DSpace at VNU: Fair ticket pricing in public transport as a constrained cost allocation game

DSpace at VNU: Fair ticket pricing in public transport as a constrained cost allocation game tài liệu, giáo án, bài giản...

Ann Oper Res (2015) 226:51–68 DOI 10.1007/s10479-014-1698-z Fair ticket pricing in public transport as a constrained cost allocation game Ralf Borndörfer · Nam-Dung ˜ Hoang Published online: 21 August 2014 © Springer Science+Business Media New York 2014 Abstract Ticket pricing in public transport usually takes a welfare maximization point of view Such an approach, however, does not consider fairness in the sense that users of a shared infrastructure should pay for the costs that they generate We propose an ansatz to determine fair ticket prices that combines concepts from cooperative game theory and linear and integer programming The ticket pricing problem is considered to be a constrained cost allocation game, which is a generalization of cost allocation games that allows to deal with constraints on output prices and on the formation of coalitions An application to pricing railway tickets for the intercity network of the Netherlands is presented The results demonstrate that the fairness of prices can be improved substantially in this way Keywords Constrained cost allocation games · f -Nucleolus · (f, r)-Least core · Fair ticket prices Mathematics Subject Classification 90C90 · 91A80 · 91-08 Introduction Public transport ticket prices are well studied in the economic literature on welfare optimization as well as in the mathematical optimization literature on certain network design A preliminary version of this paper appeared in the Proceedings of HPSC 2009 (Borndörfer and Hoang 2012) This journal article introduces better model and algorithms R Borndörfer Zuse Institute Berlin, Takustr 7, 14195 Berlin, Germany e-mail: borndoerfer@zib.de N.-D Hoang (B) Faculty of Mathematics, Mechanics, and Informatics, Vietnam National University, 334 Nguyen Trai Str., Hanoi, Vietnam e-mail: hoangnamdung@hus.edu.vn 123 52 Ann Oper Res (2015) 226:51–68 problems, see, e.g., the literature survey in Borndörfer et al (2008) To the best of our knowledge, however, the fairness of ticket prices has not been investigated yet The point is that typical pricing schemes are not related to infrastructure operation costs and, in this sense, favor some users who not fully pay for the costs they incur For example, we will show in this paper’s example of the Dutch IC railway network that the current distance tariff results in a situation where some passengers in the central Randstad region of the country pay over 25 % more than the costs they incur, and these excess payments subsidize operations elsewhere One can argue that this is not fair We therefore ask whether it is possible to construct ticket prices that better reflect operation costs Ticket pricing can be seen as a cost allocation problem, see Young (1994) for a survey/an introduction Cost allocation problems are widespread They come up whenever it is necessary or desirable to divide a common cost among several users or items Some examples of applications where cost allocations have been determined using methods from cooperative game theory are, e.g, aircraft landing fees (Littlechild and Thompson 1977), water resource planning (Straffin and Heaney 1981), water resource development (Young et al 1982), distribution cost of gas and oil transportation (Engevall et al 1998), and investment in electric power (Gately 1974) The cost allocation problems in the literature are considered as cost allocation games, which require only that the output prices must be non-negative and the total prices that the players are asked to pay have to cover the total costs exactly However, real world applications often have more requirements on the output prices and on the formation of coalitions Our ticket pricing problem is one example as it stipulates that the ticket price p AC for a trip from station A to station C via station B should fulfill the monotonicity conditions ≤ pAB , pBC ≤ pAC ≤ pAB + pBC , where pAB and pBC are ticket prices from A to B, and B to C, respectively The cooperative game theory literature has already considered games where coalition formation and payoff vectors are required to fulfill additional constraints A very general approach in (Maschler et al 1992) considers (profit allocation) games given by a pair (Π, F), where Π is a topological space and F is a finite set of real and continuous functions defined on Π It can be shown that a general nucleolus can be computed using an algorithm that iteratively shrinks the least core such that the general nucleolus is, in particular, non-empty under natural conditions In the special case of so-called truncated games F is given by the coalition excesses and coalition payoffs are constrained by lower bounds This concept is easily extended to our constrained cost allocation setting, which explicitly allows for arbitrary linear constraints, in order to derive, in particular, the non-emptiness of the general nucleolus We show in this paper how such a computation can be performed efficiently in our constrained cost allocation setting by a cutting plane algorithm The key idea is to combine linear independence and complementary slackness arguments in order to fix crucial coalition excesses and to rule out redundant ones We show that, in this way, large-scale instances of a ticket pricing constrained cost allocation game can be solved, substantially improving fairness In this paper, we model ticket pricing as a constrained cost allocation game in order to deal with pricing constraints We present an ( f, r )-least core and argue that the ( f, r )-least core of this game can be used to determine fair prices The ( f, r )-least core can be computed by solving several linear programs, whose numbers of constraints are exponential in the number