Coalitional Games with Interval-Type Payoffs: A Survey
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Coalitional Games with Interval-Type Payoffs: A Survey

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AACIMP 2011 Summer School. Operational Research Stream. Lecture by Sırma Zeynep Alparslan Gok.

AACIMP 2011 Summer School. Operational Research Stream. Lecture by Sırma Zeynep Alparslan Gok.

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    Coalitional Games with Interval-Type Payoffs: A Survey Coalitional Games with Interval-Type Payoffs: A Survey Presentation Transcript

    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative Game Theory. Operations Research Games. Applications to Interval Games Lecture 6: Coalitional Games with Interval-Type Payoffs: A Survey Sırma Zeynep Alparslan G¨k o S¨leyman Demirel University u Faculty of Arts and Sciences Department of Mathematics Isparta, Turkey email:zeynepalparslan@yahoo.com August 13-16, 2011
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011Outline Introduction Cooperative interval games Classes of cooperative interval games Economic and OR situations with interval data Handling interval solutions Final remarks and outlook References
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionIntroduction This lecture is based on the book Cooperative Interval Games: Theory and Applications by Alparslan G¨k published by o Lambert Academic Publishing (LAP). For more information please see: http://www.morebooks.de/store/gb/book/cooperative- interval-games/isbn/978-3-8383-3430-1 The book is the PhD thesis of Alparslan G¨k entitled o Cooperative interval games from Middle East Technical University, Ankara-Turkey.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionMotivation Game theory: Mathematical theory dealing with models of conflict and cooperation. Many interactions with economics and with other areas such as Operations Research (OR) and social sciences. Tries to come up with fair divisions. A young field of study: The start is considered to be the book Theory of Games and Economic Behaviour by von Neumann and Morgernstern (1944). Two parts: non-cooperative and cooperative.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionMotivation continued... Cooperative game theory deals with coalitions which coordinate their actions and pool their winnings. The main problem: How to divide the rewards or costs among the members of the formed coalition? Generally, the situations are considered from a deterministic point of view. Basic models in which probability and stochastic theory play a role are: chance-constrained games and cooperative games with stochastic/random payoffs.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionMotivation continued... Idea of interval approach: In most economic and OR situations rewards/costs are not precise. Possible to estimate the intervals to which rewards/costs belong. Why cooperative interval games are important? Useful for modeling real-life situations. Aim: generalize the classical theory to intervals and apply it to economic situations and OR situations. In this study, rewards/costs taken into account are not random variables, but just closed and bounded intervals of real numbers with no probability distribution attached.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionInterval calculus I (R): the set of all closed and bounded intervals in R I , J ∈ I (R), I = I , I , J = J, J , |I | = I − I , α ∈ R+ addition: I + J = I + J, I + J multiplication: αI = αI , αI subtraction: defined only if |I | ≥ |J| I − J = I − J, I − J weakly better than: I J if and only if I ≥ J and I ≥ J I J if and only if I ≤ J and I ≤ J better than: I J if and only if I J and I = J I J if and only if I J and I = J
