Concave Utility Research Articles

On the Mitra–Wan forest management problem in continuous time

The paper provides a continuous-time version of the discrete-time Mitra–Wan model of optimal forest management, where trees are harvested to maximize the utility of timber flow over an infinite time horizon. The available trees and the other parameters of the problem vary continuously with respect to both time and age of the trees, so that the system is ruled by a partial differential equation. The behavior of optimal or maximal couples is classified in the cases of linear, concave or strictly concave utility, and positive or null discount rate. All sets of data share the common feature that optimal controls need to be more general than functions, i.e. positive measures. Formulas are provided for golden-rule configurations (uniform density functions with cutting at the ages that solve a Faustmann problem) and for Faustmann policies, and their optimality/maximality is discussed. The results do not always confirm the corresponding ones in discrete time.

Normalized Nash Equilibrium for Power Allocation in Cognitive Radio Networks

We consider a cognitive radio system consisting of several secondary networks and primary user-terminals (primary-UTs). In a secondary network, a secondary-base station (secondary-BS) transmits to a secondary-user terminal (secondary-UT) with certain power. Secondary-BSs are constrained to allocate transmitting powers such that the total interference at each primary-UT is below a given threshold. We formulate the power allocation problem as a concave noncooperative game with secondary-BSs as players and multiple primary-UTs enforcing coupled constraints. The equilibrium selection is based on the concept of normalized Nash equilibrium (NNE). When the interference at a secondary-UT from adjacent secondary-BSs is negligible, the NNE is shown to be unique for any strictly concave utility. The NNE is also shown to be the solution of a concave potential game. We propose a distributed algorithm, which converges to the unique NNE. When the interference at a secondary-UT from adjacent secondary-BSs is not negligible, an NNE may not be unique and the computation of the NNE has exponential complexity. To avoid these drawbacks, we introduce the concept of weakly normalized Nash equilibrium (WNNE), which keeps the most of NNEs’ interesting properties but, in contrast to the latter, the WNNE can be determined with low complexity. We show the usefulness of the WNNE when the utility function is the Shannon capacity.

Self‐Insurance, Self‐Protection, and Saving: On Consumption Smoothing and Risk Management

This article studies the effect of risk preferences on self‐insurance and self‐protection in a two‐period expected utility framework. Here the investment to reduce risk precedes its effect. In contrast to single‐period models, self‐insurance and self‐protection react similarly when the agent's utility function becomes more concave. Effort is increased if and only if current consumption is sufficiently large. However, if we introduce endogenous saving, an agent with more concave utility always selects more self‐insurance, but will select more self‐protection if and only if the probability of loss is small enough. These latter results concur with those in standard monoperiodic models with no saving.

Construction of Discrete Time Shadow Price

In the paper expected utility from consumption over finite time horizon for discrete time markets with bid and ask prices and strictly concave utility function is considered. The notion of weak shadow price, i.e. an illiquid price, depending on the portfolio, under which the model without bid and ask price is equivalent to the model with bid and ask price is introduced. Existence and the form of weak shadow price is shown. Using weak shadow price usual (called in the paper strong) shadow price is then constructed.

On the risk-averse procurement strategy under unreliable supply

This study investigates an effective procurement/inventory strategy for a risk-averse retailer facing unreliable supply and stochastic demand. By using an increasing and concave utility function to describe risk aversion, we construct a basic newsvendor (single-period) model and its multi-period extension. Both models are found to have unique solutions, as the optimized expected utility is strictly concave in initial inventory level. As a result, there is a unique optimal order quantity for the effective control of supply risk. For the single-period model, the optimal order quantity is derived in its analytical form. We then show by numerical analysis that the value of the optimized expected utility is a function of the initial inventory level when the retailer is risk-averse, becomes less sensitive to initial inventory level when the degree of risk aversion decreases, and is insensitive for the risk-neutral case. This finding suggests that in our setting the inventory holding matters only when the retailer is risk-averse. For the multi-period model, we propose a solution procedure using backward induction since a direct extension of the single-period solution is impossible. We also conduct a sensitivity analysis of demand and supply with the aim of giving some managerial suggestions for demand risk control and supplier selection.

