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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

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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

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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

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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

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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

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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

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Definition (Congestion Model)

M = (N, R, S, (di)i∈N, (ci,r)r∈R;i∈N),

◮ N = {1, . . . , n} players with demands di > 0 ◮ R = {1, . . . , m} resources ◮ Strategies S = ×Si, with Si ⊆ 2R ◮ Strategy distribution (xS)S∈Si satisfying S∈Si xS = di. ◮ ci,r : R+ → R+, increasing and convex

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Definition (Congestion Model)

M = (N, R, S, (di)i∈N, (ci,r)r∈R;i∈N),

◮ N = {1, . . . , n} players with demands di > 0 ◮ R = {1, . . . , m} resources ◮ Strategies S = ×Si, with Si ⊆ 2R ◮ Strategy distribution (xS)S∈Si satisfying S∈Si xS = di. ◮ ci,r : R+ → R+, increasing and convex ◮ Complete strategy profile x = ((xS)S∈Si)i∈N ◮ Load on resource r: xr = S∈S;r∈S xS ◮ Load of a player i: xi,r = S∈Si;r∈S xS

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Example

s t 1 5 6 2 4 3

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Example

s t 1 5 6 2 4 3

◮ Player cost πi(x) = r∈R xi,rci,r(xr)

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Example

s t 1 5 6 2 4 3

◮ Player cost πi(x) = r∈R xi,rci,r(xr) ◮ Marginal cost resource: µi,r(x) = ci,r(xr) + xi,rc′ 1(xr)

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Example

s t 1 5 6 2 4 3

◮ Player cost πi(x) = r∈R xi,rci,r(xr) ◮ Marginal cost resource: µi,r(x) = ci,r(xr) + xi,rc′ 1(xr) ◮ Marginal cost strategy: µi,S(x) = r∈S µi,r(x)

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Example

s t 1 5 6 2 4 3

◮ Player cost πi(x) = r∈R xi,rci,r(xr) ◮ Marginal cost resource: µi,r(x) = ci,r(xr) + xi,rc′ 1(xr) ◮ Marginal cost strategy: µi,S(x) = r∈S µi,r(x) ◮ Equilibrium condition: If xi,S > 0, then µi,S(x) ≤ µi,S′(x)

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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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

rder-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

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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

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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

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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

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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

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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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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 Strong exchange property Let B be the set of all bases of a matroid M = (S, I). Let I and J be two elements of B. Then for all x ∈ I \ J, there exists y ∈ J \ I such that I − x + y ∈ B and J − y + x ∈ B.

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Perfect matching in the symmetric difference

We define the class of Two Sided Matching Matroids:

Definition (Two Sided Matching Matroids)

A matroid M = (S, I) is a Two Sided Matching Matroid if for every pair (I, J) of bases of M there exists a perfect matching W (I, J) in the symmetric difference I∆J. This matching has the property that for an edge (x, y) ∈ W (I, J) with x ∈ I \ J and y ∈ J \ I both: I − x + y ∈ I and J + x − y ∈ I

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

1 2 3 4 5 1 2 5 1 3 4

Figure: Two bases I (red) and J (blue) of a graphic matroid

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

1 2 3 4 5 1 2 5 1 3 4

Figure: Two bases I (red) and J (blue) of a graphic matroid

I \ J 2 5 J \ I 3 4

Figure: Perfect matching in the symmetric difference

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

1 2 3 4 5 1 2 5 1 3 4

Figure: Two bases I (red) and J (blue) of a graphic matroid

I \ J 2 5 J \ I 3 4

Figure: Perfect matching in the symmetric difference

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

1 2 3 4 5 1 2 5 1 3 4

Figure: Two bases I (red) and J (blue) of a graphic matroid

I \ J 2 5 J \ I 3 4

Figure: Perfect matching in the symmetric difference

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

1 2 3 4 5 1 2 5 1 3 4

Figure: Two bases I (red) and J (blue) of a graphic matroid

I \ J 2 5 J \ I 3 4

Figure: Perfect matching in the symmetric difference

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

1 2 3 4 5 1 2 5 1 3 4

Figure: Two bases I (red) and J (blue) of a graphic matroid

I \ J 2 5 J \ I 3 4

Figure: Perfect matching in the symmetric difference

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

1 2 3 4 5 1 2 5 1 3 4

Figure: Two bases I (red) and J (blue) of a graphic matroid

I \ J 2 5 J \ I 3 4

Figure: Perfect matching in the symmetric difference

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

1 2 3 4 5 1 2 5 1 3 4

Figure: Two bases I (red) and J (blue) of a graphic matroid

I \ J 2 5 J \ I 3 4

Figure: Perfect matching in the symmetric difference

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

1 2 3 4 5 1 2 5 1 3 4

Figure: Two bases I (red) and J (blue) of a graphic matroid

I \ J 2 5 J \ I 3 4

Figure: Perfect matching in the symmetric difference

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Theorem

Two Sided Matching-matroid congestion games have a unique Nash-equilibrium. Proof by contradiction. Assume that there are two different Nash equibria x and y on Two Sided Matching matroid M = (S, I).