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesClassical cooperative games versus cooperative intervalgames < N, v >, N := {1, 2, ..., n}: set of players v : 2N → R: characteristic function, v (∅) = 0 v (S): worth (or value) of coalition S G N : the class of all coalitional games with player set N < N, w >, N: set of players w : 2N → I (R): characteristic function, w (∅) = [0, 0] w (S) = [w (S), w (S)]: worth (value) of S IG N :the class of all interval games with player set N Example (LLR-game): Let < N, w > be an interval game with w ({1, 3}) = w ({2, 3}) = w (N) = J [0, 0] and w (S) = [0, 0] otherwise.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesArithmetic of interval games w1 , w2 ∈ IG N , λ ∈ R+ , for each S ∈ 2N w1 w2 if w1 (S) w2 (S) < N, w1 + w2 > is defined by (w1 + w2 )(S) = w1 (S) + w2 (S). < N, λw > is defined by (λw )(S) = λ · w (S). Let w1 , w2 ∈ IG N such that |w1 (S)| ≥ |w2 (S)| for each S ∈ 2N . Then < N, w1 − w2 > is defined by (w1 − w2 )(S) = w1 (S) − w2 (S). Classical cooperative games associated with < N, w > Border games: < N, w > and < N, w > Length game: < N, |w | >, where |w | (S) = w (S) − w (S) for each S ∈ 2N .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesPreliminaries on classical cooperative games < N, v > is called a balanced game if for each balanced map λ : 2N {∅} → R+ we have λ(S)v (S) ≤ v (N). S∈2N {∅} The core (Gillies (1959)) C (v ) of v ∈ G N is defined by C (v ) = x ∈ RN | xi = v (N); xi ≥ v (S), ∀S ∈ 2N . i∈N i∈S Theorem (Bondareva (1963), Shapley (1967)): Let < N, v > be an n-person game. Then, the following two assertions are equivalent: (i) C (v ) = ∅. (ii) < N, v > is a balanced game.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesSelection-based solution concepts Let < N, w > be an interval game. v is called a selection of w if v (S) ∈ w (S) for each S ∈ 2N . Sel(w ): the set of selections of w The core set of an interval game < N, w > is defined by C (w ) := ∪ {C (v )|v ∈ Sel(w )} .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesSelection-based solution concepts An interval game < N, w > is strongly balanced if for each balanced map λ it holds that λ(S)w (S) ≤ w (N). S∈2N {∅} Proposition: Let < N, w > be an interval game. Then, the following three statements are equivalent: (i) For each v ∈ Sel(w ) the game < N, v > is balanced. (ii) For each v ∈ Sel(w ), C (v ) = ∅. (iii) The interval game < N, w > is strongly balanced. Proof: (i) ⇔ (ii) follows from Bondareva-Shapley theorem. (i) ⇔ (iii) follows using w (N) ≤ v (N) ≤ w (N) and λ(S)w (S) ≤ λ(S)v (S) ≤ λ(S)w (S) S∈2N {∅} S∈2N {∅} S∈2N {∅} for each balanced map λ.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesInterval solution concepts I (R)N : set of all n-dimensional vectors with elements in I (R). The interval imputation set: I(w ) = (I1 , . . . , In ) ∈ I (R)N | Ii = w (N), Ii w (i), ∀i ∈ N . i∈N The interval core: C(w ) = (I1 , . . . , In ) ∈ I(w )| Ii w (S), ∀S ∈ 2N {∅} . i∈S Example (LLR-game) continuation: C(w ) = (I1 , I2 , I3 )| Ii = J, Ii w (S) , i∈N i∈S C(w ) = {([0, 0], [0, 0], J)} .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesClassical cooperative games (Part I in Branzei, Dimitrovand Tijs (2008)) < 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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesConvex and concave interval games < N, w > is supermodular if w (S) + w (T ) w (S ∪ T ) + w (S ∩ T ) for all S, T ∈ 2N . < N, w > is convex if w ∈ IG N is supermodular and |w | ∈ G N is supermodular (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 ∈ IG N is submodular and |w | ∈ G N is submodular (or concave).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesIllustrative examples Example 1: Let < N, w > be the two-person interval game with w (∅) = [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 with w ({i}) = [1, 1] for each i ∈ N, w (N) = w ({1, 3}) = w ({1, 2}) = w ({2, 3}) = [2, 2] and w (∅) = [0, 0]. Here, < N, w > is not convex, but < N, |w | > is supermodular, since |w | (S) = 0, for each S ∈ 2N .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesExample (unanimity interval games): Let J ∈ I (R) such that J [0, 0] and let T ∈ 2N {∅}. The unanimity interval game based on T is defined for each S ∈ 2N by J, T ⊂S uT ,J (S) = [0, 0] , otherwise. < N, |uT ,J | > is supermodular, < N, uT ,J > is supermodular: uT ,J (A ∪ B) uT ,J (A ∩ B) uT ,J (A) uT ,J (B) T ⊂ A, T ⊂B J J J J T ⊂ A, T ⊂B J [0, 0] J [0, 0] T ⊂ A, T ⊂B J [0, 0] [0, 0] J T ⊂ A, T ⊂B J or [0, 0] [0, 0] [0, 0] [0, 0].
