Multiplier Preferences Research Articles

Beyond uncertainty aversion

Although much of the theoretical and applied literature involving decision under ambiguity works under the assumption of uncertainty aversion, experimental evidence suggests that it is not a universal behavioural trait. This paper introduces and axiomatises the family of α-UA (for α-Uncertainty Attitude) preferences: a simple extension of uncertainty averse preferences with a Hurwicz-style mixing coefficient, so as to admit a richer range of uncertainty attitudes. The parameters of the model are uniquely identified in our characterisation. It provides, in the Hurwicz α-maxmin EU special case, a new resolution of a long-standing identification problem. It also yields novel models, including extensions of variational and multiplier preferences. Comparative statics support the interpretation of the mixing coefficient as an index of imprecision aversion. In a standard portfolio problem, the model yields the intuitive relationship between imprecision aversion and investment in an uncertain asset: as the former increases, the latter decreases.

Rank-Dependent Utility Under Multiple Priors

Nonlinear perception of risk is integrated into models of decision under ambiguity, thereby providing a theoretical basis for behavioral analyses that involve both risk and ambiguity. The main model replaces the risk front of maxmin expected utility by rank-dependent utility, and it is derived from simple, observable axioms that are not restricted to expected utility under risk. The model highlights the interaction between risk attitude and ambiguity attitude, and it is compatible with the paradoxical choices hypothesized by Machina. Following the same idea, nonlinear perception of risk can also be integrated into variational and multiplier preferences. When applied to comparative ambiguity attitudes, the idea enables more comparisons by allowing for both heterogeneous taste and perception of risk. This paper was accepted by Ilia Tsetlin, behavioral economics and decision analysis.

Salesforce contracting under model uncertainty

This paper considers a firm's salesforce contracting problem under model uncertainty. Based on the notion of multiplier preferences, we capture model uncertainty and explicitly characterize the structure of the optimal contract. Our findings provide guidelines on the design of salesforce compensation contracts in practical situations.

Robust experimentation in the continuous time bandit problem

We study the experimentation dynamics of a decision maker (DM) in a two-armed bandit setup (Bolton and Harris (1999)), where the agent holds ambiguous beliefs regarding the distribution of the return process of one arm and is certain about the other one. The DM entertains Multiplier preferences a la Hansen and Sargent (2001), thus we frame the decision making environment as a two-player differential game against nature in continuous time. We characterize the DM value function and her optimal experimentation strategy that turns out to follow a cut-off rule with respect to her belief process. The belief threshold for exploring the ambiguous arm is found in closed form and is shown to be increasing with respect to the ambiguity aversion index. We then study the effect of provision of an unambiguous information source about the ambiguous arm. Interestingly, we show that the exploration threshold rises unambiguously as a result of this new information source, thereby leading to more conservatism. This analysis also sheds light on the efficient time to reach for an expert opinion.

Welfare Cost of Model Uncertainty in a Small Open Economy.

This paper extends the canonical small open-economy real-business-cycle model, when considering model uncertainty. Domestic households have multiplier preferences, which leads them to take robust decisions in response to possible model misspecification for the economy’s aggregate productivity. Using perturbation methods, the paper extends the literature on real business cycle models by deriving a closed-form solution for the combined welfare effect of the two sources of uncertainty, namely risk and model uncertainty. While classical risk has an ambiguous effect on welfare, the addition of model uncertainty is unambiguously welfare-deteriorating. Hence, the overall effect of uncertainty on welfare is ambiguous, depending on consumers preferences and model parameters. The paper provides numerical results for the welfare effects of uncertainty measured by units of consumption equivalence. At moderate (high) levels of risk aversion, the effect of risk on household welfare is positive (negative). The addition of model uncertainty—for all levels of concern about model uncertainty and most risk aversion values—turns the overall effect of uncertainty on household welfare negative. It is important to remark that the analytical decomposition and combination of the effects of the two types of uncertainty considered here and the resulting ambiguous effect on overall welfare have not been derived in the previous literature on small open economies.

Risk, ambiguity, and Giffen assets

We provide necessary and sufficient conditions for the demand for financial assets to satisfy the law of demand (prices and quantity demanded move in opposite directions) when preferences take the expected utility, multiplier preferences, Choquet expected utility, and the maxmin expected utility forms. For the first three models the law of demand holds when the variation in the coefficient of relative risk aversion stays within a specified bound. Ensuring the law of demand holds in the maxmin expected utility model requires constant relative risk aversion.

