Int J Game Theory (2009) 38:291–304
DOI 10.1007/s00182-009-0154-9
ORIGINAL PAPER
On the core of routing games with revenues
Arantza Estévez-Fernández · Peter Borm ·
Marc Meertens · Hans Reijnierse
Accepted: 2 February 2009 / Published online: 11 March 2009
© The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract Traveling salesman problems with revenues form a generalization of
traveling salesman problems. Here, next to travel costs an explicit revenue is generated
by visiting a city. We analyze routing problems with revenues, where a predetermined
route on all cities determines the tours along subgroups. Corresponding routing games
with revenues are analyzed. It is shown that these games have a nonempty core and a
complete description of the core is provided.
Keywords
Routing problems · Revenues · Core
JEL Classification
C71
The authors thank two referees for their valuable suggestions for improvement. Special thanks go to
Herbert Hamers for his helpful discussion and comments.
A. Estévez-Fernández (B)
Department of Econometrics and OR, VU University Amsterdam,
De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
e-mail: mestevez@feweb.vu.nl
P. Borm · H. Reijnierse
Center and Department of Econometrics and OR, Tilburg University,
P.O. Box 90153, 5000 LE Tilburg, The Netherlands
M. Meertens
General Insurance Actuarial Risk Management (GARM), Reaal Verzekeringen,
Croeselaan 1 B03.24, P.O. Box 8444, 3503 RK Utrecht, The Netherlands
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1 Introduction
In a traveling salesman (TS) situation a salesman, starting in his home city, has to visit
a set of cities exactly once and has to come back to his home city at the end of the
journey. Associating travel costs to connections the problem is how to find a tour with
minimal cost. It is known that TS problems are NP-hard in general. For a survey on
TS problems we refer to Lawler et al. (1997).
Fishburn and Pollak (1983) introduced the cost allocation problem that arises when
each city (except the home city) corresponds to a player. The cost allocation is concerned with a fair allocation of the joint costs of the cheapest tour. This cost allocation
problem was first studied within the framework of game theory by Potters et al. (1992)
by introducing TS games. In a TS game, the value of a coalition of cities is the value
of the cheapest tour in the TS problem associated to the coalition. Here, only (and
exactly) the cities in the coalition will be visited. If the triangular inequalities are
satisfied every 3-, 4-, and 5-person TS game has a nonempty core (Potters et al. 1992;
Tamir 1989; Kuipers 1993). In the same setting however Tamir (1989) provides an
example of a 6-person TS game with an empty core. In Okamoto (2004) it is shown
that if the associated cost matrix is symmetric and has the Monge property, then the
corresponding TS game is totally balanced (i.e. every subgame has a nonempty core).
In Estévez-Fernández et al. (2006) it is seen that these results can be generalized to
multiple (longest) traveling salesman (M(L)TS) games. In an MTS problem the salesman has to visit each city exactly once except for the home city which can be revisited
as many times as desired. In a longest traveling salesman (LTS) problem there are
profits associated to connections instead of travel costs. Hence, the objective of an
LTS problem is to find a tour with maximal profit.
In Potters et al. (1992) also the class of fixed routing games is introduced. Here,
the route along all cities is predetermined (e.g. by restrictions in the agenda of the
salesman) and this tour determines the tours along all possible coalitions. Since the
chosen tour and its associated cost are provided by the salesman, coalitions do not
have any control on the tour that the salesman follows to visit them. Hence, the value
of a coalition of a routing game is defined as the cost associated to the tour that visits
the members of the coalition in the same relative order as in the predetermined tour.
Potters et al. (1992) show that routing games have a nonempty core if the predetermined tour is an optimal tour for the related TS problem. Derks and Kuipers (1997)
give a time efficient algorithm to provide core elements of a routing game. For a more
extensive motivation of the model of routing games we refer to Derks and Kuipers
(1997).
This paper studies routing problems with revenues: next to travel costs and a predetermined route on all cities, revenues of a visit are explicitly modeled and taken into
account. Note that since the revenues obtained by the visit of a salesman are explicitly
given, it might be the case that some of the cities will not be visited by the salesman if
the objective is to maximize total joint profit. We will assume that the predetermined
route is optimal in this sense and indeed visits all cities. Still, it might be optimal for a
coalition not to visit all its cities in the prescribed relative order. Hence, the value of the
associated routing game for a specific coalition is defined as the maximum attainable
profit by one of its subcoalitions if the salesman visits all cities in this subcoalition
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On the core of routing games with revenues
293
in the relative order given by the predetermined route on all cities. We will show
that every routing game with revenues has a nonempty core. Moreover, a complete
description of the core is provided.
