The Query Complexity of Correlated Equilibria
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1 The Query Complexity of Correlated Equilibria Sergiu Hart September 2015 SERGIU HART c 2015 p. 1
2 The Query Complexity of Correlated Equilibria Sergiu Hart Center for the Study of Rationality Dept of Mathematics Dept of Economics The Hebrew University of Jerusalem SERGIU HART c 2015 p. 2
3 Paper SERGIU HART c 2015 p. 3
4 Paper Sergiu Hart and Noam Nisan The Query Complexity of Correlated Equilibria Center for Rationality 2013 Revised September SERGIU HART c 2015 p. 3
5 Correlated Equilibria SERGIU HART c 2015 p. 4
6 Correlated Equilibria n-person games SERGIU HART c 2015 p. 4
7 Correlated Equilibria n-person games Each player has 2 actions SERGIU HART c 2015 p. 4
8 Correlated Equilibria n-person games Each player has 2 actions CORRELATED EQUILIBRIUM : SERGIU HART c 2015 p. 4
9 Correlated Equilibria n-person games Each player has 2 actions CORRELATED EQUILIBRIUM : 2 n unknowns 0 SERGIU HART c 2015 p. 4
10 Correlated Equilibria n-person games Each player has 2 actions CORRELATED EQUILIBRIUM : 2 n unknowns 0 2n + 1 linear inequalities SERGIU HART c 2015 p. 4
11 Correlated Equilibria n-person games Each player has 2 actions CORRELATED EQUILIBRIUM : 2 n unknowns 0 2n + 1 linear inequalities There is an algorithm for computing CORRELATED EQUILIBRIA with COMPLEXITY = POLY(2 n ) = EXP(n) SERGIU HART c 2015 p. 4
12 Correlated Equilibria SERGIU HART c 2015 p. 5
13 Correlated Equilibria BUT: Regret-based dynamics yield ǫ-correlated EQUILIBRIA with high probability (Hart & Mas-Colell 2000, 2001;...) SERGIU HART c 2015 p. 5
14 Correlated Equilibria BUT: Regret-based dynamics yield ǫ-correlated EQUILIBRIA with high probability (Hart & Mas-Colell 2000, 2001;...) in O(log(n)/ǫ 2 ) steps (Cesa-Bianchi & Lugosi 2003) SERGIU HART c 2015 p. 5
15 Correlated Equilibria BUT: Regret-based dynamics yield ǫ-correlated EQUILIBRIA with high probability (Hart & Mas-Colell 2000, 2001;...) in O(log(n)/ǫ 2 ) steps (Cesa-Bianchi & Lugosi 2003) QUERY COMPLEXITY (QC) := maximal number of pure payoff QUERIES (out of n 2 n ) SERGIU HART c 2015 p. 5
16 Correlated Equilibria BUT: Regret-based dynamics yield ǫ-correlated EQUILIBRIA with high probability (Hart & Mas-Colell 2000, 2001;...) in O(log(n)/ǫ 2 ) steps (Cesa-Bianchi & Lugosi 2003) QUERY COMPLEXITY (QC) := maximal number of pure payoff QUERIES (out of n 2 n ) There are randomized algorithms for computing ǫ-correlated EQUILIBRIA with QC = POLY(n) SERGIU HART c 2015 p. 5
17 Correlated Equilibria SERGIU HART c 2015 p. 6
18 Correlated Equilibria Surprise? SERGIU HART c 2015 p. 6
19 Correlated Equilibria Surprise?... perhaps not so much... SERGIU HART c 2015 p. 6
20 Correlated Equilibria Surprise?... perhaps not so much... There are CORRELATED EQUILIBRIA with support of size 2n + 1 basic solutions of Linear Programming SERGIU HART c 2015 p. 6
21 Correlated Equilibria Surprise?... perhaps not so much... There are CORRELATED EQUILIBRIA with support of size 2n + 1 basic solutions of Linear Programming There are ǫ-correlated EQUILIBRIA with support of size O(log n/ǫ 2 ) SERGIU HART c 2015 p. 6
22 Correlated Equilibria Surprise?... perhaps not so much... There are CORRELATED EQUILIBRIA with support of size 2n + 1 basic solutions of Linear Programming There are ǫ-correlated EQUILIBRIA with support of size O(log n/ǫ 2 ) use Lipton and Young 1994 (support of approximate optimal strategies) SERGIU HART c 2015 p. 6
23 Correlated Equilibria Surprise?... perhaps not so much... There are CORRELATED EQUILIBRIA with support of size 2n + 1 basic solutions of Linear Programming There are ǫ-correlated EQUILIBRIA with support of size O(log n/ǫ 2 ) use Lipton and Young 1994 (support of approximate optimal strategies) regret-based dynamics: O(log n/ǫ 2 ) steps SERGIU HART c 2015 p. 6
24 Correlated Equilibria Surprise?... perhaps not so much... There are CORRELATED EQUILIBRIA with support of size 2n + 1 basic solutions of Linear Programming There are ǫ-correlated EQUILIBRIA with support of size O(log n/ǫ 2 ) use Lipton and Young 1994 (support of approximate optimal strategies) regret-based dynamics: O(log n/ǫ 2 ) steps NOTE: Regret-based dynamics converge as fast as possible (up to a constant factor) SERGIU HART c 2015 p. 6
