Increasing Utility Function Research Articles

Revealed preference and revealed preference cycles: A survey

Afriat’s Theorem (1967) states that a dataset can be thought of as being generated by a consumer maximizing a continuous and increasing utility function if and only if it is free of revealed preference cycles containing a strict relation. The latter property is often known by its acronym, GARP (for generalized axiom of revealed preference). This paper surveys extensions and applications of Afriat’s seminal result. We focus on those results where the consistency of a dataset with the maximization of a utility function satisfying some property can be characterized by a suitably modified version of GARP.

An inter-temporal CAPM based on First order Stochastic Dominance

The inter-temporal Capital Asset Pricing Model (CAPM) assumes that investors are risk-averse. However, there is a very large body of empirical and experimental evidence documenting that many investors are not globally risk-averse: Prospect Theory and aspiration-level models are two well-known examples of this literature. This paper employs Stochastic Dominance criteria to generalize the inter-temporal CAPM for all investors with increasing utility functions. Another advantage of the proposed approach is its simplicity: it does not require dynamic programming, and it allows for ambiguous investment horizons.

Order Relations and Operations on the Set of Probabilistic Linguistic Term Sets

Probabilistic linguistic term sets (PLTSs) provide a flexible manner to model complex linguistic expressions in qualitative decision making. Some fundamental aspects of PLTSs, such as order relations and basic operations, are not built in a solid way. The lack of required properties make their applications be not sufficiently reliable. In this study, a PLTS is mapped to a stochastic variable. A partial order of PLTSs is defined based on the idea of stochastic order. The concept of a total order is then provided by refining the partial order. Moreover, aggregation functions of PLTSs are proposed by defining an aggregation function in the set of linguistic terms. Operations and aggregation operators of PLTSs are further carried out based on the aggregation functions. The above theoretical developments own some merits: (1) the partial order is reasonable because the ranking results coincide with rational decision-makers who consider increasing utility functions; (2) the operations preserve desirable properties which are satisfied by the linguistic aggregation functions; and (3) the operations and aggregation operators of PLTSs could be specified according to practical applications based on the proposed theoretical framework.

Optimal Financing and Disclosure

I study an entrepreneur who finances a project with uncertain cash flows and who jointly chooses the disclosure and financing policies. In the Bayesian persuasion framework, I show that it is optimal to truthfully reveal whether the project’s cash flows are above a threshold. This class of threshold policies is optimal for any prior belief, monotone security, and increasing utility function of the entrepreneur. I characterize how the disclosure threshold depends on the underlying security, the prior, and the cost of investment. The financing choice of the entrepreneur is determined by a new trade-off between the likelihood of persuading investors and relinquishing cash flow rights. Absent further frictions, the optimal security is indeterminate. If there is adverse selection after the entrepreneur has disclosed information, the unique outcome is a pooling equilibrium in which the entrepreneur pledges the entire cash flow. This paper was accepted by Tomasz Piskorski, finance.

Online Fixed Fraction Policies in Energy Harvesting Communication Systems

We consider power scheduling policies for single-user energy harvesting communication systems, where the goal is to characterize online policies that maximize the long term average utility, for general concave and monotonically increasing utility functions. The transmitter relies on energy harvested from nature to send its messages to the receiver, and is equipped with a finite-sized battery to store its harvested energy. Energy packets are independent and identically distributed (i.i.d.) over time slots, and are revealed causally to the transmitter. We first characterize the optimal solution for the case of Bernoulli arrivals. Then, for general i.i.d. arrivals, we first show that fixed fraction policies , in which a fixed fraction of the battery state is consumed in each time slot, are within a constant multiplicative gap from the optimal solution for all energy arrivals and battery sizes. We then derive a set of sufficient conditions on the utility function to guarantee that fixed fraction policies are within a constant additive gap as well from the optimal solution. We then apply these results to a specific scenario where a sensor node collects samples from a Gaussian source and sends them to a destination node over a Gaussian channel, and the goal is to minimize the long term average distortion of the source samples received at the destination. We study two problem settings for this case: the first is when sampling is cost-free, and the second is when there is a sampling cost incurred whenever samples are collected. For the problem with sampling costs, the transmission policy can be bursty; the sensor may collect samples and transmit for only a portion of the time. Finally, we present an alternative analysis approach that is more tailored to these distortion problems to show that fixed fraction policies achieve an additive gap that is independent of the sampling cost.

