RATIONAL AND SOCIAL CHOICE Part 5 potx

230 edi karni A weaker version of this approach, based on restricting consistency to a subset of hypothetical lotteries that have the same marginal distribution on S, due to Karni, Schmeidler, and Vind (1983), yields a subjective expected utility representation with state-dependent preferences. However, the subjective probabilities in this represen- tation are arbitrary, and the utility functions, while capturing the decision-maker’s state-dependent risk attitudes, do not necessarily represent his evaluation of the consequences in the different states. Wakker (1987)extendsthetheoryofKarni, Schmeidler, and Vind to include the case in which the set of consequences is a connected topological space. Other theories yielding subjective expected utility representations with state- dependent utility functions invoke preferences on conditional acts (i.e. preference relations over the set of acts conditional on events). Fishburn (1973), Drèze and Rustichini (1999), and Karni (2007) advance such theories. Skiadas (1997)proposes a model, based on hypothetical preferences, that yields a representation with state- dependent preferences. In this model, acts and states are primitive concepts, and preferences are defined on act–event pairs. For any such pair, the consequences (utilities) represent the decision-maker’s expression of his holistic valuation of the act. The decision-maker is not supposed to know whether the given event occurred; hence his evaluation of the act reflects, in part, his anticipated feelings, such as disappointment aversion. 9.3.2 Subjective Expected Utility with Moral Hazard and State-Dependent Preferences Adifferent, choice-based approach to modeling expected utility with state- dependent utility functions presumes that decision-makers believe that they possess the means to affect the likelihood of the states. This idea was originally proposed by Drèze (1961, 1987). Departing from Anscombe and Aumann’s (1963)“reversalof order in compound lotteries” axiom, Drèze assumes that a decision-maker who strictly prefers that the uncertainty of the lottery be resolved before that of the acts does so because the information allows him to affect the likely realization of the outcome of the underlying states (the outcome of a horse race, for ex- ample). The means by which the decision-maker may affect the likelihoods of the events are not an explicit aspect of the model. Drèze’s axiomatic structure implies a unique separation of state-dependent utilities from a set of probability distributions over the set of states of nature. Choice is represented as expected utility-maximizing behavior in which the expected utility associated with any given act is itself the maximal expected utility with respect to the probabilities in the set. Karni (2006b) pursues the idea that observing the choices over actions and bets of decision-makers who believe they can affect the likelihood of events by their state-dependent utility 231 actions provides information that reveals their beliefs. Unlike Drèze, Karni treats the actions by which a decision-maker may influence the likelihood of the states as an explicit ingredient of the model. Because Savage’s notion of states requires that this likelihood be outside the decision-maker’s control, to avoid confusion, Karni uses the term effects instead of states to designate phenomena on which decision- makers can place bets and whose realization, they believe, can be influenced by their actions. Like states, effects resolve the uncertainty of bets; unlike states, their likelihood is affected by the decision-maker’s choice of action. Let » be a finite set of effects, and denote by A an abstract set whose elements are referred to as actions. Actions correspond to initiatives a decision-maker may undertake that he believes affect the likely realization of alternative effects. Let Z(Ë) be a finite set of prizes that are feasible if the effect Ë obtains; denote by L(Z(Ë)) the set of lotteries on Z(Ë). Bets are analogous to acts and represent effect-contingent lottery payoffs. Formally, a bet,b, is a function on » such