cooperative interval games

6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011

Cooperative Game Theory. Operations ResearchGames. Applications to Interval Games

Lecture 4: Cooperative Interval Games

Sırma Zeynep Alparslan GokSuleyman Demirel University

Faculty of Arts and Sciences

Department of Mathematics

Isparta, Turkey

email:[email protected]

August 13-16, 2011


Outline

Introduction

Cooperative interval games

Interval solutions for cooperative interval games

Big boss interval games

Handling interval solutions

References


Introduction

Introduction

This lecture is based on the papersCooperative interval games: a survey by Branzei et al., which waspublished in Central European Journal of Operations Research(CEJOR),Set-valued solution concepts using interval-type payoffs for intervalgames by Alparslan Gok et al., which will appear in Journal ofMathematical Economics (JME) andConvex interval games by Alparslan Gok, Branzei and Tijs, whichwas published in Journal of Applied Mathematics and DecisionSciences.


Introduction

Motivation

Game theory:

I Mathematical theory dealing with models of conflict andcooperation.

I Many interactions with economics and with other areas suchas Operations Research (OR) and social sciences.

I Tries to come up with fair divisions.

I A young field of study:The start is considered to be the book Theory of Games andEconomic Behaviour by von Neumann and Morgernstern(1944).

I Two parts: non-cooperative and cooperative.


Introduction

Motivation

Cooperative game theory deals with coalitions who coordinate theiractions and pool their winnings.The main problem: Dividing the rewards/costs among themembers of the formed coalition.The situations are considered from a deterministic point of view.Basic models in which probability and stochastic theory play a roleare: chance-constrained games and cooperative games withstochastic/random payoffs.In this research, rewards/costs taken into account are not randomvariables, but just closed and bounded intervals of real numberswith no probability distribution attached.


Introduction

Motivation

Idea of interval approach: In most economic and OR situationsrewards/costs are not precise.

Possible: Estimating the intervals to which rewards/costs belong.

Why cooperative interval games are important?Useful for modeling real-life situations.

Aim: generalize and extend the classical theory to intervals andapply it to economic situations, popular OR games.


Introduction

Interval calculus

I (R): the set of all closed and bounded intervals in RI , J ∈ I (R), I =

[I , I], J =

[J, J], |I | = I − I , α ∈ R+

I addition: I + J =[I + J, I + J

]I multiplication: αI =

[αI , αI

]I subtraction: defined only if |I | ≥ |J|

I − J =[I − J, I − J

]I weakly better than: I < J if and only if I ≥ J and I ≥ J

I I 4 J if and only if I ≤ J and I ≤ J

I better than: I � J if and only if I < J and I 6= J

I I ≺ J if and only if I 4 J and I 6= J


Introduction

Classical cooperative games

A cooperative game < N, v >

I N = {1, 2, ..., n}:set of players

I v : 2N → R: characteristic function, v(∅) = 0

I v(S): worth (or value) of coalition S .

I x ∈ RN : payoff vector

GN : class of all cooperative games with player set NThe core (Gillies (1959)) of a game < N, v > is the set

C (v) =

{x ∈ RN |

∑i∈N

xi = v(N);∑i∈S

xi ≥ v(S) for each S ∈ 2N

}.

The idea: Giving every coalition S at least their worth v(S) so thatno coalition protests




I A cooperative interval game is an ordered pair < N,w >,where N is the set of players and w is the characteristicfunction of the game.

I N = {1, 2, ..., n}, w : 2N → I (R) is a map, assigning to eachcoalition S ∈ 2N a closed interval, such that w(∅) = [0, 0].

I w(S) = [w(S),w(S)]: worth (value) of S .

I w(S): lower bound, w(S): upper bound

IGN : class of all interval games with player set NExample (LLR-game): Let < N,w > be an interval game withw({1, 3}) = w({2, 3}) = w(N) = J < [0, 0] and w(S) = [0, 0]otherwise.



