In this paper, we consider a multistage expected utility maximization problem where the decision maker's utility function at each stage depends on historical data and the information on the true utility function is incomplete. To mitigate adverse impact arising from ambiguity of the true utility, we propose a maximin robust model where the optimal policy is based on the worst-case sequence of utility functions from an ambiguity set constructed with partially available information about the decision maker's preferences. We then show that the multistage maximin problem is time consistent when the utility functions are state-dependent and demonstrate with a counter example that the time consistency may not be retained when the utility functions are state-independent. With the time consistency, we show the maximin problem can be solved by a recursive formula whereby a one-stage maximin problem is solved at each stage beginning from the last stage. Moreover, we propose two approaches to construct the ambiguity set: a pairwise comparison approach and a zeta-ball approach where a ball of utility functions centered at a nominal utility function under zeta-metric is considered. To overcome the difficulty arising from solving the infinite dimensional optimization problem in computation of the worst-case expected utility value, we propose piecewise linear approximation of the utility functions and derive error bound for the approximation under moderate conditions. Finally, we use the SDDP method and the nested Benders' decomposition method to solve the multistage state-dependent preference robust problem and the scenario tree method to solve the state-independent problem, and carry out comparative analysis on the efficiency of the computational schemes as well as out-of-sample performances of the state-dependent and state-independent models.
Real-world engineering systems are typically compared and contrasted using multiple metrics. For practical machine learning systems, performance tuning is often more nuanced than minimizing a single expected loss objective, and it may be more realistically discussed as a multi-objective optimization problem. We propose a novel generative model for scalar-valued utility functions to capture human preferences in a multi-objective optimization setting. We also outline an interactive active learning system that sequentially refines the understanding of stakeholders ideal utility functions using bi…
In this paper, we consider a multi-attribute decision making problem where the decision maker's (DM's) objective is to maximize the expected utility of outcomes but the true utility function which captures the DM's risk preference is ambiguous. We propose a maximin multi-attribute utility preference robust optimization (UPRO) model where the optimal decision is based on the worst-case utility function in an ambiguity set of plausible utility functions constructed using partially available information such as the DM's specific preferences between some lotteries. Specifically, we consider a UPRO…
We study the learning problem of revealed preference in a stochastic setting: a learner observes the utility-maximizing actions of a set of agents whose utility follows some unknown distribution, and the learner aims to infer the distribution through the observations of actions. The problem can be viewed as a single-constraint special case of the inverse linear optimization problem. Existing works all assume that all the agents share one common utility which can easily be violated under practical contexts. In this paper, we consider two settings for the underlying utility distribution: a Gauss…
Continuous-Time Heterogeneous Agent Models with Recursive Utility and Preference for Late Resolution
Université de Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions（巴黎西岱大学与索邦大学，法国国家科学研究中心，雅克-路易·利翁斯实验室）；School of Mathematics and Physics, China University of Geosciences (Wuhan)（中国地质大学（武汉）数学与物理学院）
Stochastic dominance is a preference relation of uncertain prospect defined over a class of utility functions. While this utility class represents basic properties of risk aversion, it includes some extreme utility functions rarely characterizing a rational decision maker's preference. In this paper we introduce reference-based almost stochastic dominance (RSD) rules which well balance the general representation of risk aversion and the individualization of the decision maker's risk preference. The key idea is that, in the general utility class, we construct a neighborhood of the decision make…
Maximum Utility Split Method for Utility Preference Elicitation
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港中文大学系统工程与工程管理学系）
Ben-Tal and Teboulle \cite{BTT86} introduce the concept of optimized certainty equivalent (OCE) of an uncertain outcome as the maximum present value of a combination of the cash to be taken out from the uncertain income at present and the expected utility value of the remaining uncertain income. In this paper, we consider two variations of the OCE. First, we introduce a modified OCE by maximizing the combination of the utility of the cash and the expected utility of the remaining uncertain income so that the combined quantity is in a unified utility value. Second, we consider a situation where…
For incomplete preference relations that are represented by multiple priors and/or multiple -- possibly multivariate -- utility functions, we define a certainty equivalent as well as the utility buy and sell prices and indifference price bounds as set-valued functions of the claim. Furthermore, we motivate and introduce the notion of a weak and a strong certainty equivalent. We will show that our definitions contain as special cases some definitions found in the literature so far on complete or special incomplete preferences. We prove monotonicity and convexity properties of utility buy and se…
We characterize optimal consumption policies in a recursive intertemporal utility framework with local substitution. We establish existence and uniqueness and a version of the Kuhn-Tucker theorem characterizing the optimal consumption plan. An explicit solution is provided for the case when the felicity function is of the Epstein-Zin's type.
