Some characterizations of non-additive multi-period models

doi:10.1016/j.mathsocsci.2004.03.005

Mathematical Social Sciences

Volume 48, Issue 3, November 2004, Pages 235-250

https://doi.org/10.1016/j.mathsocsci.2004.03.005 Get rights and content

Abstract

Building upon the works of Gilboa [Econometrica 57 (1989) 1153], Shalev [Math. Soc. Sci. 33 (1997) 203], and De Waegenaere and Wakker [J. Math. Econ. 36 (2001) 45], we show that a simple version of variation aversion, jointly with a myopia axiom allows to derive in an infinite setting a meaningful expression for evaluating income streams. Furthermore, we prove that the usual additive discounted expectation introduced by Koopmans [Koopmans, T.C., 1972. Representations of preference orderings over time. In: McGuire, C.B., Radner, R. (Eds.), Decisions and Organizations. North-Holland, Amsterdam, pp. 79–100] can be accommodated in a non-additive way.

Introduction

Most of the models that aim at representing preferences over income streams are based on an independence axiom, for instance the discounted expectation of an utility stream (DE): $∑ n=1 ∞ δ^{n} u(x_{n})$ , axiomatized in Koopmans (1972), is the reference to evaluate income streams. In the simpler case where utility is linear in outcomes such an axiom becomes an additivity axiom. This axiom has since been challenged in the field of decision theory under uncertainty, and new tools have been developed to restrict the range of additivity (Schmeidler (1986)).

If our concern is preferences over sequences of outcomes, it seems that the particular structure of time should be taken into account. Indeed, the independence between two distinct time periods is questionable, in particular for two successive periods. Therefore, some complementarity across successive periods should be integrated in the evaluation function.

We first describe the setting, then building upon Gilboa (1989), Shalev (1997) and De Waegenaere and Wakker (2001), we introduce the way how can complementarities be taken into account. In Section 4 we provide an axiomatization infinite horizon. Then via a myopia axiom, we derive in Section 5 a tractable formula in the infinite case. Strengthening the myopia axiom, we obtain in Section 6 a truncated horizon, where the finiteness of the horizon is determined solely in a subjective way. Section 7 provides a generalization of the discounted expectation and examine the close relationship of the generalized discounted expectation and the classical discounted expectation. All the proofs are gathered in Appendix.

Section snippets

The setting

Let the set S equal either a finite-horizon set of periods {0,…, n} (for short [[0, n]]) or an infinite-horizon set {0, 1,…}.

B_∞(S) denotes the set of bounded real-valued functions defined on S, and represents sequence of consequences that a decision maker (DM for short) will have to rank; one can also give a welfarist approach, where a social planner has to evaluate streams of incomes (e.g. gross income per capita).

We assume that the DM's preferences over B_∞(S) are given through a binary

Multi-period model

Usual models of representation of ⪰ are based on additivity, where no complementarity across time can be considered. In terms of preferences, it is embodied in the following axiom, $(Additivity) ∀x,y,z∈B_{∞} (S),(x∼y)⇒(x+z∼y+z).$

Additivity stipulates that if a DM is indifferent between x and y, then adding z to both of them will not affect the indifference. A.1–A.4 and Additivity give for constant equivalent an expectation with respect to an additive (not necessarily σ-additive when S is infinite)

Multi-period model with finite horizon

We consider the finite case where S=[[0, n]], therefore B_∞(S) can be identified with $R$ ⁿ⁺¹. Under the axioms A.1–A.5 we get a specific function that evaluates every profile according to the DM preferences.

