Preference Robust Modified Optimized Certainty Equivalent

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with $\sigma$ algebra $\mathcal{F}$ and probability measure $\mathbb{P}$ and $\xi:(\Omega,\mathcal{F},\mathbb{P})\to{\rm I\!R}$ be a random variable representing future income of a decision maker (DM). The optimized certainty equivalent of $\xi$ is defined as

where $u:{\rm I\!R}\to{\rm I\!R}$ is the decision maker’s utility function and $P:=\mathbb{P}\circ\xi^{-1}$ is the probability measure on ${\rm I\!R}$ induced by $\xi$. The concept is first introduced by Ben-Tal and Teboulle [6] and closely related to other notions of certainty equivalent and risk measures, see [7] for a comprehensive discussion. The economic interpretation of this notion is that the decision maker may need to consume part of $\xi$ at present, denoted by $x$, the sure present value of $\xi$ under the consumption plan becomes $x+{\mathbb{E}}_{P}[u(\xi-x)]$, and the optimized certainty equivalent $S_{u}(\xi)$ gives rise to the optimal allocation of the consumption which maximizes the sure present value of $\xi$. As a measure, it enjoys a number of nice properties including constancy ($S_{u}(C)=C$ for constant $C$), risk aversion ($S_{u}(\xi)\leq{\mathbb{E}}_{P}[\xi]$) and translation invariance ($S_{u}(\xi+C)=S_{u}(\xi)+C$). In particular, if $u$ is a normalized exponential utility function, it coincides with the classical certainty equivalent $u^{-1}({\mathbb{E}}_{P}[u(\xi)])$ in the literature of economics. Moreover, if $u(t)=-\frac{1}{\alpha}(-t)_{+}$ where $\alpha\in(0,1)$ and $(t)_{+}=\max\{t,0\}$ for $t\in{\rm I\!R}$, then the OCE effectively recovers the conditional value-at-risk (CVaR):

The last equality is Rockafellar and Uryasev’s formulation of CVaR, see [7, 41]. Since CVaR is average of quantile, it is also known as average value-at-risk (AVaR), tail value-at-risk (TVaR) and expected shortfall, see [38, 37].

In this paper, we revisit the subject OCE from two perspectives. One is to consider a modified version of the optimized certainty equivalent

The modification is motivated to align the sure present value of $\xi$ to the expected utility theory [35] by considering the utility of present consumption $u(x)$ instead of the monetary value $x$. Recall that in Von Neumann-Morgenstern expected utility theory [35], the utility function is used to represent the decision maker’s preference relation over a prospect space including both random and deterministic prospects, and such representation is unique up to positive linear transformation. This means we can use both $u$ and $100u$ to represent the DM’s preference. However, the two utility functions would lead to completely different optimal values and optimal solutions in the OCE model. In contrast, the optimal solution is not affected in the MOCE model, and the optimal value is only affected by the same scale of the utility function. In our view, this kind of “invariance” of the optimal allocation $x^{*}$ and “scalability” w.r.t. the utility function is important because the optimal decision on the allocation/consumption $x$ should be determined by the DM’s risk preference irrespective of its equivalent representations.

The modified OCE model may be regarded as a special case of the well known consumption/investment models in economics [35, 19, 12] where $u(x)$ is the utility of the current consumption/investment whereas ${\mathbb{E}}[u(\xi-x)]$ is the expected utility of the remaining asset to be consumed/invested in future. In these models, the utility functions for the current consumption and future consumption are identical. It is also possible to use different utility functions when the consumption at present is used for a new investment or production.

The other is to consider a situation where the decision maker’s utility function $u(\cdot)$ is ambiguous, in other words, there is incomplete information to identify a utility function $u$ which captures the decision maker’s true utility preference. Consequently we propose to consider a robust optimized certainty equivalent measure

where ${\cal U}$ is a set of plausible utility functions consistent with the observed utility preferences of the decision maker. The definition is in line with the philosophy of robust optimization where the optimized certainty equivalent value is based on the worst case utility function from set ${\cal U}$ to mitigate the risk arising from potential inaccurate use or misuse of the utility function. By convention, we call ${\cal U}$ the ambiguity set. In the case that ${\cal U}$ is a singleton, RMOCE reduces to MOCE. Note that RMOCE should be differentiated from the distributionally robust formulation of OCE by Wisemann et al. [50] where the focus is on the ambiguity of $P$. In decision analysis, $P$ is known as a decision maker’s belief of the state of nature whereas $u$ characterizes the decision maker’s taste for risk/utility. The RMOCE model concerns the ambiguity of decision maker’s taste rather than belief.