Arithmetic of interval games

w1,w2 ∈ IGN , λ ∈ R+, for each S ∈ 2N

I w1 4 w2 if w1(S) 4 w2(S)

I < N,w1 + w2 > is defined by (w1 + w2)(S) = w1(S) + w2(S)

I < N, λw > is defined by (λw)(S) = λ · w(S)

I < N,w1 − w2 > is defined by (w1 − w2)(S) = w1(S)− w2(S)with |w1(S)| ≥ |w2(S)|

Classical cooperative games associated with < N,w >:

I Border games < N,w >, < N,w >

I Length game < N, |w | >, where |w | (S) = w(S)− w(S) foreach S ∈ 2N .

w = w + |w |



Interval coreI (R)N : set of all n-dimensional vectors with elements in I (R).The interval imputation set:

I(w) =

{(I1, . . . , In) ∈ I (R)N |

∑i∈N

Ii = w(N), Ii < w(i), ∀i ∈ N

}.

The interval core:

C(w) =

{(I1, . . . , In) ∈ I(w)|

∑i∈S

Ii < w(S), ∀S ∈ 2N \ {∅}

}.

Example (LLR-game) continuation:

C(w) =

{(I1, I2, I3)|

∑i∈N

Ii = J,∑i∈S

Ii < w(S)

},

C(w) = {([0, 0], [0, 0], J)} .



Classical cooperative games

< N, v > is convex if and only if the supermodularity condition

v(S ∪ T ) + v(S ∩ T ) ≥ v(S) + v(T )

for each S ,T ∈ 2N holds.< N, v > is concave if and only if the submodularity condition

v(S ∪ T ) + v(S ∩ T ) ≤ v(S) + v(T )

for each S ,T ∈ 2N holds.For details on classical cooperative game theory we refer toBranzei, Dimitrov and Tijs (2008).



Convex and concave interval games

< N,w > is supermodular if

w(S) + w(T ) 4 w(S ∪ T ) + w(S ∩ T ) for all S ,T ∈ 2N .

< N,w > is convex if w ∈ IGN is supermodular and |w | ∈ GN issupermodular (or convex).< N,w > is submodular if

w(S) + w(T ) < w(S ∪ T ) + w(S ∩ T ) for all S ,T ∈ 2N .

< N,w > is concave if w ∈ IGN is submodular and |w | ∈ GN issubmodular (or concave).



Illustrative examples

Example 1: Let < N,w > be the two-person interval game withw(∅) = [0, 0], w({1}) = w({2}) = [0, 1] and w(N) = [3, 4].Here, < N,w > is supermodular and the border games are convex,but |w | ({1}) + |w | ({2}) = 2 > 1 = |w | (N) + |w | (∅).Hence, < N,w > is not convex.Example 2: Let < N,w > be the three-person interval game withw({i}) = [1, 1] for each i ∈ N,w(N) = w({1, 3}) = w({1, 2}) = w({2, 3}) = [2, 2] andw(∅) = [0, 0].Here, < N,w > is not convex, but < N, |w | > is supermodular,since |w | (S) = 0, for each S ∈ 2N .



Example (unanimity interval games):

Let J ∈ I (R) such that J � [0, 0] and let T ∈ 2N \ {∅}. Theunanimity interval game based on T is defined for each S ∈ 2N by

uT ,J(S) =

{J, T ⊂ S[0, 0] , otherwise.

< N, |uT ,J | > is supermodular, < N, uT ,J > is supermodular:

T ⊂ A,T ⊂ BT ⊂ A,T 6⊂ BT 6⊂ A,T ⊂ BT 6⊂ A,T 6⊂ B

uT ,J(A ∪ B) uT ,J(A ∩ B) uT ,J(A) uT ,J(B)J J J JJ [0, 0] J [0, 0]J [0, 0] [0, 0] J

J or [0, 0] [0, 0] [0, 0] [0, 0].



