On the Complexity of Nash Equilibria in Anonymous Games
On the Complexity of Nash Equilibria in Anonymous Games Anthi Orfanou (Columbia University)
06/16/2015 (STOC ’15) Joint work with Xi Chen (Columbia University) and David Durfee (Georgia Institute of Technology)
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
0 / 16
Nash Equilibria
Every game has an equilibrium [Nash 50]. Games with bounded number of players: 2 Players: PPAD-complete [Pap94, DGP09, CDT09] ≥ 3 Players: FIXP-complete [EY10]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
1 / 16
Multiplayer Games
E.g. n players, 2 actions I
“The STOC Game” - actions: {“go”, “don’t go”}: O(2n )
Focus on games with succinct representation I
e.g. Anonymous, (bounded degree) Graphical, Polymatrix ...
“The Anonymous STOC Game”: I
If we care only about how many players go: O(n2 )
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
2 / 16
Anonymous games
n players, α pure strategies (actions) Player’s payoff depends on: I I
Her action Number of the other players choosing each action (partition)
Succinctly representable (for constant α): O(αnα )
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
3 / 16
Approximate Equilibria
-approximate NE: Mixed strategy profile X =(xi : ∀ player i) xi -best responce: (Expected payoff of Player i from xi ) ≥ (Expected payoff of Player i from any other x0 ) −
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
4 / 16
Expected payoffs in Anonymous games
n players, 2 actions Expected payoff of player i: vi (action + vi (action + vi (action ... + vi (action
1; h0, n − 1i)×Prob[i “sees” hk1 = 0, k2 = n − 1i] 1; h1, n − 2i)×Prob[i “sees” hk1 = 1, k2 = n − 2i] 1; h2, n − 3i)×Prob[i “sees” hk1 = 2, k2 = n − 3i] 1; hn − 1, 0i)×Prob[i “sees” hk1 = n − 1, k2 = 0i]
linear expression of Partition probabilities
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
5 / 16
Partition probabilities vs Mixed strategies
NE X = (x1 , x2 , . . . , xn ) Mixed strategy xi : probability of action 1 Player i observes partition probabilities: I I I
Q Prob[k1 = 0, k2 = n − 1] = Pj6=i (1 − Qxj ) Prob[k1 = 1, k2 = n − 2] = j6=i xj `∈{i,j} (1 − x` ) / ...
Symmetric polynomials of X I I
helpful in approximation algorithms obstacle for hardness proof
No change from swapping mixed strategies
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
6 / 16
Previous Work
Approximation algorithms for Anonymous games: PTAS: Daskalakis and Papadimitriou [DP14] I I I
2 actions [DP07,Das08] “oblivious”, “non-oblivious” [DP09] α actions [DP08] approximate pure NE for Lipschitz games [DP07]
2 actions: Query efficient algorithm [Goldberg and Turchetta 14] 2 actions (Lipschitz) Best response dynamics - O(n log n) steps: approximate pure NE [Babichenko 13]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
7 / 16
Hardness of Anonymous Games
Theorem Computing an 1/exp(n)-approximate NE in anonymous games, with α ≥ 7 strategies is PPAD-complete. + Reduction from Polymatrix Games
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
8 / 16
Polymatrix Games
Multiplayer games Players play Bimatrix against each other - Sum of payoffs -NE in Polymatrix is PPAD-hard [DGP09,CDT09]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
9 / 16
Expected Payoffs in Polymatrix
n players, 2 actions: NE Y Q1 Q2 action 1 y1 y2 action 2 1 − y1 1 − y2
··· ··· ···
Qn yn 1 − yn
Expected Payoffs: linear expressions of NE Y I I I
ui (action 1) = 2y1 + 1y2 + 4y3 + . . . + 2yn ui (action 2) = 3y1 + 2y2 + 1y3 + . . . + 5yn -NE: F F
If ui (1) > ui (2) + ⇒ yi = 1 If ui (2) > ui (1) + ⇒ yi = 0
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
10 / 16
The Reduction
Embed Polymatrix payoffs in an Anonymous game Anonymous game s.t. in NE X expected payoffs of Player i compare: I
2x1 + 1x2 + . . . + 2xn
vs
3x1 + 2x2 + . . . + 5xn
But: Expected payoffs in Anonymous - Symmetric Polynomials of X * “break” the symmetries: approximate xi by Prob[k]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
11 / 16
Playing in different scales
Game where players play at different scales in the NE: I I I
2 actions {s, t} xi = δ i δ = 1/2n
s t
player 1 player 2 δ δ2 1−δ 1 − δ2
X. Chen, D. Durfee, A. Orfanou
... ... ...
