝜗)|(;<= ?/+ )

Jan 16, 2017

Yu Cheng (USC)

PMDs with large variance

? 1~.••-ANE ? 1G/.-ANE

𝑌S = 1 − 𝑞 𝒇𝒌 + 𝑞𝒇0𝒌

Jan 16, 2017

𝑌 = ∑𝑌S

𝑇 ≤ 𝑜)0? 𝑇 = 𝑜), G/(GH~.•• )

Conclusion

Yu Cheng (USC) Jan 16, 2017

𝜗 = 1 𝜗 = 1/21L 𝜗 = 1/𝑜L 𝜗 = 1/𝑜

Computing 𝜗-ANE of 𝑜-player anonymous games

• First poly-time algorithm when 𝜗 =

? 1GHI

• New moment-matching lemma for PMDs
• A poly-time algorithm for 𝜗 =

? 1GJI ⟹ FPTAS

Open Problems

Yu Cheng (USC) Jan 16, 2017

FPTAS ?

𝜗 = 1 𝜗 = 1/21L 𝜗 = 1/𝑜L 𝜗 = 1/𝑜