Download The Shapley Value: Essays in Honor of Lloyd S. Shapley by Alvin E. Roth PDF

By Alvin E. Roth

Composed in honor of the sixty fifth birthday of Lloyd Shapley, this quantity makes available the big physique of labor that has grown out of Shapley's seminal 1953 paper. all of the twenty essays issues a few element of the Shapley worth.

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Extra info for The Shapley Value: Essays in Honor of Lloyd S. Shapley

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For the numbers of elements in R, S, TV, . . respectively. Lemma 2. For c ^ 0, 0 < r < oo, we have \c/r if / G R, Proof: Take / andj in i^, and choose TT G Il( f/) so that nR = R and 7r/ = / Then we have nvR = f^, and hence, by Axiom 1, By Axiom 2, c= c for any / G i^. This, with Lemma 1, completes the proof. 5 Any game with finite carrier is a linear combination of symmetric games vR: v= 2 cR(v)vR, (7) RCN TV being any finite carrier oft;. The coefficients are independent of N, and are given by cR(v) = 2 ( - lY-'v(T) (0 < r < oo).

If TV is a finite carrier of v, then, for / ^ TV, Proof: Take / ^ N. Both TV and NU(i) are carriers of v\ and v(N) = v(NU (/)). Hence

Second, there are some arbitrary choices involved in specifying a utility function, so the information contained in an individual's utility function is really represented by an equivalence class of functions. When preferences are defined over riskless prospects without any information about relative intensities of preference, then the class of utility functions equivalent to a given utility function u consists of all monotone transformations of u. When preferences are defined over risky prospects as well, then the 51 52 Alvin E.

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