Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Uniqueness of Nash Equilibria in Atomic Splittable Congestion Games Veerle Timmermans Tobias Harks Maastricht University Dagstuhl 2015 Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Motivation and related work ”In general, a game may have several equilibria. Yet uniqueness is crucial . . . . Nash equilibrium makes sense only if each player knows which strategies the others are playing; if the equilibrium recommended by the theory is not unique, the players will not have this knowledge.” - Robert J. Aumann (foreword to Harsanyi & Seltens book) Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Motivation and related work ”In general, a game may have several equilibria. Yet uniqueness is crucial . . . . Nash equilibrium makes sense only if each player knows which strategies the others are playing; if the equilibrium recommended by the theory is not unique, the players will not have this knowledge.” - Robert J. Aumann (foreword to Harsanyi & Seltens book) ◮ Non-atomic players: unique NE [Milchtaich, 2000] Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Motivation and related work ”In general, a game may have several equilibria. Yet uniqueness is crucial . . . . Nash equilibrium makes sense only if each player knows which strategies the others are playing; if the equilibrium recommended by the theory is not unique, the players will not have this knowledge.” - Robert J. Aumann (foreword to Harsanyi & Seltens book) ◮ Non-atomic players: unique NE [Milchtaich, 2000] ◮ Standard game with atomic players: often multiple equilibria Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Motivation and related work ”In general, a game may have several equilibria. Yet uniqueness is crucial . . . . Nash equilibrium makes sense only if each player knows which strategies the others are playing; if the equilibrium recommended by the theory is not unique, the players will not have this knowledge.” - Robert J. Aumann (foreword to Harsanyi & Seltens book) ◮ Non-atomic players: unique NE [Milchtaich, 2000] ◮ Standard game with atomic players: often multiple equilibria ◮ Atomic splittable games → ? Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Definition (Congestion Model) M = ( N , R , S , ( d i ) i ∈ N , ( c i , r ) r ∈ R ; i ∈ N ), ◮ N = { 1 , . . . , n } players with demands d i > 0 ◮ R = { 1 , . . . , m } resources ◮ Strategies S = ×S i , with S i ⊆ 2 R ◮ Strategy distribution ( x S ) S ∈S i satisfying � S ∈S i x S = d i . ◮ c i , r : R + → R + , increasing and convex Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Definition (Congestion Model) M = ( N , R , S , ( d i ) i ∈ N , ( c i , r ) r ∈ R ; i ∈ N ), ◮ N = { 1 , . . . , n } players with demands d i > 0 ◮ R = { 1 , . . . , m } resources ◮ Strategies S = ×S i , with S i ⊆ 2 R ◮ Strategy distribution ( x S ) S ∈S i satisfying � S ∈S i x S = d i . ◮ c i , r : R + → R + , increasing and convex ◮ Complete strategy profile x = (( x S ) S ∈S i ) i ∈ N ◮ Load on resource r : x r = � S ∈S ; r ∈ S x S ◮ Load of a player i : x i , r = � S ∈S i ; r ∈ S x S Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Example 1 5 3 s t 2 4 6 Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Example 1 5 3 s t 2 4 6 ◮ Player cost π i ( x ) = � r ∈ R x i , r c i , r ( x r ) Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Example 1 5 3 s t 2 4 6 ◮ Player cost π i ( x ) = � r ∈ R x i , r c i , r ( x r ) ◮ Marginal cost resource: µ i , r ( x ) = c i , r ( x r ) + x i , r c ′ 1 ( x r ) Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Example 1 5 3 s t 2 4 6 ◮ Player cost π i ( x ) = � r ∈ R x i , r c i , r ( x r ) ◮ Marginal cost resource: µ i , r ( x ) = c i , r ( x r ) + x i , r c ′ 1 ( x r ) ◮ Marginal cost strategy: µ i , S ( x ) = � r ∈ S µ i , r ( x ) Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Example 1 5 3 s t 2 4 6 ◮ Player cost π i ( x ) = � r ∈ R x i , r c i , r ( x r ) ◮ Marginal cost resource: µ i , r ( x ) = c i , r ( x r ) + x i , r c ′ 1 ( x r ) ◮ Marginal cost strategy: µ i , S ( x ) = � r ∈ S µ i , r ( x ) ◮ Equilibrium condition: If x i , S > 0, then µ i , S ( x ) ≤ µ i , S ′ ( x ) Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Related work atomic splittable games On cost function: ◮ When players experience congestion the same: Polynomials with degree at most 3 [Altman et al., 2002] On strategy space in network congestion games: ◮ If players are of the same type [Orda et al., 1993] Using a potential function [Cominetti, Correa and Stier-Moses, 2009] ◮ If the network is a two-terminal nearly-parallel graph [Richman and Shimkin, 2007] ◮ If the network is a ring graph, with some properties on the order-destination pairs [Meunier and Pradeau, 2014] ◮ When players experience congestion the same, full characterization by [Bhaskar et al., 2015] Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Can we find a sufficient condition that guarantees us a unique Nash equilibrium, no matter how strategy spaces are interweaved? Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Can we find a sufficient condition that guarantees us a unique Nash equilibrium, no matter how strategy spaces are interweaved? → Two-sided matching matroids are sufficient. Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Can we find a sufficient condition that guarantees us a unique Nash equilibrium, no matter how strategy spaces are interweaved? → Two-sided matching matroids are sufficient. → Currently working on the generalization to polymatroids Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Definition Matroid A matroid is a pair M = ( S , I ) where S is a set of resources, and I is a family of subsets of S such that: ◮ I � = ∅ ◮ If I ⊂ J and J ∈ I , then I ∈ I ◮ Let I , J ∈ I and | I | < | J | , then there exists an x ∈ S such that I + s ∈ I Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids Definition Matroid A matroid is a pair M = ( S , I ) where S is a set of resources, and I is a family of subsets of S such that: ◮ I � = ∅ ◮ If I ⊂ J and J ∈ I , then I ∈ I ◮ Let I , J ∈ I and | I | < | J | , then there exists an x ∈ S such that I + s ∈ I Bases are sets in I of maximal cardinality, denoted by B Veerle Timmermans , Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games
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