Coincidence of Cooperative Game Theoretic Solutions in the Appointment Problem. Youngsub Chun Nari Parky Duygu Yengin

Transcription

1 School of Economics Working Papers ISSN Coincidence of Cooperative Game Theoretic Solutions in the Appointment Problem Youngsub Chun Nari Parky Duygu Yengin Working Paper No March 2015 Copyright the authors

2 Coincidence of Cooperative Game Theoretic Solutions in the Appointment Problem Youngsub Chun Nari Park Duygu Yengin March 30, 2015 Abstract The fixed-route traveling salesman problem with appointments, simply the appointment problem, is concerned with the following situation. Starting from home, a traveler makes a scheduled visit to a set of sponsors and returns home. If a sponsor in the route cancels her appointment, the traveler returns home and waits for the next appointment. We are interested in finding a way of dividing the total traveling cost among sponsors in the appointment problem by applying solutions developed in the cooperative game theory. We show that the well-known solutions of the cooperative game theory, the Shapley value, the nucleolus (or the prenucleolus), and the τ-value, coincide under a mild condition on the traveling cost. JEL Classification: C71. Keywords: Fixed-route traveling salesman problem, appointment problem, Shapley value, prenucleolus, nucleolus, τ-value, coincidence. 1 Introduction Starting from home, a traveler makes a scheduled visit to a given set of sponsors and returns home (for example, a salesman visiting her customers from the main offi ce, a professor visiting several universities, etc.). We are interested in finding a way of dividing the total traveling cost of this route among sponsors. The traveling salesman problem, a well-known combinatorial optimization problem in the operations research literature, deals with finding the cheapest route of visiting a group of sponsors and returning to the starting point. The corresponding cost allocation problem can be solved by applying solutions developed in the cooperative game theory after transforming the problem into the traveling salesman game (TSG) in which the worth of a coalition of sponsors is defined to be the minimum traveling cost of the route that visits only the sponsors in the coalition. Potters et al. (1992) introduced a variant of the traveling salesman game, the fixed route traveling salesman game (or the routing game). In this game, the travel starts from and ends Department of Economics, Seoul National University, Seoul , Korea. ychun@snu.ac.kr. Chun s work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2013S1A3A ) and the Institute of Economic Research, Seoul National University. Department of Economics, Seoul National University, Seoul , Korea. flight21cj@gmail.com School of Economics, University of Adelaide, Adelaide, SA, 5000, Australia. duygu.yengin@adelaide.edu.au. 1

3 at home, but the order of visiting sponsors is fixed and therefore, the route may not be the least costly one. The worth of a coalition is the traveling cost when the traveler follows the original fixed route while skipping all sponsors who are not members of the coalition. When a sponsor is skipped, the traveler goes directly to the next sponsor in the coalition if such a sponsor exists on the fixed route; otherwise, she goes back home. Recently, Yengin (2012) introduced another variant of the traveling salesman game, the fixed-route traveling salesman game with appointments (or the appointment game). Each sponsor is supposed to make an appointment with the traveler. These appointments fix the route that the traveler has to follow. If an appointment is canceled, then the traveler returns home