A Complementary Pivot Algorithm for Market Equilibrium under Separable, Piecewise-Linear Concave Utilities

Using Lemke's scheme, we give a complementary pivot algorithm for computing an equilibrium for Arrow--Debreu markets under separable, piecewise-linear concave (SPLC) utilities. Despite the polynomial parity argument on directed graphs (PPAD) completeness of this case, experiments indicate that our algorithm is practical---on randomly generated instances, the number of iterations it needs is linear in the total number of segments (i.e., pieces) in all the utility functions specified in the input. Our paper settles a number of open problems: (1) Eaves (1976) gave an LCP formulation and a Lemke-type algorithm for the linear Arrow--Debreu model. We generalize both to the SPLC case, hence settling the relevant part of his open problem. (2) Our path following algorithm for SPLC markets, together with a result of Todd (1976), gives a direct proof of membership of such markets in PPAD and settles a question of Vazirani and Yannakakis (2011). (3) We settle a question of Devanur and Kannan (2008) of obtaining a “sy...

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].

A Decision-based Perspective on Assessing System Robustness

Although robustness often is cited as a desirable property of a system, a widely-accepted approach for quantifying it remains elusive. Informally, a system is robust if it avoids the downside consequences associated with uncertainty or perturbations. Prior approaches to quantifying robustness rely on the standard deviation of system responses or the expected ratio of performance under perturbation to nominal performance. Because the principal usefulness of any system metric is to inform decisions about the design of that system, we turn to rigorous decision theory for guidance on how to deal with the concept of robustness. We find that existing proposals for robustness quantification are inconsistent with accepted theory for rational decision making. Rather than propose an alternative quantification scheme, we argue that fundamentally there is no need to quantify robustness as an independent figure of merit. Instead, systems engineers can make decisions that favor system robustness using established methods based on expected utility theory. A key to this is formulating the decision problem in terms of fundamental objectives rather than means objectives. One's preference to favor robustness is associated with being risk averse over the fundamental objectives and is captured by using a concave utility function.

Savage in the Market

We develop a behavioral axiomatic characterization of subjective expected utility (SEU) under risk aversion. Given is an individual agent's behavior in the market: assume a finite collection of asset purchases with corresponding prices. We show that such behavior satisfies a “revealed preference axiom” if and only if there exists a SEU model (a subjective probability over states and a concave utility function over money) that accounts for the given asset purchases.

The Stochastic Cash Balance Problem with Fixed Costs: The Risk-averse Case

Abstract This paper discusses multi-period stochastic cash balance problem with fixed costs when the decision maker is risk averse. By using the consumption model introduced by Chen et al, we characterize the structure of the optimal policy for the stochastic cash balance problem under the general increasing concave utility function and exponential utility function, respectively. We show that the structure of the optimal policy for a decision maker with exponential utility function is almost identical to the structure of the optimal risk-neutral operations policy. Furthermore, we extend the results for the exponential utility function to the ambiguity aversion case.

Optimal Pricing and Order-Up-To S Inventory Policy with Expected Utility of the Present Value Criterion

The article determines pricing and order-up-to level S inventory decisions over an infinite planning horizon from the point of view of a risk-averse decision maker. The demand is assumed to be stochastic but influenced by the selling price which is a decision variable. Shortages are allowed and backordered partially. We calculate the present value of the cash flow over the entire planning horizon and incorporate the notion of risk aversion into the model using a concave utility function. We numerically demonstrate the model and investigate the impact of different model-parameters on the optimal decisions. It is observed that the optimal selling price for a risk-averse decision maker is not less than the optimal selling price of a risk-neutral decision maker while the optimal order level for the risk-averse decision maker is always less than that of the risk-neutral decision maker.