Convex preference cone-based approach for many objective optimization problems

Many objective optimization problems have turned out to be a considerable challenge for evolutionary algorithms due to the difficulty of finding and visualizing high-dimensional Pareto frontiers. Fortunately, however, the task can be simplified whenever an interaction with a human decision maker is possible. Instead of finding the entire Pareto frontier, the evolutionary search can be guided to the parts of the space that are most relevant for the decision maker. In this paper, we propose an interactive method for solving many objective optimization problems. Drawing on the recent developments in multiple criteria decision making, we introduce an effective strategy for leveraging polyhedral preference cones within an evolutionary algorithm. The approach is mathematically motivated and is applicable to situations, where the user’s preferences can be assumed to follow an unknown quasi-concave and increasing utility function. In addition to considering the preference cones as a tool for eliminating non-preferred solution candidates, we also present how the the cones can be leveraged in approximating the directions of steepest ascent to guide the subsequent search done by the evolutionary algorithm through a proposed merit function. To evaluate the performance of the algorithm, we consider well known test problems as well as a practical facility location problem.

Weakly Monotonic Preference Relations Representable by Concave Utility Functions - It's Easy

In this paper we provide conditions under which a preference relation can be represented by concave utility functions. The condition follows naturally from a proof of a theorem about representability of continuous and weakly monotonic preference relations by continuous and weakly increasing utility functions due to Wold (1943). For continuous and homothetic (hence weakly monotonic) preference relations our sufficient condition for the existence of a numerical representation by a concave utility function is also a necessary condition for such a numerical representation. In fact we are able to obtain a necessary condition for numerical representation of a homothetic preference relation by a concave utility function which we call global concavifiability and which is much stronger than concavifiability. On the way, we pick up the result, that all homothetic and continuous preference relations which are convex are numerically representable by concave utility functions.

Heterogeneous risk/loss aversion in complete information all-pay auctions

We extend previous theoretical work on n-player complete information all-pay auctions to incorporate heterogeneous risk- and loss-averse utility functions. We provide sufficient and necessary conditions for the existence of equilibria with a given set of active players with any strictly increasing utility functions and characterize the players’ equilibrium mixed strategies. Assuming that players can be ordered by their risk aversion (player a is more risk-averse than player b, if whenever player b prefers a certain payment over a given lottery, so does player a), we find that in equilibrium, the more risk-averse players either bid higher than the less risk-averse players and win with higher ex-ante probability – or they drop out. Furthermore, while each player’s expected bid decreases with the other players’ risk aversion, her expected bid increases with her own risk aversion. Thus, increasing a player’s risk aversion creates two opposing effects on total expected bid. A sufficient condition for the total expected bid to decrease with a player’s risk aversion is that this player is relatively more risk-averse compared to the rest of the players. Our findings have important implications for the literature on gender differences in competitiveness and for gender diversity in firms that use personnel contests for promotions.

Heterogeneous Risk/Loss Aversion in Complete Information All-Pay Auctions

We extend previous theoretical work on n-players complete information all-pay auction to incorporate heterogeneous risk and loss averse utility functions. We provide sufficient and necessary conditions for the existence of equilibria with a given set of active players with any strictly increasing utility functions and characterize the players' equilibrium mixed strategies. Assuming that players can be ordered by their risk aversion (player a is more risk averse than player b if whenever player b prefers a certain payment over a given lottery so will player a), we find that, in equilibrium, the more risk averse players either bid higher (in terms of first order stochastic dominance of their mixed strategy cumulative distribution) than the less risk averse players and win with higher ex-ante probability — or they drop out. Furthermore, while each player's expected bid decreases with the other players' risk aversion, her expected bid increases with her own risk aversion. Thus, increasing a player's risk aversion creates two opposing effects on total expected bid. A sufficient condition for the total expected bid to decrease with a player's risk aversion is that this player is relatively more risk averse compared to the rest of the players. Our findings have important implications for the literature on gender differences in competitiveness and for gender diversity in firms that use personnel contests for promotions.