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesSize monotonic interval games < N, w > is size monotonic if < N, |w | > is monotonic, i.e., |w | (S) ≤ |w | (T ) for all S, T ∈ 2N with S ⊂ T . SMIG N : the class of size monotonic interval games with player set N. For size monotonic games, w (T ) − w (S) is defined for all S, T ∈ 2N with S ⊂ T . CIG N : the class of convex interval games with player set N. CIG N ⊂ SMIG N because < N, |w | > is supermodular implies that < N, |w | > is monotonic.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesI-balanced interval games < N, w > is I-balanced if for each balanced map λ λS w (S) w (N). S∈2N {∅} IBIG N : class of interval balanced games with player set N. CIG N ⊂ IBIG N CIG N ⊂ (SMIG N ∩ IBIG N ) Theorem: Let w ∈ IG N . Then the following two assertions are equivalent: (i) C(w ) = ∅. (ii) The game w is I-balanced.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesSolution concepts for cooperative interval games Π(N): set of permutations, σ : N → N, of N Pσ (i) = r ∈ N|σ −1 (r ) < σ −1 (i) : set of predecessors of i in σ The interval marginal vector mσ (w ) of w ∈ SMIG N w.r.t. σ: miσ (w ) = w (Pσ (i) ∪ {i}) − w (Pσ (i)) for each i ∈ N. Interval Weber set W : SMIG N 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 ({1, 2}) = [2, 3 1 ]. This game is not size monotonic. 2 m(12) (w )is not defined. w (N) − w ({1}) = [1, 1 ]: undefined since |w (N)| < |w ({1})|. 2
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesThe interval Shapley value The interval Shapley value Φ : SMIG N → I (R)N : 1 Φ(w ) = mσ (w ), for each w ∈ SMIG N . n! σ∈Π(N) Example: N = {1, 2}, w ({1}) = [0, 1], w ({2}) = [0, 2], w (N) = [4, 8]. 1 Φ(w ) = (m(12) (w ) + m(21) (w )); 2 1 Φ(w ) = ((w ({1}), w (N) − w ({1})) + (w (N) − w ({2}), w ({2}))) ; 2 1 1 1 Φ(w ) = (([0, 1], [4, 7]) + ([4, 6], [0, 2])) = ([2, 3 ], [2, 4 ]). 2 2 2
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesProperties of solution concepts W(w ) ⊂ C(w ), ∀w ∈ CIG N and W(w ) = C(w ) is possible. Example: N = {1, 2}, w ({1}) = w ({2}) = [0, 1] and w (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 ). ([ 2 , 1 4 ], [1 1 , 2 4 ]) ∈ C(w ) 1 3 2 1 no α ∈ [0, 1] exists satisfying αm(1,2) (w ) + (1 − α)m(2,1) (w ) = ([ 1 , 1 4 ], [1 1 , 2 1 ]). 2 3 2 4 Φ(w ) ∈ W(w ) for each w ∈ SMIG N . Φ(w ) ∈ C(w ) for each w ∈ CIG N .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesThe square operator Let a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) with a ≤ b. 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 . Let A, B ⊂ RN . Then, we denote by A B the subset of I (R)N defined by A B := {a b|a ∈ A, b ∈ B, a ≤ b} . For a multi-solution F : G N RN we define F : IG N I (R)N by F = F(w ) F(w ) for each w ∈ IG N .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesSquare solutions and related results C (w ) = C (w ) C (w ) for each w ∈ IG N . Example: N = {1, 2}, w ({1}) = [0, 1], w ({2}) = [0, 2], w (N) = [4, 8]. 1 1 (2, 2) ∈ C (w ), (3 , 4 ) ∈ C (w ). 2 2 1 1 1 1 (2, 2) (3 , 4 ) = ([2, 3 ], [2, 4 ]) ∈ C (w ) C (w ). 2 2 2 2 C(w ) = C (w ) for each w ∈ IBIG N . W (w ) = W (w ) W (w ) for each w ∈ IG N . C(w ) ⊂ W (w ) for each w ∈ IG N . C (w ) = W (w ) for each w ∈ CIG N . W(w ) ⊂ W (w ) for each w ∈ CIG N .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesClassical cooperative games Theorem (Shapley (1971) and Shapley-Weber-Ichiishi (1981, 1988)): Let v ∈ G N . The following five assertions are equivalent: (i) < N, v > is convex. (ii) For all S1 , S2 , U ∈ 2N with S1 ⊂ S2 ⊂ N U v (S1 ∪ U) − v (S1 ) ≤ v (S2 ∪ U) − v (S2 ). (iii) For all S1 , S2 ∈ 2N and i ∈ N such that S1 ⊂ S2 ⊂ N {i} v (S1 ∪ {i}) − v (S1 ) ≤ v (S2 ∪ {i}) − v (S2 ). (iv) Each marginal vector mσ (v ) of the game v with respect to the permutation σ belongs to the core C (v ). (v) W (v ) = C (v ).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesBasic characterizations for convex interval games Theorem: Let w ∈ IG N be such that |w | ∈ G N is supermodular. Then, the following three assertions are equivalent: (i) w ∈ IG N is convex. (ii) For all S1 , S2 , U ∈ 2N with S1 ⊂ S2 ⊂ N U w (S1 ∪ U) − w (S1 ) w (S2 ∪ U) − w (S2 ). (iii) For all S1 , S2 ∈ 2N and i ∈ N such that S1 ⊂ S2 ⊂ N {i} w (S1 ∪ {i}) − w (S1 ) w (S2 ∪ {i}) − w (S2 ).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesBasic characterizations of convex interval games Proposition: Let w ∈ IG N . Then the following assertions hold: (i) A game < N, w > is supermodular if and only if its border games < N, w > and < N, w > are convex. (ii) A game < N, w > is convex if and only if its length game < N, |w | > and its border games < N, w >, < N, w > are convex. (iii) A game < N, w > is convex if and only if its border game < N, w > and the game < N, w − w > are convex.