that b(Ë) ∈ L (Z(Ë)). Denote by B the set of all bets, and suppose that it is a convex set, with a convex operation defined by (·b +(1− ·)b  )(Ë)=·b(Ë)+(1− ·)b  (Ë), for all b, b  ∈ B, · ∈ [0, 1], and Ë ∈ ». The choice set is the product set C:=A × B whose generic element, (a, b), is an action–bet pair. Action–bet pairs represent conceivable alternatives among which decision-makers may have to choose. The set of consequences C consists of prize–effect pairs; that is, C = {(z, Ë) | z ∈ Z(Ë), Ë ∈ »}. Decision-makers are supposed to be able to choose among action–bet pairs— presumably taking into account their beliefs regarding the influence of their choice of actions on the likelihood of alternative effects—and, consequently, on the desirability of the corresponding bets and the intrinsic desirability of the ac- tions. For instance, a decision-maker simultaneously chooses a health insurance policy and an exercise and diet regimen. The insurance policy is a bet on the effects that correspond to the decision-maker’s states of health; adopting an ex- ercise and diet regimen is an action intended to increase the likelihood of good states of health. A decision-maker is characterized by a preference relation  on C. Bets that, once accepted, render the decision-maker indifferent among all the actions are referred to as constant valuation bets. Such bets entail compensating variations in the decision-maker’s well-being due to the direct impact of the actions and the impact of these actions on the likely realization of the different effects and the corresponding payoff of the bet. To formalize this idea, let I (b; a)={b  ∈ B | (a, b  ) ∼ (a, b)} and I (p; Ë, b, a)={q ∈ L (Z(Ë)) | (a, b −Ë q) ∼ (a, b −Ë p)}. A bet ¯ b ∈ B is said to be a constant valuation bet according to  if (a, ¯ b) ∼ (a  , ¯ b) for all a, a  ∈ ˆ A, and b ∈∩ a∈ ˆ A I ( ¯ b; a)ifandonlyifb(Ë) ∈ I ( ¯ b(Ë); Ë, ¯ b, a) for all Ë ∈ » and a ∈ ˆ A. Let B cv denote the subset of constant valuation bets. Given p ∈ L (Z(Ë)), Idenoteby b −Ë p the constant valuation bet whose Ëth coordinate is p. 232 edi karni An effect Ë ∈ » is null given the action a if (a, b −Ë p) ∼ (a, b −Ë q) for all p, q ∈ L(Z(Ë)) and b ∈ B, otherwise it is nonnull given the action a. In general, an effectmaybenullundersomeactionsandnonnullunderothers.Twoeffects, Ë and Ë  , are said to be elementarily linked if there are actions a, a  ∈ A such that Ë, Ë  ∈ »(a) ∩ »(a  ), where »(a) denotes, the subset of effects that are nonnull given a. Two effects are said to be linked if there exists a sequence of effects Ë = Ë 0 , ,Ë n = Ë  such that every Ë j is elementarily linked with Ë j +1 . The preference relation  on C is nontrivial if the induced strict preference rela- tion, , is nonempty. Henceforth, assume that the preference relation is nontrivial, every pair of effects is linked, and every action–bet pair has an equivalent constant valuation bet. For every a, define the conditional preference relation  a on B by: b  a b  if and only if (a, b)  (a, b  ). The next axiom requires that, for every given effect, the ranking of lotteries be independent of the action. In other words, conditional on the effects, the risk attitude displayed by the decision-maker is independent of his actions. Formally, (A6) (Action-independent risk attitudes) For all a, a  ∈ A, b ∈ B, Ë ∈ »(a) ∩ »(a  ) and p, q ∈ L(Z(Ë)), b −Ë p  a b −Ë q if and only if b −Ë p  a  b −Ë q. The next theorem, due to Karni (2006), gives necessary and sufficient conditions for the existence of representations of preference relations over the set of action– bet pairs with effect-dependent utility functions and action-dependent subjective probability measures on the set of effects. Theorem 2. Let  be a preference relation on C that is nontrivial, every pair of effects is linked, and every action–bet pair has an equivalent constant val- uation bet. Then {  a | a ∈ A} are weak orders satisfying the Archimedean, in- dependence, and action-independent risk attitudes axioms if and only if there exists a family of probability measures {(·; a) | a ∈ A} on »; a family of effect-dependent, continuous, utility