Quasi-concave Utility Functions Research Articles

A Utility Theory Based Interactive Approach to Robustness in Linear Optimization

We treat uncertain linear programming problems by utilizing the notion of weighted analytic centers and notions from the area of multi-criteria decision making. After introducing our approach, we develop interactive cutting-plane algorithms for robust optimization, based on concave and quasi-concave utility functions. In addition to practical advantages, due to the flexibility of our approach, we are able to prove that under a theoretical framework due to Bertsimas and Sim [14], which establishes the existence of certain convex formulation of robust optimization problems, the robust optimal solutions generated by our algorithms are at least as desirable to the decision maker as any solution generated by many other robust optimization algorithms in the theoretical framework. We present some probabilistic bounds for feasibility of robust solutions and evaluate our approach by means of computational experiments.

Evolutionary quasi-variational inequality for a production economy

This paper is focused on the analysis of a time-dependent general equilibrium problem for a production economy by using the variational inequality approach. More precisely, the time-dependent allocation-price equilibrium consists in searching the optimal allocation by households and firms and in searching the market clearing condition. Firstly, a new formulation of the economic equilibrium problem by means of an evolutionary quasi-variational inequality is proven in the general context of semistrictly quasiconcave utility functions. Finally, an existence result for equilibrium solutions is given.

Turnpike theorem for terminal functionals in infinite horizon optimal control problems

An optimal control problem for continuous time systems described by a special class of multi-valued mappings and quasi-concave utility functions is considered. The objective is defined as an analogue of the terminal functional defined over an infinite time horizon. An upper bound of this functional over all solutions to the system is established. The turnpike property is proved which states that all optimal solutions converge to some unique optimal stationary point.

A Characterization of Rationalizable Consumer Behavior

For an arbitrary finite or infinite dataset D of prices and corresponding chosen bundles, it is shown that the following three conditions are equivalent. (i) D satisfies GARP; (ii) D can be rationalized by a utility function; (iii) D can be rationalized by a strictly increasing, quasiconcave utility function. Examples of infinite datasets satisfying GARP are provided for which every utility rationalization fails to be lower semicontinuous, upper semicontinuous, or concave. Thus condition (iii) cannot be substantively improved upon.

Revealed preference analysis for convex rationalizations on nonlinear budget sets

We present necessary and sufficient revealed preference conditions to verify whether a finite data set on nonlinear budget sets is consistent with the maximization of a quasi-concave utility function. When budget sets are finite unions of polyhedral convex sets, these conditions require feasibility of a set of linear inequalities, which makes them easy to use in practical applications.

A systemic approach to modelling and estimating demand for money(ies)

This article uses a consumer theory-based systemic approach to model the demand for monetary liquid asset holdings in Chile. We implement the suggestions and caveats of aggregation theory for the estimation of a demand system for liquid assets (monies) in static, dynamic and time-varying parameters setups. Our results are robust and theoretically consistent with consumer theory restrictions, as a system derived from a utility maximizing framework and a quasi concave utility function. In our estimations, we find stability of interest rate elasticities, in contrast to previous related literature. We also document evidence that long (short) maturity rates are associated to less (more) liquid assets.

Revealed Preference Analysis for Convex Rationalizations on Nonlinear Budget Sets

We present necessary and sufficient revealed preference conditions to verify whether a finite data set on nonlinear budget sets is consistent with the maximization of a quasi–concave utility function. Our results can be used to test for convexity of the underlying preference relation. We also show that in many settings, our conditions are easy to use in practical applications.

Quasiconcave preferences on the probability simplex: A nonparametric analysis

A nonparametric approach is presented to test whether decisions on a probability simplex could be induced by quasiconcave preferences. Necessary and sufficient conditions are presented which can easily be tested. If the answer is affirmative, the methods developed here allow us to reconstruct bounds on indifference curves. Furthermore we can construct quasiconcave utility functions in analogy to the utility function constructed in the proof of Afriat’s Theorem. The approach is of interest for ex ante fairness considerations when a dictator is asked to choose probabilities to win an indivisible prize. It is also of interest for decisions under risk and stochastic choice. It allows nonparametric interpersonal comparisons.