The idea of analyzing cost problems arising from a general service facility by taking
explicitly into account the profits that the service will generate is not new. It was first
studied in Littlechild and Owen (1976) within the framework of airport problems, with
a more recent follow up in Brânzei et al. (2006). Okamoto (2004) briefly discusses
prize-collecting traveling salesman problems where a reward is associated to each city
and it is obtained only if the salesman visits the city. Meertens and Potters (2006)
consider fixed tree games with revenues. Suijs et al. (2005) study the sharing of costs
and revenues within a public network communication structure.
2 Routing games with revenues
Let N = {1, 2, . . . , n} denote the set of cities that a traveling salesman has to visit. Let
N0 := N ∪{0} where 0 denotes the traveling salesman’s home city. Let C = (ci j ) be an
N0 × N0 -matrix where ci j represents the costs to go from city i to city j. Throughout
this article we assume that:
(i) cii = 0 for all i ∈ N0 ,
(ii) ci j = c ji for all i, j ∈ N0 ,
(symmetry)
(iii) ci j ≤ ci0 + c0 j for all i, j ∈ N . (triangle inequalities)
Note that the triangular inequalities are only required with respect to the home city to
ensure willingness to cooperate among players: going back to the home city is always
at least as expensive as going directly from city i to city j. Potters et al. (1992) already
pointed out that only this type of triangular inequalities are important for the analysis
of nonemptiness of the core of routing games.
Whenever city i ∈ N is visited by the traveling salesman, a revenue bi ≥ 0 is
obtained. Due to the explicit modeling of the revenues we assume that the salesman,
who starts from city 0, visits each city at most once, and only returns to city 0 at the
end of the journey.
Let R ⊂ N and set R0 := R ∪ {0}. A bijection π : R0 → R0 is called a cyclic
permutation if min{t ∈ N | π t (i) = i} = |R| + 1 for every i ∈ R0 . We denote by
(R) the set of all cyclic permutations on R0 . A cyclic permutation π corresponds to
a tour along R: it starts in 0 and visits each city in R exactly once returning to 0 at the
end of the trip. Here, π(i) is the city immediately visited after city i for all i ∈ R0 . For
convenience, we sometimes denote city 0 also by n + 1 and in particular π(i) = n + 1
means that i is the last city on the tour. For π ∈ (R),
we denote by c(π, R) the cost
associated to the tour induced by π , i.e. c(π, R) = i∈R0 ciπ(i) . Consequently, the
minimal cost c(R) of a tour along R is given by
c(R) := min {c(π, R)}.
π ∈(R)
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The total profit p(R) obtained when the salesman has visited all cities in R according
to a tour with minimal cost is
p(R) := b(R) − c(R),
where b(R) := i∈R bi .
Due to the revenue structure it may be more profitable for N not to make a (complete) tour on N itself but on a subset R ⊂ N , leaving N \R unvisited. Therefore, the
optimization problem for N boils down to finding a subset of cities R such that p(R)
is maximal. We denote the maximal profit for N by v(N ), i.e.
v(N ) := max {p(R)}.
R⊂N
From now on we assume that it is optimal to visit all cities in N via the cyclic
permutation π̂ ∈ (N ). We also assume without loss of generality that
the optimal order π̂ for N is given by 0 − 1 − 2 − · · · − n − 0.
(2.1)
Hence, v(N ) = p(N ).
Associating each city in N with a player, the question we would like to address
is how to share v(N ) among the players. For this we choose the “routing” approach,
where π̂ determines the order in which potential subcoalitions are visited.
For S ⊂ N , the cyclic permutation π̂ S ∈ (S) induced by π̂ is obtained from π̂ by
skipping the cities in N \S and leaving the order of the remaining cities unchanged.
Formally, π̂ S is given by
π̂ S (i) = π̂ t (i) (i) for every i ∈ S0
where t (i) := min{t ∈ N | π̂ t (i) ∈ S0 }. With a minor abuse of notation we denote
c(π̂ S , S) by c(π̂ , S).
