Best response polyhedron 1 T x= 1, > < 1 { ( , ) | x u x 0 , A x 1 u } u 2 1 1 2 1 2 1 A = 2 2 0 2 x 2 2 equilibrium 1 1 (completely labeled) (2/3, 1/3) x 0 1 1
Projective transformation 1 T x= 1, > < { ( , ) | x u x 0 , A x 1 u } u 1 2 A = 2 0 x 2 > < { ( , ) | x 1 x 0 , A x 1 } 1 x 1
Best response polytope z 2 1 2 A = 2 0 } { | z z > 0 , A z < 1 1 1 2 z 1 2
Symmetric Lemke−Howson algorithm z 3 1 (back) 2 2 1 z 2 3 3 (bottom) z 1
Symmetric Lemke−Howson algorithm z 3 missing label 1 1 (back) 2 2 1 z 2 3 3 (bottom) z 1
Symmetric Lemke−Howson algorithm z 3 missing label 1 1 (back) 2 2 1 z 2 3 3 (bottom) z 1
Symmetric Lemke−Howson algorithm z 3 missing label 1 1 (back) 2 2 1 z 2 3 3 (bottom) z 1
Symmetric Lemke−Howson algorithm z 3 missing label 1 1 (back) 2 2 1 z 2 3 3 (bottom) z 1
Symmetric Lemke−Howson algorithm z 3 missing label 1 1 (back) 2 2 1 z 2 3 3 (bottom) z 1
Symmetric Lemke−Howson algorithm z 3 found label 1 1 (back) 2 2 1 z 2 3 3 (bottom) z 1
Why Lemke-Howson works LH finds at least one Nash equilibrium because • finitely many "vertices" for nondegenerate (generic) games: • unique starting edge given missing label • unique continuation ⇒ precludes "coming back" like here:
END OF LINE (Papadimitriou 1991) start Given a graph G of indegree/outdegree at most 1, and a start vertex of indegree 0 and outdegree 1, find another vertex of degree 1 end
END OF LINE (Papadimitriou 1991) start Catch: 0000 graph is exponentially large defined by two boolean circuits 0101 S , P that take a vertex in { 0 , 1 } n and output its successor and predecessor S ( 0000 ) = 0101 P ( 0101 ) = 0000 end
END OF LINE (Papadimitriou 1991) start A problem belongs to PPAD if it is reducible in poly-time to END OF LINE; and PPAD -complete if END OF LINE is reducible to it. end
END OF LINE (Papadimitriou 1991) start A problem belongs to PPAD if it is reducible in poly-time to END OF LINE; and PPAD -complete if END OF LINE is reducible to it. Not to be confused with OTHER END OF THIS LINE output unique vertex end found by “following the line” end from the start – this is PSPACE -hard
PPAD-hardness for bimatrix games Theorem (DGP06, CDT06 [5, 6]) It is PPAD -complete to compute an exact Nash equilibrium of a bimatrix game. Later we will see PPAD-hardness for approximate equilibria of bimatrix and polymatrix games
Outline 1 Nash equilibria of bimatrix games 2 Linear Complementarity Problems (LCPs) 3 The Lemke–Howson Algorithm and the class PPAD 4 Lemke’s algorithm 5 PLS-hardness of pure equilibria, Graph Transduction 6 Reduction from Polymatrix Game to LCP 7 Descent method for ǫ -Nash equilibria of polymatrix games 8 Other recent work on polymatrix games
Costs instead of payoffs 1 2 2 1 → 2 0 1 3 3 − a ik a ik payoff cost with new cost matrix A > 0 : z ≥ 0 ⊥ Az ≥ 1 equilibrium z ⇔
Polyhedral view z 2 1 z ≥ 0 1 1 2 + 1 z z ≥ 1 1 2 2 z 1 2 1 + 3 z z ≥ 1 1 2 z ≥ 0 2
Lemke's algorithm given LCP z ≥ 0 ⊥ w = q + Mz ≥ 0
Lemke's algorithm augmented LCP z ≥ 0 ⊥ w = q + Mz + d z 0 ≥ 0 z 0 ≥ 0
Lemke's algorithm augmented LCP z ≥ 0 ⊥ w = q + Mz + d z 0 ≥ 0 z 0 ≥ 0 where d > 0 covering vector z 0 extra variable z ⊥ w solves original LCP z 0 = 0 ⇔
Lemke's algorithm augmented LCP z ≥ 0 ⊥ w = q + Mz + d z 0 ≥ 0 z 0 ≥ 0 Initialization: z = 0 ⊥ w = q + d z 0 ≥ 0 z 0 ≥ 0 minimal ⇒ w i = 0 for some i pivot z 0 in, w i out, ⇒ can increase z i while maintaining z ⊥ w .
