Abstract
One of the main goals of this paper is to improve the understanding of the way in which the core of a specific cooperative game, the airport game (Littlechild and Owen, Manag Sci 20:370–372, 1973), responds to monotonicity properties. Since such properties are defined for single-valued allocation rules, we use the core-center (González-Díaz and Sánchez-Rodríguez, Int J Game Theory 36:27–46, 2007) as a proxy for the core. This is natural, since the core-center is the center of gravity of the core and its behavior with respect to a given property can be interpreted as the “average behavior” of the core. We also introduce the lower-cost increasing monotonicity and higher-cost decreasing monotonicity properties that reflect whether a variation in a particular agent’s cost is beneficial to the other agents.
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Notes
Similar studies for the bargaining set and the kernel go even further back, to Megiddo (1974).
Given an airport problem \(c \in \mathcal {C}^N\) denote by \(\hat{C}(c)\) the projection of the core \(C(c)\) onto \(\mathbb {R}^{n-1}\) that simply “drops” the \(n\)-th coordinate. Figure 1 shows \(\hat{C}(c)\) for a \(3\)-agent and \(4\)-agent problem. The face \(F_i=\hat{C}(c) \cap \{x\in \mathbb {R}^{n-1} : x_1+ \dots + x_i =c_i\}\), \(i\in N{\setminus } \{n\}\), of the polytope \(\hat{C}(c)\) is a cross product of the cores of two airport games: \(F_i=C(c_1, \dots , c_i) \times \hat{C}(c_{i+1}-c_i, \dots , c_{n}-c_i)\).
In general, let \(0<c_1 \le \dots \le c_k\), \(k \in \mathbb {N}\), \(x_1\le c_1\), and denote \(u_k(x_1)=V_k(c_1-x_1, \dots , c_k-x_1)\). Then, \(\tfrac{du_k}{dx_1}(x_1)=-V_{k-1}(c_2-x_1, \dots , c_k-x_1)\).
In fact, this equality is the key result to express the core-center as a ratio of volumes, see González-Díaz et al. (2014).
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Acknowledgments
We want to thank William Thomson for his encouragement and insight during the writing process of this manuscript. Authors acknowledge the financial assistance of the Spanish Ministry for Science and Innovation through projects MTM2011-27731-C03, ECO2009-14457-C0401ECON, and from the Xunta de Galicia through project INCITE09-207-064-PR.
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González-Díaz, J., Mirás Calvo, M.Á., Sandomingo, C.Q. et al. Monotonicity of the core-center of the airport game. TOP 23, 773–798 (2015). https://doi.org/10.1007/s11750-014-0358-4
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DOI: https://doi.org/10.1007/s11750-014-0358-4