25 Correlated Equilibria (recall) SERGIU HART c 2015 p. 7
26 Correlated Equilibria (recall) There are randomized algorithms (regret-based dynamics) for computing approximate CORRELATED EQUILIBRIA with QC = POLY(n) SERGIU HART c 2015 p. 7
27 Correlated Equilibria There are randomized algorithms (regret-based dynamics) for computing approximate CORRELATED EQUILIBRIA with QC = POLY(n) exact CORRELATED EQUILIBRIA? SERGIU HART c 2015 p. 7
28 Correlated Equilibria There are randomized algorithms (regret-based dynamics) for computing approximate CORRELATED EQUILIBRIA with QC = POLY(n) exact CORRELATED EQUILIBRIA? deterministic algorithms? SERGIU HART c 2015 p. 7
29 Query Complexity of CE SERGIU HART c 2015 p. 8
30 Query Complexity of CE Algorithm Randomized Deterministic ε-ce exact CE SERGIU HART c 2015 p. 8
31 Query Complexity of CE Randomized Algorithm Deterministic ε-ce POLY(n) [1] exact CE [1] = regret-based dynamics SERGIU HART c 2015 p. 8
32 Query Complexity of CE Randomized Algorithm Deterministic ε-ce POLY(n) [1] exact CE EXP(n) [2] [1] = regret-based dynamics [2] = Babichenko and Barman 2013 SERGIU HART c 2015 p. 8
33 Query Complexity of CE Randomized Algorithm Deterministic ε-ce exact CE POLY(n) [1] EXP(n) [3] EXP(n) [2] [1] = regret-based dynamics [2] = Babichenko and Barman 2013 [3] = this paper SERGIU HART c 2015 p. 8
34 Query Complexity of CE Randomized Algorithm Deterministic ε-ce exact CE POLY(n) [1] EXP(n) [3] EXP(n) [3] EXP(n) [2] [1] = regret-based dynamics [2] = Babichenko and Barman 2013 [3] = this paper SERGIU HART c 2015 p. 8
35 Query Complexity of CE SERGIU HART c 2015 p. 9
36 Query Complexity of CE Theorem A. Every DETERMINISTIC algorithm that finds a 1/2 approximate correlated equilibrium in every n-person bi-strategy games with payoffs in {0, 1} requires 2 Ω(n) QUERIES in the worst case. SERGIU HART c 2015 p. 9
37 Query Complexity of CE Theorem A. Every DETERMINISTIC algorithm that finds a 1/2 approximate correlated equilibrium in every n-person bi-strategy games with payoffs in {0, 1} requires 2 Ω(n) QUERIES in the worst case. Theorem B. Every algorithm (randomized or deterministic) that finds an EXACT correlated equilibrium in every n-person bi-strategy games with payoffs specified as b-bit integers with b = Ω(n) requires 2 Ω(n) expected COST in the worst case. SERGIU HART c 2015 p. 9
38 Query Complexity of CE Theorem A. Every DETERMINISTIC algorithm that finds a 1/2 approximate correlated equilibrium in every n-person bi-strategy games with payoffs in {0, 1} requires 2 Ω(n) QUERIES in the worst case. Theorem B. Every algorithm (randomized or deterministic) that finds an EXACT correlated equilibrium in every n-person bi-strategy games with payoffs specified as b-bit integers with b = Ω(n) requires 2 Ω(n) expected COST in the worst case. COST = # of QUERIES + size of support of output SERGIU HART c 2015 p. 9
39 Query Complexity of CE SERGIU HART c 2015 p. 10
40 Query Complexity of CE Randomized Algorithm Deterministic ε-ce exact CE POLY(n) [1] EXP(n) [3] EXP(n) [3] EXP(n) [2] SERGIU HART c 2015 p. 10
41 Query Complexity of COARSE CE Randomized Algorithm Deterministic ε-ce exact CE POLY(n) [1] EXP(n) [3] EXP(n) [3] EXP(n) [2] SERGIU HART c 2015 p. 10
42 Query Complexity of COARSE CE Randomized Algorithm Deterministic ε-ce exact CE POLY(n) [1] EXP(n) [3] EXP(n) [3] EXP(n) [2] When every player has 2 strategies: COARSE CORRELATED EQUILIBRIUM = CORRELATED EQUILIBRIUM SERGIU HART c 2015 p. 10
43 Idea of Proof of Theorem A SERGIU HART c 2015 p. 11
44 Idea of Proof of Theorem A The set of strategy combinations = the n-dimensional hypercube SERGIU HART c 2015 p. 11
45 Idea of Proof of Theorem A The set of strategy combinations = the n-dimensional hypercube Each edge is labelled with the regret of the player whose strategy changes SERGIU HART c 2015 p. 11