of players They can be solved for large-scale instances using a constraint generation approach 123 Ann Oper Res (2015) 226:51–68 53 The article is structured as follows Section presents some concepts from cooperative game theory Section considers several computational approaches in order to calculate game theoretical solutions concepts A model that treats ticket pricing as a constrained cost allocation game is presented in Sect The final Sect is devoted to the Dutch IC railway example Game theoretical setting We present in this section the notion of constrained cost allocation game as a generalization of the classical cost allocation game It deals with the determination of fair prices subject to restrictions on the set of possible output prices and on the set of possible coalitions in a way that is very similar to truncated games, see Maschler et al (1992) Such a game can be formally defined as follows We are given a finite set of players N = {1, 2, , n} that can form a family of possible coalitions ⊆ N We denote the set of non-empty coalitions by + := \{∅} and assume N ∈ + (the grand coalition N is possible), + = {N } (there are non-empty coalitions other than the grand coalition) For S ⊆ N , let χ S denote the incidence vector of S, i.e., χ Si is if i ∈ S and else For a set family Ω ⊆ N , we denote χΩ := {χ S | S ∈ Ω} The family Ω is called full dimensional if dim span χΩ = n Associated with each coalition S is a cost c(S) ≥ for operating the service on its own, and a weight f (S) ≥ for averaging purposes, satisfying c(S) > and f (S) > for all non-empty coalitions S ∈ + ; typical weights are f (S) = |S|, f (S) = c(S), or f (S) ≡ To operate the service together in the grand coalition N , the players will be asked to pay prices in an imputation set P := {x ∈ Rn≥0 | Ax ≤ b}, that is defined by polyhedral constraints Ax ≤ b, x ≥ We assume P to be non-empty and to imply the cost recovery condition xi = c(N ) i∈N Note that P is then bounded, i.e., a polytope For each imputation x = (x , x2 , , xn ) ∈ P and each non-empty coalition S ∈ + , we define the price of coalition S as x(S): = i∈S xi and the f -excess of S at x as e f (S, x) := c(S) − x(S) f (S) The f -excess represents the (weighted average) gain (or loss, if it is negative) of coalition S, if its members accept to pay x(S) instead of operating some service on their own at cost c(S) The f -excess measures price acceptability: the smaller e f (S, x), the less favorable is price x for coalition S, and for e f (S, x) < 0, i.e., in case of a loss, x will be seen as unfair by the members of S The constrained cost allocation game Γ = (N , c, P, ) is to determine a “fair imputation” x ∈ P of the common cost c(N ) among the players in N If = 2N and P = {x ∈ Rn≥0 | x(N ) = c(N )}, then the constrained cost allocation game reduces to a (classical) cost allocation game If P = {x ∈ Rn≥0 | x(N ) = c(N ), x(S) ≤ u S ∀S ∈ }, where u ∈ R∞ is a vector of (possible infinite) upper bounds on coalition prices, then it is equivalent to a truncated game We proceed with game theoretical concepts for constrained cost allocation games 123 54 Ann Oper Res (2015) 226:51–68 Definition For a constrained cost allocation game Γ , a weight function f , and ε ∈ R, the set C ε, f (Γ ) := x ∈ P | e f (S, x) ≥ ε, ∀S ∈ + \{N } is called the (ε, f )-core of Γ In particular, C0, f (Γ ) is the f -core of Γ The f -least core of the game Γ , denoted LC f (Γ ), is the intersection of all non-empty (ε, f )-cores Equivalently, let ε f (Γ ) be the largest ε such that Cε, f (Γ ) is non-empty, i.e., ε f (Γ ) = max x∈P S∈ + \{N } e f (S, x) = max ε (1) (x,ε) s.t x(S) + ε f (S) ≤ c(S), ∀S ∈ + \{N } (2) x ∈ P, (3) then LC f (Γ ) = Cε f (Γ ), f (Γ ) In other words, the f -least core is the set of all imputations x ∈ P that maximize the minimum f -excess over all coalitions in + \{N } The number ε f (Γ ) is called the f -least core radius of Γ We call the LP (1–3) the f -least core (radius) problem LCP(Γ ) associated with Γ and (2) the coalition constraints Obviously there holds the following lemma Lemma The f -least core of a constrained cost allocation game Γ is non-empty Proof The f -least core problem LCP(Γ ) is feasible Indeed, since P = ∅ and finite, we can choose some x ∈ P and find ε sufficiently small such that x(S) + ε f (S) ≤ c(S), ∀S ∈ + + \ {N } is \{N } LCP(Γ ) is also bounded, because c(S) , ∀S ∈ f (S) ε≤ + \{N } Therefore, LCP(Γ ) has an optimal solution Let ε ∗ be the optimal objective value Then ε ∗ is the f -least core radius of Γ and the f -least core of Γ is the non-empty set {x ∈ Rn | (x, ε ∗ ) is an optimal solution of LCP(Γ )} The f -least core of a constrained cost allocation game Γ may contain, in general, more than one point However, if the coalition family is full dimensional, uniqueness can be enforced by imposing a lexicographic order on the f -excesses as follows For each x ∈ Rn , let θ f (x) + be the vector in R| |−1 whose components are the f -excesses e f (S, x) of S ∈ + \{N }, arranged in increasing order, i.e., j θ if (x) ≤ θ f (x), ∀1 ≤ i < j ≤ | + | − For x, y ∈ Rn , θ f (x) is called lexicographically greater than θ f (y), denoted θ f (x) if there exists an index i such that θ f (y), θ if0 (x) > θ if0 (y) and θ if (x) = θ if (y), ∀i < i In this case, we say that x is a more acceptable price than y We write θ f (x) θ f (x) θ f (y) or θ f (x) = θ f (y) θ f (y) if Definition The f -nucleolus of a constrained cost allocation game Γ = (N , c, P, ) is the set N f (Γ ) := {x ∈ P | θ f (x) θ f (y), ∀y ∈ P} of all price vectors in P that maximize θ f with respect to the lexicographic ordering 123 Ann Oper Res (2015) 226:51–68 55 Algorithm computes the f -nucleolus of a constrained cost allocation game Γ = (N , c, , P) It considers a finite sequence of “shrinking subgames” Γk := (N , c, Pk , k+ ) + + +, P ∗ ⊆ P ∗ of Γ , k = 1, , k ∗ , i.e., k+∗ ··· k k −1 ⊆ · · · ⊆ P1 = P, = k ∗ −1 ∗ and computes their associated f -least core radii which satisfy ε ≤ ε · · · ≤ ε k The algorithm determines in iteration k a set of coalitions Bk with optimal f -excess ε k and fixes their prices by adding the constraints x(S) = c(S)−ε k f (S), S ∈ Bk , to Pk The new constraint set Pk+1 also fixes the prices (and excesses) of all coalitions S with linearly dependent incidence vectors, i.e., χ S ∈ span χ{N }∪ k Bi ; these are removed from the coalition set k+ to produce i=1 + the next coalition set k+1 (N is not removed in order to make k+ a valid coalition family) This makes the procedure computationally efficient It stops when all prices have been fixed; Pk ∗ +1 then describes the f -nucleolus of Γ The algorithm thereby maximizes gradually the attractiveness of a cooperation for all coalitions by improving their prices Algorithm is a generalization and improvement of the one in Hallefjord et al (1995), which considers the nucleolus of classical cost allocation games Algorithm Computing the f -nucleolus of Γ = (N , c, P, ) 1: k := 1, 1+ := + , P1 := P, F1 := {N } 2: Solve the f -least core problem (LPk ) associated with the subgame Γk = (N , c, Pk , k+ ) max ε (4) (x,ε) + k \ {N } s.t x(S) + ε f (S) ≤ c(S), ∀S ∈ x ∈ Pk (5) (6) Let (x k , ε k ) and (λk , μk ) be primal and dual optimal solutions of (LPk ), where λk corresponds to the constraints (5) and μk to constraints (6) Define Πk := suppλk = {S ∈ + k k \ {N } | λ S > 0} 3: Choose Bk ⊆ Πk such that k Fk+1 := Fk ∪ Bk = {N } ∪ Bi i=1 gives rise to a maximal linearly independent set χFk+1 Define + k+1 := {S ∈ + |χ ∈ S / span χFk+1 } ∪ {N }, Pk+1 := {x ∈ P | x(S) + εi f (S) = c(S), ∀S ∈ Bi , i = 1, , k} 4: If |Fk+1 | < dim span χ then k ← k + and goto 2, else set k ∗ := k and stop There holds the following result Theorem Algorithm terminates after k ∗ steps, k ∗ ≤ dim spanχ − 1, with the f nucleolus N f (Γ ) = Pk ∗ +1 of Γ The f -nucleolus is, in particular, non-empty If the coalition family is full dimensional, the f -nucleolus contains a unique point Proof The proof proceeds by induction over k We prove the following claims: Γk = (N , c, Pk , + k ), k = 1, , k ∗ , is a well defined constrained cost allocation game 123 56 Ann Oper Res (2015) 226:51–68 χ S ∈ / span χFk , S ∈ k+ \ {N }, k = 1, , k ∗ + k ≤ |Fk | ≤ dim span χ , k = 1, , k ∗ + ∀k = 1, , k ∗ : LC f (Γk ) ⊆ LC f (Γi ), ∀i : ≤ i ≤ k Due to Lemma 1, claim implies that the algorithm is well-defined, claim is technical, claim proves that it terminates, and claim helps to show that it produces the correct output In the base case k = 1, claim holds since N ∈ + = {N } and P = ∅, claim as χ + \{N } ∩ span χF1 = χ + \{N } ∩ span = ∅, claim as |F1 | = |{N }| = 1, and claim as k = i = Now consider the induction step k → k + By the induction hypothesis, Γk is well defined Then Lemma implies that (LPk ) has optimal primal and dual solutions (x k , ε k ) and (λk , μk ); in particular, ε k is the f -least core radius of Γk and x k belongs to the f -least core LC f (Γk ) of Γk By duality, we have that λk ≥ and λkS f (S) = S∈ + k \{N } + k From this it follows that λk = and Πk = ∅ As Πk ⊆ \ {N }, claim implies {S ∈ Πk | χ S ∈ span χFk } = ∅ Hence we can choose Bk = ∅ and then |Fk+1 | > |Fk | ≥ k, i.e., |Fk+1 | ≥ k + As Fk+1 ⊆ + , we have k + ≤ |Fk+1 | ≤ dim span χ + = dim span χ , + i.e., claim holds Claim follows directly from the definition of k+1 + and We next show claim that Γk+1 is well defined by checking the conditions on k+1 Pk+1 If the termination criterion in step is not fulfilled, i.e, |Fk+1 | < dim span χ = + dim span χ + , then the set {S ∈ + | χ S ∈ / span χFk+1 } = k+1 \ {N } is non-empty and + + hence k+1 = {N }; N ∈ k+1 by definition By complementary slackness, the optimal solution (x k , ε k ) of (LPk ) satisfies x k (S) + ε k f (S) = c(S), ∀S ∈ Πk , i.e., x k ∈ Pk+1 This proves claim It also proves LC f (Γk ) ⊆ Pk+1 and hence ε k+1 ≥ ε k By the induction hypothesis, in order to show claim 4, we only have to prove that LC f (Γk+1 ) ⊆ LC f (Γk ) Let x be a vector in LC f (Γk+1 ) Since x k , x ∈ Pk+1 , we have that x k (S) + εi f (S) = c(S) = x(S) + εi f (S), ∀S ∈ Bi , i = 1, , k, i.e., x k (S) = x(S), ∀S ∈ Bi , i = 1, , k Since x k (N ) = c(N ) = x(N ), it follows that x k (S) = x(S), ∀S : χ S ∈ span Fk+1 , i.e., e f (S, x k ) = e f (S, x), ∀S ∈ 123 + \ + k+1 Ann Oper Res (2015) 226:51–68 57 From this and since (x k , ε k ) is an optimal solution of (LPk ), we have + e f (S, x) = e f (S, x k ) ≥ ε k , ∀S ∈ ( \ + k+1 ) ∩ ( + k \ {N }) = + k \ + k+1 On the other hand, since x ∈ LC f (Γk+1 ), there holds S∈ + k+1 \{N } e f (S, x) = ε k+1 ≥ ε k Hence (x, ε k ) is a feasible and therefore an optimal solution of (LPk ) for every x ∈ LC f (Γk+1 ), i.e., LC f (Γk+1 ) ⊆ LC f (Γk ) k ∗ ≤ dim span χ − follows from claim by setting k = k ∗ + We now prove that ∗ N f (Γ ) = Pk ∗ +1 For every y ∈ Pk ∗ +1 , since x k ∈ Pk ∗ +1 , similar to the proof of claim 4, there holds ∗ x k (S) = y(S), ∀S : χ S ∈ span Fk ∗ +1 Since Fk ∗ +1 contains dim span χ linearly independent vectors, we have | χ S ∈ span Fk ∗ +1 } = {S ∈ Therefore ∗ x k (S) = y(S), ∀S ∈ ∗ ∗ and θ f (x k ) = θ f (y) Hence LC f (Γk ∗ ) = Pk ∗ +1 and if we can prove that θ f (x k ) θ f (z) for every z ∈ P\Pk ∗ +1 then there holds N f (Γ ) = Pk ∗ +1 Indeed, let z be a vector in P\Pk ∗ +1 The proof of claim shows that LC f (Γk ) ⊆ Pk+1 for k = 1, , k ∗ Therefore k∗ P\P k ∗ +1 = P1 \P k ∗ +1 = k∗ Pk \Pk+1 ⊆ k=1 Pk \LC f (Γk ) k=1 Hence, z ∈ Pk \LC f (Γk ) for some k, ≤ k ≤ k ∗ Since z ∈ Pk and z ∈ LC f (Γk ) S∈ + k \{N } e f (S, z) = + k \{N } S∈ c(S) − z(S) < εk f (S) (7) ∗ On the other hand, due to claim 4, we have x k ∈ Pk ∗ +1 = LC f (Γk ∗ ) ⊆ LC f (Γk ) and hence S∈ ∗ ∗ + k \{N } e f (S, x k ) = c(S) − x k (S) = εk + f (S) k \{N } S∈ (8) ∗ Since x k , z ∈ Pk , similar to the proof of claim 4, we have that ∗ e f (S, x k ) = e f (S, z), ∀S ∈ + \ + k (9) k∗ From (7)–(9) it follows that θ f (x ) θ f (z) Finally, if is full dimensional, then there holds |Fk ∗ +1 | = dim span χ = n Hence, since χFk ∗ +1 is independent, Pk ∗ +1 contains at most one vector On the other hand, ∗ ∗ since x k belongs to Pk ∗ +1 , we have Pk ∗ +1 = {x k } That means the f -nucleolus of Γ ∗ k contains a unique point, namely, x 123 58 Ann Oper Res (2015) 226:51–68 For the f -nucleolus concept, the coalitions consisting of one player have the same priority as the other coalitions The role of each individual player is, however, more important The social critic HL Mencken once quipped that, “a wealthy man is one who earns $100 a year more than his wife’s sister’s husband” That means personal objectives are quite local It does not rely on some absolute measure but is relative to what other people have It is very hard to convince someone that his price is fair, while somebody else has to pay just a fraction of his price for one unit Game theoretical fairness (or coalitional fairness) means that the price should reflect the position of each coalition and its cost by considering all possible groupings Individual fairness tends to equality, i.e., each player has to pay the same amount of money for one unit The ( f, r )-least core, which is defined below, is a compromise between coalitional fairness and individual satisfaction N be a vector that satisfies Let r ∈ R>0 ri = c(N ) i∈N Vector r is called a reference price-vector of Γ For example for the ticket pricing problem we can choose r as the distance-price vector The distance-price of a passenger is the product of the traveling distance and some base-price for a passenger for a distance unit The ratio xrii in this case is nothing else than the ratio between the price that player i is asked to pay for a distance unit and the base-price Each individual player i prefers a small ratio xrii A price xi with a big ratio xrii will be seen as unfair by player i, since in this case there exists a player j x with much smaller ratio r jj Our goal is to find a price vector in the f -least core of Γ that is as “near” as possible to r It means that from the point of view of the cooperative game theory our price is fair since it belongs to the f -least core and hence the minimal weighted benefit of the coalitions in + \{N } is as large as possible On the other hand, from the point of view of each individual player, the increment of the price of each player in comparison to its reference price is as small as possible Define Λ := {i} | i ∈ N ∪ {N } (10) and R : Λ → R>0 , R(N ) = c(N ) and R({i}) = ri , ∀i ∈ N (11) Function R is called a reference price-function of Γ We have then a new constrained cost allocation game Δ := (N , R, LC f (Γ ), Λ) This is indeed a constrained cost allocation game since LC f (Γ ) is non-empty due to Lemma and for each x ∈ LC f (Γ ) there holds x(N ) = c(N ) = R(N ) For each price vector x and player i ∈ R, the R-excess of the coalition {i} at x is e R ({i}, x) = xi R({i}) − xi =1− R({i}) ri Due to Theorem 1, the R-nucleolus of Δ is well-defined and contains a unique vector It maximizes θ R,Δ (x) in LC f (Γ ) with respect to the lexicographic ordering, where θ R,Δ (x) is the R-excess vector of Δ at x, i.e., the vector in R N whose components are the Rexcesses e R ({i}, x), i ∈ N , arranged in increasing order Let us define ϑ R,Δ (x) as the vector in R N whose components are the ratios xrii , i ∈ N , arranged in decreasing order Then, equivalently, the R-nucleolus of Δ minimizes ϑ R,Δ (x) in LC f (Γ ) with respect to the lexicographic ordering That means, by using the R-nucleolus of Δ as the price, the ratios xrii , i ∈ N , are kept as small as possible 123 Ann Oper Res (2015) 226:51–68 59 Definition Given are a constrained cost allocation game Γ = (N , c, P, ), a weight funcN of Γ The set Λ and the function R tion f : + → R>0 , and a reference price-vector r ∈ R>0 are defined in (10) and (11), respectively The R-nucleolus of Δ = (N , R, LC f (Γ ), Λ) is called the ( f, r )-least core of Γ , denoted by LC f,r (Γ ) Due to Theorem 1, there holds the following corollary Corollary Given a constrained cost allocation game Γ = (N , c, P, ), a weight function N of Γ The ( f, r )-least core of Γ is f : + → R>0 , and a reference price-vector r ∈ R>0 well-defined and contains a unique vector Computational aspects The f -nucleolus and the ( f, r )-least core belong to the f -least core Finding a vector in the f -least core of a constrained cost allocation game is NP-hard in general Faigle et al (2000) show that computing a vector in the f -least core of min-cost spanning tree games, which is a special cost allocation game, is NP-hard The biggest challenge is that there is exponential number of possible coalitions This problem can be overcome, however, by using a constraint generation approach (Hallefjord et al 1995) In this section, to apply constraint generation approaches, we only consider the so called constrained combinatorial cost allocation games This class is large enough as the cost function is often given by a minimization problem A constrained combinatorial cost allocation game Γ = (N , c, P, ) is a constrained cost allocation game where the cost function c is given by an optimization problem of the following form ∀S ∈ + : c(S) := cξ ξ s.t Bξ ≥ Cχ S Dξ ≥ d ξ j ∈ Z j , j = 1, 2, , k, (12) where χ S is the incidence vector of S, Z j is the set of either real, or integer, or binary numbers, and B, C, and D are matrices of suitable dimensions We assume that the weight function f satisfies f = α + β| · | + γ c with α, β, γ ≥ and α + β + γ > Define Q := (x, ε) ∈ Rn+1 | x(S) + ε f (S) ≤ c(S), ∀S ∈ \{∅, N } In order to construct the above mentioned constraint generation approach, we firstly consider the separation problem of Q: Given is a vector (x, ¯ ε¯ ), we ask whether (x, ¯ ε¯ ) belongs to Q If the answer is “no”, then find a valid cut that cuts