Size monotonic interval games

I < N,w > is size monotonic if < N, |w | > is monotonic, i.e.,|w | (S) ≤ |w | (T ) for all S ,T ∈ 2N with S ⊂ T .

I SMIGN : the class of size monotonic interval games withplayer set N.

I For size monotonic games, w(T )− w(S) is defined for allS ,T ∈ 2N with S ⊂ T .

I CIGN : the class of convex interval games with player set N.

I CIGN ⊂ SMIGN because < N, |w | > is supermodular impliesthat < N, |w | > is monotonic.



Generalization of Bondareva (1963) and Shapley (1967)

< N,w > is I-balanced if for each balanced map λ∑S∈2N\{∅}

λSw(S) 4 w(N).

IBIGN : class of interval balanced games with player set N.

CIGN ⊂ IBIGN

CIGN ⊂ (SMIGN ∩ IBIGN)

Theorem: Let w ∈ IGN . Then the following two assertions areequivalent:

(i) C(w) 6= ∅.(ii) The game w is I-balanced.



The interval Weber SetΠ(N): set of permutations, σ : N → N, of NPσ(i) =

{r ∈ N|σ−1(r) < σ−1(i)

}: set of predecessors of i in σ

The interval marginal vector mσ(w) of w ∈ SMIGN w.r.t. σ:

mσi (w) = w(Pσ(i) ∪ {i})− w(Pσ(i))

for each i ∈ N.

Interval Weber set W : SMIGN � I (R)N :

W(w) = conv {mσ(w)|σ ∈ Π(N)} .

Example: N = {1, 2}, w({1}) = [1, 3],w({2}) = [0, 0] and

w(N) = [2, 3 12 ]. This game is not size monotonic.

m(12)(w)is not defined.w(N)− w({1}) = [1, 1

2 ]: undefined since |w(N)| < |w({1})|.



The interval Shapley valueThe interval Shapley value Φ : SMIGN → I (R)N :

Φ(w) =1

n!

∑σ∈Π(N)

mσ(w), for each w ∈ SMIGN .

Example: N = {1, 2}, w({1}) = [0, 1],w({2}) = [0, 2],w(N) = [4, 8].

Φ(w) =1

2(m(12)(w) + m(21)(w));

Φ(w) =1

2((w({1}),w(N)− w({1})) + (w(N)− w({2}),w({2}))) ;

Φ(w) =1

2(([0, 1], [4, 7]) + ([4, 6], [0, 2])) = ([2, 3

1

2], [2, 4

1

2]).



Properties of solution concepts

I W(w) ⊂ C(w), ∀w ∈ CIGN and W(w) 6= C(w) is possible.Example: N = {1, 2}, w({1}) = w({2}) = [0, 1] andw(N) = [2, 4] (convex).W(w) = conv

{m(1,2)(w),m(2,1)(w)

}m(1,2)(w) = ([0, 1], [2, 4]− [0, 1]) = ([0, 1], [2, 3])m(2,1)(w) = ([2, 3], [0, 1]])m(1,2)(w) and m(2,1)(w) belong to C(w).([ 1

2 , 134 ], [1 1

2 , 214 ]) ∈ C(w)

no α ∈ [0, 1] exists satisfyingαm(1,2)(w) + (1− α)m(2,1)(w) = ([ 1

2 , 134 ], [1 1

2 , 214 ]).

I Φ(w) ∈ W(w) for each w ∈ SMIGN .

I Φ(w) ∈ C(w) for each w ∈ CIGN .



The square operator

I Let a = (a1, . . . , an) and b = (b1, . . . , bn) with a ≤ b.

I Then, we denote by a�b the vector

a�b := ([a1, b1] , . . . , [an, bn]) ∈ I (R)N

generated by the pair (a, b) ∈ RN × RN .

I Let A,B ⊂ RN . Then, we denote by A�B the subset ofI (R)N defined by

A�B := {a�b|a ∈ A, b ∈ B, a ≤ b} .

I For a multi-solution F : GN � RN we defineF� : IGN � I (R)N by F� = F(w)�F(w) for each w ∈ IGN .



