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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 2011




Outline

      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
  Introduction




Introduction


      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
  Introduction




Motivation

      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
  Introduction




Motivation

      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
  Introduction




Motivation


      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
  Introduction




Interval 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
  Introduction




Classical 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 games




Cooperative 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 games




Arithmetic 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 games




Interval 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 games




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 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 games




Convex 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 games




Illustrative 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 games




Example (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 games




Size 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 games




Generalization 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 games




The 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 games




The 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 games




Properties 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 games




The 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 games




Square 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 games




Properties 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 games




T -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 games




Holding 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 games




Example 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 solutions




How to use interval games and their solutions in
interactive 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 solutions




Handling 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
  References




References
      [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
  References




References
      [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
  References




References
      [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.

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Cooperative Interval Games

  • 1. 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
  • 2. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Outline Introduction Cooperative interval games Interval solutions for cooperative interval games Big boss interval games Handling interval solutions References
  • 3. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction Introduction 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.
  • 4. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction Motivation 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.
  • 5. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction Motivation 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.
  • 6. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction Motivation 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.
  • 7. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction Interval 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
  • 8. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction Classical 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
  • 9. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval games Cooperative 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.
  • 10. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative interval games Arithmetic 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 |
  • 11. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games Interval 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)} .
  • 12. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games 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 to Branzei, Dimitrov and Tijs (2008).
  • 13. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games Convex 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).
  • 14. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games Illustrative 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 .
  • 15. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games Example (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].
  • 16. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games Size 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.
  • 17. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games Generalization 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.
  • 18. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games The 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
  • 19. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games The 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
  • 20. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games Properties 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 .
  • 21. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games The 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 .
  • 22. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Interval solutions for cooperative interval games Square 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 .
  • 23. 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}).
  • 24. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Big boss interval games Properties 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.
  • 25. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Big boss interval games T -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
  • 26. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Big boss interval games Holding 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].
  • 27. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Big boss interval games Example 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
  • 28. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Handling interval solutions How to use interval games and their solutions in interactive 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.
  • 29. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Handling interval solutions Handling 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
  • 30. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 References References [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).
  • 31. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 References References [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.
  • 32. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 References References [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.