Beyond Uncertainty Aversion

Although much of the theoretical literature on ambiguity works under the assumption of uncertainty aversion, experimental evidence suggests that it is not a universal behavioral trait. This paper introduces and axiomatises the family of α-UA (for α-Uncertainty Attitude) preferences: a simple extension of the family of uncertainty averse preferences (Cerreia-Vioglio et al., 2011b) with a Hurwicz-style mixing coefficient, so as to admit a richer range of uncertainty attitudes. The parameters of the model are uniquely identified in our characterisation. This provides, in the Hurwicz α-maxmin EU special case, a new resolution of a long-standing identification problem for this model. Moreover, it yields novel models as special cases, including extensions of variational and multiplier preferences (Maccheroni et al., 2006; Hansen and Sargent, 2001) beyond the assumption of uncertainty aversion. Comparative statics support the interpretation of the mixing coefficient as an index of imprecision aversion. In a standard portfolio problem, the model yields the intuitive relationship between imprecision aversion and investment in an uncertain asset: as the former increases, the latter decreases.

Optimal Portfolio Positioning on Multiple Assets Under Ambiguity

This paper determines the optimal financial portfolio in the multidimensional setting when the investor exhibits ambiguity aversion. We consider the Maccheroni et al. (Econometrica 74(6):1447–1498, 2006) framework which includes both the Gilboa and Schmeidler’s (J Math Econ 18(2):141–153, 1989) multiple priors preferences and the (American Econ Rev 91:60–66, 2001) multiplier preferences. We determine the optimal portfolio profile under ambiguity when the investors can invest on various risky assets. We investigate in particular the CRRA case while introducing an ambiguity index based on the relative entropy criterion. Such result extends Ben Ameur and Prigent (Econ Model 34:89–97, 2013) when there is only one risky asset. Indeed, we show for example how the ambiguity on the correlations between the risky assets crucially modify the optimal payoff. Such results have important practical applications in structured portfolio management when investing on multiple financial indices and basket options.

Robust Experimentation in the Continuous Time Bandit Problem

We study the experimentation dynamics of a decision maker (DM) in a two-armed bandit setup (Bolton and Harris [1999]), where the agent holds ambiguous beliefs regarding the distribution of the return process of one arm and is certain about the other one. The DM entertains Multiplier preferences a la Hansen and Sargent [2001], thus we frame the decision making environment as a two-player differential game against nature in continuous time. We characterize the DM's value function and her optimal experimentation strategy that turns out to follow a cut-off rule with respect to her belief process. The belief threshold for exploring the ambiguous arm is found in closed form and is shown to be increasing with respect to the ambiguity aversion index. We then study the effect of provision of an unambiguous information source about the ambiguous arm. Interestingly, we show that the exploration threshold rises unambiguously as a result of this new information source, thereby leading to more conservatism. This analysis also sheds light on the efficient time to reach for an expert opinion.

Risk aversion, risk premia, and the labor margin with generalized recursive preferences

A flexible labor margin allows households to absorb shocks to asset values with changes in hours worked as well as changes in consumption. This ability to absorb shocks along both margins alters the household's attitudes toward risk, as shown by Swanson (2012). In the present paper, I extend that analysis to the case of generalized recursive preferences, as in Epstein and Zin (1989) and Weil (1989), including multiplier preferences, as in Hansen and Sargent (2001). Understanding risk aversion for these preferences is important because they are a primary mechanism being used to bring macroeconomic models into closer agreement with asset prices. Traditional, fixed-labor measures of risk aversion show no stable relationship to the equity premium in a standard macroeconomic model, while the closed-form expressions I derive here match the equity premium closely.

Measuring ambiguity attitude: (Extended) multiplier preferences for the American and the Dutch population

Empirical studies of ambiguity aversion often use measures that are not grounded in theory. This paper shows how a theoretically-founded measure of ambiguity aversion can be derived from Hansen and Sargent’s theory of multiplier preferences. Multiplier preferences are used in macroeconomics to capture model uncertainty. At the micro level, they have not been applied yet, because they do not permit ambiguity seeking, which is usually observed for a substantial proportion of subjects. We give a preference foundation for (extended) multiplier preferences accommodating both ambiguity aversion and ambiguity seeking and we propose a simple method to measure them using matching probabilities. We illustrate our method in two large representative samples (Dutch and American) and obtain the first micro estimates of multiplier preferences.

Prudence with Respect to Ambiguity

Under expected utility, prudence is equivalent to a positive third derivative of utility and plays a crucial role in precautionary saving behaviour. Eeckhoudt and Schlesinger (2006) proposed behavioural definitions of prudence and of higher order risk preferences. The present article proposes a similar definition for prudence with respect to ambiguity, i.e. situations in which objective probabilities are not available. Implications for several ambiguity models are derived. Ambiguity prudence is implied by Hansen and Sargent’s (2001) multiplier preferences, empirically correlates with financial behaviour and plays a key role in prevention behaviour.

Robust Security Design

We consider the optimal contract between an entrepreneur and investors in a single-period model when both parties have limited liability, are risk-neutral toward cash flow risk, and are ambiguity-averse. Ambiguity aversion is modeled by multiplier preferences for robustness toward model uncertainty, as in Hansen and Sargent (2001). Efficient ambiguity-sharing implies that the first-best contract consists of either convertible debt or levered equity. As is customary, in the second-best contract, moral hazard is alleviated by giving more cash to investors in low cash flow states. Under many settings in our model, the optimal security has an equity-like component in high cash flow states, providing a contrast to the results in Innes (1990).