and waits for the next appointment. This would be the case if the traveler has to follow the initial appointment schedule due to the heavy cost of rescheduling, or inflexibility of the available time slots of other sponsors. Also, it would be better to go back home if the traveler has to wait a long period of time until the next appointment (for instance, if a service provider spends a considerable length of time for service when he visits a customer, or when a professor stays some periods of time when he visits a university). The worth of a coalition is defined to be the total traveling cost of the route when all agents who do not belong to the coalition cancel their appointments. Hence, in the appointment game, when the traveler skips an agent who does not belong to the coalition, the traveler follows the original fixed route as in the routing game; however, differently from the routing game, she always goes back home before making a visit to the next sponsor in the coalition. In addition, Yengin (2012) characterized the Shapley value of the appointment game and showed that it can be calculated in a simple way. She also showed that if for each pair of sponsors who have consecutive appointments, the sum of their traveling costs to home is (weakly) greater than the traveling cost between them, then the game is convex and the Shapley value belongs to the core. One natural question is to ask what recommendations we have if other solutions of cooperative games are applied to the appointment game. In this paper, we consider three important solutions widely studied in the cooperative game theory, namely, the prenucleolus, the nucleolus (Schmeidler, 1969), and the τ-value (Tijs, 1987). A suffi cient condition for the coincidence of the Shapley value and prenucleolus is introduced in Kar et al. (2009) under the name of the PS property, which requires that the sum of a player s marginal contribution to any coalition and its complement be a player specific constant. We show that the appointment game satisfies the PS property and hence, the Shapley value and the prenucleolus coincide on the appointment game. Moreover, if the game is convex, then the Shapley value, the nucleolus, and the τ-value coincide. 2 The Model 2.1 The Appointment Problem The framework of our analysis is taken from Yengin (2012). Let N = {1,..., n} be the set of the sponsors, where n 2. Let 0 be a special agent called home and for each S N, let S 0 S {0}. A route r = (i 1, i 2,, i T ) over N is an ordered list of the agents (sponsors and home) to be visited by a traveler such that (i) the route starts from and ends at home, i.e., i 1 = i T = 0, (ii) each sponsor is visited exactly once, 2

4 (iii) home can be visited several times and is allowed to be visited right after each other, i.e., if i j = 0, then i j+1 may still be 0. For each i, j N 0, agent i is (directly) connected to agent j on route r (denoted as i r j), if agent j is visited immediately after agent i. For each i, j N 0, let c i,j R + be the cost of traveling from agent i to j. The cost of visiting home right after each other is zero, that is, c 0,0 = 0. Although it is not essential for our results, for the simplicity of exposition, we assume that for each i N, c i,0 = c 0,i. For each i N 0, let c i c 0,i. The cost of route r over N is c(r) = T 1 j=1 c i j,i j+1. Let C = {c i,j R + {i, j} N 0 } be the cost matrix for N. An appointment problem is defined as a triple a = (N, C, r), where N is the set of sponsors, C is the cost matrix for N, and r is the fixed route over N. Let A N be the class of all appointment problems for N. For each a = (N, C, r) A N where r = (i 1, i 2,..., i T ) and each