Square solutions and related resultsI C�(w) = C (w)�C (w) for each w ∈ IGN .

Example: N = {1, 2}, w({1}) = [0, 1],w({2}) = [0, 2],w(N) = [4, 8].

(2, 2) ∈ C (w), (31

2, 4

1

2) ∈ C (w).

(2, 2)�(31

2, 4

1

2) = ([2, 3

1

2], [2, 4

1

2]) ∈ C (w)�C (w).

I C(w) = C�(w) for each w ∈ IBIGN .I W�(w) = W (w)�W (w) for each w ∈ IGN .

I C(w) ⊂ W�(w) for each w ∈ IGN .I C�(w) =W�(w) for each w ∈ CIGN .I W(w) ⊂ W�(w) for each w ∈ CIGN .



Classical big boss games (Muto et al. (1988), Tijs (1990)):< N, v > is a big boss game with n as big boss if :

(i) v ∈ GN is monotonic, i.e. v(S) ≤ v(T ) if for each S ,T ∈ 2N

with S ⊂ T ;

(ii) v(S) = 0 if n /∈ S ;

(iii) v(N)− v(S) ≥∑

i∈N\S v(N)− v(N \ {i}) for all S ,T withn ∈ S ⊂ N.

Big boss interval games:

I < N,w > is a big boss interval game if < N,w > and< N,w − w > are classical big boss games.

I BBIGN : the class of big boss interval games

I marginal contribution of each player i ∈ N:Mi (w) = w(N)− w(N \ {i}).



Properties of big boss interval games

Theorem: Let w ∈ SMIGN . Then, the following conditions areequivalent:

(i) w ∈ BBIGN .

(ii) < N,w > satisfies

(a) Veto power property:w(S) = [0, 0] for each S ∈ 2N with n /∈ S .

(b) Monotonicity property:w(S) 4 w(T ) for each S ,T ∈ 2N with n ∈ S ⊂ T .

(c) Union property:

w(N)− w(S) <∑

i∈N\S

(w(N)− w(N \ {i}))

for all S with n ∈ S ⊂ N.



T -value (inspired by Tijs(1981))

I big boss interval point: B(w) = ([0, 0], . . . , [0, 0],w(N))

I union interval point:

U(w) = (M1(w), . . . ,Mn−1(w),w(N)−n−1∑i=1

Mi (w))

I The T -value T : BBIGN → I (R)N is defined by

T (w) =1

2(U(w) + B(w)).



Holding situations with interval data

Holding situations with one agent with a storage capacity andother agents have goods to stored to generate benefits.In classical cooperative game theory holding situations aremodelled by using big boss games.We refer to Tijs, Meca and Lopez (2005).We consider a holding situation with interval data and construct aholding interval game which turns out to be a big boss intervalgame.Example 1: Player 3 is the owner of a holding house which hascapacity for one container. Players 1 and 2 have each onecontainer which they want to store. If player 1 is allowed to storehis/her container then the benefit belongs to [10, 30] and if player2 is allowed to store his/her container then the benefit belongs to[50, 70].



Example 1 continues...

The situation described corresponds to an interval game as follows:

I The interval game < N,w > with N = {1, 2, 3} andw(S) = [0, 0] if 3 /∈ S , w(∅) = w({3}) = [0, 0],w({1, 3}) = [10, 30] and w(N) = w({2, 3}) = [50, 70] is a bigboss interval game with player 3 as big boss.

I B(w) = ([0, 0], [0, 0], [50, 70]) andU(w) = ([0, 0], [40, 40], [10, 30]) are the elements of theinterval core.

I T (w) = ([0, 0], [20, 20], [30, 50]) ∈ C(w).

For more details see Alparslan Gok, Branzei and Tijs (2010).