Variational methods for equilibrium problems involving quasi-concave utility functions

This paper deals with the existence of competitive equilibrium points in a Walrasian pure exchange economy with utility function. An existence result involving quasi-concave utility functions is established. In particular, a previous result by the authors involving concave utility functions is improved. The approach is variational. Indeed, competitive equilibrium points will be found out as solutions of a quasi-variational inequality in a finite dimensional space. In particular, an existence result for (generalized) quasi-variational inequalities established by Cubiotti is applied.

Quasiconcave Preferences and Choices on a Probability Simplex - A Nonparametric Analysis

A nonparametric approach is presented to test whether decisions on a probability simplex could be induced by quasi concave preferences. Necessary and sufficient conditions are presented. If the answer is affirmative, the methods developed here allow to reconstruct bounds on indifference curves. Furthermore we can construct quasi concave utility functions in analogy to the utility function constructed in the proof of Afriat’s Theorem. The approach is of interest for decisions under risk, stochastic choice, and ex-ante fairness considerations. The method is particularly suitable for data collected in a laboratory experiment.

Independent, convergent, and divergent decision behaviour for interactive multiple objectives linear programming

An interactive paired comparison approach for multiple objective linear programming (MOLP) problems is developed where the Decision Maker's preferential behaviour is presented by a utility function. The preferential behaviour can be independent, convergent, or divergent; which are mathematically presented by additive, quasi-concave, or quasi-convex utility functions respectively. The approach to find the best alternative consists of four phases: I. An additive utility function. II. A quasi-concave utility function. III. A quasi-convex utility function. IV. A feasible goal utility function. In the first three phases a paired comparison of alternatives are used and in Phase IV a feasible goal method is developed that finds the closest feasible efficient point for a given goal by the decision maker. This method also identifies resources required to make a given goal feasible. The approach can be extended to solve multiple objective integer/nonlinear optimization problems. Some experiments and examples are provided.

Testing the efficiency of two pairwise comparison methods in discrete multiple criteria problems

In so-called pairwise comparison methods, the decision-maker (DM) makes holistic binary comparisons among feasible decision alternatives. The goal is to identify the best (most preferred) alternative using a minimal amount of comparisons. In the worst case, m−1 comparisons are required to identify the best among m alternatives. The efficiency of a pairwise comparison method is measured in terms of the number of saved comparisons with respect to the worst case. We test the efficiency of two pairwise comparison methods. The methods studied are Salminen’s piecewise linear prospect theory (PLP) method and the convex cone method by Korhonen, Wallenius, and Zionts (KWZ). The PLP method models the DMs preferences by a piecewise linear difference function. The KWZ method assumes a quasi-concave utility or value function. These methods are tested using randomly generated data sets with 50, 100, 150, and 200 alternatives and from 2 to 6 criteria. The DM’s preference statements are simulated using linear, quadratic, and Chebycheff value functions and piecewise linear difference functions. The efficiency measurements can be used for evaluating whether it is feasible to apply the methods in a real-life application with given number of alternatives and criteria. The results indicate that pairwise comparison methods are not as efficient as reported in the literature. Both methods lose their efficiency very rapidly with increasing number of criteria.

Quasiconcavity and the kernel of a separable utility

In this paper I prove that a quasiconcave separable utility function defined on an atomless space is concave.

Nash Implementation of a Proportional Solution to International Pollution Control Problems

For a very general class of pollution control models involving strictly quasi-concave utility functions over consumption and environmental quality and strictly convex emission abatement cost functions, a proportional cost sharing mechanism is presented inspired by the ratio equilibrium introduced by7. It is shown that the proportional solution yields a cost efficient allocation of abatement efforts and that the resulting utility imputation always lies in the stand alone core of the cost sharing game. In order to decentralize the proportional cost sharing equilibrium, a financial compensation mechanism implementing the proportional solution in complete information Nash equilibrium is presented.

