×

The Shapley value of resale-proof trades. (English) Zbl 0823.90148

The paper studies the Shapley value of transferable utility games associated with resale-proof trades of information. In the symmetric case, an expression for the Shapley value is presented, whereas, in the non-symmetric case, some upper and lower bounds for the Shapley value are given.
In the context of a model of technological dissemination with no assumptions of patent protection, the notion of “resale-proofness” is meant to formulate self-binding trades of information, such that no agent has an incentive to resell the information even when it is freely allowed. Let \(N= \{1, 2,\dots, n\}\) be the finite set of all agents, where agent 1 is the sole seller of some information and the others are potential buyers. For each \(i\in N\) and \(S\subseteq N\), let \(E_ i(S)\) be the monetary profit to agent \(i\) when \(S\) is the set of all holders of the information (that is, the set of all agents who have acquired the information). Given the current set \(H\) of the holders, where \(1\in H\subseteq N\), a resale from a coalition \(S\subseteq H\) to a coalition \(T\subseteq N\backslash H\) is said to be profitable iff \(\sum_{i\in S\cup T} E_ i(H\cup T)> \sum_{i\in S} E_ i(H)\).
In an inductive manner on the number of agents in \(N\backslash H\), we say a coalition \(S\subseteq H\) has an enforceable resale to a coalition \(T\subseteq N\backslash H\) iff the resale is profitable and there is no coalition \(\overline T\subseteq H\cup T\) which has an enforceable resale to some coalition \(P\subseteq N\backslash (H\cup T)\) where, in case the coalition \(N\backslash H\) consists of one agent, we say \(S\subseteq H\) has an enforceable resale to coalition \(N\backslash H\) iff the resale is profitable. A coalition \(M\) satisfying \(1\in M\subseteq N\) is called resale-proof iff there does not exist a coalition \(S\subseteq M\) which has an enforceable resale to some coalition \(T\subseteq N\backslash M\). A coalition \(M\) is called a profitable resale-proof set iff \(M\) is resale- proof and \(\sum_{i\in M} E_ i(M)> E_ 1(\{1\})\).
The associated TU-game \(v: 2^ N\to \mathbb{R}\) to the resale-proof trade market is defined as follows: if \(1\not\in S\), then \(v(S):= 0\); if \(1\in S\), then \[ v(S):= \max\Biggl[ \sum_{i\in M} E_ i(M)\mid M\subseteq S\text{ and } M\text{ is profitable resale-proof set or } M= \{1\}\Biggr]. \] So, for each coalition \(S\) containing the inventor of the information, its worth equals the sum of the monetary profits to agents of a most profitable resale-proof subset of \(S\). As a matter of fact, it is stated that the worth \(v(S)\) for any coalition \(S\) containing the inventory, is determined by the minimal profitable resale-proof subset of \(S\). Further, it is established that the Shapley value of the associated TU-game belongs to the core of the game iff there is exactly one minimal resale-proof set. Next an expression is presented for the Shapley value of symmetric games (i.e. the profits depend only on the size of the set of information holders, that is for all \(S, R\in 2^ N\) with \(1\in S\cap R\) and \(| S|= | R|\), we have \(E_ 1(S)= E_ 1(R)\) and also \(E_ i(S)= E_ j(R)\) if \(i\in S\), \(j\in R\), and \(i, j\neq 1\). Finally, some upper and lower bounds for the Shapley value for non- symmetric games are given.

MSC:

91A12 Cooperative games
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arrow, K. J., Economic welfare and allocation of resources for invention, (Nelson, R. R., The rate and direction of incentive activity (1962), Princeton University Press: Princeton University Press Princeton, NJ) · Zbl 0323.90014
[2] Berheim, B. D.; Peleg, B.; Whinston, M. D., Coalition-proof Nash equilibrium I, Concept, Journal of Economic Theory, 42, 1-12 (1987) · Zbl 0619.90090
[3] Gallini, N. T., Deterrence by market sharing: A strategic incentive for licensing, American Economic Review, 74, 931-941 (1984)
[4] Kamien, M. I.; Tauman, Y., The private value of a patent: A game theoretic approach, Journal of Economics, 93-118 (1984), Suppl. 4
[5] Kamien, M. I.; Tauman, Y., Fee versus royalties and the private value of a patent, Quarterly Journal of Economics, 101, 471-491 (1986)
[6] Kamien, M. I., Patent licensing, (Aumann, R. J.; Hart, S., Handbook of game theory with economic applications (1989), North-Holland: North-Holland Amsterdam) · Zbl 0968.91519
[7] Katz, M. L.; Shapiro, C., On the licensing of innovation, Rand Journal of Economics, 16, 504-520 (1985)
[8] Muto, S., Resale-proofness and coalition-proof Nash equilibrium, Games and Economic Behavior, 2, 337-361 (1990) · Zbl 0755.90096
[9] Nakayama, M.; Quintas, L. G., Stable payoffs in resale-proof trades of information, Games and Economic behavior, 3, 339-349 (1991) · Zbl 0757.90012
[10] Nakayama, M.; Quintas, L. G.; Muto, S., Resale-proof trades of information, The Economic Studies Quarterly, 42, no. 4, 292-302 (1991)
[11] Rostoker, M., A survey of corporate licensing, IDEA, 24, 59-92 (1984)
[12] Shapley, L. S., A value for \(n\)-person games, (Annals of Mathematics Study No. 28 (1953), Princeton University Press: Princeton University Press Princeton, NJ), 307-317 · Zbl 0050.14404
[13] Shapley, L. S., Measurement of power in political systems, Proceedings of Symposia in Applied Mathematics, 24, 69-81 (1981) · Zbl 0487.90100
[14] Shapley, L. S.; Shubik, M., A method for evaluating the distribution of power in a committee system, American Political Science Review, 48, 787-792 (1954)
[15] Von Neumann, J.; Morgenstern, O., Theory of games and economic behavior (1944), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0063.05930
[16] Young, H. P., Individual contribution and just compensation, (Roth, A., Shapley value: Essays in honor of Lloyd Shapley (1988), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0726.90100
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.