Appointment Games in Fixed-Route Traveling Salesman Problems and the Shapley Value

Starting from her home, a service provider visits several customers, following a predetermined route, and returns home after all customers are visited. The problem is to find a fair allocation of the total cost of this tour among the customers served. A transferable-utility cooperative game can be associated with this cost allocation problem. We introduce a new class of games, which we refer as the fixed-route traveling salesman games with appointments. We study the Shapley Value in this class and show that it is in the core. Our first characterization of the Shapley value involves a property which requires that sponsors do not benefit from mergers, or splitting into a set of sponsors. Our second theorem involves a property which requires that the cost shares of two sponsors who get connected are equally effected. We also show that except for our second theorem, none of our results for appointment games extend to the class of routing games (Potters et al, 1992).fixed-route traveling salesman games, routing games, appointment games, the Shapley value, the core, transferable-utility games, merging and splitting proofness, equal impact, networks, cost allocation.

On the Core of Multiple Longest Traveling Salesman Games

In this paper we introduce multiple longest traveling salesman (MLTS) games. An MLTS game arises from a network in which a salesman has to visit each node (player) precisely once, except its home location, in an order that maximizes the total reward.First it is shown that the value of a coalition of an MLTS game is determined by taking the maximum of suitable combinations of one and two person coalitions.Secondly it is shown that MLTS games with ¯ve or less players have a nonempty core.However, a six player MLTS game may have an empty core.For the special instance where the reward between a pair of nodes is equal to 0 or 1, we provide relations between the structure of the core and the underlying network.game theory;traveling salesman problem;games;core

Approximately Fair Cost Allocation in Metric Traveling Salesman Games

A traveling salesman game is a cooperative game ${\mathcal{G}}=(N,c_{D})$ . Here N, the set of players, is the set of cities (or the vertices of the complete graph) andc D is the characteristic function where D is the underlying cost matrix. For all S⊆N, define c D (S) to be the cost of a minimum cost Hamiltonian tour through the vertices of S∪{0} where $0\not \in N$ is called as the home city. Define Core ({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)=c_{D}(N)\ \mbox{and}\ \forall S\subseteq N,x(S)\le c_{D}(S)\} as the core of a traveling salesman game ${\mathcal{G}}$ . Okamoto (Discrete Appl. Math. 138:349-369, [2004]) conjectured that for the traveling salesman game ${\mathcal{G}}=(N,c_{D})$ with D satisfying triangle inequality, the problem of testing whether Core $({\mathcal{G}})$ is empty or not is $\mathsf{NP}$ -hard. We prove that this conjecture is true. This result directly implies the $\mathsf{NP}$ -hardness for the general case when D is asymmetric. We also study approximately fair cost allocations for these games. For this, we introduce the cycle cover games and show that the core of a cycle cover game is non-empty by finding a fair cost allocation vector in polynomial time. For a traveling salesman game, let \epsilon\mbox{-Core}({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)\ge c_{D}(N) and ∀ S⊆N, x(S)≤ε⋅c D (S)} be an ε-approximate core, for a given ε>1. By viewing an approximate fair cost allocation vector for this game as a sum of exact fair cost allocation vectors of several related cycle cover games, we provide a polynomial time algorithm demonstrating the non-emptiness of the log 2(|N|−1)-approximate core by exhibiting a vector in this approximate core for the asymmetric traveling salesman game. We improve it further by finding a $(\frac{4}{3}\log_{3}(|N|)+c)$ -approximate core in polynomial time for some constantc. We also show that there exists an ε 0>1 such that it is $\mathsf{NP}$ -hard to decide whether ε 0-Core $({\mathcal{G}})$ is empty or no

On the Core of Multiple Longest Traveling Salesman Games

In this paper we introduce multiple longest traveling salesman (MLTS) games. An MLTS game arises from a network in which a salesman has to visit each node (player) precisely once, except its home location, in an order that maximizes the total reward.First it is shown that the value of a coalition of an MLTS game is determined by taking the maximum of suitable combinations of one and two person coalitions.Secondly it is shown that MLTS games with ¯ve or less players have a nonempty core.However, a six player MLTS game may have an empty core.For the special instance where the reward between a pair of nodes is equal to 0 or 1, we provide relations between the structure of the core and the underlying network.

Characterizing the Shapley Value in Fixed-Route Traveling Salesman Problems with Appointments

Starting from her home, a service provider visits several customers, following a predetermined route, and returns home after all customers are visited. The problem is to ?nd a fair allocation of the total cost of this tour among the customers served. A transferable-utility cooperative game can be associated with this cost allocation problem. We intro- duce a new class of games, which we refer as the fixed-route traveling salesman games with appointments. We characterize the Shapley Value in this class using a property which requires that sponsors do not bene?t from mergers, or splitting into a set of sponsors.Fixed-route travelling salesman games, routing games, appointment games, the Shapley value, the core, transferable-utility games, merging and splitting proofness, networks, cost allocation

On some approximately balanced combinatorial cooperative games

A model of taxation for cooperativen-person games is introduced where proper coalitions Are taxed proportionally to their value. Games with non-empty core under taxation at rateɛ-balanced. Sharp bounds onɛ in matching games (not necessarily bipartite) graphs are estabLished. Upper and lower bounds on the smallestɛ in bin packing games are derived and euclidean random TSP games are seen to be, with high probability,ɛ-balanced forɛ≈0.06

Cases in Cooperation and Cutting the Cake

Cooperative game;sharing problem