S N, let r S = (i S 1, is 2,..., is T ) be the route r restricted to S such that for each j {1, 2,..., T }, i S j = { ij if i j S, 0 otherwise. Starting from home, the traveler makes a scheduled visit to sponsors in S. Let j S be the first sponsor that the traveler visits on r s. On the original route r, suppose that j r i. If i S, then on the route r s, the traveler visits sponsor i right after visiting sponsor j (i.e., j rs i). If i / S, then the traveler goes back home and waits for a next appointment. A similar procedure is followed until all sponsors in S are visited and then, the traveler goes back home. 2.2 The Appointment Game To allocate the cost of the route among the sponsors in an appointment problem, we can apply solutions of cooperative games (games, for short). To do this, appointment problems should be mapped into games. First, we formally describe games. Given a set of players N, a set S N is a coalition. A game with player set N is a real-valued function v defined on all coalitions S N satisfying v( ) = 0. The number v(s) is the worth of S. Let Γ N be the class of games with player set N. For each a = (N, C, r) A N, let v a Γ N be the associated appointment game where the worth of each coalition is defined as the cost of the route restricted to that coalition, that is, for each S N, v a (S) = c(r S ). Note that for each i N, v a ({i}) = 2c i. Example 1. Let N = {1, 2, 3, 4, 5, 6}, r = (0, 1, 2, 3, 4, 0, 0, 5, 6, 0), and S = {1, 2, 4, 6}. Then, r S = (0, 1, 2, 0, 4, 0, 0, 0, 6, 0). Also, v a (S) = c(r S ) = c 0,1 + c 1,2 + c 2,0 + c 0,4 + c 4,0 + c 0,0 + c 0,0 + c 0,6 + c 6,0 = (c 1 + c 1,2 + c 2 ) + (2c 4 ) + (2c 6 ) = c(r {1,2} ) + c(r {4} ) + c(r {6} ). 2.3 Solutions of Cooperative Games For each v Γ N, let X(v) = {(x i ) i N R N i N x i = v(n)} be the set of effi cient allocations for v and I(v) = {(x i ) i N X(v) for each i N, x i v({i})} be the set of imputations for v. 1 A solution is a function ϕ which associates with each v Γ N, an effi cient allocation 1 Since we work with cost games, the inequalities are reversed in the definition of the imputation, convexity, the prenucleolus, and the nucleolus. 3

5 ϕ(v) = (ϕ i (v)) i N X(v). In an appointment game v a Γ N, ϕ i (v a ) denotes the amount of cost assigned to sponsor i and i N ϕ i(v a ) = c(r). For each v Γ N, each i N and each S N\{i}, the marginal contribution of agent i to the worth of coalition S is defined to be i v(s) = v(s {i}) v(s). (1) The Shapley value (Shapley, 1953) assigns to each sponsor a cost equal to the weighted average of her marginal contributions to all possible coalitions with weights being determined by the size of the coalitions. Shapley value, ϕ SV : For each v Γ N and each i N, ϕ SV i (v) = S N\{i} S!( N S 1)! i v(s). N! For each v Γ N, each x X(v) and each S N, let e S (v, x) = v(s) i S x i be the excess of coalition S with respect to allocation x and e(v, x) = (e S (v, x)) S N, S R 2n 1 be the excess vector. For each y R 2n 1, let ỹ R 2n 1 be the vector obtained by rearranging the coordinates of y in non-decreasing order. For each pair x, y R 2n 1, x is lexicographically greater than y (denoted by x lex y) if either x 1 > ỹ 1, or there is k > 1 such that for all i < k, x i = ỹ i and x k > ỹ k. The nucleolus (Schmeidler, 1969) chooses the unique allocation from the set of imputations that maximizes the excess of the coalitions lexicographically. On the other hand, the prenucleolus chooses the unique allocation from the set of effi cient allocations that maximizes the excess of the coalitions lexicographically. Nucleolus, ϕ Nu : For each v Γ N such that I(v), ϕ Nu (v) = {x I(v) for each x I(v)\{x}, e(v, x) lex