Ambiguity of utility preference is a well discussed topic in behavioural economics. For instances, Thurstone [44] regards such ambiguity as a lack of accurate description of human behaviour. Karmarkar [27] and Weber [49] ascribe the ambiguity to cognitive difficulty and incomplete information. The ambiguity may also arise in the decision making problems which involve several stakeholders who fail to reach a consensus. Parametric and non-parametric approaches have subsequently been proposed to assess the true utility function, including discrete choice models (Train [45]), standard and paired gambling approaches for preference comparisons and certainty equivalence (Farquhar [15]), we refer readers to Hu et al. [23] for an excellent overview on this.

In decision making under uncertainty, a decision maker may choose the worst case utility function among a set of plausible utility functions representing his/her risk preference to mitigate the overall risk. This kind of research may be traced back to Maccheroni [33]. Cerreia-Vioglio et al. [10] seem to be the first to investigate ambiguity of decision maker’s utility function in the certainty equivalent model $u^{-1}({\mathbb{E}}[u(\xi)])$ by considering the worst-case certainty equivalent from a given set of utility functions in their cautious expected utility model. They show that the DM’s risk preference can be represented by a worst-case certainty equivalent if and only if they are given by a binary relation satisfying the weak order, continuity, weak monotonicity and negative certainty independence (NCI) (NCI states that if a sure outcome is not enough to compensate the DM for a risky prospect, then its mixture with another lottery which reduces the certainty appeal, will not be more attractive than the same mixture of the risky prospect and the lottery).

Armbruster and Delage [3] give a comprehensive treatment of the topic from minimax preference robust optimization (PRO) perspective. Specifically, they propose to use available information of the decision maker’s utility preference such as preferring certain lotteries over other lotteries and being risk averse, $S$-shaped or prudent to construct an ambiguity set of plausible utility functions and then base the optimal decision on the worst case utility function from the ambiguity set. Hu and Mehrotra [24] consider a probabilistic representation of the class of increasing concave utility functions by confining them to a compact interval and normalizing them with range $[0,1]$. In doing so, they propose a moment-type framework for constructing the ambiguity set of the decision maker’s utility preference which covers a number of important approaches such as the certainty equivalent and pairwise comparison. Hu and Stepanyan [25] propose a so-called reference-based almost stochastic dominance method for constructing a set of utility functions near a reference utility which satisfies certain stochastic dominance relationship and use the set to characterize the decision maker’s preference. Over the past few years, the research on PRO has received increasing attentions in the communities of stochastic/robust optimization and risk management, see for instances [22, 21, 14, 52, 31, 32] and references therein.

In both (MOCE) and (RMOCE) models, the true probability distribution $P$ is assumed to be known. In the data driven problems, the true $P$ is unknown but it is possible to use empirical data to construct an approximation of $P$. Unfortunately, such data may be contaminated and consequently we may be concerned by the quality of the MOCE values calculated as such. This kind of issue is well studied in robust statistics [26] and can be traced down to earlier work of Hample [20]. Cont et al. [13] first study the quality of the plug-in estimators of law invariant risk measures using Hampel’s classical concept of qualitative robustness [20], that is, the plug-in estimator of a risk functional is said to be qualitatively robust if it is insensitive to the variation of sampling data. According to Hampel’s theorem, Cont et al. [13] demonstrate that the qualitative robustness of a plug-in estimator is equivalent to the weak continuity of the risk functional and that value at risk (VaR) is qualitatively robust whereas conditional value at risk (CVaR) is not. Krätschmer et al. [30] argue that the use of Hampel’s classical concept of qualitative robustness may be problematic because it requires the risk measure essentially to be insensitive with respect to the tail behaviour of the random variable and the recent financial crisis shows that a faulty estimate of tail behaviour can lead to a drastic underestimation of the risk. Consequently, they propose a refined notion of qualitative robustness that applies also to tail-dependent statistical functionals and that allows one to compare statistical functionals in regards to their degree of robustness. The new concept captures the trade-off between robustness and sensitivity and can be quantified by an index of qualitative robustness. Guo and Xu [17] take a step forward by deriving quantitative statistical robustness of PRO models. Xu and Zhang [51] extend the analysis to distributionally robust optimization models.

In this paper, we consider a situation where the decision maker has a nominal utility function but is short of complete information as to whether it is the true. Consequently we propose to use the Kantorovich ball centered at the nominal utility function as the ambiguity set. We begin with piecewise linear utility (PLU) functions defined over a convex and closed interval of ${\rm I\!R}$ and show that the inner minimization problem in the definition of RMOCE can be reformulated as a linear program when $\xi$ has a finite discrete distribution. We then propose an iterative algorithm to compute the RMOCE by solving the inner minimization problem and outer maximization problem alternatively.

To extend the scope of the proposed computational method, we extend the discussion to the cases that the utility functions are not necessarily piecewise linear and the domain of the utility function is unbounded. We derive error bounds arising from using PLU-based RMOCE to approximate the general RMOCE. Since our numerical scheme for computing the RMOCE is based on the samples of $\xi$, we study statistical robustness of the sample-based RMOCE to address the case that the sample data of $\xi$ are potentially contaminated. Finally we carry out some numerical tests on the proposed computational schemes for concave utility functions.

