Unbounded Utility Research Articles

On existence of Berk-Nash equilibria in misspecified Markov decision processes with infinite spaces

Model misspecification is a critical issue in many areas of economics. In the context of misspecified Markov Decision Processes, Esponda and Pouzo (2021) defined the notion of Berk-Nash equilibrium and established its existence with finite state and action spaces. However, many substantive applications (including two of the three motivating examples presented by Esponda and Pouzo) involve continuous state or action spaces, and are thus not covered by the Esponda-Pouzo existence theorem. We extend the existence of Berk-Nash equilibrium to compact action spaces and sigma-compact state spaces, with possibly unbounded utility functions. A complication arises because Berk-Nash equilibrium depends critically on Radon-Nikodym derivatives, which are bounded in the finite case but typically unbounded in misspecified continuous models. The proofs rely on nonstandard analysis, and draw on novel argumentation traceable to work of the second author on nonstandard representations of Markov processes.

Balanced growth and degrowth with human capital

In a simple discrete-time version of Lucas (1988) we find that the Balanced Growth Path (BGP) is always the unique optimal planner’s solution: with linear or strictly concave production functions, with unbounded utility functions, with or without human capital depreciation. When the ”speed” of human capital accumulation is high, the optimal working time is constant and below its upper bound. Capital grows at a constant factor, but degrowth is also possible when this factor is less one (under positive capital depreciation). When this speed is low, optimal working time is at its boundary and capital declines at its depreciation factor (degrowth).

Decision Theory Unbound

AbstractCountenancing unbounded utility in ethics gives rise to deep puzzles in formal decision theory. Here, these puzzles are taken as an invitation to assess the most fundamental principles relating probability and value, with the aim of demonstrating that unbounded utility in ethics is compatible with a desirable decision theory. The resulting theory frames further discussion of Expected Utility Theory and of principles concerning symmetries of utility.

A dilemma for Nicolausian discounting

Abstract Orthodox decision theory is fanatical in the way it treats small probabilities of enormous value, if unbounded utility functions are allowed. Some have suggested a fix, Nicolausian discounting, according to which outcomes with small enough probabilities should be ignored when making decisions. However, there are lotteries involving only small-probability outcomes, none of which should intuitively be ignored. So the Nicolausian discounter needs a procedure for distinguishing the problematic cases of small-probability outcomes from the unproblematic ones. In this paper, I present a dilemma for Nicolausian discounting: the view must be coupled either with a procedure that delivers fanatical verdicts of its own, as bad as those of orthodox decision theory, or with one that entails intransitive cyclic preferences.

Optimal Pricing for Peer-to-Peer Sharing With Network Externalities

In this paper, we analyse how a peer-to-peer sharing platform should price its service to maximize profit, when user participation increases the value of the service to others by causing positive externalities. Modelling the service as an excludable public good, we propose a bounded utility model to capture many infrastructure sharing applications with bounded network value, in which complete coverage generates finite user valuation (e.g., WiFi or hotspot). Unbounded utility models are used to capture the large-scale user interactions in social media, where the network value follows Metcalfe's or Zipf's law. For these utility models, we analyze the optimal pricing schemes in the case of heterogeneous users under complete and incomplete information of users' service valuations. We propose the concept of `price of information' (PoI) to characterize the profit loss due to lack of information, and present asymptotic PoI bounds for different utility models. We also show that the difficult-to-implement differentiated pricing scheme, which is optimal under incomplete user information, can be replaced by a simple uniform price scheme that is asymptotic optimal. Finally, we extend our pricing schemes to a two-sided market by including a new group of `pure' service users who do not contribute to the public good, and show that the platform may charge zero price to the original group of users in order to attract this pure user group.

ε-Contamination with Unbounded Utility

The theory of e-contamination utility builds upon the intuition that decision makers pay extra attention to the worst outcome and overweight its occurrence. Thus, the theory is highly relevant to applied research involving downside risk. The existing theory, however, assumes utility to be bounded. We relax this assumption. Using the new theory of e-contamination utility, we explore the utility-theoretic foundation of two popular downside risk measures, expected shortfall and lower partial moment.

