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For example, if U1 (U1 , U2 ) = h1 U1 + h2 U2 + h3 U1 U2 , (10) where we have deleted the aj ’s for clarity, it is evident that h1 = 1 and h2 = h3 = 0 is the only viable solution. Another way to think about the appropriate attributes for U1 is that it can be a function of u1 and U2 so individual I1 ’s altruistic utility function could be represented by (11) U1 (u1 , U2 ) = h1 u1 + h2 U2 + h3 u1 U2 , and similarly for individual I2 , U2 (U1 , u2 ) = h4 U1 + h5 u2 + h6 U1 u2 . E. L. Keeney But (11) and (12) together lead to problems of double counting.

In fact, Luce et al. (2008 b) show that the above preference pattern is compatible with EM-EU for a sufficiently large negative A value. Such use of a negative A value makes sense as it corresponds to an aversion to gambling. 2 The Ellsberg Paradox We now provide an explanation of the Ellsberg paradox in terms of the entropymodified form of SEU given in (29). The Ellsberg (1961) paradox in coalesced8 form is of the following form with the choices between f vs. g and f vs. g where9 f = (x, R; 0, G ∪Y ) ≡ (x, p; 0, 1 − p) g = (x, G; 0, R ∪Y ) f = (x, R ∪Y ; 0, G) g = (x, G ∪Y ; 0, R) ≡ (x, 1 − p; 0, p) 8 If there are two (or more) branches (x,C), (x, D) in a gamble, with the common consequence x, then their coalesced form replaces the two by the single branch (x,C ∪ D).

The last one is a generalization of subjective expected utility (SEU) which replaces subjective probabilities with nonseparable representation of comparative beliefs first discovered by Fishburn (1983a and b). Y. J. Brams et al. ), The Mathematics of Preference, Choice and Order: Essays in Honor of Peter C. Fishburn, Studies in Choice and Welfare, c Springer-Verlag Berlin Heidelberg 2009 39 40 Y. Nakamura There may be three formulations in the literature, discussed in the next section, to arrive at axiomatizations of preferences in decision making under uncertainty: pure-act formulation (Savage’s approach), lottery-act formulation, and act-lottery formulation.