Regret theory and measurable utility

19

Economics Letters 12 (1983) 19-21 North-Holland Publishing Company

REGRET THEORY

Graham Umoersity Received

LOOMES

AND MEASURABLE and Robert

UTILITY

SUGDEN

of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK 14 September

1982

This paper examines the relationship between regret theory and a novel set of axioms for choice under uncertainty, discusses the extent of complementarity and points to certain important distinctions between the two approaches.

In a recent paper, Fishburn (1981b) described a set of five axioms which generalise von Neumann-Morgenstern utility theory in a particular way. It may be interesting to note the relationship between Fishburn’s five axioms and a rather different, psychologically-based approach to choice under uncertainty which we have called ‘regret theory’. Regret theory is described more fully elsewhere [Loomes and Sugden (1982)j but its essential features, as it relates to Fishburn’s axioms, are as follows. An individual chooses between pairs of actions, A, and A,, each action being an n-tuple of state-contingent consequences. The probability of the j th state occurring is p,, and, in this event, the consequence of choosing the ith action is denoted xi,. It is proposed that if x,, is experienced other than as a result of choice (e.g., by imposition, gift, etc.) the individual derives some level of ‘choiceless’ utility. This is represented by the function C(.), unique up to an increasing linear transformation, which assigns a real-valued utility index number c,] to every conceivable x,,. However, if the consequence is experienced as a result of choice, there may be a further consideration. The choice of A, implies the simultaneous rejection of A, so that if the jth state occurs, the individual will experience x,, but, as a result of the choice, will simultaneously miss out on xk,. In such cases it is suggested that c,, may be modified - by feelings of regret if c,, < ck,, or by rejoicing if c,] > ck,. In a restricted form of the model, the decrement/increment of utility associated with regret/rejoicing is represented by the function R(.), 0165-1765/83/0000-0000/$03.00

0 1983 North-Holland

G. Loomes, R. Sugden / Regret theory and measurable utility

20

where R(c,,, ck,) is a strictly increasing three-times differentiable function of (c,, - cL ,), and where R(0) = 0. On this basis we denote the modified utility of choosing A, and rejecting A, under the jth state as WI:,, where mf, = c,, + R(c,, - ck,). Hence the expected value of this decision is E,” = Cy=, P,mf,. Following Fishburn’s notation (by which strict preference is denoted by >0 and indifference by =0) we propose that A, A, iff E,&$ EL. Fishburn’s axioms are couched in terms of prospects (i.e., probability distributions of consequences) rather than actions; but since he (1981a, p. 9) limits himself to the special case where prospects are statistically independent, it is possible to represent any pair of prospects by a unique 2 x n matrix of state-contingent consequences. Given the assumption of statistical independence, let us consider the relationship between Fishburn’s five axioms and regret theory. According to regret theory, x > a y iff E,’ > E.,“, and y > 0 z iff E,; > E,“. But if E,’ > E_t and E_,f> Ez, then aE,” + (1 - a)E.G = aE: + (1 - a)E,Y forsomeO,y,y>,z]~y=,cux+(l -a)zforsome 0 < (Y< 1, which is Fishburn’s continuity axiom A.l. A similar argument shows A.2 (linear dominance) to be entailed by regret theory. Fishburn’s symmetry axiom A.3, which perhaps does not have the same intuitive appeal as A.1 and A.2, can also be shown to be entailed. According to regret theory, y =,,fx + iz iff

so

and

~E;+~E~-~EY-_E,L‘=O, X

Az+(l

-X)x=,+z+fy

+XE,L’+f(l

-X)E;+f(l

-fhE;-+(l

Combining

(1)

iff -X)E,y

-X)E,“-;(l

-X)E;=O.

(1) and (2) and rearranging

(2)

gives

+XE,Y+f(l-X)E,Y++(l-h)E;=+hE;++(l--)E;+f(l-X)E: (3)

which is true iff Xx + (1 - X)z =,,ix + ty. Thus A.3 holds. However, E,’ = E,” and E,’ = E,’ does not logically imply that E: = EC. Nor does E,’ = E;’ imply that +E,’ + +E: + :E,’ = +E,” + +E,” + a E,‘. SO

G. Loomes, R. Sugden / Regret theory and measurable utility

21

regret theory entails neither A.4 (the transitivity of indifference) nor A.5 (the Herstein-Minor independence axiom). Experimental evidence suggests that A.5 is consistently violated by many people. Let x be the prospect (a,, II,; 0, 1 - II,), i.e., a gamble offering the increment of wealth a, with probability II,. Let y = (al, I12; 0, 1 - II,), let z = (O,l), and let a,, a2 > 0. A.5 entails that if x =0 y then II,/2; 0, 1 - II,/2). Thus indifference (a,, 1-1,/2); 0, 1 - 17,/2) =da,, between two gambles is preserved when the probabilities attached to their prizes are scaled down by a factor of one-half. This is at variance with the widely observed common ratio effect, whereby individuals become less risk averse as the probabilities attached to increments of wealth are scaled down [Kahneman and Tversky (1979)]. Given certain assumptions about R(.), regret theory predicts this effect [see Loomes and Sugd& (1982)]. Finally, we note that while axioms A. 1-A.3 are compatible with regret theory, and can provide explanations for certain violations of conventional von Neumann-Morgenstern utility theory, the requirement of statistical independence constitutes a definite restriction. For example, the isolation effect in two-stage gambles [see Kahneman and Tversky (1979)] is predicted by regret theory, but is beyond the scope of Fishburn’s five axioms.

References Fishburn, P.C., 1981a, Nontransitive measurable utility, Bell Laboratories economic discussion paper, no. 209, Sept. (Murray Hill, NJ). Fishburn, P.C., 1981 b, An axiomatic characterization of skew-symmetric bilinear functionals, with applications to utility theory, Economics Letters 8, no. 4, 311-313. Kahneman, D. and A. Tversky, 1979, Prospect theory: An analysis of decision under risk, Econometrica 47, 263-291. Loomes, G. and R. Sugden, 1982, Regret theory: An alternative theory of rational choice under uncertainty, Economic Journal, Dec., forthcoming.