Risk premiums and certainty equivalents of loss-averse newsvendors of bounded utility

Loss-averse behavior makes the newsvendors avoid the losses more than seeking the probable gains as the losses have more psychological impact on the newsvendor than the gains. In economics and decision theory, the classical newsvendor models treat losses and gains equally likely, by disregarding the expected utility when the newsvendor is loss-averse. Moreover, the use of unbounded utility to model risk attitudes fails to explain some decision-making paradoxes. In contrast, this paper deals with the utility maximization of the newsvendor using a class of bounded utility functions to study the effect of loss aversion on the newsvendor certainty equivalents and risk premiums. New formulas are introduced to find the utility-optimal order quantity of the normal distribution. The results show that when an exponential loss aversion exists, the classical newsvendor optimal quantity serves as a lower bound when the overage costs are high and as an upper bound when the underage costs are high. In addition, we show that high loss aversion entails higher risk premiums. Similar conclusion holds when the overage/underage costs increase. Higher standard deviations, on the other hand, mean lower utility-optimal quantities and higher risk premiums. The presented formulas are advantageous in finding the optimal order quantities and risk premiums of a stochastic short-shelf life inventory when the loss is a key factor in the decision-making process.

Equilibria in Altruistic Economic Growth Models

In this paper, we consider a stochastic economic growth model in the form of an intergenerational dynamic game. Both paternalistic and non-paternalistic components are present in the model. Under very general assumptions allowing for unbounded utility functions and weakly continuous transitions, we establish the existence of Markov perfect equilibria that consist of a consumption strategy and an indirect utility function. In the pure paternalistic case, we obtain new results on equilibria. An important point of our contribution is that we make no separability assumptions on the utility functions of generations.

Multiple-priors optimal investment in discrete time for unbounded utility function

This paper investigates the problem of maximizing expected terminal utility in a discrete-time financial market model with a finite horizon under nondominated model uncertainty. We use a dynamic programming framework together with measurable selection arguments to prove that under mild integrability conditions, an optimal portfolio exists for an unbounded utility function defined on the half-real line.

On symmetric stochastic games of resource extraction with weakly continuous transitions

We study stochastic games of resource extraction, in which the players have identical preferences. The transition probability is either non-atomic or a convex combination of transition probabilities depending on the investment with coefficients also dependent on the investment. Our approach covers the unbounded utility case, which was not examined in this class of games beforehand. We prove the existence of a stationary Markov perfect equilibrium in a non-randomised class of strategies.

A note on Markov perfect equilibria in a class of non-stationary stochastic bequest games

In this note, we prove the existence of a Markov perfect equilibrium in a non-stationary version of a paternalistic bequest game. The method we advocate is general and allows to study models with unbounded state space and unbounded utility functions. We cover both, the stochastic and deterministic cases. We provide a characterization of the set of all Markov perfect equilibria by means of a set-valued recursive equation involving the best response operator. In the stationary case, we show that there exists a set of strategies that is invariant under the best response mapping.

Benchmarking the utility theory: a data envelopment approach

PurposeThe purpose of this paper is to benchmark alternatives of decision problems that include risk and uncertainty considering different risk attitudes via a new data envelopment analysis (DEA) decision model.Design/methodology/approachA new utility function of strict bounds is applied in a data envelopment model to evaluate all possible stochastic alternatives (i.e. gambles). The amount of risk in the alternatives is measured by a newly introduced risk ratio (RR). Each alternative is considered as a decision making unit (DMU). The alternatives efficiency frontier is found via linear optimization of the DEA model.FindingsIn contrast to literature studies of binary decision alternatives, here, benchmarking is conducted to evaluate multiple decision alternatives with unbounded utility of the payoffs along with a new DEA decision model. Different surveys and studies have been used to validate the model. DEA could demonstrate the ability to uncover relationships that remain hidden for other methodologies. The resulting rankings remarkably conform to those elicited by subjects.Social implicationsIndividuals of different wealth backgrounds evaluate risky decision problems differently.Originality/valueThe paper contributes to the existing research by benchmarking multiple alternatives as compared to the literature research which usually assesses binary problems. Instead of using explicit utilities, the model implements the efficiencies along with a new utility function and a new RR. The introduction of DEA to such a decision field is found to be successful in benchmarking numerous alternatives under different risk attitudes.

Stochastic games of resource extraction

We study stochastic games of resource extraction in which the transition probability is a convex combination of stochastic kernels with coefficients depending on the joint investments of the players. Our approach covers the unbounded utility case which was not examined in this class of games beforehand. We give two theorems on the existence of pure stationary Markov perfect equilibria for the models of games under consideration. A detailed discussion with illustrative examples explains the meaning of our assumptions and their relation to the conditions used earlier in the literature.

