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# Operations Research Situations and Games

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

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### Operations Research Situations and Games

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 2: Operations Research Situations and 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. 2. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011Outline Introduction Cooperative games Economic and Operations Research situations and games Market situations and big boss games Bankruptcy situations and games Sequencing situations and games Airport situations and games Final Remark References
3. 3. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 IntroductionIntroduction Operations research (OR) is an interdisciplinary mathematical science that studies the eﬀective use of technology by organizations. Operations research use techniques of other sciences such as statistics, optimization, probability theory, game theory, mathematical modeling and simulation to solve the problems. We pay much attention to the modelling part; that is how to go from an economic or Operations Research situation to game theory. Economic and OR games: Cooperative games associated with several types of economic and OR problems in which various decision makers (players) are involved, who face a joint optimization problem (in trying to minimize/maximize total joint costs/rewards) and an allocation problem in how to distribute the joint costs/rewards among them.
4. 4. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative gamesPreliminaries on cooperative games A cooperative game (Transferable utility (TU-game)) is a pair < N, v >, where N = {1, 2, ..., n} is the set of players v : 2N → R is the characteristic function of the game with v (∅) = 0. v (S) - value of the coalition, S ⊂ N. G N - the family of coalitional games with player set N. We denote the size of a coalition S ⊂ N by |S|. G N is a (2|N| − 1) dimensional linear space for which unanimity games form a basis. Let S ∈ 2N {∅}. The unanimity game based on S, uS : 2N → R is deﬁned by 1, S ⊂ T uS (T ) = 0, otherwise.
5. 5. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative gamesCore How to distribute the proﬁt generated by the cooperating players? An important role is played by the allocations in the core of the game. The core (Gillies (1959)) is deﬁned by C (v ) = x ∈ RN | xi = v (N), xi ≥ v (S)for each S ∈ 2N , i∈N i∈S for each v ∈ G N . i∈N xi = v (N): Eﬃciency condition i∈S xi ≥ v (S), S ⊂ N: Coalitional rationality condition
6. 6. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative gamesThe Shapley value Π(N): the set of all permutations σ : N → N of N. P σ (i) := r ∈ N|σ −1 (r ) < σ −1 (i) : the set of all predecessors of i with respect to the permutation σ. Let v ∈ G N and σ ∈ Π(N). The marginal contribution vector mσ (v ) ∈ Rn with respect to σ and v has the i-th coordinate the value miσ (v ) := v (P σ (i) ∪ {i}) − v (P σ (i)) for each i ∈ N. The Shapley value (Shapley (1953)) φ(v ) of a game v ∈ G N is the average of the marginal vectors of the game: 1 φ(v ) := mσ (v ). n! σ∈Π(N)
7. 7. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Cooperative gamesPreliminaries on cooperative games A game < N, v > is called superadditive if v (S ∪ T ) ≥ v (S) + v (T ) for all S, T ⊂ N with S ∩ T = ∅; subadditive if v (S ∪ T ) ≤ v (S) + v (T ) for all S, T ⊂ N with S ∩ T = ∅; convex (or supermodular) if v (S ∪ T )+v (S ∩ T ) ≥ v (S) + v (T ) for all S, T ⊂ N; concave (or submodular) if v (S ∪ T )+v (S ∩ T ) ≤ v (S) + v (T ) for all S, T ⊂ N. Each convex (concave) game is also superadditive (subadditive). For a game v ∈ G N and a coalition T ∈ 2N {∅}, the subgame with player set T , (T , vT ), is the game vT deﬁned by vT (S) := v (S) for all S ∈ 2T . For details see Branzei, Dimitrov and Tijs (2008).
8. 8. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Economic and Operations Research situations and gamesEconomic and Operations Research situations and games We study some types of Economic and Operations Research situations and their relation with game theory. Market situations and big boss games Bankruptcy situations and games Sequencing situations and games Airport situations and games
9. 9. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Market situations and big boss gamesMarket situations and big boss games A class of cooperative games suitable to model market situations with two corners regarding the outcome of cooperation is big boss games. In one corner there is a powerful player called the big boss; the other corner contains players that need the big boss to beneﬁt from cooperation. In a big boss game, the big boss has veto power (i.e. the worth of each coalition which does not include the big boss is zero) and the characteristic function of the game has a monotonicity property (i.e. joining the big boss is more beneﬁcial when the size of coalitions grows larger) and a union property (expressed in terms of marginal contributions to the grand coalition of coalitions and individuals).
