Family Of Utility Functions Research Articles

Cautious stochastic choice, optimal stopping and deliberate randomization

We study cautious stochastic choice (CSC) agents facing optimal timing decisions in a dynamic setting. In an expected utility setting, the optimal strategy is always a threshold strategy—to stop/sell the first time the price process exits an interval. In contrast, we show that in the CSC setting, where the agent has a family of utility functions and is concerned with the worst case certainty equivalent, the optimal strategy may be of non-threshold form and may involve randomization. We provide some carefully constructed examples, including one where we can solve explicitly for the optimal stopping rule and show it is a non-trivial mixture of threshold strategies. Our model is consistent with recent experimental evidence in dynamic setups whereby individuals do not play cut-off or threshold strategies.

Cautious Stochastic Choice, Optimal Stopping and Deliberate Randomization

We study Cautious Stochastic Choice (CSC) agents facing optimal timing decisions in a dynamic setting. In an expected utility setting, the optimal strategy is always a threshold strategy - to stop/sell the first time the price process exits an interval. In contrast, we show that in the CSC setting, where the agent has a family of utility functions and is concerned with the worst case certainty equivalent, the optimal strategy may be of non-threshold form and may involve randomization. We provide some carefully constructed examples, including one where we can solve explicitly for the optimal stopping rule and show it is a non-trivial mixture of threshold strategies. Our model is consistent with recent experimental evidence in dynamic setups whereby individuals do not play cut-off or threshold strategies.

Asymptotically Optimal Power Allocation for Energy Harvesting Communication Networks

For a general energy harvesting (EH) communication network, i.e., a network where the nodes generate their transmit power through EH, we derive the asymptotically optimal online power allocation solution that optimizes a general utility function when the number of transmit time slots, $N$ , and the battery capacities of the EH nodes, $B_{\rm max}$ , satisfy $N\rightarrow \infty$ and $B_{\rm max}\rightarrow \infty$ . The considered family of utility functions is general enough to include the most important performance measures in communication theory, such as the average data rate, outage probability, average bit-error probability, and average signal-to-noise ratio. The proposed power allocation solution is very simple. Namely, the asymptotically optimal power allocation for the EH network is identical to the optimal power allocation for an equivalent non-EH network whose nodes have infinite energy available but their average transmit power is constrained to be equal to the average harvested power and/or the maximum average transmit power of the corresponding nodes in the EH network. Moreover, the maximum average performance of a general EH network converges to the maximum average performance of the corresponding equivalent non-EH network, when $N\rightarrow \infty$ and $B_{\rm max}\rightarrow \infty$ . Although the proposed solution is asymptotic in nature, it is applicable to EH systems transmitting in a large but finite number of time slots and having a battery capacity much larger than the average harvested power and/or the maximum average transmit power.

Markov decision process algorithms for wealth allocation problems with defaultable bonds

Abstract In this paper we are concerned with analysing optimal wealth allocation techniques within a defaultable financial market similar to Bielecki and Jang (2007). We study a portfolio optimization problem combining a continuous-time jump market and a defaultable security; and present numerical solutions through the conversion into a Markov decision process and characterization of its value function as a unique fixed point to a contracting operator. In this paper we analyse allocation strategies under several families of utility functions, and highlight significant portfolio selection differences with previously reported results.

Time-Randomized Stopping Problems for a Family of Utility Functions

This paper studies stopping problems of the form $V=\inf_{0 \leq \tau \leq T} \mathbb{E}[U(\frac{\max_{0\le s \le T} Z_s }{Z_\tau})]$ for strictly concave or convex utility functions U in a family of increasing functions satisfying certain conditions, where Z is a geometric Brownian motion and T is the time of the nth jump of a Poisson process independent of Z. We obtain some properties of $V$ and offer solutions for the optimal strategies to follow. This provides us with a technique to build numerical approximations of stopping boundaries for the fixed terminal time optimal stopping problem presented in [J. Du Toit and G. Peskir, Ann. Appl. Probab., 19 (2009), pp. 983--1014].

Generalized Family of Multi-Step Utility Functions for Adoption in UNDP's Human Development Index

UNDP introduced the human development report in 1991, a radical change in its approach in terms of multi-step fragmenting of the utility function and a concept of threshold income level, which was followed till 1998 by following an ill-structured Atkinson based multi-step utility function. Realizing the weaknesses in it formulation, the UNDP abandoned the approach abruptly from 1999 onwards. As a viable substitute for erroneous Atkinson based multi-step formulation of utility function used in UNDP's human development reports till 1998, we provided a generalized family of the utility functions under the premise of multi-step formulation while adhering to the concept of threshold income level and have showed in the present study that the earlier proposed two alternative formulations due to Bhatnagar (2001, 2002a) turn out to be particular cases.

Contract pricing and utility sharing

In an incomplete market setting, we consider two financial agents who are willing to trade as counterparties a contract that represents a non-replicable indivisible contingent claim. Market incompleteness allows for an infinity of nonarbitrage prices at which the contract can be traded. Furthermore, indivisibility of the contract does not allow for equilibrium pricing. Assuming that the agents are utility maximizers who will resort into indifference pricing, we suggest a scenario that allows for the establishment of a natural one-to-one correspondence between the agreed transaction price of the contract and the relative bargaining power of the two agents. This amounts to the solution of an optimization problem, seeking to maximize a convex combination of the indirect utilities of the two agents, weighted by their relative bargaining power and under the constraint that the trade takes place. In particular, given the utility functions of the two agents and their relative bargaining power, we obtain an optimal unique price for the contract. Its existence is proved for a large family of utility functions, and a number of its properties are stated and discussed. As an example, we analyze extensively the case where both agents have exponential utility.