off (x, ¯ ε¯ ) from Q To this, we only have to consider the following optimization problem max S∈ \{∅,N } x(S) ¯ + ε¯ f (S) − c(S) (13) If the optimal value of (13) is non-positive, then (x, ¯ ε¯ ) ∈ Q Otherwise, let T be an optimal solution of (13), then x(T ) + ε f (T ) ≤ c(T ) 123 60 Ann Oper Res (2015) 226:51–68 is a valid cut of Q which cuts off (x, ¯ ε¯ ) Due to (12), if γ ε¯ ≤ 1, then we can rewrite (13) as follows max α ε¯ + (z,ξ ) (x¯i + β ε¯ )z i + (γ ε¯ − 1)cξ i∈N s.t Bξ ≥ C z Dξ ≥ d ξj ∈ Z j, 1≤ j = 1, 2, , k, z i ≤ |N | − 1, i∈N z∈χ , (14) where the variable z represents the incidence vector of the set S in (13) If = N , then the constraint z ∈ χ is nothing else than z ∈ {0, 1} N Based on the above separation problem, we can calculate the ( f, r )-least core using the constraint generation approach presented in Algorithm Algorithm Computing the ( f, r )-least core of Γ = (N , c, P, ) Given a (small) subset Ω of satisfying Ω N and Ω\{∅, N } = ∅ 1: Compute the f -least core radius εΩ of ΓΩ := (N , c|Ω , P, Ω) The f -least core of ΓΩ is the following set LC f (ΓΩ ) = x ∈ X (ΓΩ ) | x(S) ≤ c(S) − εΩ f (S), ∀S ∈ Ω + \{N } , where Ω + := Ω\{∅} 2: Compute the ( f, r )-least core of ΓΩ , i.e., the R-nucleolus of the constrained cost allocation game (N , R, LC f (ΓΩ ), Λ), using Algorithm and obtain a vector xΩ 3: Consider the separation problem max S∈ + \{N } xΩ (S) + εΩ f (S) − c(S) If the optimal value is positive, then find a set T in (15) + \{N } that satisfies xΩ (T ) + εΩ f (T ) − c(T ) > 0, set Ω := Ω ∪ {T } and go to 4: {xΩ } is the ( f, r )-least core of Γ Remark In Algorithm 2, since εΩ is the f -least core radius of ΓΩ , one can easily prove that γ εΩ ≤ and hence we can rewrite (15) as (14) with x¯ = xΩ and ε¯ = εΩ Remark In practice, we not add only one but several violated coalitions T in each step Computational results show that our constraint generation approach works well in practice For the IC-ticket-price example in Sect 5, instead of 285 it only need to consider 781 coalitions In the case that there exists a violated coalition T , finding the optimal solution of the separation problem, i.e., the most violated coalitions, is not necessary and very time-consuming Instead we should be able to quickly find sufficiently good solutions for the separation problem In the following, we consider several heuristics for the separation problem Heuristics 123 Ann Oper Res (2015) 226:51–68 61 not solve the original problem in general, but they may speed up the process immensely We are interested in primal and dual heuristics The primal heuristic provides good starting solutions for the mixed integer program (14), while the dual heuristic give us some measure to answer the question of whether the current best found solution is good enough Primal Heuristics: During the constraint generation process, we add several constraints to the starting set A natural idea is to create a heuristical method where some z-variables in (14) are fixed to in a reliable way For this, a function evaluating the history of the added constraints is required Let h be a function defined for every finite sequence of binary numbers Typically, we can choose h equal to the sum of the values of some function eat x with a > applied to the elements of the input sequence, where t and x correspond to their indexes and their values, respectively This choice is reasonable, since the recent added constraints are more likely to provide reliable information for fixing variables than the older ones, while the ones that were added long before may have no connection anymore with the current separation problem and should be ignored As a result we can define h for any given sequence of binary numbers {ω1 , ω2 , , ωm } as h(ω1 , ω2 , , ωm ) = m aj j=1 e ω j m aj j=n−K +1 e ω j if m ≤ K otherwise, for some given number K ∈ N For our calculations in Sect 5, we choose a = 0.1 and K = 30 Let 1m denote the sequence of m numbers Define h m := h(1m ) and H (ω1 , ω2 , , ωm ) := h(ω1 , ω2 , , ωm ) hm The function H can be used to find heuristically the players which are likely to belong to a violated coalition of the current separation problem as follows We consider now the (m + 1)th separation problem Let S j be the most violated coalition found in step j, ≤ j ≤ m For each player i ∈ N , we have a sequence {χ Si , χ Si , , χ Si m } which tells us whether the player i belongs to the most violated coalition found in each separation step in the past For each player i, denote Hm (i) := H (χ Si , χ Si , , χ Si m ) Hm (i) = means that i belongs to every coalition in {S1 , S2 , , Sm } If Hm (i) is almost 1, then i may belong to a violated coalition in the present step Let ν1 be a positive number which is smaller but close to 1, e.g., ν1 = 0.97 For every player i satisfying Hm (i) ≥ ν1 , we fix z i = in (14) and solve (14) Another fixing method is based on the idea of Relaxation Induced Neighborhood Search (RINS) (Danna et al 2005) Let ν2 and ν3 be positive numbers which are smaller but close to and ν2 < ν1 Let z R be an optimal LP-relaxation solution of (14) We then fix z i = for every player i satisfying Hm (i) ≥ ν2 and z iR ≥ ν3 and solve (14) Computational results show that our heuristical fixing methods are very effective They can find violated coalitions in almost every separation step Using the violated coalitions, which are found by our heuristics, as starting solutions for the original separation problem (14), we can identify violated coalitions faster or find coalitions with bigger violation in the same given time limit than solving (14) from scratch We also use the solution polishing heuristic of CPLEX It often finds a much better solution of the separation problem from an initial one after just a few seconds Stopping Criterion and Dual Heuristic: Since we only want to find a good solution of (14) with a positive objective value, we need a suitable stopping