10. 10. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Market situations and big boss gamesExample (Find the treasure) There is a hotel with 6 rooms. In one of the rooms a treasure is hidden. The value of the treasure is 600 units. Each room’s 1 probability of having the treasure is same, i.e. 6 . Agent O is allowed to look only in one room and to keep the treasure if found. Agents A and B know something about the position of the treasure: 1. If the treasure is in room 3 or 6, then agent A knows it. 2. If the treasure is in room 4,5 or 6 then agent B knows it. The big boss game with O as a big boss. v ({O}) = 1 600 = 100, 6 v ({A}) = v ({B}) = 0, v ({O, A}) = 3 600 = 300, 6 v ({O, B}) = 4 600 = 400, v ({A, B}) = v (∅) = 0, 6 v ({O, A, B}) = 5 600 = 500. 6
11. 11. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Market situations and big boss gamesBig 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. Property (ii) expresses the veto power of the big boss: the worth of each coalition which does not include the big boss is zero. Property (i) says that joining the big boss is more beneﬁcial when the size of coalitions grows larger, while (iii) expresses the fact that the force of non big boss players is in their union. We denote the set of all big boss games with n as a big boss by BBG N .
12. 12. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Market situations and big boss gamesBig boss games Let v ∈ G N . For each i ∈ N, the marginal contribution of player i to the grand coalition N is Mi (v ) := v (N) − v (N {i}). The core C (v ) of a big boss game is always nonempty and equal to {x ∈ I (v )|0 ≤ xi ≤ Mi (v ) for each i ∈ N {n}} . For a big boss game < N, v > (with n as a big boss) of v ∈ G N two particular elements of its core are the big boss point B(v ) deﬁned by 0, if j ∈ N {n} Bj (v ) := v (N), if j = n, and the union point U(v ) deﬁned by Mj (N, v ), if j ∈ T {n} Uj (v ) := v (N) − i∈N{n} Mi (N, v ), if j = n.
13. 13. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Market situations and big boss gamesτ -value of a big boss game For big boss games τ -value, introduced by Tijs (1981), is the element in the center of the core. τ -value of a big boss game < N, v > is deﬁned by B(v ) + U(v ) τ (v ) = . 2 Note that if the game is a convex big boss game then τ (v ) = φ(v ). For details see Tijs (1981, 1990).
14. 14. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Market situations and big boss gamesExample Let < N, v > be a three-person game with v (N) = 5, v ({1, 3}) = 3, v ({2, 3}) = 4, v (S) = 0 otherwise. This game is a big boss game, with 3 as a big boss (from deﬁnition). Now, M1 (v ) = v (N) − v (N {1}) = 5 − 4 = 1, M2 (v ) = v (N) − v (N {2}) = 5 − 3 = 2. C (v ) = x ∈ R3 : x(N) = v (N), 0 ≤ x1 ≤ M1 (v ), 0 ≤ x2 ≤ M2 (v ) 3 3 C (v ) = x ∈R : xi = 5, 0 ≤ x1 ≤ 1, 0 ≤ x2 ≤ 2 i=1 C (v ) = conv {(0, 0, 5), (1, 0, 4), (0, 2, 3), (1, 2, 2)}, which is a paralellogram. B(v ) = (0, 0, 5), U(v ) = (1, 2, 2) and τ (v ) = ( 1 , 1, 7 ) (core 2 2 elements).
15. 15. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Market situations and big boss gamesBi-monotonic allocation schemes (bi-mas) We denote by Pn the set {S ⊂ N|n ∈ S} of all coalitions containing the big boss. Let v ∈ G N be a big boss game with n as a big boss. We call a scheme a = (aiS )i∈S,S∈Pn a bi-monotonic allocation scheme (bi-mas) (Branzei, Tijs and Timmer (2001)) if (i) (aiS )i∈S is a core element of the subgame < S, v > for each coalition S ∈ Pn , (ii) aiS ≥ aiT for all i ∈ S {n} with S ⊂ T and anS ≤ anT for all S, T ∈ Pn with S ⊂ T . Interpretation: In a bi-mas the big boss is weakly better oﬀ in larger coalitions, while the other players are weakly worse oﬀ. For (total) big boss games the τ -value generates a bi-mas. Further, each core element generates a bi-mas for these games.