Theorem 1

Let ⪰ be a preference relation on $R$ ⁿ⁺¹, ⪰ satisfies A.1–A.5 if and only if there exists m₀,…, m_n, m_0,1,…, m_n−1,n≥0 uniquely determined with $∑ i=0 n m_{i} + ∑ i=0 n−1 m_{i,i+1} =1$ such that, $∀x∈ R^{n+1},x∼I(x)[[0,n]]*;$ $∀x,y∈ R^{n+1},x⪰y⇔I(x)≥I(y);$ where $∀x∈ R^{n+1},I(x)= ∑ i=0 n m_{i} .x_{i} + ∑ i=0 n−1 m_{i,i+1} . Min {x_{i},x_{i+1}}.$

From Eq. (1) the

Multi-period model with infinite horizon

We shall now consider S= $N$ . In order to get a neat expression of Eq. (1), we need to impose a continuity axiom. Such axiom can in fact be stated in term of myopia, in the spirit of Brown and Lewis (1981) and Prescott and Lucas (1972).

A.6

(Myopia)∀x ∈B_∞⁺ ( $N$ ), ∀ϵ>0, ∃N∈ $N$ /∀n≥N, x^ϵ,n≻x where x_i^ϵ,n=x_i+ϵ, if i∈[[0,n]] and 0 if i>n.

A DM that exhibits myopia is willing to give up his future outcomes for some steady improvement in the short run as soon as the future “starts” late enough. Note that our axiom

Truncated time horizon

Following Olson and Bailey (1981), some individuals make no concern of the future, for instance it may be difficult for them to consider their lives beyond a certain date. This attitude can be translated in a strengthened version of myopia (A.6),

A.6*

(Rough Myopia) ∃N∈ $N$ /∀x∈B_∞⁺( $N$ ), ∀ϵ>0, x^ϵ,N≻x

With such a behavior the DM has a subjective finite horizon in mind

Theorem 3

Let ⪰ be a preference relation on B_∞( $N$ ), ⪰ satisfies A.1–A.6* if and only if there exists m₀,…m_N, m_0,1,…,m_N−1,N≥0 uniquely determined with $∑ n=0 N$

Generalized discounted expectation

In the context of a multi-period model, the usual additive functional is the discounted expectation, i.e. there is a discount factor δ∈(0, 1) such that every x∈B_∞( $N$ ) has for constant-equivalent (1−δ) $∑ n=0 ∞ δ^{n} x_{n}$ . This particular functional axiomatized by Koopmans (1972) is obtained via a stationarity and a sensitivity axiom, $(Stationarity)∀x,y∈B_{∞} (N),∀α∈ R,(x⪰y)⇔((α,x)⪰(α,y))$ where (α, x) is the shifted profile x starting with α and, $(Sensitivity)∃x∈B_{∞} (N),∃y_{0} ∈ R,/x≁(y_{0},x_{1},x_{2},…)$

This further specification

Concluding comments

In this paper we aimed at showing that simple axioms can model pertinent deviations from additivity when evaluating income streams. Building upon the works of Gilboa (1989), Shalev (1997), and De Waegenaere and Wakker (2001), we introduced in our framework a simple version of variation aversion, which jointly with a myopia axiom allowed us to derive in an infinite setting what we hope to be a meaningful expression of an earlier model introduced in an Anscombe–Aumann setting by Gilboa (1989) and

Acknowledgements

We thank Jean-Yves Jaffray andPeter Wakker for useful comments. The valuable suggestions of an anonymous referee are gratefully acknowledged.

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Schmeidler, 1972 Throughout the section the capacity under consideration will be supposed to be exact, because this is a direct consequence of aversion to variations of flows (see Chateauneuf and Rébillé, 2004b, De Waegenaere and Wakker, 2001 and Gilboa, 1989). However this is not necessary except in Propositions 3.2 and 3.10.
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Some characterizations of non-additive multi-period models

Abstract

Introduction

Section snippets

The setting

Multi-period model

Multi-period model with finite horizon

Multi-period model with infinite horizon

Truncated time horizon

Generalized discounted expectation

Concluding comments

Acknowledgements

Mathematical Social Sciences

Journal of Mathematical Economics

Mathematical Social Sciences

Myopic economic agents

Econometrica

Modeling attitudes towards uncertainty and risk through the use of Choquet integral

Annals of Operation Research