A coalition S ⊂ N need not decide on the complete tour π̂ S on S: a tour π̂ R on a
subset R ⊂ S may be more profitable. Hence, we define the value vπ̂ (S) in the routing
game (N , vπ̂ ) by
vπ̂ (S) = max b(R) − c(π̂ , R)
R⊂S
Note that vπ̂ (N ) = v(N ).
Example 2.1 Consider the routing problem with revenues represented in Fig. 1 where
the numbers at the edges represent the travel costs and the boldface numbers at the
nodes represent the revenues.
Note that assumption (2.1) is satisfied. The associated routing game with revenues
has values: vπ̂ ({1}) = 3, vπ̂ ({2}) = 0, vπ̂ ({3}) = 3, vπ̂ ({1, 2}) = 5, vπ̂ ({1, 3}) = 7,
vπ̂ ({2, 3}) = 3 and vπ̂ (N ) = 9.
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On the core of routing games with revenues
295
Fig. 1 The routing problem
with revenues in Example 2.1
Note that if the revenues are so high that the salesman visits all cities of every
coalition, then the routing game with revenues is strategically equivalent1 to a routing
game à la Potters et al. (1992) and Derks and Kuipers (1997).
3 The core
In this section we study the core of routing games with revenues. We show that routing
games with revenues have a nonempty core. Moreover, we give an intuitive interpretation of all core elements.
Let (N , v) be a cooperative game. Recall that the core of (N , v) is given by
Core(v) = x ∈ R N | x(N ) = v(N ), x(S) ≥ v(S) for all S ∈ 2 N ,
i.e. the core is the set of efficient allocations of v(N ) such that there is no coalition
with an incentive to split off.
In the following example we illustrate that taking into account revenues has a definite impact on the structure of the core.
Example 3.1 Consider the routing problem with revenues represented in Fig. 2.
It is readily checked that assumption (2.1) is satisfied and that vπ̂ (S) = 0 for every
S ⊂ N , S = N , and vπ̂ (N ) = 2. Therefore,
Core(vπ̂ ) = conv{(2, 0, 0), (0, 2, 0), (0, 0, 2)}.
Consider now the associated (cost) routing game à la Potters et al. (1992) and
Derks and Kuipers (1997) in which the revenues are explicitly not considered: they
are high enough. One readily verifies that cπ̂ ({1}) = 2, cπ̂ ({2}) = 6, cπ̂ ({3}) = 6,
1 Here, we make a slight abuse of language when we say that a routing game with revenues is strategically
equivalent to a (cost) routing game. We mean that there exist k ∈ R++ , a ∈ R N and (N , c), a (cost) routing
game, such that vπ̂ (S) = a(S) − kc(S) for every S ⊂ N .
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Fig. 2 The routing problem
with revenues in Example 3.1
cπ̂ ({1, 2}) = 6, cπ̂ ({1, 3}) = 8, cπ̂ ({2, 3}) = 8 and c(N ) = 8. Here,2
Core(cπ̂ ) = conv{(2, 0, 6), (0, 2, 6), (0, 6, 2), (2, 4, 2)}.
Hence, there is not an obvious relation between Core(cπ̂ ) and Core(vπ̂ ). Moreover, it
is readily checked that the above routing game with revenues is not strategically equivalent to any routing game à la Potters et al. (1992) and Derks and Kuipers (1997).
Next, we show that a routing game with revenues, (N , vπ̂ ), corresponding to travel
cost matrix C ∈ R N0 ×N0 and revenue vector b ∈ R N has a nonempty core.
For every S ⊂ N we define the linear programming problem LP(S) by:
maximize
n+1
n
(b j − ci j )xi j
(3.1)
i=0 j=i+1
s.t.
i−1
xki ≤ eiS
for all i ∈ {1, . . . , n};
(3.2)
xi j = 0
(3.3)
k=0
i−1
k=0
xki −
n+1
for all i ∈ {1, . . . , n};
j=i+1
xi j ≥ 0
for all i, j ∈ {0, 1, . . . , n, n + 1} with i < j.
(3.4)
with b0 = bn+1 := 0 and cin+1 := ci0 and where e S ∈ R N is a vector of zeros and
ones with eiS = 1 if i ∈ S and eiS = 0 otherwise.
