6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011Ou...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011  ...
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Cooperation under Interval Uncertainty

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

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Cooperation under Interval Uncertainty

  1. 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 3: Cooperation under Interval Uncertainty 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. 2. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011Outline Introduction Preliminaries on classical games in coalitional form Cooperative games under interval uncertainty On two-person cooperative games under interval uncertainty Some economic examples References
  3. 3. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionIntroduction This lecture is based on the paper Cooperation under interval uncertainty by Alparslan G¨k, Miquel and Tijs o which was published in Mathematical Methods of Operations Research.
  4. 4. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionIntroduction Classical cooperative game theory deals with coalitions who coordinate their actions and pool their winnings. One of the problems is 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. However, in most economical situations rewards or costs are not known precisely, but it is possible to estimate intervals to which they belong.
  5. 5. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Introduction define a cooperative interval game introduce the notion of the core set of a cooperative interval game and various notions of balancedness focus on two-person cooperative interval games and extend to these games well-known results for classical two-person cooperative games define and analyze specific solution concepts on the class of two-person interval games, e.g., mini-core set, the ψ α -values
  6. 6. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Preliminaries on classical games in coalitional formPreliminaries on classical games in coalitional form A cooperative n-person game in coalitional form is an ordered pair < N, v >, where N = {1, 2, ..., n} (the set of players) and v : 2N → R is a map, assigning to each coalition S ∈ 2N a real number, such that v (∅) = 0. v is the characteristic function of the game. v (S) is the worth (or value) of coalition S.
  7. 7. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Preliminaries on classical games in coalitional form A payoff vector x ∈ Rn is called an imputation for the game < N, v > (the set is denoted by I (v )) if x is individually rational: xi ≥ v ({i}) for all i ∈ N n x is efficient (Pareto optimal): i=1 xi = v (N) The core (Gillies (1959)) of a game < N, v > is the set C (v ) = x ∈ I (v )| xi ≥ v (S) for all S ∈ 2N {∅} . i∈S The idea of the core is by giving every coalition S at least their worth v (S) so that no coalition has an incentive to split off. For a two-person game I (v ) = C (v ).
  8. 8. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Preliminaries on classical games in coalitional form An n-person game < N, v > is called a balanced game if for each balanced map λ : 2N {∅} → R+ we have λ(S)v (S) ≤ v (N). S Theorem (Bondareva (1963) and 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.
  9. 9. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Preliminaries on classical games in coalitional form π(N): the set of all permutations σ : N → N The marginal vector mσ (v ) ∈ Rn has as i-th coordinate (i ∈ N) miσ (v ) = v (P σ (i) ∪ {i}) − v (P σ (i)) P σ (i) = r ∈ N|σ −1 (r ) < σ −1 (i) The Shapley value (Shapley (1953)) Φ(v ) of a game is the average of the marginal vectors of the game: 1 Φ(v ) = mσ (v ) n! σ∈π(N)
  10. 10. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Preliminaries on classical games in coalitional form For a two-person game < N, v >: m(12) (v ) = (v ({1}), v ({1, 2}) − v ({1})) m(21) (v ) = (v ({1, 2}) − v ({1}), v ({2})) v ({1, 2}) − v ({1}) − v ({2}) Φi (v ) = v ({i}) + (i = {1, 2}) 2 Shapley value is the standard solution, in the middle of the core. Marginal vectors are the extreme points of the core whose average gives the Shapley value.
  11. 11. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Preliminaries on classical games in coalitional form A game < N, v > is superadditive if v (S ∪ T ) ≥ v (S) + v (T ) for all S, T ∈ 2N with S ∩ T = ∅. A two-person cooperative game < N, v > is superadditive if and only if v ({1}) + v ({2}) ≤ v ({1, 2}) holds. A two-person cooperative game < N, v > is superadditive if and only if the game is balanced. For further details on Cooperative game theory see Tijs (2003) and Branzei Dimitrov and Tijs (2008).