46 Idea of Proof of Theorem A The set of strategy combinations = the n-dimensional hypercube Each edge is labelled with the regret of the player whose strategy changes A query at node v provides the n regrets of all edges adjacent to v SERGIU HART c 2015 p. 11
47 Idea of Proof of Theorem A The set of strategy combinations = the n-dimensional hypercube Each edge is labelled with the regret of the player whose strategy changes A query at node v provides the n regrets of all edges adjacent to v If the number of queries is 2 Ω(n) then we can make the sum of the queried regrets high so that no 1/2-approximate correlated equilibrium is found within the queried nodes (use the edge iso-perimetric inequality) SERGIU HART c 2015 p. 11
48 Idea of Proof of Theorem B SERGIU HART c 2015 p. 12
49 Idea of Proof of Theorem B Take a random path in the hypercube and define the regrets so that in order to get an exact correlated equilibrium one must find the endpoint of the path SERGIU HART c 2015 p. 12
50 Idea of Proof of Theorem B Take a random path in the hypercube and define the regrets so that in order to get an exact correlated equilibrium one must find the endpoint of the path To find the endpoint one must essentially follow the path (because every n log(n) steps there is "full mixing"), which requires 2 Ω(n) queries SERGIU HART c 2015 p. 12
51 Complexity of CE SERGIU HART c 2015 p. 13
52 Complexity of CE VERIFICATION of CORRELATED EQUILIBRIUM with support of size POLY(n) is POLY(n) SERGIU HART c 2015 p. 13
53 Complexity of CE VERIFICATION of CORRELATED EQUILIBRIUM with support of size POLY(n) is POLY(n) QUERIES of mixed strategies: only POLY(n) are needed Papadimitriou and Roughgarden 2008 Jing and Leyton-Brown 2011 SERGIU HART c 2015 p. 13
54 Complexity of CE SERGIU HART c 2015 p. 14
55 Complexity of CE LP of the same size as CE: randomized algorithms for approximate solutions SERGIU HART c 2015 p. 14
56 Complexity of CE LP of the same size as CE: randomized algorithms for approximate solutions require EXP(n) QUERIES SERGIU HART c 2015 p. 14
57 Complexity of CE LP of the same size as CE: randomized algorithms for approximate solutions require EXP(n) QUERIES CE is a special LP SERGIU HART c 2015 p. 14
58 Complexity of CE LP of the same size as CE: randomized algorithms for approximate solutions require EXP(n) QUERIES CE is a special LP Dual of CE decomposes into n problems SERGIU HART c 2015 p. 14
59 Complexity of CE LP of the same size as CE: randomized algorithms for approximate solutions require EXP(n) QUERIES CE is a special LP Dual of CE decomposes into n problems existence proof : Hart and Schmeidler 1989 SERGIU HART c 2015 p. 14
60 Complexity of CE LP of the same size as CE: randomized algorithms for approximate solutions require EXP(n) QUERIES CE is a special LP Dual of CE decomposes into n problems existence proof : Hart and Schmeidler 1989 uncoupled dynamics: Hart and Mas-Colell 2003 SERGIU HART c 2015 p. 14
61 Complexity of CE LP of the same size as CE: randomized algorithms for approximate solutions require EXP(n) QUERIES CE is a special LP Dual of CE decomposes into n problems existence proof : Hart and Schmeidler 1989 uncoupled dynamics: Hart and Mas-Colell 2003 algorithm: Papadimitriou and Roughgarden 2008 SERGIU HART c 2015 p. 14
62 Complexity of CE LP of the same size as CE: randomized algorithms for approximate solutions require EXP(n) QUERIES CE is a special LP SERGIU HART c 2015 p. 14
63 Complexity of CE LP of the same size as CE: randomized algorithms for approximate solutions require EXP(n) QUERIES CE is a special LP QUESTION: Why does this help ONLY for randomized algorithms for approximate CE? SERGIU HART c 2015 p. 14
64 Complexity LP of the same size as CE: randomized algorithms for approximate solutions require EXP(n) QUERIES CE is a special LP QUESTION: Why does this help ONLY for randomized algorithms for approximate CE? QUESTION: Complexity of approximate Nash Equilibria? SERGIU HART c 2015 p. 14
65 SERGIU HART c 2015 p. 15
66 "Police brutality is a thing of the past, mate, these days we apply structured query language!" c BART SERGIU HART c 2015 p. 15
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