functions {u(·; Ë):Z(Ë) → R | Ë ∈»}; and a continuous function f : R × A → R, increasing in its first argument, such that, for all (a, b), (a  , b  ) ∈ C, (a, b)  (a  , b  ) if and only if f ⎛ ⎝  Ë∈»  ( Ë; a )  z∈Z ( Ë ) u ( z; Ë ) b ( z; Ë ) , a ⎞ ⎠ ≥ f ⎛ ⎝  Ë∈»   Ë; a    z∈Z ( Ë ) u ( z; Ë ) b  ( z; Ë ) , a  ⎞ ⎠ . (5) state-dependent utility 233 Moreover, {v(·; Ë):Z(Ë) → R | Ë ∈ »} is another family of utility functions, and g is another continuous function representing the preference relation in the sense of Eq. 5 if and only if, for all Ë ∈ »,v(·, Ë)=Îu(·, Ë)+ς(Ë), Î > 0, and, for all a ∈ A, g (Îx + ς(a), a)= f (x, a), where x ∈{  Ë∈» (Ë; a)  x∈Z(Ë) u(z; Ë)b(z; Ë) | b ∈ B} and ς(a)=  Ë∈» ς(Ë)(Ë; a). The family of probability measures {(·; a) | a ∈ A} on » is unique satisfying (Ë; a) = 0 if and only if Ë is null given a. The function f (·, a)inEq.5 represents the direct impact of the action on the decision-maker’s well-being. The indirect impact of the actions, due to variations they produce in the likelihood of effects, is captured by the probability measures {(·; a)} a∈A . However, the uniqueness of utility functions in Eq. 5 is due to a normalization; it is therefore arbitrary in the same sense as the utility function in Theorem 1 is. To rid the model of this last vestige of arbitrariness, Karni (2008) shows that if a decision-maker is Bayesian in the sense that his posterior prefer- ence relation is induced by the application of Bayes’s rule to the probabilities that figure in that representation of the prior preference relation, then the represen- tation is unique, and the subjective probabilities represent the decision-maker’s beliefs. If a preference relation  on C satisfies conditional effect independence (i.e. if  a satisfies a condition analogous to (A5), with effects instead of states), then the utility functions that figure in Theorem 2 represent the same risk attitudes and assume the functional form u(z; Ë)=Û(Ë)u(z)+Í(Ë), Û(·) > 0. In other words, effect independent risk attitudes do not imply effect-independent utility functions. The utility functions are effect-independent if and only if constant bets are constant utility bets. 9.4 Risk Aversion with State-Dependent Preferences The raison d’être of many economic institutions and practices, such as insurance and financial markets, cost-plus procurement contracts, and labor contracts, is the need to improve the allocation of risk bearing among risk-averse decision-makers. The analysis of these institutions and practices was advanced with the introduction, by de Finetti (1952), Pratt (1964), and Arrow (1971), of measures of risk aversion. These measures were developed for state-independent utility functions, however, and are not readily applicable to the analysis of problems involving state-dependent utility functions such as optimal health or life insurance. Karni (1985) extends the theory of risk aversion to include state-dependent preferences. 234 edi karni 9.4.1 The Reference Set and Interpersonal Comparison of Risk Aversion A central concept in Karni’s (1985) theory of risk aversion with state-dependent preferences is the reference set. To formalize this concept, let B denote the set of real-valued function on S, where S = {1, ,n} is a set of states. Elements of B are referred to as gambles. As in the case of state-independent preferences, a state- dependent preference relation on B is said to display risk aversion if the upper contour sets {b ∈ B | b  b  }, representing the acceptable gambles at b  , b  ∈ B, are convex. It displays risk proclivity if the lower contour set, {b ∈ B | b   b}, representing the unacceptable gambles at b  are convex. It displays these attitudes in the strict sense if the corresponding sets are strictly convex. For a given preference relation, the reference set consists of the most preferred gambles among gambles of equal mean. Formally, B(c)={b ∈ B |  s ∈S b(s)p(s )=c}, and the reference set corresponding to  is defined by RS = {b ∗ (c) | c ≥ 0}, where b ∗ (c) ∈ B(c) and b ∗ (c)  b for all b ∈ B(c). If  displays strict risk aversion, then the corresponding utility functions {u(·, s)} s ∈S are strictly concave, and the reference set RS is well-defined and is characterized by the equality of the marginal utility of money across states (i.e. u  (b ∗ (s ), s)=u  (b ∗ (s  ), s  )for all s , s  ∈ S). (Figure 9.1 depicts the reference set for strictly risk-averse prefer- ences in the case S = {1, 2}.) For such preference relations, it is convenient to depict the reference set as follows: Define f s (w)=(u  ) −1 (u  (w, 1), s ), s ∈ S, w ∈ R.Bydefinition, f 1 is the identity function, and by the concavity of the utility functions, { f s } s ∈S are increasing functions. The reference set is depicted by the function F : R + → R n defined by F (w)=(f 1 (w), , f n (w)). If the utility func- tions are state-independent, the reference set coincides with the subset of constant gambles. Given a preference relation  and a gamble b, the reference equivalence of b is the element, b ∗ (b), of the reference set corresponding to  that is indifferent to b. Let ¯ b =  s ∈S b(s)p(s ); the risk premium associated with b, Ò(b), is defined by Ò(b)=  s ∈S [ ¯ b − b ∗ (b)]p(s). Clearly, if a preference relation displays risk aversion, the risk premium is nonnegative (see Figure 9.1). Broadly speaking, two preference relations  u and  v displaying strict risk aversion are comparable if they have the same beliefs and agree on the most pre- ferred gamble among gambles of the same mean. Formally, let p be a probability distribution on S representing the beliefs embodied in the two preference re- lations. Then  u and  v are said to be comparable if RS u = RS v . Note that if the utility functions are state-independent, all risk-averse preference relations are comparable. Let Ò u (b) and Ò v (b) denote the risk premiums associated with a preference re- lation  u and  v , respectively, displaying strict risk aversion. Then  u is said to state-dependent utility 235 w 2 B(c) RS u 0 Ò B(c’) w 1 Fig. 9.1. The reference set and risk premium for state-dependent preferences. display greater risk aversion than  v if Ò u (b) ≥ Ò v (b) for all b ∈ B. Given h(·, s), h = u,v, denote by h 1 , h 11 the first and second partial derivatives with respect to the first argument. The following theorem, due to Karni (1985), gives equivalent characterizations of interpersonal comparisons of risk aversion. Theorem 3. Let  u and  v be comparable preference relations displaying strict risk aversion whose corresponding state-dependent utility functions are {u ( ·, s ) } s ∈s and {v ( ·, s ) } s ∈s . Suppose that u and v are twice continuously differentiable with respect to their first argument. Then the following conditions are equivalent: (i) − u 11 (w, s ) u 1 (w, s ) ≥− v 11 (w, s ) v 1 (w, s ) for all s ∈ S and w ∈ R. (ii) For every probability distribution p on S, there exists a strictly increasing concave function T p such that  s ∈S u( f s (w), s )p(s)= T p [  s ∈S v( f s (w), s )p(s)], and T  p is independent of p. (iii) Ò u (b) ≥ Ò v (b) for all b ∈ B. In the case of state-independent preferences, the theory of interpersonal com- parisons of risk aversion is readily applicable to the depiction of changing attitudes towards risk displayed by the same preference relation at different wealth levels. In the case of state-dependent preferences, the prerequisite of comparability must be imposed. In other words, the application of the theory of interpersonal comparisons is complicated by the requirement that the preference relations be 236 edi karni comparable. A preference relation, , displaying strict risk aversion is said to be autocomparable if, for any b ∗∗ , b ∗ ∈ RS, N ε (b ∗∗ ) ∩ RS =(b ∗∗ − b ∗ )+N ε (b ∗ ) ∩ RS, where N ε (b ∗∗ )andN ε (b ∗ ) are disjoint neighborhoods in R n . The reference sets of autocomparable preference relations are depicted by F (w)=(a s w) s ∈S , where a s > 0. All preference relations that have expected utility representation with state- independent utility function are obviously autocomparable. Denote by x the constant function in R n whose value is x.Anautocomparable preference relation is said to display decreasing (increasing, constant) absolute risk aversion if Ò(b) > (<, =)Ò(b + x)foreveryx > 0. For autocomparable preference relations with state-dependent utility functions {U(·, s )} s ∈S , equivalent character- izations of decreasing risk aversion are analogous to those in Theorem 3,with u(w, s)=U (w, s) and v(w, s)=U(w + x, s). 