Convex games and bargaining sets · 2010. 9. 9. · J.M. Izquierdo & C. Rafels | Convex games and...

Introduction Preliminaries Max-payoff vectors: a necessary condition Characterization result References

Convex games and bargaining sets

J.M. Izquierdo & C. Rafels

Universitat de Barcelona

SCW2010 - Moscow

J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 1/25


Outline

1 Introduction

2 Preliminaries

3 Max-payoff vectors: a necessary condition

4 Characterization result



Outline

1 Introduction

2 Preliminaries





IntroductionConvex games

We study cooperative situations among agents (cooperativeTU games) where marginal contributions of agents grow ascoalitions players add also grow.

N = {1, 2, . . . , n} is the set of players.

v(S) is the worth of coalition S ⊆ N .

For all i ∈ N and for all S ⊆ T ⊆ N \ {i},

v(S ∪ {i})− v(S) ≤ v(T ∪ {i})− v(T ).

It is said the game is convex (Shapley, 1971)



IntroductionSome characterizations of a convex game

Some characterizations of a convex game are:

v is convex ⇔ marginal worth vectors are in the coreShapley (1971) and Ichiisi (1981)

⇔ the core and the Weber set coincideWeber (1988)

⇔ the cores of the game and subgamesare stable sets Einy and Shitovitz (1996)

⇔ the Weber set is a subset ofthe DM bargaining set (balanced games)

Izquierdo & Rafels (2008)



Introduction

Can we characterize the convexity of a game by comparing thecore and the bargaining set?



Outline

1 Introduction

2 Preliminaries





DefinitionsConvex games

A cooperative TU-game a pair (N , v) whereN = {1, 2, . . . , n} is the set of players and

v is the characteristic function, Sv→ v(S), v(∅) = 0.

A game is 0-monotonic if for all S ⊆ T ⊆ N ,

v(S) +∑

i∈T\S

v(i) ≤ v(T ).

A game is superadditive if for all S , T ⊆ N , S ∩ T ) 6= ∅v(S) + v(T ) ≤ v(S ∪ T ).

A game v is convex if, for all i ∈ N and for allS ⊆ T ⊆ N \ {i}

v(S ∪ {i})− v(S) ≤ v(T ∪ {i})− v(T ).



DefinitionsCore

The set of preimputations of a game v is

I ∗(v) = {x ∈ RN | x(N) = v(N)}

The set of imputations of a game v is

I (v) = {x ∈ RN | x(N) = v(N) and xi ≥ v(i), for all i ∈ N}

The core of a game v is

C (v) = {x ∈ I (v) | x(S) ≥ v(S), for all S ⊆ N}



DefinitionsBargaining set

A payoff vector x ∈ RN is in the bargaining set if for everyobjection to x there is a counterobjection.

Several definitions:

Davis and Maschler (1963)

Mas-Colell (1989)

Zhou (1994)

Shimomura (1997)

Granot (2010)

Sudholter and Potters (2001).



DefinitionsBargaining set

An objection to x ∈ RN is a coalition S ⊆ N and a payoffvector y ∈ RS such that

yi > xi for all i ∈ S and y(S) = v(S)

A counterobjection to some objection(S , y) is a coalitionT ⊆ N and a payoff vector z ∈ RT such that

zi > yi for all i ∈ T∩S , zi > xi , for all i ∈ T\S and z(T ) = v(T )



DefinitionsBargaining sets

DM MC Z ZSH MCSH

M(i)1 (v) MB(v) Z(v) ZSh(v) MBSh(v)

x I (v) I ∗(v) I (β) I (v) I (v)

OBJ

S yk > xk yk ≥ xk yk > xk yk > xk yk > xk

S ∈ Γij ONE STRICT

COUNT OBJ

S ∩ T zk ≥ yk zk ≥ yk zk > yk zk > yk zk > yk

T \ S zk ≥ xk zk ≥ xk zk > xk zk > xk zk > xk

ONE STRICT

T ∈ Γji S ∩ T 6= ∅ S ∩ T 6= ∅S \ T 6= ∅ S \ T 6= ∅T \ S 6= ∅ T \ S 6= ∅



Outline

1 Introduction

2 Preliminaries





Max-payoff vectors: a necessary conditionmarginal worth vectors

A marginal worth vector of the game v relative toθ = (i1, . . . , in), mθ(v), is defined as:

mθik

(v) := v({i1, . . . ik})− v({i1, . . . ik−1}), for all k = 1, . . . , n.

mθii(v) = v({i1}),

mθi2

(v) = v({i1, i2})− v({i1}),

mθi3

(v) = v({i1, i2, i3})− v({i1, i2})...mθ

in(v) = v({i1, . . . , in})− v({i1, . . . , in−1})



Max-payoff vectors: a necessary conditionmax-payoff vectors

A max-payoff vector xθ(v) of v relative to θ = (i1, . . . , in) isdefined by

xθik

:= maxQ⊆Pθ

ik

{v({ik} ∪ Q)− xθ(Q)}, for all k ∈ {1, . . . , n − 1},

xθin

:= v(N)− xθ(N \ {in}).