Some new results on allocation of coverage limits and deductibles to mutually independent risks

ABSTRACTThis article further investigates the allocation of coverage limits and deductibles to multiple independent risks from the viewpoint of policyholders with increasing utility functions. In a more general setup, we develop the usual stochastic orders on the retained loss, which either generalize or supplement the corresponding results due to Lu and Meng (2011) and Hu and Wang (2014). Also, the most unfavorable and favorable allocations of coverage limits and deductibles are developed for multiple risks with dominated reversed hazard rates and hazard rates, respectively.

Optimal Financing and Disclosure

How does a firm’s disclosure policy depend on its choice of financing? In this paper, I study a firm that finances a project with uncertain payoffs and jointly chooses its disclosure policy and the security issued. I show that it is optimal to truthfully reveal whether the project’s payoffs are above a threshold. This class of threshold policies is optimal for any prior belief, for any security, and any increasing utility function of the entrepreneur. I characterize how the optimal disclosure threshold depends on the underlying security, the prior, and the cost of investment. The optimal security design is indeterminate despite the presence of adverse selection. Among others, the optimum can be implemented with equity, debt, and options.

Jointly Optimal Rate and Power Allocation for Multilayer Transmission

In this paper we consider joint optimization of rate and power for communication systems that use multilayer source coding with successive information refinement, accompanied with a broadcast approach at the physical layer of the system. We analyze the problem under the assumption of Rayleigh fading channels where rates and power ratios of the source layers are jointly optimized based on channel statistics information, with the objective of maximizing the expected user satisfaction which is usually defined by a differentiable concave increasing utility function of the total decoded rate. As special cases, we consider two utility functions; namely, the expected total decoded rate at the receiver and the expected rate distortion of a Gaussian source. We show that the optimal solution can be obtained using a two-dimensional bisection search for any number of layers. The outer bisection search is over the Lagrangian dual variable and the inner bisection search is over the decoding threshold of the layer. Moreover, we show that with a small number of layers, we can approach the performance upper bound that is achieved by transmitting an infinite number of layers.

Utility Adjusted Cumulative Area Ratio Optimisation Technique (UCAROT)

In this paper we propose a novel portfolio optimization technique that draws its inspiration from the concept of “almost stochastic dominance”. The technique is computationally parsimonious and applicable for all return distributions and all strictly increasing utility functions. First, we develop the concept of Utility Adjusted Cumulative Area Ratios (UCAR) to make pairwise comparisons of the individual assets. This technique involves adjusting the cumulative distributions of asset returns for the investor utility function. We then input this information into an NxN Analytic Hierarchy Process (AHP) matrix to determine optimum weights. As an example we apply it to the BIST-30 Turkish Index. We form five portfolios based on various utility functions: risk averse with decreasing, increasing and constant absolute risk aversion (DARA, CARA and IARA); risk neutral; and risk seeking. We find that all these portfolios convincingly outperform the Index according to conventional mean-variance criteria as well as second order stochastic dominance.

Big House or Small House, Which One Should We Buy? Evidence from Hong Kong

Big house or small house, which one should we buy? To answer this question, we employed both mean-variance approach and stochastic dominance approach that utilizes the entire yield distribution to rank the performance of the Hong Kong housing market. While mean-variance rules cannot show a clear preference between different classes (sizes) of housing, we find, however, that smaller class housing has first-order stochastic dominance over larger class housing. That means not only risk-averse investors but also any investor with increasing utility function can maximize their expected utilities and expected wealth. We conclude that the smallest class housing provides a better return for both risk-averse and other types of investors and a non-satiation investor will increase his wealth as well as his utility if his investment is shifted to the smallest class housing. This suggests even though there may not be arbitrage opportunity, Hong Kong housing market is not efficient. The answer is simple, in Hong Kong, we should buy small house.