Multiattribute Procurement Auctions: Efficiency and Social Welfare in Theory and Practice

One of the standard assumptions in auction theory is that preferences can be represented with quasilinear utility. This assumption is of particular significance in reverse auctions, which are used in procurement. This paper presents an analysis of quasilinear utilities and their implications. Building on observations of scholars in economics and decision sciences, who note that quasilinear preferences may be rare in practice, the paper shows that in the procurement of goods and services, price is often interrelated with costs. When preferences can be represented with convex or concave utilities, the alternatives in which the buyer's surplus is maximized are different from those that maximize social welfare. The result is that reverse auctions may cause a significant loss of social welfare, which is of particular significance for public organizations. The analysis of concave efficient frontiers in the utility space shows that it is possible to determine deals that yield greater social welfare than the winning bids. If the winning seller is willing to share the increase in utility with the buyer who faces a loss, then these alternatives can produce, for both the buyer and the seller, utility values that are higher than the utility values produced by the winning bid.

Perturbation Analysis for Investment Portfolios Under Partial Information with Expert Opinions

We analyze the Merton portfolio optimization problem when the growth rate is an unobserved Gaussian process whose level is estimated by filtering from observations of the stock price. We use the Kalman filter to track the hidden state(s) of expected returns given the history of asset prices, and then use this filter as input to a portfolio problem with an objective to maximize expected terminal utility. Our results apply for general concave utility functions. We incorporate time-scale separation in the fluctuations of the returns process, and utilize singular and regular perturbation analysis on the associated partial-information HJB equation, which leads to an intuitive interpretation of the additional risk caused by uncertainty in expected returns. The results are an extension of the partially informed investment strategies obtained by the Black--Litterman model, wherein investors' views on upcoming performance are incorporated into the optimization along with any degree of uncertainty that the investor...

Voter Preferences, Polarization, and Electoral Policies

In most variants of the Hotelling-Downs model of election, it is assumed that voters have concave utility functions. This assumption is arguably justified in issues such as economic policies, but convex utilities are perhaps more appropriate in others, such as moral or religious issues. In this paper, we analyze the implications of convex utility functions in a two-candidate probabilistic voting model with a polarized voter distribution. We show that the equilibrium policies diverge if and only if voters' utility function is sufficiently convex. If two or more issues are involved, policies converge in “concave issues” and diverge in “convex issues.” (JEL D72)

Dynamic theory of preferences: Habit formation and taste for variety

We analyze individual preferences over infinite horizon consumption choices. Our axioms provide the foundation for a recursive representation of the utility function that contains as particular cases the classical Koopmans representation (Koopmans (1960)) as well as the habit formation specification.We examine some of the consequences of our axiomatization by considering a standard consumer choice problem, and show that typically in the space of concave utility functions satisfying our axioms the consumer displays a taste for variety. The latter means that such a consumer selects optimally time variant consumption programs for any given time invariant sequence of commodities’ relative prices and for all possible sequences of market discount factors. In contrast, if a concave utility function satisfies Koopmans’ axioms the consumer does not display a taste for variety.

Dopamine Reward Prediction Error Responses Reflect Marginal Utility

SummaryBackgroundOptimal choices require an accurate neuronal representation of economic value. In economics, utility functions are mathematical representations of subjective value that can be constructed from choices under risk. Utility usually exhibits a nonlinear relationship to physical reward value that corresponds to risk attitudes and reflects the increasing or decreasing marginal utility obtained with each additional unit of reward. Accordingly, neuronal reward responses coding utility should robustly reflect this nonlinearity.ResultsIn two monkeys, we measured utility as a function of physical reward value from meaningful choices under risk (that adhered to first- and second-order stochastic dominance). The resulting nonlinear utility functions predicted the certainty equivalents for new gambles, indicating that the functions’ shapes were meaningful. The monkeys were risk seeking (convex utility function) for low reward and risk avoiding (concave utility function) with higher amounts. Critically, the dopamine prediction error responses at the time of reward itself reflected the nonlinear utility functions measured at the time of choices. In particular, the reward response magnitude depended on the first derivative of the utility function and thus reflected the marginal utility. Furthermore, dopamine responses recorded outside of the task reflected the marginal utility of unpredicted reward. Accordingly, these responses were sufficient to train reinforcement learning models to predict the behaviorally defined expected utility of gambles.ConclusionsThese data suggest a neuronal manifestation of marginal utility in dopamine neurons and indicate a common neuronal basis for fundamental explanatory constructs in animal learning theory (prediction error) and economic decision theory (marginal utility).

