69 research outputs found
Large Newsvendor Games
We consider a game, called newsvendor game, where several retailers, who face a random demand, can pool their resources and build a centralized inventory that stocks a single item on their behalf. Profits have to be allocated in a way that is advantageous to all the retailers. A game in characteristic form is obtained by assigning to each coalition its optimal expected profit. A similar game (modeled in terms of costs) was considered by Muller et al. (2002), who proved that this game is balanced for every possible joint distribution of the random demands. In this paper we consider newsvendor games with possibly an infinite number of newsvendors. We prove in great generality results about balancedness of the game, and we show that in a game with a continuum of players, under a nonatomic condition on the demand, the core is a singleton. For a particular class of demands we show how the core shrinks to a singleton when the number of players increases.newsvendor games, nonatomic games, core, balanced games.
Differential information in large games with strategic complementarities
We study equilibrium in large games of strategic complementarities (GSC) with differential information. We define an appropriate notion of distributional Bayesian Nash equilibrium and prove its existence. Furthermore, we characterize order-theoretic properties of the equilibrium set, provide monotone comparative statics for ordered perturbations of the space of games, and provide explicit algorithms for computing extremal equilibria. We complement the paper with new results on the existence of Bayesian Nash equilibrium in the sense of Balder and Rustichini (J Econ Theory 62(2):385–393, 1994) or Kim and Yannelis (J Econ Theory 77(2):330–353, 1997) for large GSC and provide an analogous characterization of the equilibrium set as in the case of distributional Bayesian Nash equilibrium. Finally, we apply our results to riot games, beauty contests, and common value auctions. In all cases, standard existence and comparative statics tools in the theory of supermodular games for finite numbers of agents do not apply in general, and new constructions are required
Cores of Cooperative Games in Information Theory
Cores of cooperative games are ubiquitous in information theory, and arise
most frequently in the characterization of fundamental limits in various
scenarios involving multiple users. Examples include classical settings in
network information theory such as Slepian-Wolf source coding and multiple
access channels, classical settings in statistics such as robust hypothesis
testing, and new settings at the intersection of networking and statistics such
as distributed estimation problems for sensor networks. Cooperative game theory
allows one to understand aspects of all of these problems from a fresh and
unifying perspective that treats users as players in a game, sometimes leading
to new insights. At the heart of these analyses are fundamental dualities that
have been long studied in the context of cooperative games; for information
theoretic purposes, these are dualities between information inequalities on the
one hand and properties of rate, capacity or other resource allocation regions
on the other.Comment: 12 pages, published at
http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/318704 in EURASIP
Journal on Wireless Communications and Networking, Special Issue on "Theory
and Applications in Multiuser/Multiterminal Communications", April 200
The convexity-cone approach to comparative risk and downside risk.
We establish a calculus characterization of the core of supermodular games, which reduces the description of the core to the computation of suitable Gateaux derivatives of the Choquet integrals associated with the game. Our result generalizes to infinite games a classic result of Shapley (1971). As a secondary contribution, we provide a fairly complete analysis of the Gateaux and Frechet differentiability of the Choquet integrals of supermodular measure games.
Variance Allocation and Shapley Value
Motivated by the problem of utility allocation in a portfolio under a
Markowitz mean-variance choice paradigm, we propose an allocation criterion for
the variance of the sum of possibly dependent random variables. This
criterion, the Shapley value, requires to translate the problem into a
cooperative game. The Shapley value has nice properties, but, in general, is
computationally demanding. The main result of this paper shows that in our
particular case the Shapley value has a very simple form that can be easily
computed. The same criterion is used also to allocate the standard deviation of
the sum of random variables and a conjecture about the relation of the
values in the two games is formulated.Comment: 20page
Refinement Derivatives and Values of Games
A definition of set-wise differentiability for set functions is given through refining the partitions of sets. Such a construction is closely related to the one proposed by Rosenmuller (1977) as well as that studied by Epstein (1999) and Epstein and Marinacci (2001). We present several classes of TU games which are differentiable and study differentiation rules. The last part of the paper applies refinement derivatives to the calculation of value of games. Following Hart and Mas-Colell (1989), we define a value operator through the derivative of the potential of the game. We show that this operator is a truly value when restricted to some appropriate spaces of games. We present two alternative spaces where this occurs: the spaces pM( ) and POT2. The latter space is closely related to Myerson's balanced contribution axiom.TU games; large games; non-additive set functions; value; derivatives
Contracting for innovation under knightian uncertainty
At any given point in time, the collection of assets existing in the economy is observable.
Each asset is a function of a set of contingencies. The union taken over all assets of these contingencies is what we call the set of publicly known states. An innovation is a set of states that are not publicly known along with an asset (in a broad sense) that pays contingent on those states. The creator of an innovation is an entrepreneur. He is represented by a probability measure on the set of new states. All other agents perceive the innovation as ambiguous: each of them is represented by a set of probabilities on the new states. The agents in the economy are classified with respect to their attitude towards this Ambiguity: the financiers are (locally) Ambiguity-seeking while the consumers are Ambiguity-averse. An entrepreneur and a financier come together when the former seeks funds to implement his project and the latter seeks new profit opportunities. The resulting contracting problem does not fall within the standard theory due to the presence of Ambiguity (on the financier’s side) and to the heterogeneity in the parties’ beliefs. We prove existence and monotonicity (i.e., truthful revelation) of an optimal contract. We characterize such a contract under the additional assumption that the financiers are globally Ambiguity-seeking. Finally, we re-formulate our results in an insurance framework and extend the classical result of Arrow [4] and the more recent one of Ghossoub. In the case of an Ambiguity-averse insurer, we also show that an optimal contract has the form of a generalized deductible
Information Design in Large Anonymous Games
We consider anonymous Bayesian cost games with a large number of players,
i.e., games where each player aims at minimizing a cost function that depends
on the action chosen by the player, the distribution of the other players'
actions and an unknown parameter. We study the nonatomic limit versions of
these games. In particular, we introduce the concepts of correlated and Bayes
correlated Wardrop equilibria, which extend the concepts of correlated and
Bayes correlated equilibria to nonatomic games. We prove that (Bayes)
correlated Wardrop equilibria are indeed limits of action flow distributions
induced by (Bayes) correlated equilibria of the game with a large finite set of
small players. For nonatomic games with complete information admitting a convex
potential, we show that the set of correlated Wardrop equilibria is the set of
probability distributions over Wardrop equilibria. Then, we study how to
implement optimal Bayes correlated Wardrop equilibria and show that in games
with a convex potential, every Bayes correlated Wardrop equilibrium can be
fully implemented.Comment: 53 page
Subcalculus for set functions and cores of TU games.
This paper introduces a subcalculus for general set functions and uses this framework to study the core of TU games. After stating a linearity theorem, we establish several theorems that characterize mea- sure games having finite-dimensional cores. This is a very tractable class of games relevant in many economic applications. Finally, we show that exact games with Þnite dimensional cores are generalized linear production games.TU games; non-additive set functions; subcalculus; cores
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