Technical Papers - DO NOT EDIThttp://hdl.handle.net/10106/13542024-03-29T08:35:58Z2024-03-29T08:35:58ZOn the Computation of Semivalues for TU GamesDragan, Irinelhttp://hdl.handle.net/10106/50442023-12-05T22:55:39Z2008-12-01T00:00:00ZOn the Computation of Semivalues for TU Games
Dragan, Irinel
In an earlier paper, [Dragan,2006a] , we proved that every Least Square Value is the Shapley Value of a game obtained by rescaling from the given game. In the paper where the Least Square Values were introduced by Ruiz, Valenciano and Zarzuelo, the authors have shown that the efficient normalization of a Semivalue is a Least Square Value. In the present paper, we develop the idea suggested by these two results, and we obtain a direct relationship between the Efficient normalization of a Semivalue and the Shapley Value [see also Dragan, 2006b]. The main tools for proofs were the so called Average per capita formulas we proved earlier for the Shapley Value [Dragan, I 992] and for a Semivalue {Dragan and Martinez-Legas,2001]. Note that the connection between the Efficient normalization of a Semivalue and the Shapley Value has been used for computing a Semivalue, via a rescaling of the worth of coalitions in the given game. In this paper, the main purpose is to offer two other alternatives for the computation of a Semivalue, via the Shapley Value; beside the rescaling done before the computation, we consider rescalings within the computation. As seen above, this paper contains results from different sources, so that to make the paper self contained we shall be proving below our results together with the new results appearing here for the first time. Our proofs are algebraic, in opposition to those found in Ruiz et al., which are axiomatic. The direct connection between a Semivalue and the Shapley Value does not need any reference to the Least Square Values, which may well be unknown to the reader of the present paper. In the first section, we prove the Average per capita formula for Semivalues, (Theorem 1), from which we derive our earlier Average per capita formula for the Shapley Value, to be used later. In the second section , we derive an Average per capita formula for the Efficient normalization of a Semivalue, by computing the efficiency term, (Theorem 2), as well as the main results, showing the connection between the Efficient normalization and the Shapley Value, (Theorems 3 and 4). In the third section, we discuss a first alternative for the computation of a Semivalue via the Shapley Value, illustrated in Example 1: we compute the needed ratios from the Average per capita formula for the Shapley Value and rescaling is done only n —1 times, over these ratios. In the last section, we discuss a second alternative method for computing Shapley Values; the algorithm has been invented for computing Weighted Shapley Values and it will be adapted to the computation of the Semivalues. Some formulas are derived from this new method for computing the Weighted Shapley Values, based upon the null space of the Weighted Shapley Value operator, [Dragan,2008]. Note that the Semivalues are not Weighted Shapley Values. The Example 2 is illustrating the algorithm on the same 4-person game as before. The motivation for the present work was the fact that we know other works for computing the Shapley Value, but no computational work for Semivalues is known to us.
2008-12-01T00:00:00ZParticle Simulation in Contact Mechanics of a Bouncing Elastic BallGreenspan, Donaldhttp://hdl.handle.net/10106/48292023-12-05T22:56:12Z1999-05-01T00:00:00ZParticle Simulation in Contact Mechanics of a Bouncing Elastic Ball
Greenspan, Donald
An elastic ball is simulated by considering lumped mass molecular sets called particles. Particles are allowed to interact only locally in a fashion similar to that employed in molecular mechanics. Under local and gravity forces an n-body problem results when initial data are prescribed. By scaling the forces, the resulting n-body problem is solved numerically using only a scientific personal computer. A variety of examples of bouncing balls are described and discussed.
1999-05-01T00:00:00ZMolecular Mechanics Simulations of the Three Dimensional Cavity ProblemGreenspan, Donaldhttp://hdl.handle.net/10106/48282023-12-05T22:56:33Z1999-03-01T00:00:00ZMolecular Mechanics Simulations of the Three Dimensional Cavity Problem
Greenspan, Donald
The rapid development of computer technology has resulted in broad interest in three dimensional fluid simulation, which is known to have more complex force interactions than occur in two dimensions (see, e.g., refs. [1-6], [9], [11-15], [17], [18] and the additional references therein). In this paper we develop and analyze some molecular mechanics simulations of the three dimensional cavity problem. The fluid considered is water at 15° C.
1999-03-01T00:00:00ZTennis, Geometric Progression, Probability and BasketballGhandehari, Mostafahttp://hdl.handle.net/10106/48272023-12-05T22:56:54Z1999-03-01T00:00:00ZTennis, Geometric Progression, Probability and Basketball
Ghandehari, Mostafa
The following problem about a tennis match is well—known. See Halmos [1, 2]. Consider 2n tennis players playing a single elimination match. Ask the question: what are the number of games played? The answer can be obtained in two ways. First using the geometric progression 2n-1 + 2n-2 + • • • -2+1 we find that the answer is 2n — 1. We can also explain the answer as follows: for each game played there is a loser. Thus the total number of games played is equal to the number of losers. Since there is only one winner the total number of games played is equal to 2n — 1, the number of losers.
1999-03-01T00:00:00Z