85 research outputs found
The Shapley Values on Fuzzy Coalition Games with Concave Integral Form
A generalized form of a cooperative game with fuzzy coalition variables is proposed. The character function of the new game is described by the Concave integral, which allows players to assign their preferred expected values only to some coalitions. It is shown that the new game will degenerate into the Tsurumi fuzzy game when it is convex. The Shapley values of the proposed game have been investigated in detail and their simple calculation formula is given by a linear aggregation of the Shapley values on subdecompositions crisp coalitions
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Risk capital allocation and cooperative pricing of insurance liabilities
The Aumann–Shapley [Values of Non-atomic Games, Princeton University Press, Princeton] value, originating in cooperative game theory, is used for the allocation of risk capital to portfolios of pooled liabilities, as proposed by Denault [Coherent allocation of risk capital, J. Risk 4 (1) (2001) 1]. We obtain an explicit formula for the Aumann–Shapley value, when the risk measure is given by a distortion premium principle [Axiomatic characterisation of insurance prices, Insur. Math. Econ. 21 (2) (1997) 173]. The capital allocated to each instrument or (sub)portfolio is given as its expected value under a change of probability measure. Motivated by Mirman and Tauman [Demand compatible equitable cost sharing prices, Math. Oper. Res. 7 (1) (1982) 40], we discuss the role of Aumann–Shapley prices in an equilibrium context and present a simple numerical example
The Cores for Fuzzy Games Represented by the Concave Integral
We propose a new fuzzy game model by the concave integral by assigning subjective expected values to random variables in the interval [0
Integral Representations of Cooperative Game with Fuzzy Coalitions
Classical extensions of fuzzy game models are based on various integrals, such as Butnariu game and Tsurumi game. A new class of symmetric extension of fuzzy game with fuzzy coalition variables is put forward with Concave integral, where players’ expected values are on a partial set of coalitions. Some representations and properties of some limited models are compared in this paper. The explicit formula of characteristic function determined by coalition variables is given. Moreover, a calculation approach of imputations is discussed in detail. The new game could be regarded as a general form of cooperative game. Furthermore, the fuzzy game introduced by Tsurumi is a special case of the proposed game when game is convex
Capital allocation rules and the no-undercut property
This paper makes the point on a well known property of capital allocation rules, namely the
one called no-undercut. Its desirability in capital allocation stems from some stability game theoretical
features related to the notion of core, both for finite and infinite games. We review these aspects,
by relating them to the properties of the risk measures involved in capital allocation problems. We
also discuss some problems and possible extensions arising when we deal with non-coherent risk
measures
Suitability of Capital Allocations for Performance Measurement
Capital allocation principles are used in various contexts in which a risk
capital or a cost of an aggregate position has to be allocated among its
constituent parts. We study capital allocation principles in a performance
measurement framework. We introduce the notation of suitability of allocations
for performance measurement and show under different assumptions on the
involved reward and risk measures that there exist suitable allocation methods.
The existence of certain suitable allocation principles generally is given
under rather strict assumptions on the underlying risk measure. Therefore we
show, with a reformulated definition of suitability and in a slightly modified
setting, that there is a known suitable allocation principle that does not
require any properties of the underlying risk measure. Additionally we extend a
previous characterization result from the literature from a mean-risk to a
reward-risk setting. Formulations of this theory are also possible in a game
theoretic setting
Mathematical Game Theory
These lecture notes attempt a mathematical treatment of game theory akin to
mathematical physics. A game instance is defined as a sequence of states of an
underlying system. This viewpoint unifies classical mathematical models for
2-person and, in particular, combinatorial and zero-sum games as well as models
for investing and betting. n-person games are studied with emphasis on notions
of utilities, potentials and equilibria, which allows to subsume cooperative
games as special cases. The represenation of a game theoretic system in a
Hilbert space furthermore establishes a link to the mathematical model of
quantum mechancis and general interaction systems
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