It is readily checked that LP(S) is feasible and bounded. Here, xi j can be interpreted
as the “amount of flow that goes from i to j”. The profit obtained per unit of flow
from i to j is b j − ci j for every i and j such that 0 ≤ i < j ≤ n + 1 and the objective
2 The core of a cost game is defined as Core(c) = {x ∈ R N | x(N ) = v(N ), x(S) ≤ c(S) for all S ∈ 2 N }.
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On the core of routing games with revenues
297
function is to maximize the total profit as represented in (3.1). Equation (3.2) indicates
that the total flow “arriving” at city i can not exceed one unit of flow, i.e. one can think
of this as a capacity restriction on the nodes. Equation (3.3) makes sure that the amount
of flow “arriving” at i equals the amount of flow “leaving” i.
Note that the game (N , u) with u(S) defined as the optimal value of LP(S) for
S ⊂ N , is a linear production game and therefore it has a nonempty core (Owen
1975).
Lemma 3.1 (i) u(S) ≥ vπ̂ (S) for every S ⊂ N .
(ii) u(N ) = vπ̂ (N ).
Proof (i) Let S ⊂ N and R ⊂ S such that
vπ̂ (S) = b(R) − c(π̂ , R).
If R = ∅ , then vπ̂ (S) = 0 ≤ u(S). Otherwise, R := {i 1 , . . . , ir } with i 1 < · · · < ir .
Define
1 if i ∈ {0, i 1 , . . . , ir −1 } and j = π̂ R (i), or i = ir and j = n + 1,
xi j :=
0 otherwise.
Then, (xi j )0≤i< j≤n+1 is a feasible solution of LP(S) and
u(S) ≥
n
n+1
(b j − ci j )xi j
i=0 j=i+1
= bi1 + · · · + bir − c0i1 + ci1 i2 + · · · + cir −1 ir + c0ir
= vπ̂ (S).
(ii) By (i) it suffices to show that u(N ) ≤ vπ̂ (N ). Note that LP(N ) is a transportation
problem with {0, 1, . . . , n} the set of sources, {1, . . . , n, n + 1} the set of sinks, and
such that there are no links going from a source i to a sink j with i > j and the reward
when going from i ∈ N to itself is zero. Then, there exists an integral optimal solution, x̄ = (x̄i j )0≤i< j≤n+1 , for LP(N ) (see Nemhauser and Wolsey 1988, Chap. I.3,
Corollary 5.2). Moreover,
(i) x̄i j ∈ {0, 1} for every i, j with 0 ≤ i < j ≤ n + 1 by Eqs. (3.2) and (3.4).
(ii) If i−1
k=0 x̄ ki = 1, then there exists a unique k(i) ∈ {0, . . . , i − 1} such that
x̄k(i)i = 1 by (i).
(iii) If i−1
k=0 x̄ ki = 1, then there exists a unique j (i) ∈ {i + 1, . . . , n + 1} such that
x̄
Eq. (3.3).
i j (i) = 1 by
n
n
(iv)
i=1 x̄ 0i =
i=1 x̄ in+1 by Eq. (3.3).
Let N (x̄) := {i ∈ N | i−1
k=0 x̄ ki = 1} and let {i 1 , . . . , ir } = {i ∈ N ( x̄) | x̄ 0i = 1}.
Let t (il ) be the smallest integer such that j t (il ) (il ) = n + 1. We define
Nl (x̄) = { j t (il ) | t ∈ {1, . . . , t (il ) − 1}
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A. Estévez-Fernández et al.
for l ∈ {1, . . . , r }. It is readily checked that N1 (x̄), . . . , Nr (x̄) is a partition of N (x̄).
Moreover, note that this partition implies that r tours, π 1 ∈ (N1 (x̄)), . . . , π r ∈
(Nr (x̄)), are followed in the optimal solution, where tour π l is given by 0 − il −
· · · − j t (il )−1 (il ) − (n + 1) for l ∈ {1, . . . , r } (See Fig. 3).
Define π̃ ∈ (N (x̄)) by 0 − i 1 − · · · − j t (i1 )−1 (i 1 ) − i 2 − · · · − j t (i2 )−1 (i 2 ) − · · · −
ir − · · · − j t (ir )−1 (ir ) − (n + 1) (See Fig. 4).
Hence,
u(N ) =
=
=
n+1
n+1
(b j − ci j )x̄i j
i=0 j=i+1
r
l=1 i∈Nl (x̄)∪{0}
r
bπ l (i) − ciπ l (i)
b(Nl (x̄)) −
l=1
r
c π l , Nl (x̄)
l=1
≤ b ∪rl=1 Nl (x̄) − c π̃, ∪rl=1 Nl (x̄)
≤ b ∪rl=1 Nl (x̄) − c ∪rl=1 Nl (x̄)
≤ b(N ) − c(N ) = vπ̂ (N )
where the first inequality holds by the triangular inequalities, the second one holds
by definition of c(∪rl=1 Nl (x̄)) and the last one by assumption (2.1) and because it is
optimal to visit all cities in N by assumption.
