On Elements of Chance

One aspect of the utility of gambling may evidence itself in failures of idempotence, i.e., when all chance outcomes give rise to the same consequence the `gamble' may not be indifferent to its common consequence. Under the assumption of segregation, such gambles can be expressed as the joint receipt of the common consequence and what we call `an element of chance', namely, the same gamble with the common consequence replaced by the status quo. Generalizing, any gamble is indifferent to the joint receipt of its element of chance and a certain consequence, which is called the `kernel equivalent' of the gamble. Under idempotence, the kernel equivalent equals the certainty equivalent. Conditions are reported (Theorem 4) that are sufficient for the kernel equivalents to have the kind of utility representation first discussed by Luce and Fishburn (1991), including being idempotent. This utility representation of the kernel equivalents together with the derived form of utility over joint receipts yields a utility representation of the original structure. Possible forms for the utility of an element of chance are developed.

[1]  R. Pope.  The delusion of certainty in Savage's sure-thing principle , 1991 .

[2]  R. Pope.  Attractions to and Repulsions from Chance , 1998 .

[3]  L. J. Savage,et al.  The Foundations of Statistics , 1955 .

[4]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[5]  J. Conlisk.  The utility of gambling , 1993 .

[6]  R. Luce,et al.  When Four Distinct Ways to Measure Utility Are the Same , 1996 .

[7]  R. Luce.  Utility of Gains and Losses: Measurement-Theoretical and Experimental Approaches , 2000 .

[8]  A. Tversky,et al.  Prospect Theory : An Analysis of Decision under Risk Author ( s ) : , 2007 .

[9]  George Wu.  An empirical test of ordinal independence , 1994 .

[10]  Peter C. Fishburn,et al.  A simple model for the utility of gambling , 1980 .

[11]  M. Birnbaum,et al.  Testing Descriptive Utility Theories: Violations of Stochastic Dominance and Cumulative Independence , 1998 .

[12]  I. Ritov,et al.  Emotion-based choice , 1999 .

[13]  I. Ritov,et al.  Decision Affect Theory: Emotional Reactions to the Outcomes of Risky Options , 1997 .

[14]  R. Luce.  Coalescing, Event Commutativity, and Theories of Utility , 1998 .

[15]  Robert Sugden,et al.  Testing for juxtaposition and event-splitting effects , 1993 .

[16]  P. Fishburn,et al.  Rank- and sign-dependent linear utility models for finite first-order gambles , 1991 .

[17]  J. Quiggin.  Generalized expected utility theory : the rank-dependent model , 1994 .

[18]  Ulrich Schmidt,et al.  A Measurement of the Certainty Effect. , 1998, Journal of mathematical psychology.

[19]  P. Fishburn,et al.  A note on deriving rank-dependent utility using additive joint receipts , 1995 .

[20]  R. Pope.  Debates on the Utility of Risk. A Look Back to Move Forward , 2001 .

[21]  J. Quiggin,et al.  Generalized Expected Utility Theory. The Rank-Dependent Model , 1996 .