  12. 12. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative games under interval uncertaintyCooperative games under interval uncertainty A cooperative n-person interval game in coalitional form is an ordered pair < N, w >: N := {1, 2, . . . , n} is the set of players w : 2N → I (R) is the characteristic function which assigns to each coalition S ∈ 2N a closed interval w (S) ∈ I (R) I (R) is the set of all closed intervals in R such that w (∅) = [0, 0] The worth interval w (S) is denoted by [w (S), w (S)]. Here, w (S) is the lower bound and w (S) is the upper bound. If all the worth intervals are degenerate intervals, i.e., w (S) = w (S), then the interval game < N, w > corresponds to the classical cooperative game < N, v >, where v (S) = w (S).
  13. 13. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative games under interval uncertainty Let < N, w > be an interval game; then v is called a selection of w if v (S) ∈ w (S) for each S ∈ 2N . The set of selections of w is denoted by Sel(w ). The imputation set of an interval game < N, w > is defined by I (w ) := ∪ {I (v )|v ∈ Sel(w )}. The core set of an interval game < N, w > is defined by C (w ) := ∪ {C (v )|v ∈ Sel(w )}. An interval game < N, w > is strongly balanced if for each balanced map λ it holds that λ(S)w (S) ≤ w (N). S
  14. 14. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative games under interval uncertainty 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 and Shapley theorem. (i) ⇔ (iii) follows using the inequalities: w (N) ≤ v (N) ≤ w (N) λ(S)w (S) ≤ λ(S)v (S) ≤ λ(S)w (S)
  15. 15. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertaintyBalancedness and related topics Let < N, w > be a two-person interval game: the pre-imputation set I ∗ (w ) := x ∈ R2 |x1 + x2 ∈ w (1, 2) the imputation set I (w ) := x ∈ R2 |x1 ≥ w (1), x2 ≥ w (2), x1 + x2 ∈ w (1, 2) the mini-core set MC (w ) := x ∈ R2 |x1 ≥ w (1), x2 ≥ w (2), x1 + x2 ∈ w (1, 2) the core set C (w ) := x ∈ R2 |x1 ≥ w (1), x2 ≥ w (2), x1 + x2 ∈ w (1, 2)
  16. 16. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertaintyExample 1 The mini-core set and the core set of a strongly balanced game x2 12 d 10 d core set d © d d d d d mini-core set d ©d 5 d d d d d d 2 d d d d d d 1 3 10 12 x1
  17. 17. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertainty w (1) + w (2) = 3 + 5 ≤ w (1, 2) = 10 (a strongly balanced game) x2 12 d 10 d core set d © d d d d d mini-core set © d d 5 d d d d d d 2 d d d d d d 1 3 10 12 x1
  18. 18. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertaintyBalancedness and related topics s1 ∈ w (1) = [w (1), w (1)], s2 ∈ w (2) = [w (2), w (2)] t ∈ w (1, 2) = [w (1, 2), w (1, 2)] w s1 ,s2 ,t : the selection of w corresponding to s1 , s2 and t C (w ) = ∪ C (w s1 ,s2 ,t )|(s1 , s2 , t) ∈ w (1) × w (2) × w (1, 2) MC (w ) = ∪ C (w s1 ,s2 ,t )|s1 ∈ [w (1), w (1)], s2 ∈ [w (2), w (2)] MC (w ) ⊂ ∪ C (w s1 ,s2 ,t )|s1 ∈ w (1), s2 ∈ w (2), t ∈ w (1, 2) The mini-core set is interesting because for each s1 , s2 and t all points in MC (w ) with x1 + x2 = t are also in C (w s1 ,s2 ,t ).
  19. 19. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertaintySuperadditivity Let A and B be two intervals. We say that A is left to B, denoted by A B, if for each a ∈ A and for each b ∈ B, a ≤ b. A two-person interval game < N, w > is called superadditive, if w (1) + w (2) w (1, 2)
  20. 20. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertainty If < N, w > is a superadditive game, then for each s1 , s2 and t we have s1 + s2 ≤ t. So, each selection w s1 ,s2 ,t of w is balanced. If w (1) + w (2) ≤ w (1, 2) is satisfied, then each selection w s1 ,s2 ,t of w is superadditive. A two-person interval game < N, w > is superadditive if and only if < N, w > is strongly balanced.