Lemke's algorithm for M = 2 1 , d = 2 1 3 1 − 1 w 1 2 1 2 = + z 1 + z 2 + z 0 − 1 w 2 1 3 1 1 − 5 − 2 w 1 0 = + z 1 + z 2 + w 2 1 − 1 − 3 − 1 z 0
− 1 w 1 2 1 2 = + z 1 + z 2 + z 0 − 1 w 2 1 3 1 1 − 5 − 2 w 1 0 = + z 1 + z 2 + w 2 1 − 1 − 3 − 1 z 0 − 0.2 − 0.4 z 2 0.2 0 = + z 1 + w 1 + w 2 0.4 − 1 z 0 0.6 0.2
1 − 5 − 2 w 1 0 = + z 1 + z 2 + w 2 1 − 1 − 3 − 1 z 0 − 0.2 − 0.4 z 2 0.2 0 = + z 1 + w 1 + w 2 0.4 − 1 z 0 0.6 0.2 − 0.2 − 0.4 z 2 0.2 0 = + z 0 + w 1 + w 2 0.4 − 1 z 1 0.6 0.2
Polyhedral view of Lemke
Polyhedral view of Lemke z 2 1 1 2 z 1 2
Polyhedral view of Lemke z 2 1 1 2 z 1 2 z 0
Polyhedral view of Lemke z 2 1 1 2 z 1 2 z 0
Polyhedral view of Lemke z 2 1 1 2 z 1 2 z 0
Polyhedral view of Lemke z 2 1 1 2 z 1 2 z 0
Polyhedral view of Lemke z 2 1 1 z = 0 0 2 z 1 2 z 0
Outline 1 Nash equilibria of bimatrix games 2 Linear Complementarity Problems (LCPs) 3 The Lemke–Howson Algorithm and the class PPAD 4 Lemke’s algorithm 5 PLS-hardness of pure equilibria, Graph Transduction 6 Reduction from Polymatrix Game to LCP 7 Descent method for ǫ -Nash equilibria of polymatrix games 8 Other recent work on polymatrix games
The class PLS (Polynomial Local Search) s Given a starting solution s ∈ S = Σ n a P-time algorithm that computes the cost c ( s ) a P-time function that computes a neighbouring solution s ′ ∈ N ( s ) with lower cost, i.e. s.t. c ( s ′ ) < c ( s ) , or reports that no such neighbour exists: find a local optimum of the cost function c “every DAG has a sink”
Local Max Cut Find local optimum of Max Cut with the FLIP-neighbourhood (exactly one node can change sides) Sch¨ affer and Yannakakis [22] showed that Local Max Cut is PLS-complete (via an extremely involved reduction) Local Max Cut is to PLS what 3-SAT is to NP
Local Max Cut Find local optimum of Max Cut with the FLIP-neighbourhood (exactly one node can change sides) Sch¨ affer and Yannakakis [22] showed that Local Max Cut is PLS-complete (via an extremely involved reduction) Local Max Cut is to PLS what 3-SAT is to NP 1 1 2 − 4 1 1 3 3 4 − 2
Local Max Cut Find local optimum of Max Cut with the FLIP-neighbourhood (exactly one node can change sides) Sch¨ affer and Yannakakis [22] showed that Local Max Cut is PLS-complete (via an extremely involved reduction) Local Max Cut is to PLS what 3-SAT is to NP 1 1 2 − 4 Solutions : {{ 1 , 3 , 4 } , { 2 }} (actual Max Cut) 1 1 3 3 4 − 2
Local Max Cut Find local optimum of Max Cut with the FLIP-neighbourhood (exactly one node can change sides) Sch¨ affer and Yannakakis [22] showed that Local Max Cut is PLS-complete (via an extremely involved reduction) Local Max Cut is to PLS what 3-SAT is to NP 1 1 2 − 4 Solutions : {{ 1 , 3 , 4 } , { 2 }} (actual Max Cut) 1 1 3 {{ 3 } , { 1 , 2 , 4 }} 3 4 − 2
Pure Equilibrium in Polymatrix Game 2 1 2 − 1 2 3
Pure Equilibrium in Polymatrix Game a b a 0 , 0 2 , 2 b 2 , 2 0 , 0 1 2 a b a b a 0 , 0 -1 , -1 a 0 , 0 2 , 2 b -1 , -1 0 , 0 b 2 , 2 0 , 0 3