criterion We stop the solver 123 62 Ann Oper Res (2015) 226:51–68 whenever a time limit or a gap limit is exceeded The gap limit is given by a decreasing function depending on the objective value of the current best found solution of (14) It is hard to improve not only the primal bound of (14) but also its dual bound However, the optimal values of successive separation problems are only slightly different Therefore, since we not solve the separation problem to optimality, an exact dual bound is not required and we can use the best dual bound of the separation problems in the past to evaluate the obtained solutions in the current separation step The dual heuristic works as follows Define a number called heuristical dual bound and set it to infinite at the beginning of the constraint generation process Whenever it is larger than the dual bound of the current separation problem or smaller than the current best found feasible solution, we set it to the current dual bound The latter can happen, but in practice it only occurs a few (say, less than five) times in the first steps of the constraint generation process The heuristical gap is defined as |heuristical dual bound − best objective value| 10−10 + |best objective value| Ticket pricing as a constrained cost allocation game The ticket pricing problem gives rise to a constrained cost allocation game Γ = (N , c, P, N ) in the following way Consider a railway network as a graph G = (V, E), and let N ⊆ V × V be a set of origin-destination (OD) pairs, between which passengers want to travel, i.e., we consider each (set of passengers of an) OD-pair as a player The price of each passenger of an OD-pair is equal to the price of the corresponding player divided by the number of passengers of the OD-pair Note that one can also build a game model where the individual trips are the players, but then the number of coalitions is enormous, which makes the problem unsolvable We next define the cost c(S) of a coalition S ⊆ N as the minimum operation cost of a network of railway lines in G that service S Using the classical line planning model of Bussieck (1998), c(S) can be computed by solving the integer program c(S) := (ξ,ρ) (cr, f ξr, f + cr, f ρr, f ) (r, f )∈R×F ccap f (mξr, f + ρr, f ) ≥ s.t r ∈R,r e f ∈F Pei , ∀e ∈ E i∈S f ξr, f ≥ Fei , ∀(i, e) ∈ S × E r ∈R,r e f ∈F ρr, f − (M − m)ξr, f ≤ 0, ∀(r, f ) ∈ R × F ξr, f ≤ 1, ∀r ∈ R f ∈F |R×F | ξ ∈ {0, 1}|R×F | , ρ ∈ Z≥0 (16) The model assumes that the Pi passengers of each OD-pair i travel on a unique shortest path P i (with respect to some distance in space or time) through the network, such that demands Pei on transportation capacities on edges e arise, and, likewise, demands Fei on frequencies of edges These demands can be covered by a set R of possible routes (or lines) in G, which can be operated at a (finite) set of possible frequencies F , and with a minimal and maximal number of wagons m and M in each train ccap is the capacity of a wagon, 123 Ann Oper Res (2015) 226:51–68 63 and c2 , (r, f ) ∈ R × F , are cost coefficients for the operation of route r at frequency cr, f r, f f The variable ξr, f equals if route r is operated at frequency f , and otherwise, while variable ρr, f denotes the number of wagons in addition to m on route r with frequency f The constraints guarantee sufficient capacity and frequency on each edge, link the two types of route variables, and ensure that each route is operated at a single frequency Model (16) has originally been proposed by Bussieck (1998) and is often used as a benchmark model in the line planning literature; the data is publicly available Though the model is NP-hard, it can be solved in reasonable time with a standard IP-solver for the problem sizes that we consider We remark that more sophisticated line planning models exist, that could also be used in our approach (at increased computational cost) Finally, we define the polyhedron P, which gives conditions on the prices x that the players are asked to pay, as follows For each OD-pair (s, t), let (u j−1 , u j ), j = 1, , l, be OD-pairs such that u j , j = 0, , l, belong to the travel path P st associated with the OD-pair (s, t), u = s, and u l = t, and let (u, v) be an arbitrary OD-pair such that u and v also lie on the travel path P st from s to t We then stipulate that the prices xi /Pi that individual passengers of OD-pair i have to pay must satisfy the monotonicity properties 0≤ xuv xst ≤ ≤ Puv Pst l j=1 xu j−1 u j Pu j−1 u j (17) Moreover, we can require that the prices should have the following property max st xst xst ≤ K , st dst Pst dst Pst (18) where dst is the distance of the route (s, t) This inequality guarantees that the price difference per unit of length, say one kilometer, is bounded by a factor of K Finally, the prices should be non-negative and cover exactly the total cost c(N ) The tuple Γ = (N , c, P, N ) defines a constrained cost allocation game to determine fair cost-covering prices for using the railway network G, in which coalitions S consider the option to bail out of the common system and set up their own, private ones Fair inter-city ticket prices We now use our ansatz to compute ticket prices for the intercity network of the Netherlands, which is shown in Fig Our data is a simplified version of that published in Bussieck (1998), namely, we consider all 23 cities, but reduce the number of OD-pairs to 85 by removing pairs with small demand However, with 285 −1 possible coalitions, the problem is still very large Since there is only one train type, the distance price depends only on the traveling distance As reported in Borndörfer et al (2006), the distance price, which has been used by the railway operator NS Reizigers for this network, is piecewise linear depending on the traveling distance, where the average price for one kilometer decreases However, since the data of this academic example and the real