Objective rationality and uncertainty averse preferences

We provide a bridge between Bewley preferences [2] and Uncertainty averse preferences [4]. In doing this, we generalize the ?ndings of Gilboa, Maccheroni, Marinacci, and Schmeidler [11]. To exemplify this new framework, we then study a class of preferences that we call Constrained Multiplier preferences and that was ?rst proposed by Wang [19].

Dynamically stable preferences

Abstract In the framework of dynamic choice under uncertainty, we define dynamic stability as a combination of two assumptions prevalent in the literature: dynamic consistency and the requirement that updated preferences have the same “structure” as ex ante ones. Dynamic stability also turns out to be a defining characteristic of the multiplier preferences of Hansen and Sargent (2001) [24] within the scope of variational preferences. Generally, for any class of invariant preferences, dynamic stability is shown to be connected to another independent property — consequentialism.

The (im)possibility of collective risk measurement: Arrovian aggregation of variational preferences

This paper studies collective decision making when individual preferences can be represented by convex risk measures. It addresses the question whether there exist non-dictatorial aggregation functions of convex risk measures satisfying the following Arrow-type rationality axioms: a weak form of universality, systematicity (a strong variant of independence), and the Pareto principle. Herein, convex risk measures are identified with variational preferences on account of the Maccheroni et al. (Econometrica 74(6):1447–1498, 2006) axiomatisation of variational preference relations and Föllmer’s and Schied’s (Finance Stoch 6(4):429–447, 2002; Stochastic Finance. An Introduction in Discrete Time. de Gruyter Studies in Mathematics, vol. 27, 2nd edn. de Gruyter, Berlin, 2004) representation theorem for concave monetary utility functionals. The cases of both finite and infinite electorates are considered. For finite electorates, we prove a variational analogue of Arrow’s impossibility theorem. For infinite electorates, the possibility of rational aggregation depends on an equicontinuity condition for the variational preference profiles; we prove a variational analogue of Fishburn’s possibility theorem and point to potential analogues of Campbell’s impossibility theorem. The proof methodology is based on a model-theoretic approach to aggregation theory inspired by Lauwers and Van Liedekerke (J Math Econ 24(3):217–237, 1995). Diverse applications of these results are conceivable, in particular (a) the constitutional design of panels of risk managers and (b) microfoundations for (macroeconomic) multiplier preferences.

Optimal portfolio positioning under ambiguity

This paper analyzes the optimality of financial portfolios when the investor has a utility with ambiguity aversion. It provides a general result about the optimal portfolio profile under ambiguity, in the Anscombe–Aumann framework, using the Maccheroni et al. (2006) approach which includes Gilboa and Schmeidler's (1989) multiple prior preferences and Hansen and Sargent's (2011) multiplier preferences. The paper then details the CRRA case with an ambiguity index based on relative entropy. Such findings have practical applications in structured portfolio management. Indeed, it is important to take account of uncertainty about the true values of financial parameters when determining the best portfolio profile.

Optimal stopping with dynamic variational preferences

We solve optimal stopping problems in uncertain environments for agents assessing utility by virtue of dynamic variational preferences as in Maccheroni, Marinacci and Rustichini (2006) [16] or, equivalently, assessing risk in terms of dynamic convex risk measures as in Cheridito, Delbaen and Kupper (2006) [4]. The solution is achieved by generalizing the approach in Riedel (2009) [21] introducing the concept of variational supermartingales and variational Snell envelopes with an accompanying theory. To illustrate results, we consider prominent examples: dynamic multiplier preferences and a dynamic version of generalized average value at risk introduced in Cheridito and Tianhui (2009) [5].

Axiomatic Foundations of Multiplier Preferences

This paper axiomatizes the robust control criterion of multiplier preferences introduced by Hansen and Sargent (2001). The axiomatization relates multiplier preferences to other classes of preferences studied in decision theory. Some properties of multiplier preferences are generalized to the broader class of variational preferences, recently introduced by Maccheroni, Marinacci and Rustichini (2006). The paper also establishes a link between the parameters of the multiplier criterion and the observable behavior of the agent. This link enables measurement of the parameters on the basis of observable choice data and provides a useful tool for applications.

Axiomatic Foundations of Multiplier Preferences

This paper axiomatizes the robust control criterion of multiplier preferences introduced by Hansen and Sargent (2001). The axiomatization relates multiplier preferences to other classes of preferences studied in decision theory, in particular the variational preferences, recently introduced by Maccheroni, Marinacci and Rustichini (2006). The paper also establishes a link between the parameters of the multiplier criterion and the observable behavior of the agent. This link enables measurement of the parameters on the basis of observable choice data and provides a useful tool for applications.