Pure Equilibrium in Polymatrix Game a b a 0 , 0 2 , 2 b 2 , 2 0 , 0 1 2 a b a b a 0 , 0 -1 , -1 a 0 , 0 2 , 2 b -1 , -1 0 , 0 b 2 , 2 0 , 0 3 The bimatrix games ( A , B ) we used are examples of team games because A = B ; also called coordination games
Proof that the reduction is correct Define potential function for “team” polymatrix games Φ( S ) = 1 � u i ( S ) 2 i This is an exact potential function : when i changes strategy then the potential function changes by exactly i ’s change in utility Fact: in exact potential games, pure equilibria ↔ local optima of exact potential function Our exact potential function value equals value of the cut for all strategy profiles �
Summary on PLS and polymatrix games In contrast to bimatrix games , computing a pure equilibrium in polymatrix games is PLS -hard Next, an application of team polymatrix games
Application: Graph Transduction semi-supervised learning : estimate a classification function defined over graph of labeled and unlabeled nodes ie. propagate labels to unlabelled nodes in consistent way
Application: Graph Transduction semi-supervised learning : estimate a classification function defined over graph of labeled and unlabeled nodes ie. propagate labels to unlabelled nodes in consistent way INPUT: Weighted graph , where some nodes are labelled ; edge weights represent similarities one approach is to use global optimization an alternative approach is to use a polymatrix game
Application: Graph Transduction semi-supervised learning : estimate a classification function defined over graph of labeled and unlabeled nodes ie. propagate labels to unlabelled nodes in consistent way INPUT: Weighted graph , where some nodes are labelled ; edge weights represent similarities one approach is to use global optimization an alternative approach is to use a polymatrix game Note: without the labelled examples, this is a clustering problem; also see e.g., “Hedonic Clustering Games” [12, 2]
Application: Graph Transduction a b a 2 , 2 0 , 0 b 0 , 0 2 , 2 1 2 a b a b a -1 , -1 0 , 0 a 2 , 2 0 , 0 b 0 , 0 -1 , -1 b 0 , 0 2 , 2 3
Application: Graph Transduction a b a 2 , 2 0 , 0 b 0 , 0 2 , 2 1 2 a b a b a -1 , -1 0 , 0 a 2 , 2 0 , 0 b 0 , 0 -1 , -1 b 0 , 0 2 , 2 3 Note: asymmetric similarity measures have also been considered. Then we may no longer have pure equilibria, but mixed equilibria are still considered meaningful
Open question for team polymatrix games Can we compute a mixed Nash equilibrium of a team polymatrix game in polynomial-time? [7] Note that this problem lies in PPAD ∩ PLS so is unlikely to be hard for either of them
Open question for team polymatrix games Can we compute a mixed Nash equilibrium of a team polymatrix game in polynomial-time? [7] Note that this problem lies in PPAD ∩ PLS so is unlikely to be hard for either of them Question: Can anyone think of an easy mixed equilibrium for the local max cut game?