9.4.2 Application: Disability Insurance The following disability insurance scheme illustrates the applicability of the theory of risk aversion with state-dependent preferences. Let the elements of S correspond to potential states of disability (including the state of no disability). Suppose that an insurance company offers disability insurance policies (–, I) according to the formula –(I )=‚ ¯ I , where I is a positive, real-valued function on S representing the indemnities corresponding to the different states of disability; ¯ I represents the actuarial value of the insurance policy; – is the insurance premium corresponding to I ;and‚ ≥ 1 is the loading factor. The insurance scheme is actuarially fair if ‚ =1. Let p be a probability measure on S representing the relative frequencies of the various disabilities in the population. Consider a risk-averse, expected-utility- maximizing decision-maker whose risk attitudes depend on his state of disability. Let  w = {w(s )} s ∈S represent the decision-maker’s initial wealth corresponding to the different states of disability. The decision-maker’s problem may be stated as follows: Choose I ∗ so as to maximize  s ∈S u(w(s ) − I (s ) − –(I ), s )p(s)subject to the constraints –(I )=‚ ¯ I and I (s ) ≥ 0 for all s . If the insurance is actuarially fair, the optimal distribution of wealth,  w ∗ = {w(s )} s ∈S is the element of the reference set whose mean value is ¯ w =  s ∈S w(s ) p(s ). Consequently, the optimal insurance is given by I ∗ (s )=w ∗ (s ) − w(s ), s ∈ S. Thus comparable individuals, and only comparable individuals, choose the same coverage under fair insurance for every given w. If the insurance is actuarially unfair (that is, ‚ > 1), the optimal disability insur- ance requires that the indemnities be equal to the total loss above state-dependent minimum deductibles (see Arrow 1974). In other words, there is a subset T of dis- ability states and Î > 0suchthatu  ( ˆ w(s ), s)=Î for all s ∈ T and u  (w(s ), s) < Î otherwise, and I ∗ (s )= ˆ w(s ) − w(s)ifs ∈ T and I ∗ (s ) = 0 otherwise. The values state-dependent utility 237 RS E u E v A w 1 w 2 Fig. 9.2. Optimal disability insurance coverage with different degrees of risk aversion. { ˆ w(s )} s ∈T are generalized deductibles. Karni (1985) shows that if  u and  v are comparable preference relations displaying strict risk aversion in the sense of The- orem 3,then,ceteris paribus,if  u displays a greater degree of risk aversion than  v ,then ˆ w u (s ) ≥ ˆ w v (s ) for all s ∈ S, where ˆ w i (s ), i ∈{u,v} are the optimal de- ductibles corresponding to  i .Thus,ceteris paribus, the more risk-averse decision- maker takes out a more comprehensive disability insurance. For the two-states case in which 1 is the state with no disability and 2 is the disability state, the situation is depicted in Figure 9.2. The point A indicates the initial (risky) endowment, and the points E u and E v indicate the equilibrium positions of decision-makers whose preference relations are  u and  v , respectively. The preference relation  u displays greater risk aversion than  v and its equilibrium position, E u , entails a more comprehensive coverage. References Anscombe,F.J.,andAumann,R.J.(1963). A Definition of Subjective Probability. Annals of Mathematical Statistics, 43, 199–205. Arrow,K.J.(1971). Essays in the Theory of Risk Bearing. Chicago: Markham Publishing Co. (1974). Optimal Insurance and Generalized Deductibles. Scandinavian Actuarial Jour- nal, 1–42. Cook,P.J.,andGraham,D.A.(1977). The Demand for Insurance and Protection: The Case of Irreplaceable Commodities. Quarterly Journal of Economics, 91, 143–56. 238 edi karni de Finetti,B.(1952). Sulla preferibilità. Giornale degli Economisti e Annali di Economia, 11, 685–709. Drèze,J.H.