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 group 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 \( \tau \)-value, coincide under a mild condition on the traveling cost.
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Notes
Since we work with cost games, the inequalities are reversed in the definition of the imputation, convexity, the prenucleolus, and the nucleolus.
Alternatively, \(\gamma _i\) can be defined as sum of \(\Delta _i v_a(\{h\})\) and \(\Delta _i v_a(\{k\})\).
The nucleolus belongs to the core whenever the core is non-empty.
We are extremely grateful to an anonymous referee for suggesting this implication.
A solution \(\varphi \) satisfies zero independence if for each \(v,\) \(v'\in \Gamma ^{N}\) and each \(\beta \in \mathbb {R}^{N}\) such that for each \(S\subseteq N,\) \(v'(S) = v(S) + \sum _{j\in S} \beta _{j},\) \(\varphi (v') = \varphi (v) + \beta \).
Under the name of 2-game. They also show that weak 2-additivity implies the PS property.
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Acknowledgments
Chun’s work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2013S1A3A2055391) and the Institute of Economic Research, Seoul National University.
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Chun, Y., Park, N. & Yengin, D. Coincidence of cooperative game theoretic solutions in the appointment problem. Int J Game Theory 45, 699–708 (2016). https://doi.org/10.1007/s00182-015-0478-6
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DOI: https://doi.org/10.1007/s00182-015-0478-6
Keywords
- Fixed-route traveling salesman problem
- Appointment problem
- Shapley value
- Prenucleolus
- Nucleolus
- \(\tau \)-value
- Coincidence