1. 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)
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2. 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]
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3. Multiplayer Games
E.g. n players, 2 actions
“The STOC Game” - actions: {“go”, “don’t go”}: O(2n
)
Focus on games with succinct representation
e.g. Anonymous, (bounded degree) Graphical, Polymatrix ...
“The Anonymous STOC Game”:
If we care only about how many players go: O(n2
)
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4. Anonymous games
n players, α pure strategies (actions)
Player’s payoff depends on:
Her action
Number of the other players choosing each action (partition)
Succinctly representable (for constant α): O(αnα)
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5. 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 x ) −
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6. Expected payoffs in Anonymous games
n players, 2 actions
Expected payoff of player i:
vi (action 1; 0, n − 1 )×Prob[i “sees” k1 = 0, k2 = n − 1 ]
+ vi (action 1; 1, n − 2 )×Prob[i “sees” k1 = 1, k2 = n − 2 ]
+ vi (action 1; 2, n − 3 )×Prob[i “sees” k1 = 2, k2 = n − 3 ]
. . .
+ vi (action 1; n − 1, 0 )×Prob[i “sees” k1 = n − 1, k2 = 0 ]
linear expression of Partition probabilities
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7. Partition probabilities vs Mixed strategies
NE X = (x1, x2, . . . , xn)
Mixed strategy xi : probability of action 1
Player i observes partition probabilities:
Prob[k1 = 0, k2 = n − 1] = j=i (1 − xj )
Prob[k1 = 1, k2 = n − 2] = j=i xj /∈{i,j}(1 − x )
. . .
Symmetric polynomials of X
helpful in approximation algorithms
obstacle for hardness proof
No change from swapping mixed strategies
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8. Previous Work
Approximation algorithms for Anonymous games:
PTAS: Daskalakis and Papadimitriou [DP14]
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]
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9. Hardness of Anonymous Games
Theorem
Computing an 1/exp(n)-approximate NE in anonymous games, with α ≥ 7
strategies is PPAD-complete.
10. Reduction from Polymatrix Games
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11. Polymatrix Games
Multiplayer games
Players play Bimatrix against each other - Sum of payoffs
-NE in Polymatrix is PPAD-hard [DGP09,CDT09]
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12. Expected Payoffs in Polymatrix
n players, 2 actions: NE Y
Q1 Q2 · · · Qn
action 1 y1 y2 · · · yn
action 2 1 − y1 1 − y2 · · · 1 − yn
Expected Payoffs: linear expressions of NE Y
ui (action 1) = 2y1 + 1y2 + 4y3 + . . . + 2yn
ui (action 2) = 3y1 + 2y2 + 1y3 + . . . + 5yn
-NE:
If ui (1) > ui (2) + ⇒ yi = 1
If ui (2) > ui (1) + ⇒ yi = 0
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13. The Reduction
Embed Polymatrix payoffs in an Anonymous game
Anonymous game s.t. in NE X expected payoffs of Player i compare:
2x1 + 1x2 + . . . + 2xn vs 3x1 + 2x2 + . . . + 5xn
But: Expected payoffs in Anonymous - Symmetric Polynomials of X
“break” the symmetries: approximate xi by Prob[k]
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14. Playing in different scales
Game where players play at different scales in the NE:
2 actions {s, t}
xi = δi
δ = 1/2n
player 1 player 2 . . . player n
s δ δ2 . . . δn
t 1 − δ 1 − δ2 . . . 1 − δn
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15. 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}
xi = δi
xi,s1 + xi,s2 = δi
δ = 1/2n
16. Goal: Perturb the payoffs of s1, s2: Embed the Polymatrix
player 1 player 2 . . . player n
s1 x1 x2 . . . xn
s2 δ − x1 δ2 − x2 . . . δn − xn
t 1 − δ 1 − δ2 . . . 1 − δn
with xi ∈ [0, δi ]
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20. Using the “Scaling” Property: xi,s1 + xi,s2 ≈ δi
E.g. Estimate x1:
– Prob[k1 = 1, k2 = 0] = x1 · j=1(1 − δj ) + . . . + xn · j=n(1 − δj ) ≈ x1 ± O(δ2)
E.g. Estimate x2:
— Prob[k1 = 2, k2 = 0] = x1x2 · j=1,2(1 − δj ) + x1x3 · j=1,3(1 − δj ) + . . .
≈ x1x2 ± O(δ4)
˜ Prob[k1 = 1, k2 = 1]= · · · ≈ x1 δ2 + x2 δ − 2 x1x2 ± O(δ4)
x2: linear combination of –,—,˜
The Estimation lemma
Every xi can be written as linear form of Prob[k] with poly-time
computable coefficients (± error).
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21. Concluding the reduction - the Radix Game
n+2 players, 6 actions → 7 after the “splitting”
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
Reduction recovers a 1/n-NE of Polymatrix
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22. Open Problems
PPAD-hardness for ≤ 6 actions?
2 actions?
FPTAS for 2 actions?
Faster PTAS for α 2 actions?
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23. The end
Thank you!
Questions?
“On the Complexity of NE in Anonymous Games”
Xi Chen, David Durfee, Anthi Orfanou
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![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](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)