Necessary and possible preference structures

A classical approach to model a preference on a set A of alternatives uses a reflexive, transitive and complete binary relation, i.e. a total preorder. Since the axioms of a total preorder do not usually hold in many applications, preferences are often modeled by means of weaker binary relations, dropping either completeness (e.g. partial preorders) or transitivity (e.g. interval orders and semiorders). We introduce an alternative approach to preference modeling, which uses two binary relations–the necessary preference ≿N and the possible preference ≿P–to fulfill completeness and transitivity in a mixed form. Formally, a NaP-preference (necessary and possible preference) on A is a pair (≿N,≿P) such that ≿N is a partial preorder on A and ≿P is an extension of ≿N satisfying mixed properties of transitivity and completeness. We characterize a NaP-preference (≿N,≿P) by the existence of a nonempty set R of total preorders such that ⋂R=≿N and ⋃R=≿P. In order to analyze the representability of NaP-preferences via families of utility functions, we generalize the notion of a multi-utility representation of a partial preorder by that of a modal utility representation of a pair of binary relations. Further, we give a dynamic view of the family of all NaP-preferences on a fixed set A by endowing it with a relation of partial order, which is defined according to the stability of the information represented by each NaP-preference.

Stochastic skylines

In many applications involving multiple criteria optimal decision making, users may often want to make a personal trade-off among all optimal solutions for selecting one object that fits best their personal needs. As a key feature, the skyline in a multidimensional space provides the minimum set of candidates for such purposes by removing all points not preferred by any (monotonic) utility/scoring functions; that is, the skyline removes all objects not preferred by any user no matter how their preferences vary. Driven by many recent applications with uncertain data, the probabilistic skyline model is proposed to retrieve uncertain objects based on skyline probabilities. Nevertheless, skyline probabilities cannot capture the preferences of monotonic utility functions. Motivated by this, in this article we propose a novel skyline operator, namely stochastic skylines. In the light of the expected utility principle, stochastic skylines guarantee to provide the minimum set of candidates to optimal solutions over a family of utility functions. We first propose the lskyline operator based on the lower orthant orders . lskyline guarantees to provide the minimum set of candidates to the optimal solutions for the family of monotonic multiplicative utility functions. While lskyline works very effectively for the family of multiplicative functions, it may miss optimal solutions for other utility /scoring functions (e.g., linear functions). To resolve this, we also propose a general stochastic skyline operator, gskyline , based on the usual orders . gskyline provides the minimum candidate set to the optimal solutions for all monotonic functions. For the first time regarding the existing literature, we investigate the complexities of determining a stochastic order between two uncertain objects whose probability distributions are described discretely . We firstly show that determining the lower orthant order is NP-complete with respect to the dimensionality; consequently the problem of computing lskyline is NP-complete. We also show an interesting result as follows. While the usual order involves more complicated geometric forms than the lower orthant order, the usual order may be determined in polynomial time regarding all the inputs, including the dimensionality; this implies that gskyline can be computed in polynomial time. A general framework is developed for efficiently and effectively retrieving lskyline and gskyline from a set of uncertain objects, respectively, together with efficient and effective filtering techniques. Novel and efficient verification algorithms are developed to efficiently compute lskyline over multidimensional uncertain data, which run in polynomial time if the dimensionality is fixed, and to efficiently compute gskyline in polynomial time regarding all inputs. We also show, by theoretical analysis and experiments, that the sizes of lskyline and gskyline are both quite similar to that of conventional skyline over certain data. Comprehensive experiments demonstrate that our techniques are efficient and scalable regarding both CPU and IO costs.

A Comment on Reis

This note gives a counterexample on Reis [1]. Using a certain family of utility functions, this note not only gives a sharper representation than that of Reis but also demonstrates that interest rate inelastic money demand does not lead to superneutrality. This implies that superneutrality does not exist when uncerinty is introduced.

A Tractable Utility-Based Approach to Consistent Model Selection and Asset Allocation

Investors, such as traders and portfolio managers, who face the practical problem of how to allocate their assets in financial markets, often use probabilistic models, in conjunction with some quantification of their preferences. In this paper, we explore a consistent, tractable framework, based on utility theory, for both selecting models and allocating assets. This framework leads us to a particular family of utility functions, the generalized logarithmic family, which is consistent with model selection via likelihood. We also discuss drawdown and optimality consequences of applying this framework as well as related calibration issues.

On the Independence of Irrelevant Assets: McEntire's Conjecture

In his 1984 paper in Management Science, McEntire conjectured that all concave utility functions satisfy his independence-from-irrelevant-assets (IIA) property. This paper shows that this conjecture is false. The paper provides several families of utility functions, all of whose members violate the IIA property; moreover, it is shown that the set of utility functions violating the IIA property is quite large, and that it is widespread throughout the entire space of utility functions.

Minimal and maximal product differentiation in Hotelling's duopoly

Hotelling's (1929) model of duopolistic competition is re-examined. A family of utility functions is used which has as a special case Hotelling's original utility function. In a two-stage location-price game it is shown that an equilibrium exists when the curvature of the utility functions in the space of characteristics is sufficiently high. The (subgame-perfect) equilibrium never exhibits minimum product differentiation. On the other hand, not all equilibria are at maximal product differentiation.

Stochastic dominance with pair-wise risk aversion

A class of stochastic orders is defined on the set of bivariate distribution functions. This class of orders is linearly orderable by inclusion. A family of utility functions, coherent with each of the stochastic orders previously defined, is determined. These utility functions represent pair-wise risk aversion. The relations with univariate stochastic orders are examined.