16. 16. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Market situations and big boss gamesExample Let < N, v > be a big boss game (with 3 as big boss) and v ({i}) = 0 for i ∈ N, v ({1, 2}) = 0, v ({1, 3}) = 6, v ({2, 3}) = 5, v (N) = 9. The (total) τ -value is a bi-monotonic allocation scheme:   1 2 3 (1, 2, 3)  2 1 1 5 2   2 1  (1, 3)  3 ∗ 3 .   (2, 3)  ∗ 21 21  2 2 (3) ∗ ∗ 0 In larger coalitions player 3 is better oﬀ, other players worse oﬀ.
17. 17. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Bankruptcy situations and gamesBankruptcy situations and games Bankruptcy situations and bankruptcy games have been intensively studied in literature (O’Neill (1982), Aumann and Maschler (1985)). The story is based on a certain amount of money (estate) which has to be divided among some people (claimants) who have individual claims on the estate, and the total claim is weakly larger than the estate. A bankruptcy situation with set of claimants N is a pair (E , d), where E ≥ 0 is the estate to be divided and d ∈ RN is the vector + of claims such that i∈N di ≥ E .
18. 18. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Bankruptcy situations and gamesBankruptcy situations and games We assume that 0 ≤ d1 ≤ d2 ≤ . . . ≤ dn and denote by BR N the set of bankruptcy situations with player set N. The total claim is denoted by D = i∈N di . A bankruptcy rule is a function f : BR N → RN which assigns to each bankruptcy situation (E , d) ∈ BR N a payoﬀ vector f (E , d) ∈ RN such that 0 ≤ f (E , d) ≤ d (reasonability) and i∈N fi (E , d) = E (eﬃciency).
19. 19. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Bankruptcy situations and gamesBankruptcy situations and games The proportional rule (PROP) is one of the most often used in real life, deﬁned by di PROPi (E , d) = E j∈N dj for each bankruptcy problem (E , d) and all i ∈ N. Another interesting rule the rights-egalitarian (f RE ) rule is deﬁned 1 by fi RE (E , d) = di + n (E − i∈N di ), for each division problem (E , d) and all i ∈ N.
20. 20. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Bankruptcy situations and gamesBankruptcy situations and games To each bankruptcy situation (E , d) ∈ BR N one can associate a bankruptcy game vE ,d deﬁned by vE ,d (S) = (E − i∈NS di )+ for each S ∈ 2N , where x+ = max {0, x}. The game vE ,d is convex and the bankruptcy rules PROP and f RE provide allocations in the core of the game (Aumann and Maschler (1985)).
21. 21. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Bankruptcy situations and gamesExample A bankruptcy situation (E , d) is given by E = 500 and d = (100, 200, 300). The associated bankruptcy game (by deﬁnition) vE ,d is as follows S ∅ {1} {2} {3} {1, 2} {1, 3} {2, 3} N . vE ,d (S) 0 0 100 200 200 300 400 500 For example vE ,d ({1, 2}) = max 0, E − i∈N{1,2} di = max {0, 500 − 300} = 200. I (vE ,d ) = conv {(0, 100, 400), (0, 300, 200), (200, 100, 200)} , C (vE ,d ) = conv {(0, 200, 300), (100, 200, 200), (100, 100, 300)} .
22. 22. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Bankruptcy situations and gamesExample continues... Marginal vectors of the bankruptcy game can be observed from the following table.   σ σ σ σ  m1 (vE ,d ) m2 (vE ,d ) m3 (vE ,d )    (123)   0 200 300   (132)   0 200 300  . (213)   100 100 300   (231)   100 100 300   (312)  100 200 200  (321) 100 200 200 200 500 800 φ(vE ,d ) = ( , , ) ∈ C (vE ,d ) 3 3 3 (bankruptcy games are convex).