⊔
⊓
Note that if two games (N , v) and (N , u) are such that v(S) ≤ u(S) for every
S ⊂ N , v(N ) = u(N ), and Core(u) = ∅, then Core(v) = ∅ and Core(u) ⊂ Core(v).
Hence, as a direct consequence of Lemma 3.1 we have that a routing game with
revenues has a nonempty core.
Theorem 3.2 Any routing game with revenues has a nonempty core.
The following result gives a full description of the core of a routing game with revenues.3 It states that an allocation x belongs to the core of the game if each coordinate
xi can be written as xi = bi −ci−1i +z i−1 −z i . This can be interpreted in the following
way: First of all, player i obtains the revenue bi when the salesman visits its city and
has to pay the travel costs ci−1i from city i − 1 to city i. Next, since player i − 1
also gets revenues from the visit, it helps player i with the travel costs by paying a
compensation z i−1 . In a similar way, player i helps player i + 1 with the travel costs
of the trip from city i to city i + 1 with z i . Equation (3.6) below reflects that player i
never compensates i + 1 more than the total amount he gets once i − 1 has paid the
compensation. Equation (3.7) reflects the fact that player j + 1 indeed prefers that the
salesman comes from player j instead of another player i(< j).
3 Since routing games with revenues generalize routing games, this description of the core can also be
applied to routing games.
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On the core of routing games with revenues
299
Fig. 3 Representation of the
tours given by π l
Fig. 4 Representation of π̃
Theorem 3.3 Let (N , vπ̂ ) be a routing game with revenues corresponding to travel
cost matrix C ∈ R N0 ×N0 and revenue vector b ∈ R N . Then, the following three
assertions are equivalent.
(i) x ∈ Core(vπ̂ ).
(ii) x ≥ 0, x(N ) = vπ̂ (N ), and x(N \S) ≥ vπ̂ (N \S) for every S ⊂ N with
S = {i, i + 1, . . . , j} and i ≤ j.
(iii) xi := bi − ci−1i + z i−1 − z i for all i ∈ {1, . . . , n} with
z i − z i−1
z 0 := 0, z n := c0n
≤ bi − ci−1i for all i ∈ {1, . . . , n}
(3.5)
(3.6)
z j − z i ≥ c j j+1 − ci j+1 for all i, j ∈ {0, 1, . . . , n} with i < j.
(3.7)
Proof (i)⇒(ii) is immediate and therefore omitted.
(ii)⇒(iii) Let x ∈ R N satisfy the conditions mentioned in assertion (ii) of the theorem.
Define the vector z ∈ R N0 as follows:
z 0 := 0
z i := bi − ci−1i + z i−1 − xi for every i ∈ {1, . . . , n}.
It is readily checked that z i =
{1, . . . , n}. Hence,
i
k=1 bk
−
i
k=1 ck−1k
−
i
k=1 x k
for all i ∈
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A. Estévez-Fernández et al.
z n = b(N ) −
= b(N ) −
=
n
k=1
n
ck−1k − x(N )
ck−1k − b(N ) −
n+1
ck−1k
k=1
k=1
cnn+1 = c0n
where the second equality follows from x(N ) = vπ̂ (N ). Consequently, Eq. (3.5)
holds.
With respect to Eq. (3.6), clearly x ≥ 0 implies z i − z i−1 ≤ bi − ci−1i .
Next, we show Eq. (3.7), i.e. z j − z i ≥ c j j+1 − ci j+1 for all i, j ∈ {0, 1, . . . , n}
with i < j. Suppose that there exist i, j ∈ {0, 1, . . . , n} with i < j be such that
z j − z i < c j j+1 − ci j+1 . Define S = {i + 1, . . . , j}. Clearly, if we can show that
x(N \S) < vπ̂ (N \S) we arrive at a contradiction with one of the assumptions in (ii).