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesBasic characterizations for convex interval games Theorem: Let w ∈ IBIG N . Then, the following assertions are equivalent: (i) w is convex. (ii) |w | is supermodular and C(w ) = W (w ). Proof: By (ii) of Proposition, w is convex if and only if |w | , w and w are convex. Clearly, the convexity of |w | is equivalent with its supermodularity. Further, w and w are convex if and only if W (w ) = C (w ) and W (w ) = C (w ). These equalities are equivalent with W (w ) = C (w ). Finally, since w is I-balanced by hypothesis, we have C(w ) = W (w ).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesBasic characterizations for convex interval games Theorem: Let w ∈ IG N . Then, the following assertions are equivalent: (i) w is convex. (ii) |w | is supermodular and mσ (w ) ∈ C(w ) for all σ ∈ Π(N). Proposition: Let w ∈ CIG N . Then, W(w ) ⊂ C(w ). Proof: By the above theorem we have mσ (w ) ∈ C(w ) for each σ ∈ Π(N). Now, we use the convexity of C(w ).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesPopulation interval monotonic allocation schemes (pmias)(inspired by Sprumont (1990)) For a game w ∈ IG N and a coalition S ∈ 2N {∅}, the interval subgame with player set T is the game wT defined by wT (S) := w (S) for all S ∈ 2T . T IBIG N : class of totally I-balanced interval games (interval games whose all subgames are I-balanced) with player set N. We say that for a game w ∈ T IBIG N a scheme A = (AiS )i∈S,S∈2N {∅} with AiS ∈ I (R)N is a pmias of w if (i) i∈S AiS = w (S) for all S ∈ 2N {∅}, (ii) AiS AiT for all S, T ∈ 2N {∅} with S ⊂ T and for each i ∈ S.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesPopulation interval monotonic allocation schemes A pmias allocates a larger payoff to each player as the coalitions grow larger. In order to take the possibility of partial cooperation a pmias specifies not only how to allocate w (N) but also how to allocate w (S) of every coalition S ∈ 2N {∅}. We say that for a game w ∈ CIG N an imputation I = (I1 , . . . , In ) ∈ I(w ) is pmias extendable if there exists a pmas A = (AiS )i∈S,S∈2N {∅} such that AiN = Ii for each i ∈ N. Theorem: Let w ∈ CIG N . Then each element I of W(w ) is extendable to a pmias of w .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesPopulation interval monotonic allocation schemes Example: Let w ∈ CIG N with w (∅) = [0, 0], w ({1}) = w ({2}) = w ({3}) = [0, 0], w ({1, 2}) = w ({1, 3}) = w ({2, 3}) = [2, 4] and w (N) = [9, 15]. It is easy to check that the interval Shapley value for this game generates the pmias depicted as     1 2 3   N   [3, 5] [3, 5] [3, 5]   {1, 2}   [1, 2] [1, 2] ∗   {1, 3}   [1, 2] ∗ [1, 2] .  {2, 3}   ∗ [1, 2] [1, 2]   {1}   [0, 0] ∗ ∗   {2}  ∗ [0, 0] ∗  {3} ∗ ∗ [0, 0]
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesClassical big boss games versus big boss interval games 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 ∈ G N 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∈NS (v (N) − v (N {i})) for all S, T with n ∈ S ⊂ N. Big boss interval games: < N, w > is a big boss interval game if < N, w > and < N, w − w > are classical (total) big boss games. BBIG N : the class of big boss interval games. Marginal contribution of each player i ∈ N to the grand coalition: Mi (w ) := w (N) − w (N {i}).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesProperties of big boss interval games Theorem: Let w ∈ SMIG N . Then, the following conditions are equivalent: (i) w ∈ BBIG N . (ii) < N, w > satisfies (a) Veto power property: w (S) = [0, 0] for each S ∈ 2N with n ∈ S. / (b) Monotonicity property: w (S) w (T ) for each S, T ∈ 2N with n ∈ S ⊂ T . (c) Union property: w (N) − w (S) (w (N) − w (N {i})) i∈NS for all S with n ∈ S ⊂ N.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesT -value (inspired by Tijs(1981)) the big boss interval point: B(w ) := ([0, 0], . . . , [0, 0], w (N)); the union interval point: n−1 U(w ) := (M1 (w ), . . . , Mn−1 (w ), w (N) − Mi (w )). i=1 The T -value T : BBIG N → I (R)N is defined by 1 T (w ) := (U(w ) + B(w )). 2
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesHolding situations with interval data Holding situations: one agent has a storage capacity and other agents have goods to store to generate benefits. In classical cooperative game theory, holding situations are modeled by using big boss games (Tijs, Meca and L´pez (2005)). o For a holding situation with interval data one can construct a holding interval game which turns out to be a big boss interval game. Example: Player 3 is the owner of a holding house which has capacity for one container. Players 1 and 2 have each one container which they want to store. If player 1 is allowed to store his/her container, then the benefit belongs to [10, 30] and if player 2 is allowed to store his/her container, then the benefit belongs to [50, 70].