How to use interval games and their solutions ininteractive situations

Stage 1 (before cooperation starts):with N = {1, 2, . . . , n} set of participants with interval data ⇒interval game < N,w > and interval solutions ⇒ agreement forcooperation based on an interval solution ψ and signing a bindingcontract (specifying how the achieved outcome by the grandcoalition should be divided consistently with Ji = ψi (w) for eachi ∈ N).

Stage 2 (after the joint enterprise is carried out):The achieved reward R ∈ w(N) is known; apply the agreed uponprotocol specified in the binding contract to determine theindividual shares xi ∈ Ji .Natural candidates for rules used in protocols are bankruptcy rules.




Example 2:w(1) = [0, 2], w(2) = [0, 1] and w(1, 2) = [4, 8].Φ(w) = ([2, 4 1

2 ], [2, 3 12 ]). R = 6 ∈ [4, 8]; choose proportional rule

(PROP) defined by

PROPi (E , d) :=di∑j∈N dj

E

for each bankruptcy problem (E , d) and all i ∈ N.(Φ1(w),Φ2(w)) +

PROP(R − Φ1(w)− Φ2(w); Φ1(w)− Φ1(w),Φ2(w)− Φ2(w))= (2, 2) + PROP(6− 2− 2; (2 1

2 , 112 ))

= (3 14 , 2

34 ).

For more details see Branzei, Tijs and Alparslan Gok (2010).


References

References

[1] Alparslan Gok S.Z., Branzei O., Branzei R. and Tijs S.,Set-valued solution concepts using interval-type payoffs for intervalgames, to appear in Journal of Mathematical Economics (JME).[2] Alparslan Gok S.Z., Branzei R. and Tijs S., Convex intervalgames, Journal of Applied Mathematics and Decision Sciences,Vol. 2009, Article ID 342089, 14 pages (2009) DOI:10.1115/2009/342089.[3] Alparslan Gok S.Z., Branzei R., Tijs S., Big Boss IntervalGames, International Journal of Uncertainty, Fuzziness andKnowledge-Based Systems (IJUFKS), Vol: 19, no.1 (2011)pp.135-149.[4] Bondareva O.N., Certain applications of the methods of linearprogramming to the theory of cooperative games, ProblemlyKibernetiki 10 (1963) 119-139 (in Russian).


References

References

[5] Branzei R., Branzei O., Alparslan Gok S.Z., Tijs S.,Cooperative interval games: a survey, Central European Journal ofOperations Research (CEJOR), Vol.18, no.3 (2010) 397-411.[6] Branzei R., Dimitrov D. and Tijs S., Models in CooperativeGame Theory, Springer, Game Theory and Mathematical Methods(2008).[5] Branzei R., Tijs S. and Alparslan Gok S.Z., How to handleinterval solutions for cooperative interval games, InternationalJournal of Uncertainty, Fuzziness and Knowledge-based Systems,Vol.18, Issue 2, (2010) 123-132.[8] Gillies D. B., Solutions to general non-zero-sum games. In:Tucker, A.W. and Luce, R.D. (Eds.), Contributions to the theoryof games IV, Annals of Mathematical Studies 40. PrincetonUniversity Press, Princeton (1959) pp. 47-85.


References

References

[9] Muto S., Nakayama M., Potters J. and Tijs S., On big bossgames, The Economic Studies Quarterly Vol.39, No. 4 (1988)303-321.[10] Shapley L.S., On balanced sets and cores, Naval ResearchLogistics Quarterly 14 (1967) 453-460.[11] Tijs S., Bounds for the core and the τ -value, In: MoeschlinO., Pallaschke D. (eds.), Game Theory and MathematicalEconomics, North Holland, Amsterdam(1981) pp. 123-132.[12] Tijs S., Big boss games, clan games and information marketgames. In:Ichiishi T., Neyman A., Tauman Y. (eds.), Game Theoryand Applications. Academic Press, San Diego (1990) pp.410-412.[13]Tijs S., Meca A. and Lopez M.A., Benefit sharing in holdingsituations, European Journal of Operational Research 162(1)(2005) 251-269.

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