We consider black-box global optimization of time-consuming-to-evaluate functions on behalf of a decision-maker (DM) whose preferences must be learned. Each feasible design is associated with a time-consuming-to-evaluate vector of attributes and each vector of attributes is assigned a utility by the DM's utility function, which may be learned approximately using preferences expressed over pairs of attribute vectors. Past work has used a point estimate of this utility function as if it were error-free within single-objective optimization. However, utility estimation errors may yield a poor sugg…
This paper analyzes popular time-nonseparable utility functions that describe "habit formation" consumer preferences comparing current consumption with the time averaged past consumption of the same individual and "catching up with the Joneses" (CuJ) models comparing individual consumption with a cross-sectional average consumption level. Few of these models give reasonable optimum consumption time series. We introduce theoretically justified utility specifications leading to a plausible consumption behavior to show that habit formation preferences must be described by a power CRRA utility fun…
We perform a stability analysis for the utility maximization problem in a general semimartingale model where both liquid and illiquid assets (random endowments) are present. Small misspecifications of preferences (as modeled via expected utility), as well as views of the world or the market model (as modeled via subjective probabilities) are considered. Simple sufficient conditions are given for the problem to be well-posed, in the sense the optimal wealth and the marginal utility-based prices are continuous functionals of preferences and probabilistic views.
We consider Bayesian optimization of expensive-to-evaluate experiments that generate vector-valued outcomes over which a decision-maker (DM) has preferences. These preferences are encoded by a utility function that is not known in closed form but can be estimated by asking the DM to express preferences over pairs of outcome vectors. To address this problem, we develop Bayesian optimization with preference exploration, a novel framework that alternates between interactive real-time preference learning with the DM via pairwise comparisons between outcomes, and Bayesian optimization with a learne…
Cash-invariant hull representation of divergence preferences
Bayes Business School, City St George’s, University of London（伦敦大学城市圣乔治学院贝斯商学院）；London School of Economics and Political Science（伦敦政治经济学院）；ETH Zurich（苏黎世联邦理工学院）
Income- and price-elasticity of demand quantify the responsiveness of markets to changes in income, and in prices, respectively. Under the assumptions of utility maximization and preference-independence (additive preferences), mathematical relationships between income-elasticity values and uncompensated own and cross price-elasticity of demand are here derived for bundle of goods, using the differential approach to demand analysis. Key parameters are: the elasticity of the marginal utility of income, and the average budget-share. The proposed method can be applied to forecast the direct and in…
We study the problem of online learning (OL) from revealed preferences: a learner wishes to learn a non-strategic agent's private utility function through observing the agent's utility-maximizing actions in a changing environment. We adopt an online inverse optimization setup, where the learner observes a stream of agent's actions in an online fashion and the learning performance is measured by regret associated with a loss function. We first characterize a special but broad class of agent's utility functions, then utilize this structure in designing a new convex loss function. We establish th…
Traditional network models encapsulate travel behavior among all origin-destination pairs based on a simplified and generic utility function. Typically, the utility function consists of travel time solely and its coefficients are equated to estimates obtained from stated preference data. While this modeling strategy is reasonable, the inherent sampling bias in individual-level data may be further amplified over network flow aggregation, leading to inaccurate flow estimates. This data must be collected from surveys or travel diaries, which may be labor intensive, costly and limited to a small t…
Decision maker's preferences are often captured by some choice functions which are used to rank prospects. In this paper, we consider ambiguity in choice functions over a multi-attribute prospect space. Our main result is a robust preference model where the optimal decision is based on the worst-case choice function from an ambiguity set constructed through preference elicitation with pairwise comparisons of prospects. Differing from existing works in the area, our focus is on quasi-concave choice functions rather than concave functions and this enables us to cover a wide range of utility/risk…