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesExample continues ... The situation described corresponds to an interval game as follows: The interval game < N, w > with N = {1, 2, 3} and w (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 big boss interval game with player 3 as big boss. B(w ) = ([0, 0], [0, 0], [50, 70]) and U(w ) = ([0, 0], [40, 40], [10, 30]) are the elements of the interval core. T (w ) = ([0, 0], [20, 20], [30, 50]) ∈ C(w ).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesBi-monotonic interval allocation schemes (inspired byBranzei, Tijs and Timmer (2001)) Pn : the set {S ⊂ N|n ∈ S} of all coalitions containing the big boss. Take a game w ∈ BBIG N with n as a big boss. We call a scheme B := (BiS )i∈S,S∈Pn an (interval) allocation scheme for w if (BiS )i∈S is an interval core element of the subgame < S, w > for each coalition S ∈ Pn . Such an allocation scheme B = (BiS )i∈S,S∈Pn is called a bi-monotonic (interval) allocation scheme (bi-mias) for w if for all S, T ∈ Pn with S ⊂ T we have BiS BiT for all i ∈ S {n}, and BnS BnT . Remark: In a bi-mias the big boss is weakly better off in larger coalitions, while the other players are weakly worse off.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Classes of cooperative interval gamesBi-monotonic interval allocation schemes We say that for a game w ∈ BBIG N with n as a big boss, an imputation I = (I1 , . . . , In ) ∈ I(w ) is bi-mias extendable if there exists a bi-mas B = (BiS )i∈S,S∈Pn such that BiN = Ii for each i ∈ N. Theorem: Let w ∈ BBIG N with n as a big boss and let I ∈ C(w ). Then, I is bi-mias extendable. Example continues: The T -value generates a bi-mias represented by the following matrix:   1 2 3 N  [0, 0] [20, 20] [30, 50]    {1, 3}  [5, 15]  ∗ [5, 15]  .  {2, 3}  ∗ [25, 35] [25, 35]  {3} ∗ ∗ [0, 0]
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataAirport situations with interval data In airport situations, the costs of the coalitions are considered (Driessen (1988)): One runway and m types of planes (P1 , . . . , Pm pieces of the runway: P1 for type 1, P1 and P2 for type 2, etc.). Tj [0, 0]: the interval cost of piece Pj . Nj : the set of players who own a plane of type j. nj : the number of (owners of) planes of type j. < N, d > is given by N = ∪m Nj : the set of all users of the runway; j=1 d(∅) = [0, 0], d(S) = j Ti i=1 if S ∩ Nj = ∅, S ∩ Nk = ∅ for all j + 1 ≤ k ≤ m. S needs the pieces P1 , . . . , Pj of the runway. The interval cost of the used pieces of the runway is j Ti . i=1
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataAirport situations with interval data m ∗ Formally, d = k=1 Tk u∪m Nr , where r =k ∗ 1, K ∩ S = ∅ uK (S) := 0, otherwise. Interval Baker-Thompson allocation for a player i of type j: j m γi := ( nr )−1 Tk . k=1 r =k Proposition: Interval Baker-Thompson allocation agrees with the interval Shapley value Φ(d).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataAirport situations with interval data Proposition: Let < N, d > be an airport interval game. Then, < N, d > is concave. Proof: It is well known that non-negative multiples of classical dual unanimity games are concave (or submodular). By formal definition of d the classical games d = m T k uk,m and k=1 ∗ m ∗ |d| = k=1 |Tk | uk,m are concave because T k ≥ 0 and |Tk | ≥ 0 for each k, implying that < N, d > is concave. Proposition: Let (N, (Tk )k=1,...,m ) be an airport situation with interval data and < N, d > be the related airport interval game. Then, the interval Baker-Thompson rule applied to this airport situation provides an allocation which belongs to C(d).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataExample: < N, d > airport interval game interval costs: T1 = [4, 6], T2 = [1, 8], d(∅) = [0, 0], d(1) = [4, 6], d(2) = d(1, 2) = [4, 6] + [1, 8] = [5, 14], ∗ ∗ d = [4, 6]u{1,2} + [1, 8]u{2} , Φ(d) = ( 1 ([4, 6] + [0, 0]), 2 ([1, 8] + [5, 14])) = ([2, 3], [3, 11]), 2 1 1 1 γ = ( 2 [4, 6], 2 [4, 6] + [1, 8]) = ([2, 3], [3, 11]) ∈ C(d).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataSequencing situations with interval data Sequencing situations with one queue of players, each with one job, in front of a machine order. Each player must have his/her job processed on this machine, and for each player there is a cost according to the time he/she spent in the system (Curiel, Pederzoli and Tijs (1989)). A one-machine sequencing interval situation is described as a 4-tuple (N, σ0 , α, p), σ0 : a permutation defining the initial order of the jobs α = ([αi , αi ])i∈N ∈ I (R+ )N , p = ([p i , p i ])i∈N ∈ I (R+ )N : vectors of intervals with αi , αi representing the minimal and maximal unitary cost of the job of i, respectively, p i , p i being the minimal and maximal processing time of the job of i, respectively.