On the differentiability of the consumer demand function

For strictly quasiconcave utility functions, demand is shown to be differentiable almost everywhere if 1. (i) utility is separable 2. (ii) utility is concave and quasilinear 3. (iii) utility is differentiable and marginal utilities are pointwise Lipschitzian.

How to Select a Dual Nash Equilibrium

The paper analyzes situations, generalizing the duopoly problem, where two identical players are allowed with two control variables each, all of them linked through two non-strategic private constraints. Four dual equilibria are then obtained when each agent selects one leading variable to optimize and adjusts the other, and these equilibria are compared in a meta-game. For a simplified class of continuous games with linear constraints, it is shown that one symmetric dual equilibrium dominates the others and is the only perfect equilibrium of the metagame. The latter result holds locally for all quasi-concave utility functions and globally for all homogeneous ones, always keeping linear constraints. However, it is no longer valid in discrete games where the implicit constraints are not linear. Journal of Economic Literature Classification Numbers: C72, D43.

Interactive Approaches for Discrete Alternative Multiple Criteria Decision Making with Monotone Utility Functions

In this paper we develop interactive approaches for the discrete alternative multiple criteria decision making problem. We develop an algorithm that finds the most preferred alternative of a decision maker (DM) assuming only that the DM has a monotonic utility function. The algorithm divides the criteria space into a number of smaller subspaces and then uses the ideal points of these subspaces to eliminate alternatives. We also develop a more efficient version of the algorithm for the more restrictive case of a monotonic quasiconcave utility function. We present favorable computational results in terms of the required number of pairwise comparisons for both versions of the algorithm. We then develop a general algorithm that first identifies the type of the DM's utility function and then employs the approach that is compatible with the identified utility function type. We also present computational results for the general algorithm.

An Interactive Approach for Choosing the Best of a Set of Alternatives

In this paper we address the problem of choosing the most preferred alternative among a large number of alternatives where each alternative is defined by multiple criteria. We assume that the decision maker has a quasiconcave utility function. We develop an exact approach that combines the ideas that have appeared in the literature regarding the use of different types of dummy alternatives in conjunction with real alternatives. Our experimental results indicate that the new approach is comparable to leading existing approaches.

Testing strictly concave rationality

We prove that the Strong Axiom of Revealed Preference tests the existence of a strictly quasiconcave (or strictly concave, strictly monotone) utility function generating finitely many demand observations. This sharpens earlier results of Afriat, Diewert, and Varian that tested (“nonparametrically”) existence of a piecewise linear utility function. For finite data sets, one implication of our result is that even some weak types of rational behavior—maximization of pseudotransitive or semitransitive preferences—are observationally equivalent to maximization of continuous, strictly concave, and strictly monotone utility functions. And for infinitely many observations, our result is the basis of several new rationality theorems.

Preference Structure Representation Using Convex Cones in Multicriteria Integer Programming

A new efficient system of representing the decision-maker's preference structure in solving multicriteria integer programming problems is developed. The problem is solved by an interactive branch-and-bound method that employs the procedure of Zionts and Wallenius (Zionts, S., J. Wallenius. 1983. An interactive multiple objective linear programming method for a class of underlying nonlinear utility functions. Management Sci. 29(5).) for multicriteria linear programming. The decision-maker's underlying utility function is assumed to be pseudoconcave, and his pairwise comparisons of decision alternatives are used to determine his preference structure in terms of certain convex cones in the objective function space and constraints on the weights on the objectives in the weight space. The two forms of preference structure representation are interrelated, and their underlying theory is developed. The primary objective of a representation scheme is exactness, and, in this respect, it is shown that the constraints on the weights are not adequate for representing nonlinear utility functions. On the other hand, the convex cones exactly represent any quasiconcave utility function and clearly avoid the approximations and inaccuracies in other utility assessment systems. Accordingly, an efficient ordered representation using convex cones is developed. An algorithmic framework for multicriteria integer programming that integrates the representation using convex cones with the branch-and-bound solution procedure is developed. Computational experience with bicriteria problems having up to 80 variables and 40 constraints is presented.