Hierarchical Interference Mitigation for Massive MIMO Cellular Networks

We propose a hierarchical interference mitigation scheme for massive MIMO cellular networks. The MIMO precoder at each base station (BS) is partitioned into an inner precoder and an outer precoder. The inner precoder controls the intra-cell interference and is adaptive to local channel state information (CSI) at each BS (CSIT). The outer precoder controls the inter-cell interference and is adaptive to channel statistics. Such hierarchical precoding structure reduces the number of pilot symbols required for CSI estimation in massive MIMO downlink and is robust to the backhaul latency. We study joint optimization of the outer precoders, the user selection, and the power allocation to maximize a general concave utility which has no closed-form expression. We first apply random matrix theory to obtain an approximated problem with closed-form objective. We show that the solution of the approximated problem is asymptotically optimal with respect to the original problem as the number of antennas per BS grows large. Then using the hidden convexity of the problem, we propose an iterative algorithm to find the optimal solution for the approximated problem. We also obtain a low complexity algorithm with provable convergence. Simulations show that the proposed design has significant gain over various state-of-the-art baselines.

Does it pay to be ethical? Evidence from the FTSE4Good

The empirical mean–variance evidence comparing the performance of Socially Responsible Investments (SRI) and conventional investments suggests that there is no significant difference between the two. This paper re-examines the problem in the context of Marginal Conditional Stochastic Dominance (MCSD), which can accommodate any return distribution or concave utility function. Our results provide strong evidence that there is a financial price to be paid for socially responsible investing. Indices composed of socially responsible firms are MCSD dominated by trademarked indices composed of conventional firms as well as by indices carefully matched by size and industry with the firms in the SRI indices. Zero cost portfolios created by shorting the SRI index and using the proceeds to invest in the conventional index generate higher average returns, lower variance and higher skewness than either of the two indices standing alone. They also MCSD dominate the SRI and conventional indices standing alone.

The Robust Stability of Every Equilibrium in Economic Models of Exchange Even Under Relaxed Standard Conditions

In an economic model of exchange of goods, the structure can be specified by utility functions. Under utility conditions identified here even more broadly than usual, except for concavity in place of quasi-concavity, every equilibrium will simultaneously be stable with respect to shifts in the associated holdings of the agents and with respect to the Walrasian tâtonnement process of price adjustment. This fact, seemingly contrary to widespread belief, is revealed by paying close attention not only to prices but also to the proximal status of initial holdings.The conditions on the concave utility functions are standard for stability investigations, in that they invoke properties coming from second derivatives, but significantly relaxed in not forcing all goods to be held only in positive amounts. For a given agent, some goods are allowed to have no effect at all on utility, whereas other goods, although insatiably interesting, may anyway end up at zero in equilibrium. Recent advances in variational analysis provide the support needed for working in that context. The stability results also point the way toward further developments in which an equilibrium might evolve in response to exogenous inputs to the agents' holdings, or extractions.

Eliciting Prospect Theory When Consequences Are Measured in Time Units: “Time Is Not Money”

We elicited the prospect theory components (utility, probability weighting, and loss aversion) when consequences are expressed as the time dedicated to a specific task or activity. A similar elicitation was performed for monetary consequences to allow an across-attribute (time/money) comparison of the elicited components (at the individual level). We obtained less concave utility and smaller loss aversion for time than for money. Moreover, while the probability weighting was predominantly inverse S-shaped for both attributes, it was less sensitive to probabilities and more elevated for time than for money. This finding implies more optimism for gains and more pessimism for losses. This paper was accepted by Peter Wakker, decision analysis.