Elsevier

Games and Economic Behavior

Volume 131, January 2022, Pages 264-274
Games and Economic Behavior

Note
A note on topological aspects in dynamic games of resource extraction and economic growth theory

https://doi.org/10.1016/j.geb.2021.11.011Get rights and content

Abstract

We show that right-continuous monotone strategies used in Markov perfect equilibria for economic growth models and related dynamic games can be recognised as members of the Hilbert space of square integrable functions of the state variable. We provide an application of this result to a bequest game and point out that this result also holds for the class of left-continuous monotone functions. The result is fundamental for using the Schauder fixed point theorem. Furthermore, it considerably simplifies the classical approach, where such strategies are represented by non-negative measures on the state space.

Introduction

Monotone and right-continuous or left-continuous functions on the state space comprise a natural class of strategies in several models of dynamic games of resource extraction or economic growth theory. Existence of Markov perfect equilibria for such models is an infinite dimensional fixed point problem. Therefore, if D denotes the class of non-decreasing right-continuous (or left-continuous) functions on the state space, say S=[0,r], being the strategy space of a player, it is important to view D as a compact convex subset of a locally convex linear topological Hausdorff space. This is connected with the assumptions of the Schauder-Tychonoff fixed point theorem, see Corollary 17.56 in Aliprantis and Border (2006).

First results on discontinuous equilibria in altruistic economic growth models (bequest games) were given by Bernheim and Ray (1983) and Leininger (1986). They studied deterministic games with a compact state space and applied various sophisticated techniques based on different fixed point theorems. For example, Leininger (1986) used a so-called “leveling” method to obtain a fixed point mapping in the space of continuous functions on S endowed with the supremum norm.

Ray (1987) and Sundaram (1989a) recognised the monotone right-continuous functions FD as distribution functions of some non-negative bounded measures on the state space S. Thus, D could be viewed (by the Riesz representation theorem, see Theorem 13.25 in Royden (1988)) as a compact convex subset of the locally convex space of all finite signed measures on S endowed with the weak*-topology. However, their proofs of the continuity of the best response mappings involved in the equilibrium problem needed some corrections, see Balbus et al., 2015c, Balbus et al., 2016 and Sundaram (1989b) for the details. It is worthy to emphasise that the right-continuity of the functions in the family D is very important in the proof.1 The approaches presented by Ray (1987), Sundaram, 1989a, Sundaram, 1989b and Dutta and Sundaram (1992) refer to classical results in measure theory and functional analysis, which are precisely described in Subsection 3.1 to provide a perspective for our result and also to show an extension to an unbounded state space case given in Subsection 3.2.

In the economic growth models or games of resource extraction, it is desirable to allow the state space S to be unbounded from above. Then, the strategies are unbounded, monotone and semi-continuous. Obviously, the measures induced by these functions are unbounded. There are two methods to obtain an equilibrium in such game models. The first approach is based on the Schauder-Tychonoff fixed point theorem. Note that now one cannot use directly the Riesz representation theorem for a homeomorphic embedding D into a locally convex linear topological Hausdorff space. The idea is to use approximation techniques by models with bounded state spaces. Although this concept was used in Balbus et al. (2015b), it has not been precisely explained. Balbus et al. (2015b) tacitly make use of the approach for the bounded state spaces and the “diagonal method” of selecting convergent subsequences as in Helly's selection theorem, see Chapter 17.3 in Williams (1991). Some remarks (without proof) on this issue are also given in Leininger (1986). Subsection 3.2 fills in this gap. We describe how to use the classical approach of embedding homeomorphically the set D in the Cartesian product of the spaces of signed measures on growing bounded intervals [0,r] as r. The whole material of Section 3 has a methodological character and shows that this embedding of the set D requires an application of various classical results in measure theory and functional analysis. In Subsection 3.2 it is precisely shown how to homeomorphically embed D into a locally convex Fréchet space.

The second approach relies on the use of the Schauder fixed point theorem, see Aliprantis and Border (2006), and is described in Section 4. We provide a very simple homeomorphic embedding of the set D into the Hilbert space of square integrable functions on the half line. The monotonicity of functions in D plays a crucial role in our approach (see Lemma 5). Compared to the methods from Section 3, this idea appears to be elementary. Moreover, it allows in future to think about techniques for constructing approximate equilibria. It can be used for applications of the fixed point theorem of Schauder (1930) in previous and future studies of Markov perfect equilibria instead of using the fixed point theorem of Tychonoff (1935).

Section 2 provides a stochastic bequest game with unbounded state space. In this model the current generation derives its utility from its own consumption level and consumption of its immediate descendent. Within such a framework it is more convenient to work with the set of left-continuous non-decreasing strategies. Although our results are stated for right-continuous strategies, it is easy to reformulate them for left-continuous functions. This fact was pointed out in Remark 4, Remark 6. Moreover, we also state Proposition 3 on the homeomorphic embedding of the set of non-decreasing left-continuous functions into the Hilbert space. This result is essential to apply our second approach and to prove existence of a stationary Markov perfect equilibrium in this model (Theorem 1). More details on the models of games mentioned in this note and related ones can be found in Jaśkiewicz and Nowak (2018).

Section snippets

Stochastic bequest games and economic growth, dynamic games of resource extraction and related models

In this section, we state an extension of Theorem 1 in Balbus et al. (2015c) on equilibria in a class of stochastic bequest games by allowing the state space and utility functions to be unbounded. An equilibrium is obtained in a class of left-continuous monotone saving functions of the state variable. We also briefly overview the literature on monotone (left- or right-continuous) equilibria in related models with compact state space, including dynamic decision processes with quasi-hyperbolic

Monotone strategies and the spaces of signed measures

Let Y be an interval in the real line R. By B(Y) we denote the σ-algebra of all Borel subsets of Y.

Monotone strategies in the Hilbert space of square integrable functions

Let η:SR be a continuous and positive function such that Sβ2(s)η(s)ds<. We can choose η such that it is a density function of some probability measure λ on S.

If F:SR is a Borel function, then by F˜ we denote the class of functions G such that F=G λ-almost surely. Let L2(S) be the Hilbert space of all classes F˜ of Borel functions F such that SF2(s)λ(ds)<. Because 0F2(s)β2(s) for all sS and FD, the classes of functions in D belong to L2(S). The distance m(F˜,G˜) of two elements F˜, G˜L

Proof of Theorem 1

Take β(s)=s in Section 4 and λ(ds)=esds. Recall that F is the set of all non-decreasing and lower semicontinuous functions F:SS such that 0F(s)β(s)=s for all sS. Note that every FF is left-continuous and is continuous at s0=0. The weak convergence FnwF in F is understood in the same manner as in the set D.

For each FF, denote by F˜ its class in L2(S). DefineK2:={F˜:FF}andψ˜(F)=F˜L2(S). Observe that Lemma 5 holds true, when D is replaced by F. The proof is very similar. Therefore, we

Concluding remarks

As Proposition 1, Proposition 2 show, an embedding of the sets of monotone and left- or right-continuous functions used in economic growth models or dynamic games of resource extraction is not obvious and straightforward.

Remark 8

The simplest way for viewing D as a subset of locally convex linear topological vector space is the following. Consider any vector space V containing the set D. The topology on V can be defined by the seminorms ps(F):=|F(s)|, sS and, in fact, it is a topology of pointwise

Acknowledgement

We thank Łukasz Balbus for useful discussions.

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    The authors acknowledge the financial support from the National Science Centre, Poland: Grant 2016/23/B/ST1/00425.

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