On the Complexity of Anonymous Games
player n δn 1 − δn
06/16/2015
12 / 16
Radix: A “scaling” anonymous game Game where players play at different scales in the NE: “Split” action s → s1 , s2 2 actions {s, t} 3 actions {s1 , s2 , t} i I x = δi x + x i i,s1 i,s2 = δ n I δ = 1/2 + Goal: Perturb the payoffs of s1 , s2 : Embed the Polymatrix I I
s1 s2 t
player 1 player 2 x1 x2 2 δ − x1 δ − x2 1−δ 1 − δ2
... ... ... ...
player n xn n δ − xn 1 − δn
with xi ∈ [0, δ i ]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
13 / 16
Estimating xi from Prob[k]
+ Using the “Scaling” Property: xi,s1 + xi,s2 ≈ δ i E.g. Estimate x1 : Ê Prob[k1 = 1, k2 = 0] = x1 ·
X. Chen, D. Durfee, A. Orfanou
Q
j6=1 (1
− δ j ) + . . . + xn ·
Q
j6=n (1
On the Complexity of Anonymous Games
− δ j ) ≈ x1 ± O(δ 2 )
06/16/2015
14 / 16
Estimating xi from Prob[k] + Using the “Scaling” Property: xi,s1 + xi,s2 ≈ δ i E.g. Estimate x1 : Ê Prob[k1 = 1, k2 = 0] = x1 ·
Q
j6=1 (1
− δ j ) + . . . + xn ·
Q
j6=n (1
− δ j ) ≈ x1 ± O(δ 2 )
E.g. Estimate x2 : Ë Prob[k1 = 2, k2 = 0] = x1 x2 · ≈ x1 x2 ± O(δ 4 )
Q
j6=1,2 (1
− δ j ) + x1 x3 ·
Q
j6=1,3 (1
− δj ) + . . .
Ì Prob[k1 = 1, k2 = 1]= · · · ≈ x1 δ 2 + x2 δ − 2 x1 x2 ± O(δ 4 )
x2 : linear combination of Ê,Ë,Ì
The Estimation lemma Every xi can be written as linear form of Prob[k] with poly-time computable coefficients (± error).
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
14 / 16
Concluding the reduction - the Radix Game
n+2 players, 6 actions → 7 after the “splitting” I I
n “main” players, 2 actions: simulate Polymatrix players 2 special players + 4 extra actions: enforce the “scaling” NE property
PPAD-hardness for 7 actions, 1/exp(n)-NE I
Reduction recovers a 1/n-NE of Polymatrix
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
15 / 16
Open Problems
PPAD-hardness for ≤ 6 actions? I
2 actions?
FPTAS for 2 actions? Faster PTAS for α > 2 actions?
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
16 / 16
The end
Thank you! Questions? “On the Complexity of NE in Anonymous Games” Xi Chen, David Durfee, Anthi Orfanou
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
16 / 16
06/16/2015 (STOC ’15) Joint work with Xi Chen (Columbia University) and David Durfee (Georgia Institute of Technology)
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
0 / 16
Nash Equilibria
Every game has an equilibrium [Nash 50]. Games with bounded number of players: 2 Players: PPAD-complete [Pap94, DGP09, CDT09] ≥ 3 Players: FIXP-complete [EY10]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
1 / 16
Multiplayer Games
E.g. n players, 2 actions I
“The STOC Game” - actions: {“go”, “don’t go”}: O(2n )
Focus on games with succinct representation I
e.g. Anonymous, (bounded degree) Graphical, Polymatrix ...
“The Anonymous STOC Game”: I
If we care only about how many players go: O(n2 )
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
2 / 16
Anonymous games
n players, α pure strategies (actions) Player’s payoff depends on: I I
Her action Number of the other players choosing each action (partition)
Succinctly representable (for constant α): O(αnα )
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
3 / 16
Approximate Equilibria
-approximate NE: Mixed strategy profile X =(xi : ∀ player i) xi -best responce: (Expected payoff of Player i from xi ) ≥ (Expected payoff of Player i from any other x0 ) −
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
4 / 16
Expected payoffs in Anonymous games
n players, 2 actions Expected payoff of player i: vi (action + vi (action + vi (action ... + vi (action
1; h0, n − 1i)×Prob[i “sees” hk1 = 0, k2 = n − 1i] 1; h1, n − 2i)×Prob[i “sees” hk1 = 1, k2 = n − 2i] 1; h2, n − 3i)×Prob[i “sees” hk1 = 2, k2 = n − 3i] 1; hn − 1, 0i)×Prob[i “sees” hk1 = n − 1, k2 = 0i]
linear expression of Partition probabilities
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
5 / 16
Partition probabilities vs Mixed strategies
NE X = (x1 , x2 , . . . , xn ) Mixed strategy xi : probability of action 1 Player i observes partition probabilities: I I I
Q Prob[k1 = 0, k2 = n − 1] = Pj6=i (1 − Qxj ) Prob[k1 = 1, k2 = n − 2] = j6=i xj `∈{i,j} (1 − x` ) / ...