Indeed,
x(N \ S) = b(N \ S) − [c01 + · · · + ci−1i + c j j+1 + · · · + cn−1n ]
+ [z 0 + · · · + z i−1 + z j + · · · + z n−1 ]
− [z 1 + · · · + z i + z j+1 + · · · + z n ]
= b(N \ S) − [c01 + · · · + ci−1i + c j j+1 + · · · + cn−1n ] + z j − z i − z n
= b(N \ S) − [c01 + · · · + ci−1i + c j j+1 + · · · + cn−1n + cn0 ] + z j − z i
< b(N \ S) − [c01 + · · · + ci−1i + c j j+1 + · · · + cn−1n + cn0 ]
+ c j j+1 − ci j+1
= b(N \ S) − [c01 + · · · + ci−1i + ci j+1 + · · · + cn−1n + c0n ]
= b(N \ S) − c π̂ N\S , N \ S
≤ vπ̂ (N \S),
where the second equality follows from z 0 = 0, the third one is a consequence of
z n = c0n , the strict inequality follows from the assumption and the weak inequality is
by definition of vπ̂ .
(iii)⇒(i) Let z ∈ R N0 satisfy the conditions (3.5), (3.6), and (3.7) mentioned in assertion (iii) of the theorem. Define xi := bi −ci−1i +z i−1 −z i for all 1 ≤ i ≤ n. It is readily
checked that x(N ) = vπ̂ (N ). Let S ⊂ N be a coalition, and let R := {i 1 , . . . , ir } ⊂ S
be such that vπ̂ (S) = b(R) − c(π̂ R , R). It suffices to prove that x(S) ≥ vπ̂ (S). For
this, note that
x(S) ≥ x(R) = b(R) −
r
cik −1ik +
r
k=1
123
cik −1ik +
z ik −1 −
r
k=2
z ik −1 −
r
zik
k=1
k=1
k=1
= b(R) −
r
r −1
k=1
z ik − z 0 + z i1 −1 − z ir + z n − c0n
On the core of routing games with revenues
≥ b(R) −
= b(R) −
= b(R) −
≥ b(R) −
= b(R) −
r
k=1
r
k=1
r
k=2
r
k=2
r
−1
cik −1ik +
cik −1ik +
cik −1ik +
cik −1ik +
301
r
k=2
r
−1
z ik −1 −
r −1
zik
k=1
r −1
z ik + ci1 −1i1 − c0i1 − c0ir
z ik+1 −1 −
k=1
r
−1
+ ci1 −1i1 − c0i1 − c0ir
k=1
[z ik+1 −1 − z ik ] − c0i1 − c0ir
k=1
r
−1
[cik+1 −1ik+1 − cik ik+1 ] − c0i1 − c0ir
k=1
cik ik+1 − c0i1 − c0ir
k=1
= b(R) − c(π̂ R , R)
= vπ̂ (S).
Here, the first inequality is a consequence of Eq. (3.6) which implies xi ≥ 0 for every
i ∈ N , the second inequality follows by applying Eq. (3.7) to j = i 1 − 1, i = 0 and
to j = n, i = ir and the second one is also an immediate consequence of Eq. (3.7).
⊔
⊓
Consider the vector x ∈ R N defined recursively by
xi = vπ̂ (N ) − max {vπ̂ (N \{k, . . . , i}) + x({k, . . . , i − 1})}
k≤i
(3.8)
for i ∈ {1, . . . , n}. This allocation can be interpreted as follows. Assume only
connected coalitions (i.e. coalitions {k, k + 1, . . . , i}) are allowed to step out of
the negotiations on the allocation of vπ̂ (N ) and stepping out is decided recursively
by the individual players. Consider that player i wants to step out. If the coalition
{k, k + 1, . . . , i} decides to step out, the players in N \{k, k + 1, . . . , i} further negotiate the allocation of vπ̂ (N \{k, k + 1, . . . , i}) and each player j ∈ {k, k + 1, . . . , i − 1}
already got x j . Hence, player i is left with vπ̂ (N )−[vπ̂ (N \{k, . . . , i})+x({k, . . . , i −
1})]. Having no influence on “earlier” stepping out player i can only claim the minimum compensation over the set of all possible connected coalitions {k, . . . , i} with
1 ≤ k ≤ i which is reflected in (3.8).
It turns out that the allocation x defined by Eq. (3.8) is a core element of vπ̂ . This
result is an immediate consequence of Theorem 4 in Derks and Kuipers (1997) and
the description of the core by coalitions N \ {k, . . . , i} given in Theorem 3.3. Hence,
the proof is omitted.