23. 23. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Bankruptcy situations and gamesExample continues... Let us calculate PROP and f RE PROP1 (E , d) = d1 dj E = 100 500 = 600 250 3 , j∈N d2 200 500 PROP2 (E , d) = dj E = 600 500 = 3 , j∈N d3 300 PROP3 (E , d) = dj E = 600 500 = 250. j∈N f1RE (E , d) = d1 + 3 (E − i∈N di ) = 100 + 3 (500 − 600) = 200 , 1 1 3 RE (E , d) = d + 1 (E − 1 500 f2 2 3 i∈N di ) = 200 + 3 (500 − 600) = 3 f3RE (E , d) = d3 + 3 (E − i∈N di ) = 300 + 3 (500 − 600) = 800 1 1 3 250 500 Note that PROP(E , d) = ( 3 , 3 , 250) and f RE (E , d) = ( 200 , 500 , 800 ) are also core elements, because the 3 3 3 game is convex. Note that f RE (E , d) = φ(vE ,d ).
24. 24. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesSequencing situations and games Waiting lines are part of everyday life (people standing in line, service jobs, manufacturing jobs, machines to be repaired, telecommunication transmissions etc..). We consider the sequencing situations with one queue of players, each with one job, in front of a machine order. Each player must have his/her job processed on this machine, and for each player there is a cost according to the time he/she spents in the system.
25. 25. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesSequencing situations and games First problem is to ﬁnd an optimal order of the jobs taking into account the individual processing times and the costs per unit of time because it is useful for reducing the cost connected with the time spent in the system. After an optimal order is found, second problem is to motivate the agents to switch their positions according to the new order since agents are basically interested in their individual beneﬁt . First problem: An optimal order may be obtained simply reordering the jobs for weakly decreasing urgency indices (Smith (1956)). Second problem: Cooperative games arising from such sequencing situations is useful to solve this problem which is done by oﬀering fair shares of the gain generated by cooperation(Curiel, Pederzoli and Tijs (1989)).
26. 26. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesSequencing situations and games Formally, a one-machine sequencing situation is a 4-tuple (N, σ0 , α, p) where: N = {1, ..., n} is the set of jobs; σ0 : N → {1, ..., n} is a permutation that deﬁnes the initial order of the jobs; α = (αi )i∈N ∈ Rn is a non-negative real vector, where αi is + the cost per unit of time of job i; p = (pi )i∈N ∈ Rn is a positive real vector, where pi is the + processing time of job i.
27. 27. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesSequencing situations and games Given a sequencing situation and an ordering σ of the jobs, for each i ∈ N we denote by P(σ, i) the set of jobs preceding i, according to the order σ. The time spent in the system by job i is the sum between the waiting time that jobs in P(σ, i) need to be processed and the processing time of job i yielding the related cost αi j∈P(σ,i) pj + pi . The (total) cost associated with σ: Cσ , is given by Cσ = i∈N αi j∈P(σ,i) pj + pi .
28. 28. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesSequencing situations and games The optimal order of the jobs σ ∗ produces the minimum cost Cσ∗ = i∈N αi j∈P(σ ∗ ,i) pj + pi or the maximum cost saving Cσ0 − Cσ∗ . Smith (1956) proved that an optimal order can be obtained reordering the jobs according to decreasing urgency indices, where α the urgency index of job i ∈ N is deﬁned as ui = p i . i The following question arises: Is it possible to share this cost savings Cσ0 − Cσ∗ among the agents in such a way that no agent will protest? This question can be answered by using cooperative games.
29. 29. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesExample Let us ﬁnd the optimal order of the three agents, where α1 = 20, α2 = 60, α3 = 100 and p1 = 2, p2 = 3, p3 = 4. The urgency indices are u1 = α1 = 20 = 10, u2 = p 1 2 α2 p2 = 60 3 = 20 and u3 = α3 p3 = 100 4 = 25. Optimal order of service is σ ∗ = (3, 2, 1).
30. 30. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesSequencing situations and games A sequencing game is a pair < N, v > where N is the set of players, that coincides with the set of jobs, and the characteristic function v assigns to each coalition S the maximal cost savings that the members of S can obtain by reordering only their jobs. We say that a set of jobs T is connected according to an order σ if for all i, j ∈ T and k ∈ N, σ(i) < σ(k) < σ(j) implies k ∈ T . A switch of two connected jobs i and j, where i precedes j, generates a change in cost equal to αj pi − αi pj . We denote the gain of the switch as gij = (αj pi − αi pj )+ = max{0, αj pi − αi pj }.
