4,549 research outputs found
Linking Strategic Interaction and Bargaining Theory. The Harsanyi - Schelling Debate on the Axiom of Symmetry
This paper analyses the early contributions of John Harsanyi and Thomas C. Schelling to bargaining theory. In his work, Harsanyi (1956) draws Nash’s solution to two-person cooperative games from the bargaining model proposed by Zeuthen (1930). Whereas Schelling (1960) proposes a multi-faceted theory of conflict that, without dismissing the assumption of rational behaviour, points out some of its paradoxical consequences. Harsanyi and Schelling’s contrasting views on the axiom of symmetry, as postulated by Nash (1950), are then presented. The analysis of this debate illustrates that, although in the early 1960s two different approaches to link strategic interaction and bargaining theory were proposed, only Harsanyi’s insights were fully developed later. Lastly, the causes of this evolution are assessed.bargaining, game theory, symmetry
Games on lattices, multichoice games and the Shapley value: a new approach
Multichoice games have been introduced by Hsiao and Raghavan as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a multichoice game, the first one is called the precore and is a direct generalization of the classical definition. We show that the precore coincides with the definition proposed by Faigle, and that it contains unbounded imputations, which makes its application questionable. A second definition is proposed, imposing normalization at each level, causing the core to be a convex closed set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their equality remain valid. A last section makes a comparison with the core defined by van den Nouweland et al.multichoice game ; lattice ; core
Weighted Banzhaf power and interaction indexes through weighted approximations of games
The Banzhaf power index was introduced in cooperative game theory to measure
the real power of players in a game. The Banzhaf interaction index was then
proposed to measure the interaction degree inside coalitions of players. It was
shown that the power and interaction indexes can be obtained as solutions of a
standard least squares approximation problem for pseudo-Boolean functions.
Considering certain weighted versions of this approximation problem, we define
a class of weighted interaction indexes that generalize the Banzhaf interaction
index. We show that these indexes define a subclass of the family of
probabilistic interaction indexes and study their most important properties.
Finally, we give an interpretation of the Banzhaf and Shapley interaction
indexes as centers of mass of this subclass of interaction indexes
Measuring the interactions among variables of functions over the unit hypercube
By considering a least squares approximation of a given square integrable
function by a multilinear polynomial of a specified
degree, we define an index which measures the overall interaction among
variables of . This definition extends the concept of Banzhaf interaction
index introduced in cooperative game theory. Our approach is partly inspired
from multilinear regression analysis, where interactions among the independent
variables are taken into consideration. We show that this interaction index has
appealing properties which naturally generalize the properties of the Banzhaf
interaction index. In particular, we interpret this index as an expected value
of the difference quotients of or, under certain natural conditions on ,
as an expected value of the derivatives of . These interpretations show a
strong analogy between the introduced interaction index and the overall
importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a
few applications of the interaction index
Cooperative game theory and its application to natural, environmental, and water resource issues : 1. basic theory
Game theory provides useful insights into the way parties that share a scarce resource may plan their use of the resource under different situations. This review provides a brief and self-contained introduction to the theory of cooperative games. It can be used to get acquainted with the basics of cooperative games. Its goal is also to provide a basic introduction to this theory, in connection with a couple of surveys that analyze its use in the context of environmental problems and models. The main models (bargaining games, transfer utility, and non-transfer utility games) and issues and solutions are considered: bargaining solutions, single-value solutions like the Shapley value and the nucleolus, and multi-value solutions such as the core. The cooperative game theory (CGT) models that are reviewed in this paper favor solutions that include all possible players and ignore the strategic stages leading to coalition building. They focus on the possible results of the cooperation by answering questions such as: Which coalitions can be formed? And how can the coalitional gains be divided to secure a sustainable agreement? An important aspect associated with the solution concepts of CGT is the equitable and fair sharing of the cooperation gains.Environmental Economics&Policies,Economic Theory&Research,Livestock&Animal Husbandry,Education for the Knowledge Economy,Education for Development (superceded)
Approximations of Lovasz extensions and their induced interaction index
The Lovasz extension of a pseudo-Boolean function is
defined on each simplex of the standard triangulation of as the
unique affine function that interpolates at the
vertices of the simplex. Its degree is that of the unique multilinear
polynomial that expresses . In this paper we investigate the least squares
approximation problem of an arbitrary Lovasz extension by Lovasz
extensions of (at most) a specified degree. We derive explicit expressions of
these approximations. The corresponding approximation problem for
pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then
solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an
alternative definition of Banzhaf interaction index. Similarly we introduce a
new interaction index from approximations of and we present some of
its properties. It turns out that its corresponding power index identifies with
the power index introduced by Grabisch and Labreuche (2001).Comment: 19 page
Strategic Power Revisited
Traditional power indices ignore preferences and strategic interaction. Equilibrium analysis of particular non-cooperative decision procedures is unsuitable for normative analysis and assumes typically unavailable information. These points drive a lingering debate about the right approach to power analysis. A unified framework that works both sides of the street is developed here. It rests on a notion of a posteriori power which formalizes players' marginal impact to outcomes in cooperative and non-cooperative games, for strategic interaction and purely random behaviour. Taking expectations with respect to preferences, actions, and procedures then defines a meaningful a priori measure. Established indices turn out to be special cases.power indices, spatial voting, equilibrium analysis, decision procedures
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