The rest of the paper are organized as follows. Section 2 discusses the basic properties of MOCE and RMOCE. Section 3 presents numerical schemes for computing the RMOCE when the utility functions in the ambiguity set are piecewise linear. Section 4 details approximation of the ambiguity set of general utility functions by the ambiguity set of piecewise linear utility functions and its effect on RMOCE. Section 5 discusses the RMOCE model with utility function having unbounded domain and streamlines the potential extensions of the MOCE model to multi-attribute decision making. Section 6 discusses statistical robustness of RMOCE when it is calculated with contaminated data. Section 7 reports numerical results and finally Section 8 concludes with a brief summary of the main contributions of the paper.

We begin by discussing the well-definedness of MOCE and RMOCE. Let $L_{p}(\Omega,{\cal F},\mathbb{P})$ denote the space of random variables mapping from $(\Omega,{\cal F},\mathbb{P})$ to ${\rm I\!R}$ with finite $p$-th order moments and $\xi\in L_{p}(\Omega,{\cal F},\mathbb{P})$. Let $\mathscr{U}:{\rm I\!R}\to{\rm I\!R}$ be the set of nondecreasing concave utility functions. Throughout this paper, we make a blanket assumption to ensure the well-definedness of the expected utility in the definitions of MOCE and RMOCE.

The condition stipulates the interaction between the tails distribution of $\xi$ and tails of the utility function. We refer readers to Guo and Xu [18] for more detailed discussions on this. To facilitate the forthcoming discussions, we let $\mathscr{P}(\Xi)$ denote the set of probability measures on $\Xi\subset{\rm I\!R}$, and for each fixed $x$, define

for $i=1,2$. Let ${\cal C}_{\Xi}^{\phi_{i}}$ denote the class of continuous functions $h:\Xi\to{\rm I\!R}$ such that $|h(t)|\leq C(\phi_{i}(t)+1)$ for all $t\in\Xi$. The $\phi_{i}$-topology, denoted by $\tau_{\phi_{i}}$, is the coarsest topology on ${\cal M}^{\phi_{i}}$ for which the mapping $g_{h}:=\int_{\Xi}h(z)P(dz),\;h\in{\cal C}_{\rm I\!R}^{\phi_{i}}$ is continuous. A sequence $\{P_{N}\}\subset{\cal M}^{\phi_{i}}$ is said to converge $\phi_{i}$-weakly to $P\in{\cal M}^{\phi_{i}}$ written ${P_{N}}\xrightarrow[]{\phi_{i}}P$ if it converges w.r.t. $\tau_{\phi_{i}}$. Note that in the case that when the support set of $\xi$ is a compact set in ${\rm I\!R}$, then the $\phi_{i}$-topology reduces to ordinary topology of weak convergence.

Our first technical result is on the attainability of the optimum in the definition of MOCE.

The second part of the claim follows directly from the first part in that the interval $[\xi_{\min}^{N}/2,\\ \xi_{\max}^{N}/2]$ converges to a single point $\hat{\xi}/2$ and $\tau_{\phi_{1}}$-convergence coincides with the weak convergence because of the restriction of the range of $\xi$ to a compact subset of ${\rm I\!R}$.  

Ben-Tal and Teboulle [7] derive a similar result to the first part of the proposition for the optimized certainty equivalent and demonstrate that $S_{u}(\xi)\in[\xi_{\min},\xi_{\max}]$ under the conditions that $u$ is concave and $1\in\partial u(0)$ rather than strictly concave. Strict concavity is needed to ensure the optimum in (2.5) to be achieved in $[\xi_{\min}/2,\xi_{\max}/2]$. We can find a counter example otherwise, see 2.3.

Like the optimized certainty equivalent, the newly defined modified optimized certainty equivalent enjoys a number of properties as stated in the next proposition.

Part (iv). “$\Longleftarrow$”. By the definition of certainty equivalent, $C_{u}(\xi_{1})\geq C_{u}(\xi_{2})$ implies ${\mathbb{E}}_{P}[u(\xi_{1})]\geq{\mathbb{E}}_{P}[u(\xi_{2})]$ for all concave utility functions. The latter implies $\xi_{1}$ dominates $\xi_{2}$ in second order, which in turn guarantees $\xi_{1}-x$ dominates $\xi_{2}-x$ in second order for any fixed $x\in{\rm I\!R}$. Consequently ${\mathbb{E}}_{P}[u(\xi_{1}-x)]\geq{\mathbb{E}}_{P}[u(\xi_{2}-x)]$ for any $x\in{\rm I\!R}$. Adding both sides of the inequality by $u(x)$ and taking the maximum, we obtain $M_{u}(\xi_{1})\geq M_{u}(\xi_{2})$.