Theorem 3.4 Let (N , vπ̂ ) be a routing game with revenues corresponding to travel
cost matrix C ∈ R N0 ×N0 and revenue vector b ∈ R N . Let x be defined as in Eq. (3.8).
Then, x ∈ Core(vπ̂ ).
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A. Estévez-Fernández et al.
4 Final remarks
The dual (N , cd ) of a TU-game (N , c) is defined by
cd (S) = c(N ) − c(N \S) for every S ⊂ N .
In Derks and Kuipers (1997) the core of routing games is analyzed by studying
dual routing games. It turns out that dual routing games have the connectedness property (i.e. possibly after reindexing the players, all essential coalitionsare of the form
u
{i, . . . , j}, with i ≤ j, where a coalition S is essential if cd (S) > l=1
cd (Rl ) for
every (non-trivial) partition < R1 , . . . , Ru > of S). This together with the fact that
routing games are N -subadditive (i.e. c(N ) ≤ c(S) + c(N \S) for any S ⊂ N ) suffices
to show nonemptiness of the core for routing games. The following example shows
that this type of result cannot be directly extended in general if one additionally considers revenues in the way of this paper. Additional information on the underlying cost
structure that is not captured in the game is required.
Example 4.1 Consider the cost game (N , c) described in Table 1 and let b = (30, 9,
30). The corresponding dual game (N , cd ) and cost-revenue game (N , v) are also
given in Table 1. Here, following the ideas of this paper, (N , v) is defined by
v(S) = max{b(R) − c(R)}
R⊂S
It is readily checked that the dual game has the connectedness property and that
(N , c) is N -subadittive. Consequently, c is balanced, but (N , v) has an empty core
since v({1}) + v({2}) + v({3}) = 20 > 19 = v(N ).
Note that (N , c) corresponds to a routing game generated by the routing problem
given in Fig. 5. This routing problem however violates the triangular inequalities since:
c13 = 31 > 20 = c10 + c03 . Hence, the triangular inequalities are essential for our
result on nonemptiness of the core.
We have analyzed the core of routing games with revenues in which the
predetermined route is optimal for the associated combinatorial problem and visits
all cities in N . Next, we provide an example that illustrates that our assumption (2.1)
(i.e. the salesman visits all cities) is not restrictive. It turns out that if the salesman
only visits some of the cities, those that are unvisited receive a payoff of zero in any
core allocation and the various results provided in the previous sections are still valid.
Table 1 Coalitional values of
the cost game, dual game, and
cost-revenue game in
Example 4.1
123
S
{1}
{2}
{3}
{1, 2}
{1, 3}
{2, 3}
{1, 2, 3}
c(S)
20
20
20
35
51
35
50
cd (S)
15
−1
15
30
30
30
50
v(S)
10
0
10
10
10
10
19
On the core of routing games with revenues
303
Fig. 5 The routing problem
behind the cost game in
Example 4.1
Fig. 6 The routing problem
with revenues in Example 4.2
Example 4.2 Consider the routing problem with revenues represented in Fig. 6 where
the numbers at the edges represent the traveling costs and the boldface numbers at the
nodes represent the revenues.
It is readily seen that the optimal tour for this situation only visits the cities 1, 2, and
3 in the order 0 − 1 − 2 − 3 − 0 denoted by π̂ . Hence the coalitional values of the
routing game are: vπ̂ ({1}) = 1, vπ̂ ({2}) = 1, vπ̂ ({3}) = 0, vπ̂ ({4}) = 0, vπ̂ ({1, 2}) =
4, vπ̂ ({1, 3}) = 2, vπ̂ ({1, 4}) = 1, vπ̂ ({2, 3}) = 3, vπ̂ ({2, 4}) = 1, vπ̂ ({3, 4}) =
0, vπ̂ ({1, 2, 3}) = 6, vπ̂ ({1, 2, 4}) = 4, vπ̂ ({1, 3, 4}) = 2, vπ̂ ({2, 3, 4}) = 3 and
vπ̂ (N ) = 6. Here, player 4 is a zero player and the core of the game is Core(vπ̂ ) =
conv{(3, 3, 0, 0), (3, 1, 2, 0), (2, 4, 0, 0), (1, 4, 1, 0), (1, 3, 2, 0)}. Note that the core
can still be described by means of the cost of the tour, the vector of revenues, and a
vector of compensations as in Theorem 3.3.
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium,
provided the original author(s) and source are credited.
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