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2010 | OriginalPaper | Chapter

Cooperation Under Ambiguity

Author : Sjur Didrik Flåm

Published in: Energy, Natural Resources and Environmental Economics

Publisher: Springer Berlin Heidelberg

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Abstract

Exchange of contingent claims is construed here as a cooperative game with transferable utility. Solutions are sought in the core. The novelty is that agents, being uncertainty averse, may use distorted, subjective probabilities. Choquet integrals therefore replace expected utility. When convoluted payoff is concave at the aggregate endowment, there is a price-generated, explicit core solution.

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previous chapter An Overview of Models and Solution Methods for Pooling Problems

next chapter The Perpetual American Put Option for Jump-Diffusions

Typically, \(\mathcal{X} = {\mathbb{R}}^{G}\), where G denotes a finite, fixed set of goods. Finance and insurance deals with instances that feature merely one good, namely money. More general examples include state-contingent ownership of different natural resources – or of permits to emit diverse pollutants. It is tacitly assumed that the state, unknown ex ante, becomes common knowledge ex post. Otherwise information is asymmetric; see Flåm and Koutsougeras (2010).

A producer’s endowment is identified with his state-dependent bundle of output commitments or claims to production factors.

The cooperative game at hand has player setA and so-called characteristic functionC↦Π(e _C).

In case of risk, typically described by common or objective probabilities, several studies have already dealt with Pareto efficient allocations. But clearly, besides being efficient, participation had better be voluntary as well. Thus the core naturally occupies center stage; see Baton and Lemaire (1981); Borch (1960a,b).

The paradigm of expected utility still holds dominant sway in economic theory. It has, however, been provoked by empirical paradoxes and challenged by concerns with the foundations of Bayesian decision making; see Dempster (1968), Machina (1987), and Schmeidler (1989). To mitigate matters, numerous generalized criteria have come up during the last decades; see Fishburn (1988) for review.

This extension retains the elegance and tractability of the von Neumann–Morgenstern–Savage model. In addition, it can accommodate bid-ask spreads, belief functions, distorted probabilities, evidence weights, non-linear pricing, sunspots, and omitted states; see Gilboa (1987), Gilboa and Schmeidler (1994), Karni and Schmeidler (1991), Schmeidler (1986), Schmeidler (1989), and Shafer (1976). It has already been applied to Pareto optimal sharing; see Chateauneuf et al. (2000) and insurance premia; see Wang et al. (1997), but not to cooperative contracts as here.

More precisely: until Sect. 5, posit \(\mathcal{X} = \mathbb{R}\).

For simplicity or computation, one may envisage S as finite - and let Σ comprise all its subsets. Then everym : S → ℝ is Σ-measurable.

The integrands on the right hand side of (9) are monotone decreasing. The integral there is the classical one of Riemann–Stiltjes.

For set functions these notions originated in cooperative game theory; consult Osborne and :̧def :̧def Rubinstein (1994).

That is, the Choquet integral is superadditive iff the capacity is supermodular.

When c is convex, there does indeed exist a probability measure P ≥ c; consult Shapley (1971).

Uncertainty aversion, as introduced here, bears resemblance to variance aversion, meaning that m ≿ m + m ^′′ whenever \(\mathbb{E}{m}^{{\prime\prime}} = 0\) and cov(m, m ^′′) = 0.

Both sorts of aversion affect portfolio choice; see Gollier (2006), Maenhout (2004). In general, uncertainty or ambiguity enters when a model must be specified. On such occasions, a prudent agent had better worry about making robust decisions. For studies of such issues in dynamic settings see Anderson et al. (2003), Cagetti et al. (2002).

When S is finite, this is trivial. Otherwise, compactness refers to the w ^∗-topology.

And, of course, non-concave objectives would render the computation harder.

The issue resembles that of time consistency Obstfeld and Rogoff (1996). But the setting here involves only before and after the state is unveiled - and not time proper.

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Title: Cooperation Under Ambiguity
Author: Sjur Didrik Flåm
Publisher: Springer Berlin Heidelberg
Book: Energy, Natural Resources and Environmental Economics
Print ISBN: 978-3-642-12066-4

Electronic ISBN: 978-3-642-12067-1

Copyright Year: 2010
DOI: https://doi.org/10.1007/978-3-642-12067-1_27

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