Bequest games with unbounded utility functions

The existence of Markov perfect equilibria for a deterministic bequest game with bounded from below utility function was established independently by Bernheim and Ray [2] and Leininger [8]. The aim of this paper is to provide a simpler proof that, in contrast to [2,8], also works for models with unbounded, e.g., isoelastic utility functions. Our approach employs the concept of strict single crossing property, introduced by Milgrom and Shannon in [10], and includes supermodular functions as special cases.

On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function

We consider a class of infinite-horizon optimal control problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of that kind the initial state is fixed, no constraints are imposed on the behavior of the admissible trajectories at infinity, and the objective functional is given by a discounted improper integral. Earlier, for such problems, S.M. Aseev and A.V. Kryazhimskiy in 2004–2007 and jointly with the author in 2012 developed a method of finite-horizon approximations and obtained variants of the Pontryagin maximum principle that guarantee normality of the problem and contain an explicit formula for the adjoint variable. In the present paper those results are extended to a more general situation where the instantaneous utility function need not be locally bounded from below. As an important illustrative example, we carry out a rigorous mathematical investigation of the transitional dynamics in the neoclassical model of optimal economic growth.

RECURSIVE EQUILIBRIA IN AN AIYAGARI‐STYLE ECONOMY WITH PERMANENT INCOME SHOCKS

We prove existence of a recursive competitive equilibrium (RCE) for an Aiyagari‐style economy with permanent income shocks and derive important economic implications. We show that there exist equilibria where borrowing constraints are never binding and establish a nontrivial lower bound on the equilibrium interest rate. These results imply distinct consumption dynamics compared to existing studies. We present a new approach to solve the agent's problem that uses lattices of consumption functions to deal with permanent income shocks and an unbounded utility function. The approach provides a theoretical foundation for convergence of the time iteration algorithm widely used in applied work.

Time Resolution of the St. Petersburg Paradox: A Rebuttal

Peters (2011) claims to provide a resolution of the three century old St. Petersburg paradox by using time averages and thereby avoiding the use of utility theory completely. Peters also claims to have found an error in Menger (1934, 1967) who established the vulnerability of any unbounded utility function to the St. Petersburg paradox. This paper argues that both these claims in Peters (2011) are incorrect. The time average argument can be circumvented by using a single random number (between zero and one) to represent the entire infinite sequence of coin tosses, or alternatively by applying a time reversal to the coin tossing. Menger’s proof can be reinstated by comparing the utility of playing the Super St. Petersburg game to the utility of an arbitrarily large sure payoff.

Translation invariance when utility streams are infinite and unbounded

The axiom translation invariance consists in asserting the invariance of the ranking of two utility streams if one applies the same translation to both. This axiom is significant in the characterization of utilitarian criteria in finite dimension. This characterization is achieved thanks to the weak weighted utilitarianism theorem.The objective here is to propose a generalization of this theorem in a space of infinite and unbounded utility streams. A consequence of the suggested generalization is that, in the context of intergenerational choice, every maximal point with respect to a paretian utilitarian order granting comparable considerations to the present and the future, is also a maximal point with respect to some future-oriented criterion.

On variable discounting in dynamic programming: applications to resource extraction and other economic models

This paper generalizes the classical discounted utility model introduced in Samuelson (Rev. Econ. Stud. 4:155–161, 1937) by replacing a constant discount rate with a function. The existence of recursive utilities and their constructions are based on Matkowski’s extension of the Banach Contraction Principle. The derived utilities go beyond the class of recursive utilities studied in the literature and enable a discussion on subtle issues concerning time preferences in the theory of finance, economics or psychology. Moreover, our main results are applied to the theory of optimal economic growth related with resource extraction models with unbounded utility function of consumption.

Prospect theory for continuous distributions: A preference foundation

Preference foundations give necessary and sufficient conditions for a decision model, stated directly in terms of the empirical primitive: the preference relation. For the most popular descriptive model for decision making under risk and uncertainty today, prospect theory, preference foundations have as yet been provided only for prospects taking finitely many values. In applications, however, prospects often are complex and involve infinitely many values, as in normal and lognormal distributions. This paper provides a preference foundation of prospect theory for such complex prospects. We allow for unbounded utility and only require finite additivity of the underlying probability distributions, leaving the restriction to countably additive distributions optional. As corollaries, we generalize previously obtained preference foundations for special cases of prospect theory (rank-dependent utility and Choquet expected utility) that all required countable additivity. We now obtain genuine generalizations of de Finetti’s and Savage’s finitely additive setups to unbounded utility.