Open question for team polymatrix games Can we compute a mixed Nash equilibrium of a team polymatrix game in polynomial-time? [7] Note that this problem lies in PPAD ∩ PLS so is unlikely to be hard for either of them Question: Can anyone think of an easy mixed equilibrium for the local max cut game? Suggested reading: Daskalakis & Papadimitriou Continuous local search SODA 2011
Outline 1 Nash equilibria of bimatrix games 2 Linear Complementarity Problems (LCPs) 3 The Lemke–Howson Algorithm and the class PPAD 4 Lemke’s algorithm 5 PLS-hardness of pure equilibria, Graph Transduction 6 Reduction from Polymatrix Game to LCP 7 Descent method for ǫ -Nash equilibria of polymatrix games 8 Other recent work on polymatrix games
Polymatrix games → LCPs At least three different reductions to LCP; each gives an almost-complementarity algorithm 1 Howson 1972 [15] 2 Eaves 1973 [9] (more general) 3 Miller and Zucker 1991 [19] Instead we are going to present bilinear games which appeared in Ruta Mehta’s thesis [18, 13], and which are a specialization of Eave’s games
Bilinear Games Inspired by sequence form of Koller, Megiddo, von Stengel (1996) [17] They turn out to be are a special case of Eaves’ polymatrix games with joint constraints [9], where we restrict to: two players polytopal strategy constraint sets
Bilinear Games A bilinear game is given by: two m × n dimensional payoff matrices A and B polytopal strategy constraint sets : X = { x ∈ R m | Ex = e , x ≥ 0 } Y = { y ∈ R n | Fy = f , y ≥ 0 } With payoffs x T Ay and x T By for the strategy profile ( x , y ) ∈ X × Y
Bilinear Games A bilinear game is given by: two m × n dimensional payoff matrices A and B polytopal strategy constraint sets : X = { x ∈ R m | Ex = e , x ≥ 0 } Y = { y ∈ R n | Fy = f , y ≥ 0 } ( x , y ) ∈ X × Y is a Nash equilibrium iff x T Ay ≥ ¯ x T A for all ¯ x ∈ X and x T By ≥ x T B ¯ y for all ¯ y ∈ Y
An LCP for Bilinear Games Encode best response condition via an LP : x ⊤ ( Ay ) max x x ⊤ E ⊤ = e ⊤ , s.t. x ≥ 0
An LCP for Bilinear Games Encode best response condition via an LP : x ⊤ ( Ay ) max x x ⊤ E ⊤ = e ⊤ , s.t. x ≥ 0 The dual LP has an unconstrained vector p : e ⊤ p min y E ⊤ p ≥ Ay s.t. We will again use complementary slackness :
An LCP for Bilinear Games Encode best response condition via an LP : x ⊤ ( Ay ) max x x ⊤ E ⊤ = e ⊤ , s.t. x ≥ 0 The dual LP has an unconstrained vector p : e ⊤ p min y E ⊤ p ≥ Ay s.t. We will again use complementary slackness : Feasible x , p are optimal iff x ⊤ ( Ay ) = e ⊤ p = x ⊤ E ⊤ p , i.e.,
An LCP for Bilinear Games Encode best response condition via an LP : x ⊤ ( Ay ) max x x ⊤ E ⊤ = e ⊤ , s.t. x ≥ 0 The dual LP has an unconstrained vector p : e ⊤ p min y E ⊤ p ≥ Ay s.t. We will again use complementary slackness : x ⊤ ( − Ay + E ⊤ p ) = 0
An LCP for Bilinear Games Find: z , w ∈ R n so that Given: q ∈ R n , M ∈ R n × n E ⊤ − E ⊤ − A 0 − B ⊤ F ⊤ − F ⊤ 0 − E e M = q = E − e − F f F − f z = ( x , y , p ′ , p ′′ , q ′ , q ′′ ) ⊤ where p = p ′ − p ′′ , q = q ′ − q ′′
Lemke’s algorithm for Bilinear Games Theorem 4.1 in [17] says: If we have 1 z ⊤ Mz ≥ 0 for all z ≥ 0 , and 2 z ≥ 0 , Mz ≥ 0 and z ⊤ Mz = 0 imply that z ⊤ q ≥ 0 then Lemke’s algorithm computes an solution to the LCP M , q
Polymatrix games as Bilinear Games Polymatrix game (with complete interaction graph): players i = 1 , . . . , n , with pure strategy sets S i and payoff matrices for player i , A ij ∈ R | S i |×| S j | for pairs of players ( i , j ) let ( x 1 , . . . , x n ) in ∆( S i ) × · · · × ∆( S n ) be a mixed strategy profile, then the payoff to player i is � ( x i ) ⊤ A ij x j u i ( x 1 , . . . , x n ) = i � j
Polymatrix games as Bilinear Games (Symmetric) bilinear game: ( A , A ⊤ , E , E , e , e ) payoff matrices ( A , A ⊤ ) strategy constraints Ex = e where e = ✶ n , and ✶ ⊤ 0 · · · 0 A 12 A 1 n 0 · · · | S 1 | ✶ ⊤ A 21 A 2 n 0 · · · 0 0 | S 2 | A = E = . ... . ... . . . . A n 1 A n 2 · · · 0 ✶ ⊤ 0 0 · · · | S n |
Reductions for sparse polymatrix games Existing reductions apply to polymatrix games on complete interaction graphs For other interactions graphs, missing edges are replaced with games with all 0 payoffs Can we come up with more space efficient reductions for non-complete interaction graphs ?
Recommend
More recommend