data of NS Reizigers are different, we not know the coefficients of the distance price function for our application Hence, instead of using a piecewise linear function, we choose the linear distance price function for pricing That means each passenger has to pay the same amount of money for one traveling distance unit, which is called the base price The distance price of each passenger is then the product of the base price and the traveling distance The base price is so chosen that the total distance 123 64 Ann Oper Res (2015) 226:51–68 Fig The intercity network of the Netherlands prices cover exactly the common cost c(N ), i.e., base price = common cost total traveling kilometers of all passengers The distance price x¯i of an OD-pair i is the product of the distance price for one passenger in this OD-pair and the number of its passengers For our ( f, r )-least core price, we choose f = c and r = x ¯ We start with a “pure fairness scenario” where the prices are only required to have the monotonicity property (17), be non-negative, and cover exactly the total cost c(N ), i.e., we ignore property (18) for the moment By using Algorithm 2, we determine the (c, x)-least ¯ core, which contains a unique point x ∗ , and define the (c, x)-least ¯ core ticket price (lc-price) for each passenger in an OD-pair i as pi∗ := xi∗ /Pi Before starting with the comparison between different pricing approaches, we want to briefly deal with the question of how one can evaluate the fairness of a given price vector and/or visually compare two different price vectors For games with many players, it is impossible to calculate the cost and profit of every coalition with a given price vector Therefore, we should only consider some representative coalitions, whose f -profits sample and represent the f profits of all coalitions in \{∅, N } These coalitions are chosen heuristically as described in Chapter in Hoang (2010).1 To compare the distance and (c, x)-least ¯ core prices, a pool of 7084 representative coalitions is (heuristically) created, which also contains coalitions having the worst relative profits In Hoang (2010) those coalitions are called essential However this name exists already in cooperative game theory We therefore use another name to avoid confusion 123 65 0.3 0.8 0.2 relative profit relative profit Ann Oper Res (2015) 226:51–68 0.6 0.4 0.2 -0.4 0.1 -0.1 -0.2 lc prices distance prices -0.2 -0.3 2000 4000 6000 lc prices distance prices 8000 20 40 60 80 100 coalition coalition Fig Distance versus unbounded (c, x)-least ¯ core prices (1) lc-prices/distance-prices relative profit 0.8 0.6 0.4 0.2 lc prices distance prices -0.2 -0.4 2000 4000 6000 14 distribution 12 10 8000 coalition 20000 40000 60000 80000 number of passengers Fig Distance versus unbounded (c, x)-least ¯ core prices (2) with these prices The coalitions in the pool are sorted in non-decreasing order regarding their relative profits with the distance prices Figure compares the lc-price vector with the distance price vector The picture on the left side plots the relative profits c(S)−x(S) of every c(S) coalition S in the pool with x = x ∗ and x = x, while the picture on the right side considers only the 100 first coalitions The picture on the left side of Fig plots also these c-profits but the values are sorted in non-decreasing order for both prices x ∗ and x, i.e., a point k in the horizontal axis does not represent the kth coalition in the pool but the two coalitions which have the kth smallest c-profits with the two prices x ∗ and x Note that the core of this particular game is empty and therefore with any price vector there exist coalitions which have to pay more than their costs The maximum c-loss of any coalition with the lc-prices is a mere 1.1 % This hardly noticeable unfairness is in contrast with the 25.67 % maximum c-loss of the distance prices In other words, the subsidized amount with the distance prices, that high demand routes have to subsidize the lower demand ones, is too high, while the lc-prices reduce this amount significantly In fact, there are 10 other coalitions in our pool with losses of more than 20 % Even worse, the coalition with the maximum loss is a large coalition of passengers traveling in the center of the country It is the coalition of the following OD-pairs: Amsterdam CS—Den Haag HS, Rotterdam CS—Schiphol, Amsterdam CS—Rotterdam CS, Den Haag HS—Rotterdam CS, Roosendaal Grens—Schiphol, Amsterdam CS—Roosendaal Grens, Den Haag HS—Roosendaal Grens, Roosendaal Grens—Rotterdam CS Table lists several major coalitions that would earn a substantial benefit from shrinking the network This table demonstrates the unfairness of the distance price vector and the instability of the grand coalition if the distance price vector is used Figure shows that the c-profits of the 123 66 Ann Oper Res (2015) 226:51–68 Table Unfairness of the distance price vector Coalition ID Relative profit (%) 26 56 88 133 191 −25.67 −18.39 −16.36 −14.12 −11.90 −10.43 unbounded lc-prices bounded lc-prices 0.4 relative profit relative profit 15.34 18.35 30.81 34.02 57.34 78.14 0.5 0.5 0.4 Percentage of all passengers (%) 0.3 0.2 0.1 unbounded lc-prices bounded lc-prices 0.3 0.2 0.1 -0.05 -0.05 2000 4000 6000 coalition 8000 2000 4000 6000 8000 coalition Fig Unbounded versus bounded lc-prices 100 coalitions which have the worst c-profit (the largest c-lost) with the distance prices are increased significantly with the the lc-prices Many of them have even a relative profit of more than 10 % with the lc-prices The picture on the right side of Fig plots the distribution of the ratio between the lcprices and the distance prices A point (Π, ρ) in this graph means that there are exactly Π passengers who have to pay at least ρ times their distance prices It can be seen that lc-prices are lower, equal, or slightly higher than the distance prices for most passengers However, some