(1961). Les fondements logiques de l’utilité cardinale et de la probabilité subjective. La Décision. Colloques Internationaux de CNRS. (1987). Decision Theory with Moral Hazard and State-Dependent Preferences. In Es- says on Economic Decisions under Uncertainty, 23–89. Cambridge: Cambridge University Press. and Rustichini,A.(1999). Moral Hazard and Conditional Preferences. Journal of Mathematical Economics, 31, 159–81. (2004). State-Dependent Utility and Decision Theory. In S. Barbera, P. Diamon, and C. Seidl (eds.), Handbook of Utility Theory, ii. 839–92.Dordrecht:Kluwer. Eisner,R.,andStrotz,R.H.(1961). Flight Insurance and the Theory of Choice. Journal of Political Economy, 69, 355–68. Fishburn,P.C.(1973). A Mixture-Set Axiomatization of Conditional Subjective Expected Utility. Econometrica, 41, 1–25. Grant,S.,andKarni,E.(2004). A Theory of Quantifiable Beliefs. Journal of Mathematical Economics, 40, 515–46. Karni,E.(1985). Decision Making under Uncertainty: The Case of State-Dependent Prefer- ences. Cambridge, MA: Harvard University Press. (2006). Subjective Expected Utility Theory without States of the World. Journal of Mathematical Economics, 42, 325–42. (2007). A Foundations of Bayesian Theory. Journal of Economic Theory, 132, 167–88. (2008). A Theory of Bayesian Decision Making. Unpublished MS. and Mongin,P.(2000). On the Determination of Subjective Probability by Choice. Management Science, 46, 233–48. and Schmeidler,D.(1981). An Expected Utility Theory for State-Dependent Preferences. Working Paper 48–80, Foerder Institute for Economic Research, Tel Aviv University. and Vind,K.(1983). On State-Dependent Preferences and Subjective Probabili- ties. Econometrica, 51, 1021–31. Machina,M.J.,andSchmeidler,D.(1992). A More Robust Definition of Subjective Probability. Econometrica, 60, 745–80. Pratt,J.W.(1964). Risk Aversion in the Small and in the Large. Econometrica, 32, 122–36. Savage,L.J.(1954). The Foundations of Statistics. New York: John Wiley. Skiadas,C.(1997). Subjective Probability under Additive Aggregation of Conditional Pref- erences. Journal of Economic Theory, 76, 242–71. von Neumann,J.,andMorgenstern,O.(1947). Theory of Games and Economic Behavior, 2nd edn. Princeton: Princeton University Press. Wakker,P.P.(1987). Subjective Probabilities for State-Dependent Continuous Utility. Mathematical Social Sciences, 14, 289–98. chapter 10 CHOICE OVER TIME paola manzini marco mariotti 10.1 Introduction Many economic decisions have a time dimension—hence the need to describe how outcomes available at future dates are evaluated by individual agents. The history of the search for a “rational” model of preferences over (and choices between) dated outcomes bears some interesting resemblances and dissimilarities to the corresponding search in the field of risky outcomes. First, a standard and widely accepted model was settled upon. This is the exponential discounting model (EDM) (Samuelson 1937), for which the utility from a future prospect is equal to the present discounted value of the utility of the prospect. That is, an outcome x available at time t is evaluated now, at time t =0,as‰ t u(x), with ‰ aconstantdiscountfactor and u an (undated) utility function on outcomes. So, according to the EDM, x at time t is preferred now to y at time s if ‰ t u(x) > ‰ s u(y). We wish to thank Steffen Andersen, Glenn Harrison, Michele Lombardi, Efe Ok, Andreas Ortmann, and Daniel Read for useful comments and guidance to the literature. We are also grateful to the ESRC for their financial support through grant n. RES000221636.Anyerrorisourown. [...]... above, and let a ∼ b if and only if neither a b nor b a Assume that P and I are the asymmetric and symmetric parts, respectively, of a complete order on the set of pure outcomes X Finally, let ∗ (with ∗ and ∼∗ the corresponding symmetric and asymmetric parts, respectively) denote a complete preference relation (not necessarily transitive) on the set 9 See Pigou (1920), p 25 choice over time 251 of... three consecutive eight-month periods 15 See also Benzion, Rapoport, and Yagil (1989) for an example in the case of hypothetical choices, and Pender (1996) for actual choices choice over time 257 Rather, his evidence is consistent with subadditive discounting, as discussed in Section 10.3.3 10 .5. 