  21. 21. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertaintyψ α -values and their axiomatization α = (α1 , α2 ) ∈ [0, 1] × [0, 1]: the optimism vector α s1 1 (w ) := α1 w (1) + (1 − α1 )w (1) α s2 2 (w ) := α2 w (2) + (1 − α2 )w (2)
  22. 22. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertainty α α If α = (1, 1), then s1 1 (w ) = w (1), s2 2 (w ) = w (2) which are the optimistic (upper) points. α α If α = (0, 0), then s1 1 (w ) = w (1), s2 2 (w ) = w (2) which are the pessimistic (lower) points. If α = ( 1 , 1 ), then s1 1 (w ) = w (1)+w (1) , s2 2 (w ) = w (2)+w (2) 2 2 α 2 α 2 which are the middle points of the intervals w (1) and w (2), respectively.
  23. 23. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertainty A map F : IG {1,2} → K(R2 ) assigning to each interval game w a unique curve F (w ) : [w (1, 2), w (1, 2)] → R2 for t ∈ [w (1, 2), w (1, 2)], i ∈ {1, 2}, in K(R2 ) is called a solution. IG {1,2} : the family of all interval games with player set {1, 2}
  24. 24. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertainty F has the following properties: (i) efficiency (EFF), if for all w ∈ IG {1,2} and t ∈ [w (1, 2), w (1, 2)]; i∈N F (w )(t)i = t. (ii) α-symmetry (α-SYM), if for all w ∈ IG {1,2} and α α t ∈ [w (1, 2), w (1, 2)] with s1 1 (w ) = s2 2 (w ); F (w )(t)1 = F (w )(t)2 . (iii) covariance with respect to translations (COV), if for all w ∈ IG {1,2} , t ∈ [w (1, 2), w (1, 2)] and a = (a1 , a2 ) ∈ R2 F (w + ˆ)(a1 + a2 + t) = F (w )(t) + a. a
  25. 25. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertaintyψ α -values and their axiomatization ˆ ∈ IG {1,2} is defined by a ˆ({1}) = [a1 , a1 ], ˆ({2}) = [a2 , a2 ] a a ˆ({1, 2}) = [a1 + a2 , a1 + a2 ] a w + ˆ ∈ IG {1,2} is defined by a (w + ˆ)(s) = w (s) + ˆ(s) a a for s ∈ {{1} , {2} , {1, 2}}
  26. 26. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertainty For each w ∈ IG {1,2} and t ∈ [w (1, 2), w (1, 2)] we define the map ψ α : IG {1,2} → K(R2 ) such that α α ψ α (w )(t) := (s1 1 (w ) + β, s2 2 (w ) + β), where 1 α α β = (t − s1 1 (w ) − s2 2 (w )). 2
  27. 27. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertainty Proposition: The ψ α -value satisfies the properties EFF, α-SYM and COV. Proof: (i) EFF property is satisfied since α α ψ α (w )(t)1 + ψ α (w )(t)2 = s1 1 (w ) + s2 2 (w ) + 2β = t. α α (ii) α-SYM property is satisfied since s1 1 (w ) = s2 2 (w ) implies α α ψ α (w )(t)1 = s1 1 (w ) + β = s2 2 (w ) + β = ψ α (w )(t)2 .
  28. 28. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertainty (iii) COV property is satisfied since α ˆ α ˆ ψ α (w + ˆ)(a1 + a2 + t) = (s1 1 (w + ˆ) + β, s2 2 (w + ˆ) + β) a a a Then, α α ψ α (w +ˆ)(a1 +a2 +t) = (s1 1 +β, s2 2 +β)+(a1 , a2 ) = ψ α (w )(t)+a. a Note that ˆ 1 t α α β = β = (ˆ − s1 1 (w + ˆ) − s2 2 (w + ˆ)), a a 2 where ˆ = a1 + a2 + t. t
  29. 29. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertainty Theorem: The ψ α -value is the unique solution satisfying EFF, α-SYM and COV properties. Proof: Suppose F satisfies the properties above. Show F = ψ α . α α Let a = (s1 1 (w ), s2 2 (w )). Then, s α (w − ˆ) = (0, 0). a By α-SYM and EFF, for each ˜ = t − a1 − a2 t F (w − ˆ)(˜) = ( 1 ˜, 2 ˜) = ψ α (w − ˆ)(˜). a t 2t 1 t a t By COV of F and ψ α , F (w )(t) = F (w − ˆ)(˜) + a = ψ α (w − ˆ)(˜) + a = ψ α (w )(t). a t a t From Proposition it follows ψ α satisfies EFF, α-SYM and COV.
