# Schedule - Parallel Session 6 - Axiomatic Foundations of Utility Theory

WMG IMC Auditorium 0.02 - 15:40 - 17:10#### Subjective Expected Utility Representations for Savage Preferences on Topological Spaces

Vassili Vergopoulos; Marcus Pivato

#### Abstract

Economic decisions under uncertainty are often constrained by the restrictions imposed by their environment. For instance, consider a farmer who must make planting decisions without knowing the future temperature, rainfall, or sunshine. The outcomes of his various planting strategies, in terms of crop yields, are thus uncertain at the moment of choice. But furthermore, slight variations of the meteorological conditions will only result in slight variations in yields. Put differently, technological feasibility constraints impose continuous production plans: the farmer can only supply crop quantities that depend continuously on the meteorological conditions. Other examples where continuity with respect to uncertainty emerges as a technological constraint include the production of drinking water, hydroelectricity or solar power; the design of supply networks; the adoption of climate change policies or pollution abatement technologies. We study such situations of decision-making under uncertainty where feasibility constraints impose continuity restrictions on the available alternatives. More specifically, we provide an axiomatic treatment of Subjective Expected Utility (SEU) that is adapted to these situations. The uncertainty that affects a decision problem is given by a topological space of states of the world. The various possible consequences of decisions are given by a topological space of outcomes. Furthermore, we assume an exogenously given collection of continuous functions, or acts, from the state space onto the outcome space. This collection describes the set of all the theoretically conceivable alternatives that conform to the continuous technological constraints. The preferences of a decision-maker only rank these feasible acts. Thus, the classical Savage (1954) approach to SEU does not apply to such a restricted domain of acts. In this context, we identify a system of axioms on these preferences that characterize the existence and uniqueness of an SEU representation. In the representation, behavior is explained in terms of tastes and beliefs. Tastes take the form of a continuous utility function on the space of outcomes, thereby capturing the intuition that “similar outcomes” are assigned “similar utility levels”. Meanwhile, the specific form of beliefs depends on the topological assumptions made on the state space. For instance, if it is Hausdorff and compact, then beliefs are represented by a Borel probability measure. But, in general, beliefs take the form of what we call a credence. A credence on a topological state space is a finitely additive probability measure on a specific subdomain of the topology; namely, the Boolean algebra of regular events. Despite the restrictions we impose on both acts and events, we obtain an SEU representation where the utility function is unique up to positive affine transformation, and the credence (or Borel probability measure) is uniquely defined.

**Vassili Vergopoulos**

#### Costly Self-Control and Limited Willpower

Simon Grant

#### Abstract

We present a representation theorem for individual choice among sets (that is, “menus”) of lotteries, from which the individual will later choose a single lottery. Our axioms, building on those in Gul and Pesendorfer (2001), allow for a preference for commitment and self-control subject to sufficient willpower. Four of the five axioms of our characterization are as in Theorem 3 of Gul and Pesendorfer except that the independence axiom is restricted to singleton menus and those two-element menus in which any failure of self-control in the second period arises from the individual being unwilling to incur the cost of exercising such self-control rather than from being unable to exert self-control because of limited willpower. We add one new axiom to regulate willpower as a limited cognitive resource in which the available “stock” of willpower does not vary across menus. In our characterization, agents with insufficient willpower to resist temptations are bound to choose an option with lower “compromise utility” while the behaviors of agents who are able to resist temptations remain unchanged.

**Simon Grant**

#### Multiple Sources of Uncertainty and Varying Risk Confidence

Fabio Maccheroni; Veronica Cappelli; Simone Cerreia-Vioglio; Massimo Marinacci

#### Abstract

There is by now solid empirical evidence on the dependence of risk attitudes of decision makers on the risk source they are facing (Heath and Tversky, 1991, Fox and Tversky, 1995, Slovic, 1999). For example, human casualties generated by different catastrophic events (such as earthquakes, epidemics, terror attacks, nuclear accidents) may be evaluated in very different ways by policy makers taking prevention measures. Analogously, consumption at future dates is obviously discounted in different ways, but an investor may also take into account the fact that in different future dates he will be more on less affected by outcomes’ variability (older people are more vulnerable to consumption shocks than younger ones). In this paper, we provide a framework to describe source dependent prospects and we obtain a general axiomatic foundation for the representation of preferences between these prospects. Specifically, prospects depending on different sources are represented by vectors of source dependent random variables, and we characterize the evaluation of prospects by means of a two stage process: in the first stage, decision makers compute the certainty equivalents of the different components of the vector using source-specific utility functions, probability measures, and probability distortion functions; in the second stage, they take a suitably weighted mean of these certainty equivalents. In particular, when prospects are portfolios of bets on two different urns, like in the classic example of Ellsberg (1961), we are able to characterize evaluation by means of the sum of urn-dependent certainty equivalents, thus immediately rationalizing the eponymous paradox in the spirit of Smith (1969). Analogously, when prospects are consumption streams and sources correspond to dates, we are able to pin down the behavioural assumptions underlying evaluation by means of the sum of discounted date-dependent certainty equivalents. The features of the axiomatic foundation we provide guarantee testability of the model, the possibility of estimating its components, and the sensibility of performing comparative statics exercises.

**Fabio Maccheroni**

#### Uncertainty and Binary Stochastic Choice

Matthew Ryan

#### Abstract

Experimental evidence suggests that decision-making has a stochastic element and is better described through choice probabilities than preference relations. Binary choice probabilities admit a strong utility representation (SUR) if there exists a utility function *u* such that the probability of choosing *a* over *b* is a strictly increasing function of the utility difference *u(a)-u(b).* Debreu (1958) obtained a simple set of sufficient conditions for the existence of a SUR when alternatives are drawn from a suitably rich domain. Dagsvik (2008) specialised Debreu’s result to the domain of risky prospects (lotteries) and provided axiomatic foundations for a SUR in which the underlying utility function conforms to expected utility. This paper presents a general SUR theorem for mixture set domains, along with several applications of this general result. We first strengthen Dagsvik’s theorem by weakening one of his axioms. We then consider binary choices between uncertain prospects of the Anscombe-Aumann variety. We give sufficient conditions for a SUR with respect to a utility function for invariant biseparable preferences (Ghirardato *et al*., 2004). The SEU, CEU and MEU models all fall within this class, and a specialised SUR theorem is provided for each.

**Matthew Ryan**