xθii

(v) = v({i1}),

xθi2

(v) = max{v({i2}), v({i1, i2})− xθi1

(v)},

xθi3

(v) = max{v({i3}), v({i1, i3})− xθi1

(v), v({i2, i3})− xθi2

(v),

v({i1, i2, i3})− xθi1

(v)− xθi2

(v)}...xθin

(v) = v(N)− xθ(N \ {in})



Max-payoff vectors: a necessary conditionAn example

Consider the 2× 2 glove market defined by matrix

3 41 1 12 1 1

v({i}) = 0v({1, 3}) = v({1, 4}) = 1v({2, 3}) = v({2, 4}) = 1v(S) = 1, if |S | = 3v(N) = 2

Take the ordering θ∗ = (1, 3, 4, 2),

1 3 4 2mθ∗(v) 0 1 0 1xθ∗(v) 0 1 1 0



Max-payoff vectors: a necessary conditionProperties of max-payoff vectors

Max-payoff vector. Property 1

Given θ = (i1, i2, . . . , in) and S ⊆ N ,

if xθ(S) < v(S), then in ∈ S

Max-payoff vector. Property 2

If v is convex xθ(v) = mθ(v), for all ordering θ = (i1, i2, . . . , in)



Max-payoff vectors: a necessary conditionThe Theorem

Theorem 1 (Izquierdo and Rafels, 2010a)

For any arbitrary balanced game v ∈ BN we have:

1. If C (v) = ZSh(v), then xθ(v) ∈ C (v), for all θ ∈ ΘN

2. If C (v) =MBSh(v), then xθ(v) ∈ C (v), for all θ ∈ ΘN .

Hint: If not, by Property 1, we can raise an objection to xθ(v)

through a coalition of maximal excess S∗ with in ∈ S∗, giving as

much as possible to player in.

The condition is not sufficient: for the game associated tothe 2× 2 glove market, xθ(v) ∈ C (v), for all θ ∈ ΘN , but thecore is strictly included in the Shimonura bargaining sets.



Outline

1 Introduction

2 Preliminaries





Characterization resultBargaining sets and the core

The coincidence of the core with the bargaining set

CONVEX GAMES AVERAGE MONOTONIC GAMES ASSIGNMENT GAMES

DAVIS-MASCHLER X X XMAS-COLELL X X X

MAS-COLELLSh X x xZHOU , ZHOUSh X x x



CharacterizationBargaining sets and the core

Theorem 2 (Izquierdo and Rafels, 2010b)

Let v ∈ GN . Then,

1. v is convex ⇔ ZSh(v) = C (v) and v is superadditive;

2. v is convex ⇔MBSh(v) = C (v) and v is 0-monotonic.



CharacterizationBargaining sets and the core

sketch of the proof

if v is convex then C (v) = ZSH(v) =MBSh(v))

Let x ∈ I (v) \ C (v) and let S∗ ⊆ N be a minimal coalition of thelargest excess,

S∗ ∈ argmax{v(S)− x(S)}

where, if S S∗, v(S)− (S) < v(S∗)− x(S∗).

Compute the excess game

wx(S) := maxR⊆S{v(R)− x(R)}, for all S ⊆ N

and take the Shapley value of wx , namely Φ(wx).

define the objection (S∗, y) with yi = xi + Φi , for all i ∈ S∗.

this objection cannot be countered 2




sketch of the proof

If C (v) =MBSH(v) and v is 0-monotonic ⇒ v is convex)

By Theorem 1 xθ(v) ∈ C (v), for all θ.

Since the game is not convex there exists

θ∗ = (i1, . . . , ik∗−1, ik∗ , ik∗+1, . . . , in)

such that

? xθ∗(v) 6= mθ∗(v), (Property 2)

xθ∗ik∗> mθ∗

ik∗≥ v({ik∗}), and xθ∗

ik= mθ∗

ik, k = 1, . . . k∗−1.

? For all S ⊆ {i1, . . . , ik∗} such that |S | ≤ k ∗ −1, thesubgame vS is convex




The vector x 6∈ C (v) but we prove x ∈MBSH(v). 2J.M. Izquierdo & C. Rafels — Convex games and bargaining sets 24/25


References

Davis, M., Maschler, M., 1963. Existence of stable payoffconfigurations for cooperative games. B. Am. Math Soc. 69,106–108.

Einy, E., Shitovitz, B., 1996. Convex Games and Stable Sets,Games Econ. Behav. 16(2), 192–201.

Ichiishi, T., 1981. Super-modularity: applications to convexgames and to the greedy algorithm in LP, J Econ. Theory 25,283–286.

Izquierdo, J.M., Rafels, C., 2010a. On the coincidence betweenthe Shimomura’s bargaining sets and the core, Working paperseries E10/241, Faculty of Economics, University of Barcelona.

Shimomura, K., 1997. Quasi-Cores in Bargaining Sets. Int. J.Game Theory 26, 283–302.


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