e(v, x )}. Prenucleolus, ϕ P N : For each v Γ N, ϕ P N (v) = {x X(v) for each x X(v)\{x}, e(v, x) lex e(v, x )}. For each v Γ N and each i N, let M i (v) v(n) v(n\{i}), R i (S, v) v(s) j S\{i} M j(v), and m i (v) min S N, i S {R i (S, v)}. A game v Γ N is quasi-balanced if i N m i(v) v(n) i N M i(v) and for each i N, m i (v) M i (v). Let Γ N QB be the class of quasi-balanced games. For each quasi-balanced game v Γ N QB, the τ-value (Tijs, 1987) chooses the unique effi cient allocation on the line connecting M(v) = (M i (v)) i N and m(v) = (m i (v)) i N, where M(v) and m(v) are interpreted as the lower and upper bounds on costs assigned to the players, respectively. τ value, ϕ τ : For each quasi-balanced game v Γ N QB and each i N, ϕ τ i (v) = λm i (v) + (1 λ)m i (v), 4

6 where λ [0, 1] is chosen to satisfy j N [λm j(v) + (1 λ)m j (v)] = v(n). A game v Γ N is convex (Shapley, 1971) if for each i N and each S T N\{i}, v(s {i}) v(s) v(t {i}) v(t ). A game v Γ N QB is semi-convex (Driessen and Tijs, 1985) if for each i N, m i (v) = v({i}). Let Γ N Con and ΓN SC be the classes of convex and semi-convex games, respectively. Note that Γ N Con ΓN SC ΓN QB (Driessen and Tijs, 1985). 3 Coincidence Results 3.1 Coincidence of the Shapley value and the (pre)nucleolus Kar et al. (2009) identify a suffi cient condition for the coincidence of the Shapley value and the prenucleolus, called the PS property. It requires that the sum of a player s marginal contribution to any coalition and its complement be a player specific constant. Definition 1. A game v Γ N satisfies the PS property if for each i N, there exists γ i R such that for each S N\{i}, A game satisfying the PS property is a PS game. i v(s) + i v(n\(s {i})) = γ i. (2) We establish the coincidence of the Shapley value and the prenucleolus by showing that the appointment game satisfies the PS property. For each a = (N, C, r) A N, each i N and each h, k N 0 such that h r i r k, let γ i = (c i + c h,i c h ) + (c i + c i,k c k ). 2 Note that if h = k = 0, then γ i = 4c i ; if h = 0, then γ i = 2c i + (c i + c i,k c k ); and if k = 0, then γ i = (c i + c h,i c h ) + 2c i. Theorem 1. For each a = (N, C, r) A N, the appointment game v a satisfies the PS property. Proof : Let a = (N, C, r) A N and i N. We show that for all S N\{i}, i v a (S) + i v a (N\(S {i})) = γ i. The proof is divided into three cases depending on how agent i is located in the route r. Case 1 : 0 r i r 0. For each S N\{i}, v a (S {i}) = v a (S) + v a ({i}). Since i v a (S) = v a ({i}), for each S N\{i}, i v a (S) + i v a (N\(S {i})) = 2v a ({i}) = 4c i = γ i. Case 2 : For some l N\{i}, 0 r i r k. (The case when h r i r 0 can be handled similarly.) (2-1) For each S N\{i} such that k S, i v a (S) = c i + c i,k c k. (2-2) For each S N\{i} such that k / S, i v a (S) = v a ({i}) = 2c i. If S satisfies (2-1), then N\(S {i}) satisfies (2-2), and vice versa. Therefore, for each S N\{i}, i v a (S) + i v a (N\(S {i})) = (c i + c i,k c k ) + 2c i = γ i. Case 3 : For some {h, k} N\{i}, h r i r k. (3-1) For each S N\{i} such that {h, k} S, i v a (S) = c h,i + c i,k c h c k. (3-2) For each S N\{i} such that S {h, k} = {h}, i v a (S) = c i + c h,i c h. (3-3) For each S N\{i} such that S {h, k} = {k}, i v a (S) = c i + c i,k c k. 2 Alternatively, γ i can be defined as sum of iv a({h}) and iv a({k}). 5

