Cost sharing results

In this section we reformulate our results using the classic cost sharing termi-nology. In order to unify the dierent terminologies used in the literature, we exclusively use Thomson (2007)'s notions. First we introduce the notion of a rule, which is in practice analogous to the notion of a cost-sharing rule intro-duced by Aadland and Kolpin (1998) (see Denition 2.2). Let us consider the class of allocation problems dened on cost trees, i.e. the set of cost trees. A rule is a mapping which assigns a cost allocation to a cost tree allocation problem, providing a method by which the cost is allocated among the players. Note that the rule is a single-valued mapping. The analogy between the rule and the solution is clear, the only signicant dierence is that while the solution is a set-valued mapping, the rule is single-valued.

Let us introduce the rule known in the literature as sequential equal contribu-tions rule.

Denition 5.22 (SEC rule) For all cost trees (G, c) and for all players i the distribution according to the SEC rule is given as follows:

ξ_i^SEC(G, c) = X

j∈P_i(G)\{r}

c_j−j

|S_j(G)| .

In the case of airport games, where graph G is a chain, the SEC rule can be given as follows, for any player i:

ξ_i^SEC(G, c) = c₁

n +· · ·+ c_i−i n−i+ 1 ,

where the players are ordered according to the their positions in the chain, i.e.

player i is at the ith position of the chain.

Littlechild and Owen (1973) have shown that the SEC rule and the Shapley value are equivalent on the class of irrigation games.

Claim 5.23 (Littlechild and Owen, 1973) For any cost-tree (G, c) it holds that ξ(G, c) =φ(v_(G,c)), where v_(G,c) is the irrigation game generated by cost tree (G, c). In other words, for cost-tree allocation problems the SEC rule and the Shapley value are equivalent.

In the following we consider certain properties of rules (see Thomson, 2007).

Denition 5.24 Let G = (V, A) be a rooted tree. Rule χ dened on the set of

ˆ eciency, if for each cost function c, P

i∈N

χ_i(G, c) = P

e∈A

c_e,

ˆ equal treatment of equals, if for each cost function c and pair of players i, j ∈N, P

e∈A_Pi(G)

c_e= P

e∈A_Pj(G)

c_e implies χ_i(G, c) = χ_j(G, c),

ˆ conditional cost additivity, if for any pair of cost functions c, c⁰, χ(G, c+ c⁰) =χ(G, c) +χ(G, c⁰),

ˆ independence of at-least-as-large costs, if for any pair of cost functions c, c⁰ and player i ∈ N such that for each j ∈ P_i(G), P

The interpretations of the above properties of rules dened above are as fol-lows (Thomson, 2007). Non-negativity claims that for each problem the rule must only give a non-negative cost allocation vector as result. Cost boundedness

re-cost allocation vector. Eciency describes that coordinates of the re-cost allocation vector must add up to the maximal cost. Equal treatment of equals states that players with equal individual costs must pay equal amounts. Conditional cost ad-ditivity requires that if two cost trees are summed up (the tree is xed), then the cost allocation vector dened by the sum must be the sum of the cost allocation vectors of the individual problems. Finally, independence of at-least-as-large costs means that the sum payed by a player must be independent of the costs of the segments he does not use.

These properties are similar to the axioms we dened in Denition 5.17. The following proposition formalizes the similarity.

Claim 5.25 Let Gbe a rooted tree, χbe dened on cost trees (G, c), solution ψ be dened on GG such that χ(G, c) = ψ(v_(G,c))for any cost function c. Then, if χ satises

ˆ non-negativity and cost boundedness, then ψ isN P,

ˆ eciency, then ψ is P O,

ˆ equal treatment of equals, then ψ is ET P,

ˆ conditional cost additivity, then ψ is ADD,

ˆ independence of at-least-as-large costs, then ψ is M.

Proof: N N andCB⇒N P: Clearly, playeriisN P, if and only ifP

e∈A_Pi(G)c_e= 0. Then N N implies that χ_i(G, c) ≥ 0, and from CB: χ_i(G, c) ≤ 0. Consequently, χ_i(G, c) = 0, therefore ψ(v_(G,c)) = 0.

E ⇒P O: From the denition of irrigation games (Denition 5.1):P

e∈Ac_e =

IALC ⇒M: It is easy to check that if for cost trees (G, c),(G, c⁰) and player i ∈ N it holds that (v_(G,c))⁰_i = (v_(G,c⁰₎)⁰_i, then for each j ∈ P_i(G): P

e∈A_Pj(G)c_e = P

e∈A_Pj(G)c⁰_e, therefore χ_i(G, c) = χ_i(G, c⁰). Consequently, (v_(G,c))⁰_i = (v_(G,c⁰₎)⁰_i implies ψ_i(v_(G,c)) =ψ_i(v_(G,c⁰₎).

It is worth noting that all but the eciency point are tight, i.e. eciency and the Pareto-optimal property are equivalent, in the other cases the implication holds in only one way. Therefore the cost-sharing axioms are stronger than the game theoretical axioms.

The above results and Theorem 5.19 imply Dubey (1982)'s result as a direct corollary.

Theorem 5.26 (Dubey, 1982) Rule χ on airport the class of airport games satises non-negativity, cost boundedness, eciency, equal treatment of equals and conditional cost additivity, if and only if χ=ξ, i.e. χ is the SEC rule.

Proof: ⇒: With a simple calculation it can be shown that the SEC rule satises properties N N, CB, E, ET E, andCCA (see e.g.Thomson, 2007).

⇐: Claim 5.25 implies that we can apply Theorem 5.19 and thereby get the Shapley value. Then Claim 5.23 implies that the Shapley value and the SEC rule coincide.

Moulin and Shenker (1992)'s result can be deduced from the results above and Theorem 5.20, similarly to Dubey (1982)'s result.

Theorem 5.27 (Moulin and Shenker, 1992) Rule χ on the class of airport problems satises eciency, equal treatment of equals and independence of at-least-as-large costs, if and only if χ=ξ, i.e. χ is the SEC rule.

Proof: ⇒: It can be shown with a simple calculation that the SEC rule satises the E, ET E and IALC properties (see e.g. Thomson, 2007).

⇐: Based on Claim 5.25, Theorem 5.19 can be applied and thereby get the Shapley value. Then Claim 5.23 implies that the Shapley value and the SEC rule coincide.

Finally, in the following two theorems we extend the results of Dubey (1982) and Moulin and Shenker (1992) to any cost tree allocation problem. The proofs of these results is the same as those of the previous two theorems.

Theorem 5.28 Rule χ on cost-tree problems satises non-negativity, cost boundedness, eciency, equal treatment of equals and conditional cost additiv-ity, if and only if χ=ξ, i.e. χ is the SEC rule.

Theorem 5.29 Rule χ on cost-tree problems satises eciency, equal treat-ment of equals and independence of at-least-as-large costs, if and only if χ = ξ, i.e. χ is the SEC rule.

Chapter 6