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataSequencing situations with interval data To handle such sequencing situations, we propose to use either the approach based on urgency indices or the approach based on relaxation indices. This requires to be able to compute α p p either ui = p i , αi (for each i ∈ N) or ri = αi , αii (for each p i i i i ∈ N), and such intervals should be pair-wise disjoint. Interval calculus: Let I , J ∈ I (R+ ). We define · : I (R+ ) × I (R+ ) → I (R+ ) by I · J := [I J, I J]. Let Q := (I , J) ∈ I (R+ ) × I (R+ {0}) | I J ≤ I J . I I I We define ÷ : Q → I (R+ ) by J := [ J , J ] for all (I , J) ∈ Q.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataSequencing situations with interval data Example (a): Consider the two-agent situation with p1 = [1, 4], p2 = [6, 8], α1 = [5, 25], α2 = [10, 30]. We can compute 4 5 u1 = 5, 25 , u2 = 3 , 15 and use them to reorder the jobs as the 4 intervals are disjoint. Example (b): Consider the two-agent situation with p1 = [1, 3], p2 = [4, 6], α1 = [5, 6], α2 = [11, 12]. Here, we can compute r1 = 5 , 1 , r2 = 11 , 1 , but we cannot reorder the jobs 1 2 4 2 as the intervals are not disjoint. Example (c): Consider the two-agent situation with p1 = [1, 3], p2 = [5, 8], α1 = [5, 6], α2 = [10, 30]. Now, r1 is defined but r2 is undefined. On the other hand, u1 is undefined and u2 is defined, so no comparison is possible; consequently, the reordering cannot take place.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataSequencing situations with interval data Let i, j ∈ N. We define the interval gain of the switch of jobs i and j by αj pi − αi pj , if jobs i and j switch Gij := [0,0], otherwise. The sequencing interval game: w := Gij u[i,j] . i,j∈N:i<j Gij ∈ I (R) for all switching jobs i, j ∈ N and u[i,j] is the unanimity game defined as: 1, if {i, i + 1, ..., j − 1, j} ⊂ S u[i,j] (S) := 0, otherwise.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataSequencing situations with interval data The interval equal gain splitting rule is defined by 1 IEGSi (N, σ0 , α, p) = 2 Gij + 12 Gij , for each j∈N:i<j j∈N:i>j i ∈ N. Proposition: Let < N, w > be a sequencing interval game. Then, 1 i) IEGS(N, σ0 , α, p) = 2 (m(1,2...,n) (w ) + m(n,n−1,...,1) (w )). ii) IEGS(N, σ0 , α, p) ∈ C(w ). Proposition: Let < N, w > be a sequencing interval game. Then, < N, w > is convex. Example: Consider the interval situation with N = {1, 2}, σ0 = {1, 2}, p = (2, 3) and α = ([2, 4], [12, 21]). The urgency indices are u1 = [1, 2] and u2 = [4, 7], so that the two jobs may be switched. We have: G12 = [18, 30], IEGS(N, σ0 , α, p) = ([9, 15], [9, 15]) ∈ C(w ).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataBankruptcy situations with interval data In a classical bankruptcy situation, a certain amount of moneyd has to be divided among some people, N = {1, . . . , n}, who have individual claims ci , i ∈ N on the estate, and the total claim is weakly larger than the estate. The corresponding bankruptcy game vE ,d : vE ,d (S) = (E − i∈NS di )+ for each S ∈ 2N , where x+ = max {0, x} (Aumann and Maschler (1985)). A bankruptcy interval situation with a fixed set of claimants N = {1, 2, . . . , n} is a pair (E , d) ∈ I (R) × I (R)N , where E = [E , E ] [0, 0] is the estate to be divided and d is the vector of interval claims with the i-th coordinate di = [d i , d i ] (i ∈ N), such that [0, 0] d1 d2 . . . dn and E < n di. i=1 BRI N : the family of bankruptcy interval situations with set of claimants N.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataBankruptcy situations with interval data We define a subclass of BRI N , denoted by SBRI N , consisting of all bankruptcy interval situations such that |d(N S)| ≤ |E | for each S ∈ 2N with d(N S) ≤ E . We call a bankruptcy interval situation in SBRI N a strong bankruptcy interval situation. With each (E , d) ∈ SBRI N we associate a cooperative interval game < N, wE ,d >, defined by wE ,d (S) := [vE ,d (S), vE ,d (S)] for each S ⊂ N. SBRIG N : the family of all bankruptcy interval games wE ,d with (E , d) ∈ SBRI N .