Symmetric polynomials of X I I
helpful in approximation algorithms obstacle for hardness proof
No change from swapping mixed strategies
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
6 / 16
Previous Work
Approximation algorithms for Anonymous games: PTAS: Daskalakis and Papadimitriou [DP14] I I I
2 actions [DP07,Das08] “oblivious”, “non-oblivious” [DP09] α actions [DP08] approximate pure NE for Lipschitz games [DP07]
2 actions: Query efficient algorithm [Goldberg and Turchetta 14] 2 actions (Lipschitz) Best response dynamics - O(n log n) steps: approximate pure NE [Babichenko 13]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
7 / 16
Hardness of Anonymous Games
Theorem Computing an 1/exp(n)-approximate NE in anonymous games, with α ≥ 7 strategies is PPAD-complete. + Reduction from Polymatrix Games
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
8 / 16
Polymatrix Games
Multiplayer games Players play Bimatrix against each other - Sum of payoffs -NE in Polymatrix is PPAD-hard [DGP09,CDT09]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
9 / 16
Expected Payoffs in Polymatrix
n players, 2 actions: NE Y Q1 Q2 action 1 y1 y2 action 2 1 − y1 1 − y2
··· ··· ···
Qn yn 1 − yn
Expected Payoffs: linear expressions of NE Y I I I
ui (action 1) = 2y1 + 1y2 + 4y3 + . . . + 2yn ui (action 2) = 3y1 + 2y2 + 1y3 + . . . + 5yn -NE: F F
If ui (1) > ui (2) + ⇒ yi = 1 If ui (2) > ui (1) + ⇒ yi = 0
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
10 / 16
The Reduction
Embed Polymatrix payoffs in an Anonymous game Anonymous game s.t. in NE X expected payoffs of Player i compare: I
2x1 + 1x2 + . . . + 2xn
vs
3x1 + 2x2 + . . . + 5xn
But: Expected payoffs in Anonymous - Symmetric Polynomials of X * “break” the symmetries: approximate xi by Prob[k]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
11 / 16
Playing in different scales
Game where players play at different scales in the NE: I I I
2 actions {s, t} xi = δ i δ = 1/2n
s t
player 1 player 2 δ δ2 1−δ 1 − δ2
X. Chen, D. Durfee, A. Orfanou
... ... ...
On the Complexity of Anonymous Games
player n δn 1 − δn
06/16/2015
12 / 16
Radix: A “scaling” anonymous game Game where players play at different scales in the NE: “Split” action s → s1 , s2 2 actions {s, t} 3 actions {s1 , s2 , t} i I x = δi x + x i i,s1 i,s2 = δ n I δ = 1/2 + Goal: Perturb the payoffs of s1 , s2 : Embed the Polymatrix I I
s1 s2 t
player 1 player 2 x1 x2 2 δ − x1 δ − x2 1−δ 1 − δ2
... ... ... ...
player n xn n δ − xn 1 − δn
with xi ∈ [0, δ i ]
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
13 / 16
Estimating xi from Prob[k]
+ Using the “Scaling” Property: xi,s1 + xi,s2 ≈ δ i E.g. Estimate x1 : Ê Prob[k1 = 1, k2 = 0] = x1 ·
X. Chen, D. Durfee, A. Orfanou
Q
j6=1 (1
− δ j ) + . . . + xn ·
Q
j6=n (1
On the Complexity of Anonymous Games
− δ j ) ≈ x1 ± O(δ 2 )
06/16/2015
14 / 16
Estimating xi from Prob[k] + Using the “Scaling” Property: xi,s1 + xi,s2 ≈ δ i E.g. Estimate x1 : Ê Prob[k1 = 1, k2 = 0] = x1 ·
Q
j6=1 (1
− δ j ) + . . . + xn ·
Q
j6=n (1
− δ j ) ≈ x1 ± O(δ 2 )
E.g. Estimate x2 : Ë Prob[k1 = 2, k2 = 0] = x1 x2 · ≈ x1 x2 ± O(δ 4 )
Q
j6=1,2 (1
− δ j ) + x1 x3 ·
Q
j6=1,3 (1
− δj ) + . . .
Ì Prob[k1 = 1, k2 = 1]= · · · ≈ x1 δ 2 + x2 δ − 2 x1 x2 ± O(δ 4 )
x2 : linear combination of Ê,Ë,Ì
The Estimation lemma Every xi can be written as linear form of Prob[k] with poly-time computable coefficients (± error).
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
14 / 16
Concluding the reduction - the Radix Game
n+2 players, 6 actions → 7 after the “splitting” I I
n “main” players, 2 actions: simulate Polymatrix players 2 special players + 4 extra actions: enforce the “scaling” NE property
PPAD-hardness for 7 actions, 1/exp(n)-NE I
Reduction recovers a 1/n-NE of Polymatrix
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
15 / 16
Open Problems
PPAD-hardness for ≤ 6 actions? I
2 actions?
FPTAS for 2 actions? Faster PTAS for α > 2 actions?
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
16 / 16
The end
Thank you! Questions? “On the Complexity of NE in Anonymous Games” Xi Chen, David Durfee, Anthi Orfanou
X. Chen, D. Durfee, A. Orfanou
On the Complexity of Anonymous Games
06/16/2015
16 / 16
