Mathematical Social Sciences 40 2000 277–295 www.elsevier.nl locate econbase
Separable and additive representations of binary gambles of gains
a , b
R. Duncan Luce , A.A.J. Marley
a
Institute for Mathematical Behavioral Sciences , University of California, Irvine, CA 92697-5100, USA
b
Department of Psychology , McGill University, Montreal, Canada
Received 1 April 1999; received in revised form 1 November 1999; accepted 1 November 1999
Abstract
Two approaches are taken to a new utility representation of binary gambles that is called ‘‘ratio rank-dependent utility.’’ Both are based on known axiomatizations of a ranked-additive representa-
tion of consequence pairs x, y in binary gambles x, C; y of gains with C held fixed and of a separable one of the special gambles x, C; e, where e denotes the status quo. The axiomatized
version imposes the condition of status-quo event commutativity to get a functional equation that leads to the result. The other assumes, but does not axiomatize, a separable representation of the
C; y portion of the gamble. These assumptions lead to two difficult functional equations that are solved in the mathematical literature, but the former only under the assumption that the function is
twice differentiable. Three behavioral conditions are shown to force this new utility representation to reduce to the standard rank-dependent utility one for gains. They are co-monotonic trade-off
consistency, ranked bisymmetry, and segregation, the latter requiring the addition of an operation of joint receipt.
2000 Elsevier Science B.V. All rights reserved.
1. Introduction
We begin with the following primitives. The set consists of pure consequences involving no uncertainty. Examples are goods purchased in reputable stores, money
obtained from banks, etc. Let e [ denote the consequence in which nothing happens, the status quo. Let E denote a chance phenomenon or experiment, in the sense used by
1
probabilists and statisticians , with universal event E and an algebra of events. Thus,
E
Corresponding author. Tel.: 11-949-824-6239. E-mail address
: rdluceuci.edu R. Duncan Luce.
1
Parzen 1960 used the phrase ‘‘chance phenomenon’’ whereas Feller 1950, Hayes 1963, and Rotor 1997 speak of an ‘‘experiment.’’ This use of the term ‘‘experiment’’ is only loosely coupled to the uses in
the experimental sciences. 0165-4896 00 – see front matter
2000 Elsevier Science B.V. All rights reserved.
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278 R
. Duncan Luce, A.A.J. Marley Mathematical Social Sciences 40 2000 277 –295
if C [ , then C 7 E. We do not assume that a probability measure is associated with
E
the experiment. Examples of the kinds of chance generating experiments we typically run in laboratory experiments are spinners on a partitioned pie and dice, both of which
have a well known probability description, or drawing a colored ball from an opaque urn that contains some unknown mix of two colors of balls.
From these ingredients we may form binary first order gambles by assigning ]
] ]
consequences in to event partitions
hC, Cj where C [ and C 5 E\C. If x is
E
] ]
assigned to C and y to C, then we write the gamble as x, C; y, C or, for simplicity of writing, just as x,C; y. Let denote a weak preference order over consequences and
1 2
gambles. Let denote all such binary first-order gambles of gains , i.e., those with
xe and ye. It should be noted that the gambles cannot really be viewed as functions because we shall assume that the rank order of the consequences matters. So we will be
required to make explicit certain assumptions those below in which | connects the two sides that would be automatic were we dealing with functions.
In addition to first order gambles, we will also make use of second-order ones in
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which the experiment is run independently twice , and the consequences that are attached to the first running are themselves first-order gambles. A typical example is x,
C; y, B; u, D; v. So, in the first experiment if B occurs one receives the gamble x, C; y and that is resolved by running the experiment again yielding x if C occurs and y
] otherwise. If, however, B occurs, then the second gamble is u, D; v and it is resolved
by running the experiment a second time. There will be no reason to go to higher order gambles.
A detailed discussion of this type of framework is provided by Luce 2000.
2. Singly separable additive utility