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and OR situations with interval dataBankruptcy situations with interval data Example: Let (E , d) be a two-person bankruptcy situation. We suppose that the claims of the players are closed intervals d1 = [70, 70] and d2 = [80, 80], respectively, and the estate is E = [100, 140]. Then, the corresponding game < N, wE ,d > is given by wE ,d (∅) = [0, 0], wE ,d (1) = [20, 60], wE ,d (2) = [30, 70] and wE ,d (1, 2) = [100, 140]. This game is supermodular, but is not convex because |wE ,d | ∈ G N is not convex.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Handling interval solutionsHow 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 for cooperation based on an interval solution ψ and signing a binding contract (specifying how the achieved outcome by the grand coalition should be divided consistently with Ji = ψi (w ) for each i ∈ N. Stage 2 (after the joint enterprise is carried out): The achieved reward R ∈ w (N) is known; apply the agreed upon protocol specified in the binding contract to determine the individual shares xi ∈ Ji . Natural candidates for rules used in protocols are bankruptcy rules.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Handling interval solutionsHandling interval solutions Example: w (1) = [0, 2], w (2) = [0, 1] and w (1, 2) = [4, 8]. 1 1 Φ(w ) = ([2, 4 2 ], [2, 3 2 ]). R = 6 ∈ [4, 8]; choose proportional rule (PROP) defined by di PROPi (E , d) := E j∈N dj 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 )) 1 1 = (2, 2) + PROP(6 − 2 − 2; (2 2 , 1 2 )) 1 3 = (3 4 , 2 4 ).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final remarks and outlookConclusion and future work The State-of-the-art of interval game literature: Branzei R., Dimitrov D. and Tijs S., “Shapley-like values for interval bankruptcy games”, Economics Bulletin 3 (2003) 1-8. Alparslan G¨k S.Z., Branzei R., Fragnelli V. and Tijs S., o “Sequencing interval situations and related games”, to appear in Central European Journal of Operations Research (CEJOR). Alparslan G¨k S.Z., Branzei O., Branzei R. and Tijs S., o “Set-valued solution concepts using interval-type payoffs for interval games”, to appear in Journal of Mathematical Economics (JME).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final remarks and outlook Alparslan G¨k S.Z., Branzei R. and Tijs S., “Convex interval o games”, Journal of Applied Mathematics and Decision Sciences, Vol. 2009, Article ID 342089, 14 pages (2009a) DOI: 10.1115/2009/342089. Alparslan G¨k S.Z., Branzei R. and Tijs S., “Big boss interval o games”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems (IJUFKS), Vol. 19, no:1 (2011) pp.135-149. Branzei R. and Alparslan G¨k S.Z., “Bankruptcy problems o with interval uncertainty”, Economics Bulletin, Vol. 3, no. 56 (2008) pp. 1-10.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final remarks and outlook Branzei R., Mallozzi L. and Tijs S., “Peer group situations and games with interval uncertainty”, International Journal of Mathematics, Game Theory, and Algebra, Vol.19, Issues 5-6 (2010). Branzei R., Tijs S. and Alparslan G¨k S.Z., “Some o characterizations of convex interval games”, AUCO Czech Economic Review, Vol. 2, no.3 (2008) 219-226. Branzei R., Tijs S. and Alparslan G¨k S.Z., “How to handle o interval solutions for cooperative interval games”, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, Vol.18, Issue 2, (2010) 123-132. Branzei R., Branzei O., Alparslan G¨k S.Z., Tijs S., o “Cooperative interval games: a survey”, Central European Journal of Operations Research (CEJOR), Vol.18, no.3 (2010) 397-411.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final remarks and outlook Moretti S., Alparslan G¨k S.Z., Branzei R. and Tijs S., o “Connection situations under uncertainty and cost monotonic solutions”, Computers and Operations Research, Vol.38, Issue 11 (2011) pp.1638-1645. Branzei R., Alparslan Gk S.Z. and Branzei O., “On the Convexity of Interval Dominance Cores”, to appear in Central European Journal of Operations Research (CEJOR), DOI: 10.1007/s10100-010-0141-z. Alparslan G¨k S.Z., Branzei R. and Tijs S., “Airport interval o games and their Shapley value”, Operations Research and Decisions, Issue 2 (2009). Alparslan G¨k S.Z., Miquel S. and Tijs S., “Cooperation o under interval uncertainty”, Mathematical Methods of Operations Research, Vol. 69, no.1 (2009) 99-109.