  30. 30. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 On two-person cooperative games under interval uncertaintyMarginal curves and the Shapley-like solution The marginal curves for a two-person game < N, w > are defined by mσ,α (w ) : [w (1, 2), w (1, 2)] → R2 , where α α m(1,2),α (w )(t) = (s1 1 (w ), t − s1 1 (w )), α α m(2,1),α (w )(t) = (t − s2 2 (w ), s2 2 (w )). The Shapley-like solution ψ α is equal to 1 ψ α (w ) = (m(1,2),α (w ) + m(2,1),α (w )). 2
  31. 31. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Some economic examplesExample 2 A bankruptcy situation with two claimants with demands d1 = 70 and d2 = 90 on (uncertain) estate E = [100, 120]. w (∅) = [0, 0], w (1) = [(E − d2 )+ , (E − d2 )+ ] = [10, 30] w (2) = [(E − d1 )+ , (E − d1 )+ ] = [30, 50], w (1, 2) = [100, 120]. w (1) + w (2) = 30 + 50 ≤ w (1, 2) = 100 (a strongly balanced game)
  32. 32. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Some economic examples 1 ψ (0,0) (w )(t) = (10 + β, 30 + β) with β = (t − 40) and t ∈ [100, 120] 2   t 100 106 110 114 120 β  30 33 35 37 40  ψ (0,0) (w )(t) (40, 60) (43, 63) (45, 65) (47, 67) (50, 70)
  33. 33. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Some economic examples The mini-core set and the ψ (0,0) -values of the game < N, w >, where L = {ψ α (w )(t)|t ∈ [100, 120]}. x2 120 d 100 d d d d d d Ld d d mini-core 50 d ©d d d 30 d d d d d d d d 10 30 100 120 x1
  34. 34. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Some economic examplesExample 3 A reward game with N = {1, 2} which is not strongly balanced. w (∅) = [0, 0], w (1) = [1, 3], w (2) = [2, 5], w (1, 2) = [6, 10]. ψ (0,0) (w )(t) = (1 + β, 2 + β) with 3 + 2β = t and t ∈ [6, 10].   t 6 7 8 9 10 β  11 2 2 21 2 3 31 2  (0,0) (w )(t) 1 1 1 1 1 1 ψ (2 2 , 3 2 ) (3, 4) (3 2 , 4 2 ) (4, 5) (4 2 , 5 2 )
  35. 35. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Some economic examples x2 10 d d d 6 d © mini-core 5 d d d   d dL   d 2 d d d d d d 1 3 6 10 x1 The mini-core set and the ψ α -values of an interval game, where L = {ψ α (w )(t)|t ∈ [6, 10]}.
  36. 36. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [1] Alparslan G¨k S.Z., Miquel S. and Tijs S., “Cooperation under o interval uncertainty”, Mathematical Methods of Operations Research, Vol. 69 (2009) 99-109. [2] 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). [3] Branzei R., Dimitrov D. and Tijs S., “Models in Cooperative Game Theory”, Springer-Verlag Berlin (2008). [4] Gillies D.B., “Some Theorems on n-person Games”, Ph.D. Thesis, Princeton University Press (1953).
  37. 37. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 References [5] Shapley L.S., “A value for n-person games”, Annals of Mathematics Studies 28 (1953) 307-317. [6] Shapley L.S., “On balanced sets and cores”, Naval Research Logistics Quarterly 14 (1967) 453-460. [7] Tijs S., “Introduction to Game Theory”, SIAM, Hindustan Book Agency, India (2003).
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