7 (3-4) For each S N\{i} such that S {h, k} =, i v a (S) = 2c i. If S satisfies (3-1), then N\(S {i}) satisfies (3-4), and vice versa. Therefore, if S satisfies either (3-1) or (3-4), then i v a (S)+ i v a (N\(S {i})) = c h,i +c i,k c h c k +2c i = (c i +c h,i c h )+(c i +c i,k c k ) = γ i. On the other hand, if S satisfies (3-2), then N\(S {i}) satisfies (3-3), and vice versa. Therefore, i v a (S)+ i v a (N\(S {i})) = (c i +c h,i c h )+(c i +c i,k c k ) = γ i. Therefore, for each S N\{i}, i v a (S) + i v a (N\(S {i})) = γ i. In any case, for each S N\{i}, i v a (S)+ i v a (N\(S {i})) is a player specific constant and does not depend on the choice of S. Hence, v a is a PS game. Our first coincidence result follows from Theorem 1 and the fact that the Shapley value and the prenucleolus coincide for PS games (Theorem 3.3 in Kar et al. 2009). Corollary 1. For each a A N, ϕ SV (v a ) = ϕ P N (v a ). As shown in Yengin (Proposition 2, 2012), the appointment game is convex if for each route and each pair of connected sponsors, the sum of their traveling costs to home is weakly greater than the traveling cost between them. Formally, for each r over N and each pair {i, j} N such that i r j, c i + c j c i,j. Let A N con be the set of all problems a = (N, C, r) A N which satisfies the condition. This condition requires that if i and j are connected on r, then the total cost of r, (i.e., v a (N)) is smaller than the total cost that would be obtained if i and j were not connected on r. Note that the route r is not required to be a least costly tour for N since there may be pair of agents who are not connected on r even if connecting them would reduce the total travel cost. The core is the subset of imputations at which no coalition can be made better off on their own, that is, for each v Γ N, Core(v) = {x I(v) for each S N, i S x i v(s)}. For convex games, the Shapley value and the nucleolus belong to the core 3 and the nucleolus coincides with the prenucleolus (Theorem in Maschler et al., 2013). This leads to the next coincidence result. Corollary 2. For each a A N con, ϕ SV (v a ) = ϕ Nu (v a ). Remark 1. By Proposition 1 in Yengin (2012), the Shapley value of the appointment game can be expressed in the following simple form: for each a = (N, C, r) A N and each i N, ϕ SV i (v a ) = γ i 2. By Theorem 1, for each a = (N, C, r) AN and each i N, ϕ P i N (v a ) = γ i 2. In fact, as shown in Kar et al. (Theorem 3.3, 2009), this simple expression can be generalized to all PS games. Remark 2. Let v Γ N and v D Γ N be its dual game such that for each S N, v D (S) = v(n) v(n\s). Kar et al. (2009) shows that if v satisfies the PS property, then v + v D is an additive game. It is interesting to note that if v + v D is an additive game, then the coincidence of the Shapley value and the nucleolus is obtained for v. 4 Let w Γ N be an additive game such that for all S N, w(s) = j S γ j and that w = v + v D. By zeroindependence 5 of the Shapley value and the prenucleolus, ϕ SV (w) = ϕ P N (w) = γ. Note that ϕ SV (v) = ϕ SV (v D ) and ϕ P N (v) = ϕ P N ( v D ). From the additivity of the Shapley value, ϕ SV (w) = ϕ SV (v + v D ) = ϕ SV (v) + ϕ SV (v D ) = 2ϕ SV (v) = γ. On the other hand, by the 3 The nucleolus belongs to the core whenever the core is non-empty. 4 We are extremely grateful to an anonymous referee for suggesting this implication. 5 A solution ϕ satisfies zero independence if for each v, v N and each β R N such that for each S N, v (S) = v(s) + j S β j, ϕ(v ) = ϕ(v) + β. 6