“$\Longrightarrow$”. Let $x_{1},x_{2}$ be the points where the supremum of $M_{u}(\xi_{1})$ and $M_{u}(\xi_{2})$ are attained. Then

which yields ${\mathbb{E}}_{P}[u(\xi_{1}-x_{1})]\geq{\mathbb{E}}_{P}[u(\xi_{2}-x_{1})]$. The latter implies ${\mathbb{E}}_{P}[u(\xi_{1})]\geq{\mathbb{E}}_{P}[u(\xi_{2})]$.

Part (v). First we prove the concavity of $M_{u}$, i.e. for $\lambda\in(0,1)$ and any random variables $\xi_{1}$, $\xi_{2}$,

Since $u$ is concave, the function $f(z,x):=u(x)+u(z-x)$ is joint concave over ${\rm I\!R}\times{\rm I\!R}$. Therefore, for any $x_{1},x_{2}\in{\rm I\!R}$, with $x_{\lambda}:=\lambda x_{1}+(1-\lambda)x_{2}$ and $\xi_{\lambda}:=\lambda\xi_{1}+(1-\lambda)\xi_{2}$, one has

Since $M_{u}(\xi_{\lambda})=M_{u}(\lambda\xi_{1}+(1-\lambda)\xi_{2}))=\sup_{x\in{\rm I\!R}}{\mathbb{E}}_{P}[f(\xi_{\lambda},x)]$, it follows that

Next, we turn to prove the subhomogeneity of $M_{u}$. Let $s(\delta):=\frac{1}{\delta}M_{u}(\delta\xi)$, for $\delta>0$. Then

Let $\delta_{2}>\delta_{1}>0$. By the concavity of $u$,

Since $u(0)\geq 0$, the above inequality implies

Inequality (2.12) also implies

A combination of the two inequalities implies the objective function in (2.11) is non-increasing in $\delta$ and hence $s(\delta)$. By setting $\delta_{1}$ and $\delta_{2}$ to $1$ respectively in the inequality above, we obtain (2.10).  

Next, we discuss how the utility function $u$ may be recovered from a given modified certainty equivalent $M_{u}(\xi)$, which is an important property enjoyed by the OCE. Let

where $0<p<1$ and $z>0$. For a concave utility function $u$, the modified optimized certainty equivalent $M_{u}(\xi_{p})$ can be written as

We now move on to discuss the properties of the robust modified optimized certainty equivalent.

Part (iv). Following a similar argument to the proof of part (iv) of 2.2, we can show that $C_{u}(\xi_{1})\geq C_{u}(\xi_{2})$ implies ${\mathbb{E}}_{P}[u(\xi_{1})]\geq{\mathbb{E}}_{P}[u(\xi_{2})]$ and ${\mathbb{E}}_{P}[u(\xi_{1}-x)]\geq{\mathbb{E}}_{P}[u(\xi_{2}-x)]$ for any fixed $x\in{\rm I\!R}$ and hence

Taking infimum on both sides w.r.t. $u$ over ${\cal U}$ and then supremum w.r.t. $x$, we obtain $R(\xi_{1})\geq R(\xi_{2})$.

Part (v). Let $g_{u}(\delta,x):=\frac{1}{\delta}u(\delta x)+{\mathbb{E}}_{P}\left[\frac{1}{\delta}u(\delta(\xi-x))\right].$ We can show as in the proof of 2.2 (v) that $g_{u}(\cdot,x)$ is non-increasing over ${\rm I\!R}$. This property is preserved after taking the infimum in $u$ over ${\mathcal{U}}$ and then supremum in $x$ over ${\rm I\!R}$.  

Before concluding this section, we remark that it is possible to use a different utility function $v$ for the present consumption $x$, i.e.,

In that case, some of the properties of MOCE may be retained. For example, law invariance, monotonicity, risk aversion, concavity, positive subhomogeneity and second-order stochastic dominance are all satisfied when $v$ enjoys the same property as $u$. 2.3 also holds when $v$ satisfies the same property as $u$. However, the change will have an effect on 2.1, in which case it will be difficult to estimate the interval containing the optimal solution.

Having investigated the properties of MOCE and RMOCE in the previous section, we move on to discuss numerical schemes for computing RMOCE in this section. To this end, we need to have a concrete structure of the ambiguity set. As reviewed in the introduction, various approaches have been proposed for constructing an ambiguity set of utility functions in the literature of preference robust optimization depending on the availability of information. Here we consider a situation where the decision maker has a nominal utility function obtained from empirical data or subjective judgement but lacks of complete information to identify whether it is the true utility function which captures precisely the decision maker’s preference. Consequently we may construct a ball of utility functions centered at the nominal utility function under some appropriate metrics. Here we concentrate on the Kantorovich metric.