31. 31. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesSequencing situations and games For a connected coalition T according to an order σ is v (T ) = gij . j∈T i∈P(σ,j)∩T If S is not a connected coalition, the order σ induces a partition in connected components, denoted by S/σ. For a nonconnected coalition S, v (S) = v (T ) for each S ⊂ N. T ∈S/σ The characteristic function v of the sequencing game can be deﬁned as v = gij u[i,j] , where u[i,j] is the unanimity game i,j∈N:i<j deﬁned as: 1 if {i, i + 1, ..., j − 1, j} ⊂ S u[i,j] (S) = . 0 otherwise
32. 32. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesSequencing situations and games Curiel, Pederzoli and Tijs (1989) show that sequencing games are convex games and, consequently, their core is nonempty. Moreover, it is possible to determine a core allocation without computing the characteristic function of the game. They propose to share equally between the players i, j the gain gij produced by the switch and call this rule the Equal Gain Splitting rule (EGS-rule). EGSi (N, σ0 , α, p) = 2 k∈P(σ,i) gki + 1 j:i∈P(σ,j) gij for each 1 2 i ∈ N.
33. 33. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesExample Let us ﬁnd the optimal order of the three agents where α1 = 20, α2 = 10, α3 = 30 and p1 = 1, p2 = 1, p3 = 1. The urgency indices are u1 = α1 = 20, u2 = α2 = 10 and u3 = α3 = 30. p 1 p 2 p 3 Optimal order of service is σ ∗ = (3, 1, 2). Solution: Go from σ0 to optimal order by neigbour switches and share equally the gain. This can be done by two neigbour services. First 2 and 3 switch, gain is g23 = α3 p2 − α2 p3 = 30 − 10 = 20. Second 1 and 3 switch, gain is g13 = α3 p1 − α1 p3 = 30 − 20 = 10. EGS(N, σ0 , α, p) = (∗, 10, 10) + (5, ∗, 5) = (5, 10, 15) ∈ C (v ).
34. 34. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesExample continues... The sequencing game < N, v > (convex) with N = {1, 2, 3} can be constructed as follows: v ({1}) = v ({2}) = v ({3}) = v (∅) = 0 v ({1, 2}) = 0 (1 is more urgent than 2), v ({1, 3}) = 0 (switch is not allowed because 2 is in between), v ({2, 3}) = 20, v (N) = 20 + 10 = 30. Note that v = 20u[2,3] + 10u[1,3] . For example v ({2, 3}) = 20u[2,3] ({2, 3}) + 10u[1,3] ({2, 3}) = 20 + 0 = 20, v (N) = 20u[2,3] ({1, 2, 3}) + 10u[1,3] ({1, 2, 3}) = 20 + 10 = 30. I (v ) = conv {(0, 0, 30), (0, 30, 0), (30, 0, 0)} , W (v ) = C (v ) = conv {(0, 0, 30), (0, 30, 0), (10, 0, 20), (10, 20, 0)} .
35. 35. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Sequencing situations and gamesExample continues... Marginal vectors of the sequencing game can be observed from the following table.   σ σ σ σ  m1 (v ) m2 (v ) m3 (v )    (123)   0 0 30   (132)   0 30 0  . (213)   0 0 30   (231)  10  0 20   (312)  0 30 0  (321) 10 20 0 φ(v ) = ( 10 , 40 , 40 ) ∈ C (v ) (sequencing games are convex). 3 3 3
36. 36. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Airport situations and gamesAirport situations and games In airport situations costs of the coalitions are considered (Driessen (1988)). Airport situations lead to concave games and the Shapley value belongs to the core of the game. Baker(1965)-Thompson(1971): only users of a piece of the runway pay for that piece and they share the cost of it equally. Littlechild and Owen (1973) showed that the Baker-Thompson rule corresponds to the Shapley value.
37. 37. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Airport situations and gamesAirport situations and games Consider the aircraft fee problem of an airport with one runway. Suppose that the planes which are to land are classiﬁed into m types. For each 1 ≤ j ≤ m, denote the set of landings of planes of type j by Nj and its cardinality by nj . Then N = ∪m Nj represents the set of all landings. j=1 Let cj represent the cost of a runway adequate for planes of type j. We assume that the types are ordered such that 0 = c0 < c1 < . . . < cm .
38. 38. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Airport situations and gamesAirport situations and games We consider the runway divided into m consecutive pieces Pj , 1 ≤ j ≤ m, where P1 is adequate for landings of planes of type 1; P1 and P2 together for landings of planes of type 2, and so on. The cost of piece Pj , 1 ≤ j ≤ m, is the marginal cost cj − cj−1 . The Baker-Thompson rule is given by βi = j [ m nr ]−1 (ck − ck−1 ) whenever i ∈ Nj . That is, k=1 r =k every landing of planes of type j contributes to the cost of the pieces Pk , 1 ≤ k ≤ j, equally allocated among its users ∪m Nr . r =k The airport TU game < N, c > is given by c(S) = max {ck |1 ≤ k ≤ m, S ∩ Nk = ∅} for all S ⊂ N.
39. 39. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Airport situations and gamesExample The players 1, 2 and 3 own planes which need landing strip of length |AD|. The strip |AD| is divided into 3 pieces |AB|, |BC | and |CD|. Player 1 needs to use the strip of length |AB| with the cost k1 , player 2 needs to use the strip of length |AC | with the cost k1 + k2 , player 3 needs to use the strip of length |AD| with the cost k1 + k2 + k3 . In cooperation they can share strips leading to the cost game < N, c > with N = {1, 2, 3}, c(∅) = 0, c({1}) = k1 , c({2}) = c({1, 2}) = k1 + k2 , c(N) = c({1, 3}) = c({2, 3}) = c({3}) = k1 + k2 + k3 . The Baker Thompson rule is β = (β1 , β2 , β3 ) = ( k1 , k1 + k2 , k1 + k2 + k3 ). 3 3 2 3 2 We know that Baker-Thompson rule is equal to the Shapley value.
40. 40. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Airport situations and gamesExample continues... Marginal vectors of the airport game can be observed from the following table.   σ σ σ  m1 (c) m2 (c) σ m3 (c)    (123)  k1  k2 k3   (132)  k1  0 k2 + k3  . (213)   0 k1 + k2 k3   (231)   0 k1 + k2 k3   (312)  0 0 k1 + k2 + k3  (321) 0 0 k1 + k2 + k3 k1 1 1 φ(c) = ( , (2k1 + 3k2 ), (6k3 + 3k2 + 2k1 )). 3 6 6 φ(c) ∈ C (c) (airport games are concave).
41. 41. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 Final RemarkFinal Remark For other interesting Operations research games see Operations Research Games: A Survey by Borm, Hamers and Hendrickx published in TOP (the Operational Research journal of SEIO (Spanish Statistics and Operations Research Society)).
42. 42. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [1]Aumann R. and Maschler M., “Game theoretic analysis of a bankruptcy problem from the Talmud”, Journal of Economic Theory 36 (1985) 195-213. [2]Baker J.Jr., “Airport runway cost impact study”, Report submitted to the Association of Local Transport Airlines, Jackson, Mississippi (1965). [3]Borm P., Hamers H. Hendrickx R., (2001)Operations Research Games: A Survey, TOP 9, 139-216. [4]Branzei R., Dimitrov D. and Tijs S., “Models in Cooperative Game Theory”, Game Theory and Mathematical Methods, Springer (2008). [5]Branzei R., Tijs S. and Timmer J., “Information collecting situations and bi-monotonic allocation schemes”, Mathematical Methods of Operations Research 54 (2001) 303-313.
43. 43. 6th Summer School AACIMP - Kyiv Polytechnic Institute (KPI) - National Technical University of Ukraine, 8-20 August 2011 ReferencesReferences [6] Curiel I., Pederzoli G. and Tijs S., “Sequencing games”, European Journal of Operational Research 40 (1989) 344-351. [7] Driessen T., “Cooperative Games, Solutions and Applications”, Kluwer Academic Publishers (1988). [8] Gillies D. B., “Solutions to general non-zero-sum games”. In: Tucker, A.W. and Luce, R.D. (Eds.), Contributions to the theory of games IV, Annals of Mathematical Studies 40. Princeton University Press, Princeton (1959) pp. 47-85. [9] Littlechild S.C. and Owen G., “A simple expression for the Shapley value in a special case”, Management Science 20 (1973) 370-372. [10]Muto S., Nakayama M., Potters J. and Tijs S., “On big boss games”, The Economic Studies Quarterly Vol. 39, no. 4 (1988) 303-321.
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