8 zero independence of the prenucleolus, ϕ P N (v) = ϕ P N ( v D )+γ = ϕ P N (v)+γ. Altogether, ϕ SV (v) = ϕ P N (v) = γ 2. Remark 3. Alternatively, the coincidence can be established by checking the appointment cost savings game v c, which is defined as v c (S) = i S v a({i}) v a (S) for each S N. We note that appointment cost savings game satisfies weak 2-additivity (Kar et al., 2009), 6 another suffi cient condition for the coincidence of the Shapley value and the prenucleolus. Moreover, if the game is convex, then it satisfies 2-additivity (Deng and Papadimitriou, 1994; van den Nouweland et al., 1996), a suffi cient condition for the coincidence of the Shapley value and the nucleolus. 3.2 Coincidence of the Shapley value and the τ-value It is well-known that for games with 2 players, the τ-value coincides with the Shapley value and the nucleolus. Driessen and Tijs (Theorem 4.9, 1985) discuss conditions under which the coincidence result holds in semi-convex games with 3 players. Their result is not generalized to semi-convex games with more than 3 players. Here we show that the τ-value coincides with the Shapley value and the pre-nucleolus for semi-convex appointment games with any number of players. Theorem 2. For each a = (N, C, r) A N, if v a Γ N SC, then ϕτ (v a ) = ϕ SV (v a ) = ϕ P N (v a ). Proof : Let a = (N, C, r) A N and v a Γ N SC. Note that for each i N, m i(v a ) = v a ({i}) = i v a ( ) and M i (v a ) = i v a (N\{i}). Since v a is a PS game, for each i N, m i (v a )+M i (v a ) = i v a ( )+ i v a (N\{i}) = γ i, where γ i is defined as in Subsection 3.1. Note that for each i N, (v a ) = γ i 2 = 1 2 [m i(v a ) + M i (v a )] and that j N ϕsv j (v a ) = v a (N). Since τ-value chooses the unique effi cient allocation on the line connecting m(v a ) and M(v a ), ϕ τ (v a ) = ϕ SV (v a ). By Corollary 1, ϕ τ (v a ) = ϕ P N (v a ). ϕ SV i Our next example demonstrates that if the appointment game is not semi-convex, then Theorem 2 no longer holds. That is, for some a = (N, C, r) A N such that v a Γ N QB \ΓN SC, ϕ SV (v a ) ϕ τ (v a ). Example 2. Let a = (N, C, r) be such that N = {1, 2, 3, 4}, c 1 = 12, c 2 = 6, c 3 = c 4 = c 3,4 = 2, c 1,2 = 14, c 2,3 = 9, and r = (0, 1, 2, 3, 4, 0). Here, v a (N ) = 39, M(v a ) = (20, 9, 3, 2), and m(v a ) = (23, 12, 4, 3). Note that m(v a ) M(v a ) and j N m j(v a ) v a (N) j N M j(v a ). Thus, v a Γ N QB. On the other hand, for i {1, 4}, m i(v a ) v a ({i}). Therefore, v a is not semiconvex. One can calculate that ϕ τ (v a ) = (21.9, 10.9, 3.6, 2.6) and ϕ SV i (v a ) = (22, 10.5, 3.5, 3). We note that in an appointment problem a = (N, C, r), if the triangle inequality is not satisfied for some pair {i, k} N such that either 0 r i r k or k r i r 0, then the corresponding appointment game is not quasi-balanced and moreover, the τ-value is not welldefined. To see this, let a = (N, C, r) / A N con where for some pair {i, k} N, 0 r i r k and c i + c k < c i,k (a similar argument can be made for the case when k r i r 0). Note that M i (v a ) = v a (N) v a (N\{i}) = c i +c i,k c k. Since c i +c k < c i,k, M i (v a ) > 2c i. Note also that m i (v a ) min S N, i S {R i (S, v a )} R i ({i}, v a ) = 2c i. Hence, m i (v a ) < M i (v a ) and v a / Γ N QB. 6 Under the name of 2-game. They also show that weak 2-additivity implies the PS property. 7

9 4 Concluding Remarks In this paper, we establish the coincidence of the Shapley value, the nucleolus, and the τ-value in the appointment game. However, this coincidence result does not carry over other related games such as the routing game or the traveling salesmen game without a fixed route. In fact, Engevall et al. (1998) provided an example showing that the Shapley value, the nucleolus, and the τ-value give different allocations for traveling salesman game without a fixed route. Let a = (N, C, r) be a traveling salesman problem and S N. For the routing game (Potters et al., 1992), when a traveler skips a sponsor, she goes directly to the next sponsor in S, if such a sponsor exists on the fixed route. Let rs be the resulting route over S. Then, the worth of each coalition S N in a routing game, v r, is defined