passengers, mainly in the periphery of the country, pay much more to cover the costs that they produce The increment factor is at most 3.775 except for two OD-pairs, which face very high price increases The top of the list is the OD-pair Den Haag CS–Den Haag HS, which gets 14.4 times more expensive The reason is that the travel path of this OD-pair consists of a single edge that is not used by any other travel route The other two in the top three OD-pairs with a high increment factor are Hengelo–Oldenzaal Grens (factor 11.85) and Apeldoorn–Oldenzaal Grens (factor 3.775) The passengers of these OD-pairs travel in the periphery of the country From a game theoretical point of view, these (unbounded) lc-prices can be seen as fair It would, however, be very difficult to implement such prices in practice We, therefore, add property (18) to limit the difference in the prices for one traveling kilometer of passengers by a factor of K Considering the results from the previous computation, we set K = The (c, x)¯ least core prices with this constraint are called the bounded lc-prices Figure compares the relative profits of the coalitions in our coalitions-pool with the bounded and unbounded lcprices The picture on the left side presents the relative profits of 7,050 coalitions, while the picture on the right side also plots these c-profits but their values are sorted in non-decreasing order Here we not plot the remaining 34 coalitions in the pool which have larger c-profits in order to observe the differences better These pictures show that the c-profits of every 123 67 0.3 0.8 0.2 0.6 relative profit relative profit Ann Oper Res (2015) 226:51–68 0.4 0.2 0.1 -0.1 -0.2 lc prices distance prices -0.2 lc prices distance prices -0.3 -0.4 2000 4000 6000 8000 20 40 60 80 100 coalition coalition Fig Distance versus bounded (c, x)-least ¯ core prices (1) lc-prices/distance-prices relative profit 0.8 0.6 0.4 0.2 lc prices distance prices -0.2 distribution 1.5 0.5 -0.4 0 2000 4000 6000 8000 coalition 20000 40000 60000 80000 number of passengers Fig Distance versus bounded (c, x)-least ¯ core prices (2) coalition with the two lc-prices are relatively close to each other However, the bounded lc-prices are better than the unbounded ones Figures and give the same comparisons to the distance prices as Figs and for the bounded lc-prices The maximum c-loss of any coalition with the bounded lc-prices is 1.68 %, which is slightly worse than before But the price increments are significantly smaller as plotted in the picture on the right side of Fig Again a point (Π, ρ) in this graph says that there are exactly Π passengers who have to pay at least ρ times their distance prices With the bounded lc-prices, nobody has to pay more than 1.89 times his distance price and nobody has to pay for one unit more than times the unit price of another passenger In this way, one can come up with price systems that constitute a good compromise between fairness and enforceability The computations were done on a PC with an Intel Core2 Quad 2.83 GHz processor and 16 GB RAM CPLEX 11.2 was used as linear and integer program solver It took in average 11.4 and 18.3 h respectively in order to calculate the bounded and unbounded lc-prices Conclusion Combining an extension of cost allocation games, a game theoretical concept, and computational algorithms, we proposed a new ansatz to determine fair ticket prices in public transport that constitute a good compromise between fairness and enforceability The constrained cost allocation game is a suitable generalization of cost allocation games in order to deal with real world requirements and can be used to model a wide range of applications 123 68 Ann Oper Res (2015) 226:51–68 Acknowledgments We would like to thank three anonymous reviewers for their insightful comments on the paper The work of Nam-D˜ung Hoàng is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) References Borndörfer, R., & Hoang, N.-D (2012) Determining fair ticket prices in public transport by solving a cost allocation problem In H G Bock, X P Hoang, R Rannacher, & J P Schlöder (Eds.), Modeling, simulation and optimization of complex processes (pp 53–63) Berlin: Springer Borndörfer, R., Neumann, M., & Pfetsch, M E (2006) Optimal fares for public transport In: Operations Research Proceedings 2005, pp 591–596 Borndörfer, R., Neumann, M., & Pfetsch, M E (2008) Models for fare planning in public transport, Technical Report, ZIB Report 08–16, Zuse-Institut Berlin Bussieck, M R (1998) Optimal lines in public rail transport, Ph.D thesis TU Braunschweig Danna, E., Rothberg, E., & Le Pape, C (2005) Exploring 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(1977) Aircraft landing fees: A game theory approach Bell Journal of Economics, 8, 186–204 Maschler, M., Potters, J., & Tijs, S (1992) The general nucleolus and the reduced game property International Journal of Game Theory, 21, 85–106 Straffin, P., & Heaney, J (1981) Game theory and the tennessee valley authority International Journal of Game Theory, 10, 35–43 Young, H P (1994) Cost allocation In R J Aumann & S Hart (Eds.), Handbook of Game Theory (Vol 2) Amsterdam: North-Holland Young, H P., Okada, N., & Hashimoto, T (1982) Cost allocation in water resources development Water Resources Research, 18, 463–475 123 ... way, large-scale instances of a ticket pricing constrained cost allocation game can be solved, substantially improving fairness In this paper, we model ticket pricing as a constrained cost allocation. .. Dutch IC railway example Game theoretical setting We present in this section the notion of constrained cost allocation game as a generalization of the classical cost allocation game It deals with... computational approaches in order to calculate game theoretical solutions concepts A model that treats ticket pricing as a constrained cost allocation game is presented in Sect The final Sect is