Revealed Preference in a Discrete Consumption Space

We show that an agent maximizing some utility function on a discrete (as opposed to continuous) consumption space will obey the generalized axiom of revealed preference (GARP), so long as the agent obeys cost efficiency. Cost efficiency will hold if there is some good, outside the set of goods being studied by the modeler, that can be consumed by the agent in continuous quantities. An application of Afriat's Theorem then guarantees that there is a strictly increasing utility function on the discrete consumption space that rationalizes price and demand observations. (JEL D11)

Portfolios Resampling and International Diversification: A Non-Parametric Stochastic Dominance Approach

The aim of this paper is to consider instability and ambiguity problems on portfolio selection. We examine the impact of estimation errors on financial portfolios optimization processes. We investigate the controversy problem of international and domestic optimal diversification strategies choices using a non parametric stochastic dominance approach (NPSDA) based on Monte Carlo (MC) and Bootstrap p-values from an American point of view. Based on a data set consisting of daily closing prices of U.S. stocks and index and Asian and Latin American stock market indices for the period from August 03, 1993 to August 31, 2007, estimation errors visualisation results show that small changes in input parameters imply large changes in the optimized portfolio composition and a considerably modification of mean-variance efficient frontiers shape. Though mean-variance optimization could not be used to draw any preference between international and domestic diversification, empirical findings reveal the utility of the SD approach to define an optimal strategy. Further, results show that there exists substantial evidence of international global diversification benefits for domestic investors. More precisely, for risk levels higher than 30 percent, risk-adverse American investor having an increasing utility function prefers global international to domestic resampled diversification strategy. Besides, we find that domestic resampled diversification strategy beats all international major and emerging markets diversification strategies. For risk levels lower than 30 percent and 23.98, respectively, U.S. investor having a high risk-aversion coefficient prefers domestic to international major and emerging markets diversification and invests 77 percent and 82 percent, respectively, of his wealth’ locally. SD analysis suggests that global international diversification dominates entirely major and emerging markets diversification strategies for a U.S. risk-adverse investor having an increasing utility function. Finally, the findings of the SD tests reveal that risk-adverse U.S. investor having an increasing utility function prefers to diversify 45 percent of his wealth in major markets rather than in emerging markets.

Stochastic comparisons for allocations of policy limits and deductibles with applications

In this paper, we study the problem of comparing losses of a policyholder who has an increasing utility function when the form of coverage is policy limit and deductible. The total retained losses of a policyholder ∑ i = 1 n ( X i − l i ) + are ordered in the usual stochastic order sense when X i ( i = 1 , … , n ) are ordered with respect to the likelihood ratio order. The parallel results for the case of deductibles are obtained in the same way. It is shown that the ordering of the losses are related to the characteristics (log-concavity or log-convexity) of distributions of the risks. As an application of the comparison results, the optimal problems of allocations of policy limits and deductibles are studied in usual stochastic order sense and the closed-form optimal solutions are obtained in some special cases.

Distributed Iterative Optimal Resource Allocation With Concurrent Updates of Routing and Flow Control Variables

Consider a set of active elastic sessions over a network. Session traffic is routed at each hop (potentially through multiple network paths) based only on its destination. Each session is associated with a concave increasing utility function of its transfer rate. The transfer rates of all sessions and the routing policy define the operating point of the network. We construct a metric f of the goodness of this operating point. f is an increasing function of the session utilities and a decreasing function of the extent of congestion in the network. We define ldquogoodrdquo operating points as those that maximize f, subject to the capacity constraints in the network. This paper presents a distributed, iterative algorithm for adapting the session rates and the routing policy across the network so as to converge asymptotically to the set of ldquogoodrdquo operating points. The algorithm updates session rates and routing variables concurrently and is, therefore, amenable to distributed online implementation. The convergence of the concurrent update scheme is proved rigorously.

On convergence to the exponential utility problem

We provide a method for solving dynamic expected utility maximization problems with possibly not everywhere increasing utility functions in an L p -semimartingale setting. In particular, we solve the problem for utility functions of type − e − x (exponential problem) and − ( 1 − x 2 m ) 2 m ( 2 m -th problem). The convergence of the 2 m -th problems to the exponential one is proved. Using this result an explicit portfolio for the exponential problem is derived.

Manipulation of Preferences and Relative Utilitarianism

Given n agents with who wish to divide m commodities, consider the n-person noncooperative game with strategies consisting of concave increasing utility functions, and whose outcomes are the relative utilitarian solution. Any constrained equal-income competitive equilibrium allocation for the true utilities is shown to ba a Nash equilibrium outcome for the noncooperative game. Conditions are presented under which these are the only pure strategy equilibrium outcomes. Journal of Economic Literature Classification Numbers: C72, C78, D71.