as the cost of the route rs, that is, for each S N, v r (S) = c(rs ). For the routing game, the following example shows that the Shapley value does not coincide with either the nucleolus or the τ-value. Example 3. Let a = (N, C, r) be such that N = {1, 2, 3}, c 1 = c 1,2 = c 2,1 = 5, c 2 = c 3 = c 2,3 = c 3,2 = 3, c 1,3 = c 3,1 = 6, and r = (0, 1, 2, 3, 0). Note that for each pair {i, j} {1, 2, 3}, c i + c j c i,j = c j,i, and r is a least costly route for a. Thus, by Potters et al. (1992), this routing game v r Γ N has a non-empty core. Since Core(v r ), the prenucleolus and the nucleolus coincide and they belong to the core of v r. From the simple formula of the Shapley value in Yengin (Example 4, 2012), ϕ SV (v r ) = (8.17, 3.67, 4.17). The excess vector at ϕ SV (v r ) is e S (v r, ϕ SV (v r )) = v r (S) i S ϕ SV i (v r ) = 1.16 for S {{1, 2}, {2, 3}}, 1.66 for S = {1, 3}, 1.83 for S {{1}, {3}}, 2.33 for S = {2}. Note that since for each S N, e S (v r, ϕ SV (v r )) 0, ϕ SV (v r ) belongs to the core of v r. Now, consider x = (8.35, 3.3, 4.35) I(v r ). Then, e S (v r, x) = 1.35 if S {{1, 2}, {2, 3}}, e {1,3} (v r, x) = 1.3, e {i} (v r, x) = 1.65 if i {1, 3}, and e {2} (v r, x) = 2.7. Hence, e(v r, x) lex e(v r, ϕ SV (v r )), which implies that ϕ SV (v r ) ϕ Nu (v r ). Since v r has a non-empty core, it is quasi-balanced (Tijs and Lipperts, 1982) and thus, τ-value is well-defined. One can calculate that ϕ τ (v r ) = (8.2, 3.6, 4.2) ϕ SV (v r ). Also, e S (v r, ϕ τ (v r )) = 1.2 if S {{1, 2}, {2, 3}}, e {1,3} (v r, ϕ τ (v r )) = 1.6, e {i} (v r, ϕ τ (v r )) = 1.8 if i {1, 3}, and e {2} (v r, ϕ τ (v r )) = 2.4. Since e(v r, x) lex e(v r, ϕ τ (v r )), ϕ τ (v r ) ϕ Nu (v r ). It remains an open question how far the coincidence result can be generalized in the context of the traveling salesman problem. References [1] [2] Deng, X., Papadimitriou, C. H., 1994, On the complexity of cooperative solution concepts, Mathematics of Operations Research 19(2): [3] Driessen, T. S. H., Tijs, S. H., 1985, The τ-value, the core and semiconvex games, International Journal of Game Theory 14(4): :

10 [4] Engevall, S., Göthe-Lundgren M., Värbrand P., 1998, The traveling salesman game: an application of cost allocation in a gas and oil company, Annals of Operations Research 82: [5] Kar, A., Mitra, M., Mutuswami, S., 2009, On the coincidence of the prenucleolus and the Shapley value, Mathematical Social Sciences 57: [6] Maschler, M., Solan, E., Zamir, S., 2013, Game Theory, Cambridge University Press. [7] Potters, J.A.M., Curiel, I. J., Tijs, S. H., 1992, Traveling salesman games, Mathematical Programming 53: [8] Schemidler, D., 1969, The nucleolus of a characteristic function game, SIAM Journal on Applied Mathematics 17(6): [9] Shapley, L. S., 1953, A Value for n-person Games, in Contributions to the Theory of Games II, Annals of Mathematics Studies No.28 (H. W. Kuhn and A. W. Tucker, Eds.), pp , Princeton, NJ: Princeton University Press. [10] Shapley, L. S., 1971, Cores of convex games, International Journal of Game Theory 1:11-26; errata: 1:199. [11] Tijs, S. H., 1987, An axiomatization of the τ-value, Mathematical Social Sciences 13(2): [12] Tijs, S. H., Lipperts, F. A. S., 1982, The hypercube and the core cover of N-person cooperative games. Cahiers du Centre d Etudes de Recherche Operationelle 24: [13] van den Nouweland, A., Borm, P., Brouwers, W., van Golstein, Bruinderink, R.G., Tijs, S., 1996, A game theoretic approach to problems in telecommunication. Management Science 42: [14] Yengin, D., 2012, Characterizing the Shapley value in fixed-route traveling salesman problems with appointments, International Journal of Game Theory 41: