Cooperative Interval Games
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Cooperative Interval Games

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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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    Cooperative Interval Games Cooperative Interval Games 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 4: Cooperative Interval Games 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 Interval solutions for cooperative interval games Big boss interval games Handling interval solutions 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 papers Cooperative interval games: a survey by Branzei et al., which was published in Central European Journal of Operations Research (CEJOR), Set-valued solution concepts using interval-type payoffs for interval games by Alparslan G¨k et al., which will appear in Journal of o Mathematical Economics (JME) and Convex interval games by Alparslan G¨k, Branzei and Tijs, which o was published in Journal of Applied Mathematics and Decision Sciences.
    • 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 Cooperative game theory deals with coalitions who coordinate their actions and pool their winnings. The main problem: Dividing the rewards/costs among the members of the formed coalition. 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. In this research, 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 IntroductionMotivation Idea of interval approach: In most economic and OR situations rewards/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 and apply it to economic situations, popular OR games.
    • 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 IntroductionClassical cooperative games A cooperative game < N, v > N = {1, 2, ..., n}:set of players v : 2N → R: characteristic function, v (∅) = 0 v (S): worth (or value) of coalition S. x ∈ RN : payoff vector G N : class of all cooperative games with player set N The core (Gillies (1959)) of a game < N, v > is the set C (v ) = x ∈ RN | xi = v (N); xi ≥ v (S) for each S ∈ 2N . i∈N i∈S The idea: Giving every coalition S at least their worth v (S) so that no coalition protests
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval gamesCooperative interval games A cooperative interval game is an ordered pair < N, w >, where N is the set of players and w is the characteristic function of the game. N = {1, 2, ..., n}, w : 2N → I (R) is a map, assigning to each coalition S ∈ 2N a closed interval, such that w (∅) = [0, 0]. w (S) = [w (S), w (S)]: worth (value) of S. w (S): lower bound, w (S): upper bound IG N :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) < 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 >: Border games < N, w >, < N, w > Length game < N, |w | >, where |w | (S) = w (S) − w (S) for each S ∈ 2N . w = w + |w |
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval gamesInterval core 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 Interval solutions for cooperative interval gamesClassical 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 to Branzei, Dimitrov and Tijs (2008).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for 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 Interval solutions for 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 Interval solutions for 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 Interval solutions for 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 Interval solutions for cooperative interval gamesGeneralization of Bondareva (1963) and Shapley (1967) < 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 Interval solutions for cooperative interval gamesThe interval Weber Set 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 (N) = [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 Interval solutions for 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 Interval solutions for 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 Interval solutions for 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 Interval solutions for 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 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 big boss games. BBIG N : the class of big boss interval games marginal contribution of each player i ∈ N: Mi (w ) = w (N) − w (N {i}).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Big boss 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 Big boss interval gamesT -value (inspired by Tijs(1981)) big boss interval point: B(w ) = ([0, 0], . . . , [0, 0], w (N)) 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 Big boss interval gamesHolding situations with interval data Holding situations with one agent with a storage capacity and other agents have goods to stored to generate benefits. In classical cooperative game theory holding situations are modelled by using big boss games. We refer to Tijs, Meca and L´pez (2005). o We consider a holding situation with interval data and construct a holding interval game which turns out to be a big boss interval game. Example 1: 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 Big boss interval gamesExample 1 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 ). For more details see Alparslan G¨k, Branzei and Tijs (2010). o
    • 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 2: 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 ). For more details see Branzei, Tijs and Alparslan G¨k (2010). o
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [1] 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). [2] 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 (2009) DOI: 10.1115/2009/342089. [3] Alparslan G¨k S.Z., Branzei R., 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. [4] 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).
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [5] 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. [6] 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 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. [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.
    • 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [9] Muto S., Nakayama M., Potters J. and Tijs S., On big boss games, The Economic Studies Quarterly Vol.39, No. 4 (1988) 303-321. [10] Shapley L.S., On balanced sets and cores, Naval Research Logistics Quarterly 14 (1967) 453-460. [11] 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. [12] 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. [13]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.