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Theorem

Two Sided Matching-matroid congestion games have a unique Nash-equilibrium. Proof by contradiction. Assume that there are two different Nash equibria x and y on Two Sided Matching matroid M = (S, I). We define R+ = {r ∈ R|xr − yr > 0} We look at a player i with

r∈R+

xi,r − yi,r > 0

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B1 B2 B3 B4 Bk B1 B2 B3 B4 Bk x y

Figure: Two Nash equilibria x and y

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B1 B2 B3 B4 Bk B1 B2 B3 B4 Bk 1 2 2 3 4 2 2 1 1 x y

Figure: Two Nash equilibria x and y

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B1 B2 B3 B4 Bk B1 B2 B3 B4 Bk 1 2 2 3 4 2 2 1 1 x y underloaded → underloaded →

verloaded →
verloaded →

Figure: Two Nash equilibria x and y

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B1 B2 B3 B4 Bk B1 B2 B3 B4 Bk 1 2 2 3 4 2 2 1 1 x y underloaded → underloaded →

verloaded →
verloaded →

Bi,+ Bi,−

Figure: Two Nash equilibria x and y

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B1 B2 B3 B4 Bk B1 B2 B3 B4 Bk 1 2 2 3 4 2 2 1 1 x y underloaded → underloaded →

verloaded →
verloaded →

Bi,+ Bi,−

Figure: Two Nash equilibria x and y

We make a transshipment from Bi,+ to Bi,−.

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B B′ tB,B′

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B B′ tB,B′

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B B′ tB,B′ B B′

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B B′ tB,B′ B B′

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

B B′ tB,B′ B B′ tB,B′ tB,B′

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t Cut T R+

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t Cut T R+

r∈R+ xi,r − yi,r > 0

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t Cut T R+

r∈R+ xi,r − yi,r > 0

u v

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t Cut T R+

r∈R+ xi,r − yi,r > 0

u v µu(y) < µu(x)

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t Cut T R+

r∈R+ xi,r − yi,r > 0

u v µu(y) < µu(x) ≤ µv(x)

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t Cut T R+

r∈R+ xi,r − yi,r > 0

u v µu(y) < µu(x) ≤ µv(x) < µv(y)

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

V1 V2 s t Cut T R+

r∈R+ xi,r − yi,r > 0

u v µu(y) < µu(x) ≤ µv(x) < µv(y) ≤ µu(y)

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Laminar matroids

Definition (Laminar family)

A family F of subsets of a finite set S is called laminar if for all X, Y ∈ F one has X ⊆ Y or Y ⊆ X or X ∩ Y = ∅.

Definition (Laminar matroid)

Let F be a laminar family defined on S, such that each x ∈ S is in some set X ∈ F. For each X ∈ F, let k(X) be a positive integer associated with it. Let I = {I : I ⊆ S, |I ∩ X| ≤ k(X) ∀X ∈ F}. Then M(S, I) is a matroid. uniform matroids ⊂ partition matroids ⊂ laminar matroids

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Traversal matroids

Definition (Traversal)

Let A = (A1, . . . An) be a family of sets. A set T is called a traversal of A if there exist distict elements a1 ∈ A1, . . . , an ∈ An such that T = {a1, . . . an}. A set T ′ is called a partial transversal if it is the transversal of some subfamily

Definition (Traversal matroid)

Let X = (X1, . . . , Xn) be a family of subsets of a finite set S and let I be the collection of partial transversals of X. Then M = (S, I) is a matroid.

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Graphic Matroids on GSP graphs

Definition (GSP-graph)

A graph is generalized series parallel, if it may be turned into K2 by a sequence of the following operations:

◮ Replacement of a pair of parallel edges e1, e2 with a single

edge e that connects their common endpoints.

◮ Replacement of a pair of edges e1, e2 incident to a vertex of

degree 2 other than s or t with a single edge e.

◮ Deletion of a vertex of degree one.

⇒ ⇒ e e1 e2 e2 e1 e

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Theorem

A graph is GSP if and only if does not contain K4 as a minor. [Korneyenko,1994], [Nishizeki et al, 1988]

Theorem

GSP-graphs are the maximal structure in network congestion games where every combination of bases of the graphic matroid of this graph contain a two-sided perfect matching 1 2 3 4 5 6

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Theorem

A graph is GSP if and only if does not contain K4 as a minor. [Korneyenko,1994], [Nishizeki et al, 1988]

Theorem

GSP-graphs are the maximal structure in network congestion games where every combination of bases of the graphic matroid of this graph contain a two-sided perfect matching 1 2 3 4 5 6 1 2 6 3 4 5

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games

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Motivation The Model Two-Sided Matching Matroids Uniqueness of Nash Equilibria for TSM Matroids

Open problems

Are there more existing classes of matroids that are two sided matching matroids? Can we find an example of a matroid congestion game that has multiple equilibria? (Using the K4)

Veerle Timmermans, Tobias Harks Uniqueness of Nash Equilibria in Matroid Congestion Games