### State-dependent Preferences [chapter]

Edi Karni
1990 Utility and Probability
1 State-dependent preferences. Theories of individual decision making under uncertainty pertain to situations in which a choice of a course of action, by itself, does not determine the outcome. To formulate these theories Savage (1954) introduced what has become the standard analytical framework. It consists of three sets: a set S, of states of the world (or states, for short); an arbitrary set C, of consequences; and the set F , of all the functions from the set of states to the set of
more » ... nces. Elements of F, referred to as acts, represent courses of action, consequences describe anything that may happen to a person, and states are the resolutions of uncertainty, that is, "a description of the world so complete that, if true and known, the consequences of every action would be known" (Arrow  , p. 45). Decision makers are characterized by preference relations, <, on F . With few exceptions, preference relations are taken to be complete (that is, for all f and g in F , either f < g or g < f ) and transitive binary relations on F . The symbols f < g have the interpretation "the course of action f is preferred or indifferent to the course of action g." The strict preference relation, Â, and the indifference relation, ∼, are the asymmetric and symmetric parts of <, respectively. Loosely speaking, a preference relation is state dependent when the prevailing state of nature is itself of direct concern to the decision maker. For example, taking out a health insurance policy is choosing an act whose consequencesthe indemnities -depend on the realization of the decision maker's state of health. In this example, the state is the decision maker's state of health. It affects the decision maker's well-being directly, and indirectly, through the payoff prescribed by the health insurance policy. The preference relation may display ordinal state dependence, in which case the underlying state may affect the decision maker's preferences by altering his ordinal ranking of the consequences; or cardinal state dependence, by altering his risk attitudes; or both. To define state dependence formally, it is convenient to adopt the model of Anscombe and Aumann (1963). In this model the state space is finite, and the consequences are lotteries, that is, probability distributions that assign strictly positive probability to a finite number of outcomes. Denote by L(X) the set of lotteries on an arbitrary set, X, of possible outcomes. Given a preference relation < on F ; a state s; and f, f 0 , g, g 0 in F , define a preference relation on F conditional on s, < s , by f < s f 0 if g < g 0 whenever f (s) = g (s) , f 0 (s) = g 0 (s) and g (s 0 ) = g 0 (s 0 ) for all s 0 ∈ S − {s}. Because acts are functions, f (s) is defined uniquely. Thus < s defines a preference relation on L (X) conditional on s. This induced preference relation is also denoted by < s . A state s ∈ S is said to be null if f < s f 0 for all f, f 0 ∈ F , otherwise it is nonnull. Definition: A preference relation < on F is state dependent if < s 6 =< s 0 for some nonnull s and s 0 in S. Because consequences are lotteries, if a preference relation < on F displays state dependence, then < s and < s 0 must differ on the ranking of some lotteries in L (X). This may be due to distinct attitudes toward risk and/or distinct ordering of outcomes, that is, degenerate lotteries that assign the given outcomes probability one. Circumstances in which the dependence of the decision maker's