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final remarks and outlook Alparslan G¨k S.Z., “Cooperative interval games”, PhD o Dissertation Thesis, Institute of Applied Mathematics, Middle East Technical University, Ankara-Turkey (2009). Alparslan G¨k S.Z., Branzei R. and Tijs S., “The interval o Shapley value: an axiomatization”, Central European Journal of Operations Research (CEJOR), Vol.18, Issue 2 (2010) pp. 131-140. Future work: Promising topic (interesting open problems exist to be generalized in the theory of cooperative interval games). Other OR situations and combinatorial optimization problems with interval data can be modeled by using cooperative interval games, e.g., flow, linear production, holding situations, financial and energy markets.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [1]Alparslan G¨k S.Z., Cooperative Interval Games: Theory and o Applications, Lambert Academic Publishing (LAP), Germany (2010) ISBN:978-3-8383-3430-1. [2]Aumann R. and Maschler M., Game theoretic analysis of a bankruptcy problem from the Talmud, Journal of Economic Theory 36 (1985) 195-213. [3] Bondareva O.N., Certain applications of the methods of linear programming to the theory of cooperative games, Problemly Kibernetiki 10 (1963) 119-139 (in Russian). [4] Branzei R., Dimitrov D. and Tijs S., Models in Cooperative Game Theory, Springer, Game Theory and Mathematical Methods (2008). [5] Branzei R., Tijs S. and Timmer J., Information collecting situations and bi-monotonic allocation schemes, Mathematical Methods of Operations Research 54 (2001) 303-313.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [6]Curiel I., Pederzoli G. and Tijs S., Sequencing games, European Journal of Operational Research 40 (1989) 344-351. [7]Driessen T., Cooperative Games, Solutions and Applications, Kluwer Academic Publishers (1988). [8] Gillies D. B., Solutions to general non-zero-sum games. In: Tucker, A.W. and Luce, R.D. (Eds.), Contributions to the theory of games IV, Annals of Mathematical Studies 40. Princeton University Press, Princeton (1959) pp. 47-85. [9]Ichiishi T., Super-modularity: applications to convex games and to the greedy algorithm for LP, Journal of Economic Theory 25 (1981) 283-286.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [10] Muto S., Nakayama M., Potters J. and Tijs S., On big boss games, The Economic Studies Quarterly Vol.39, No. 4 (1988) 303-321. [11]Shapley L.S., On balanced sets and cores, Naval Research Logistics Quarterly 14 (1967) 453-460. [12] Shapley L.S., Cores of convex games, International Journal of Game Theory 1 (1971) 11-26. [13] Sprumont Y., Population Monotonic Allocation Schemes for Cooperative Games with Transferable Utility, Games and Economic Behavior 2 (1990) 378-394. [14] Tijs S., Bounds for the core and the τ -value, In: Moeschlin O., Pallaschke D. (eds.), Game Theory and Mathematical Economics, North Holland, Amsterdam (1981) pp. 123-132.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [15] Tijs S., Big boss games, clan games and information market games. In:Ichiishi T., Neyman A., Tauman Y. (eds.), Game Theory and Applications. Academic Press, San Diego (1990) pp.410-412. [16]Tijs S., Meca A. and L´pez M.A., Benefit sharing in holding o situations, European Journal of Operational Research 162 (1) (2005) 251-269. [17] von Neumann, J. and Morgernstern, O., Theory of Games and Economic Behaviour, Princeton: Princeton University Press (1944). [18] Weber R., Probabilistic values for games, in Roth A.E. (Ed.), The Shapley Value: Essays in Honour of Lloyd S. Shapley, Cambridge University Press, Cambridge (1988) 101-119.