3 Sources of Data and Other Elicitation Issues Since our focus is on the rationality or otherwise of decision-makers,... finds that “irrational” choices present a systematic pattern, not encountered previously Of these, the most striking are the association between certain types of rational choices and irrational choices (those who prefer a decreasing to a constant sequence are disproportionately concentrated among those who also prefer a constant to an increasing sequence) and the association between irrational choices of... for the possible combinations of gain, loss, and neutral frames with either a receipt or a payment For receipts, she found that implied discount rates are higher for small amounts ($40 and $200) than for large amounts of money ($1000 and $50 00), and for speedup than for delay (time horizons considered were 6 months and 1 year for the small amounts, and 2 and 4 years for the large amounts) From an economist’s... instance, Loewenstein and Prelec (1993) explain by a “preference for improving sequences” the behavior of a consistent See e.g Frederick, Loewenstein and O’Donoghue (2002), table 1 The literature on whether or not the payment of experimental subjects has an effect on response is huge See e.g Plott and Zeiler (20 05) ; Read (20 05) ; Hertwig and Ortmann (2001); Ortmann and Hertwig (20 05) , to cite just a few... the additional monetary increments x choice over time 259 information (e.g giving for each choice the implicit annual discount/interest rate implied by each choice and the prevalent market rate in the real economy) in order to reduce the extent to which subjects anchor their choices to their own experience outside the lab and unknown to the experimenters Coller and Williams (1999) found discount rates... constant and even decreasing sequences of outcomes over time (e.g Chapman 1996; Gigliotti and Sopher 1997; Guyse, Keller, and Epple 2002) The domain of choice seems also to be important (e.g there are differences in observed choices depending on whether or not the sequences are of money, or health or environmental outcomes; see e.g Chapman 1996 and Guyse, Keller and Epple 2002) Manzini, Mariotti, and Mittone... indifferent between receiving $ 15 immediately and $60 in a year, and at the same time indifferent between receiving $3000 immediately and $4000 in a year While the first choice (assuming linear utility) implies a 25 percent discount factor, the second implies a much larger implicit discount factor, of 75 percent Shelley (1993) carried out a study of both the delay/speedup asymmetry and the magnitude effect She... role when making choices over time (or under risk) He also shifts attention to the procedural aspects of decision-making He suggests that a decision procedure choice over time 249 he originally defined for choices under risk (in Rubinstein 1988) can be adapted to model choices over time, too Let ≈time and ≈outcome be similarity relations (reflexive and symmetric binary relations) on times and outcomes respectively... (y, s ) and (y, r ) ∼ (z, t), then (x, r ) ∼ (z, s ) This allows a different representation result: choice over time 243 Theorem 2 (Fishburn and Rubinstein 1982) If Order, Monotonicity, Continuity, Impatience, and Thomsen separability hold, and X is an interval, then there are continuous real-valued functions u on X and ‰ on T such that (x, t) (y, s ) ⇔ ‰(t)u(x) ≥ ‰(s )u(y) In addition, u(0) = 0 and u . e.g. Ainslie (19 75) ; Benzion, Rapoport, and Yagil (1989); Laibson (1997); Loewenstein and Prelec (1992); and Thaler (1981). It is important to stress that Harrison and Lau (20 05) have argued against. Economics, 31, 159 –81. (2004). State-Dependent Utility and Decision Theory. In S. Barbera, P. Diamon, and C. Seidl (eds.), Handbook of Utility Theory, ii. 839–92.Dordrecht:Kluwer. Eisner,R.,andStrotz,R.H.(1961) Insurance and the Theory of Choice. Journal of Political Economy, 69, 355 –68. Fishburn,P.C.(1973). A Mixture-Set Axiomatization of Conditional Subjective Expected Utility. Econometrica, 41, 1– 25. Grant,S.,andKarni,E.(2004).