Abstract
This paper extends our earlier work on reverse Bayesianism by relaxing the assumption that decision makers abide by expected utility theory, assuming instead weaker axioms that merely imply that they are probabilistically sophisticated. We show that our main results, namely, (modified) representation theorems and corresponding rules for updating beliefs over expanding state spaces and null events that constitute “reverse Bayesianism,” remain valid.
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Notes
We thank Graeme Doole for calling our attention to this paragraph in Lancaster’s article.
Fishburn (1970) Ch. 12 discusses a construction of a state space along similar lines. He does not, however, discuss an extension of the set of acts to include conceivable acts.
Formally, \(p\in {\Delta } \left ( C\right ) \) is a function \(p:C\rightarrow \left [ 0,1\right ] \) satisfying \({\Sigma }_{c\in C}p\left ( c\right ) =1.\) Notice that with this definition of Δ(C) we have that, for any \(C\subset C^{\prime } \), any p∈Δ(C) is also an element of \({\Delta } (C^{\prime } )\) with p(c)=0 for all \(c\in C^{\prime } -C\). Likewise, \(q\in {\Delta } (C^{\prime })\) is an element of Δ(C) if q(c)=0 for all \(c\in C^{\prime } -C\).
Fishburn’s (1970) notion of excluded states is analogous to our non-feasible states.
Below, \(f^{\prime } =f\) on an event E means that \(f^{\prime } (s)=f(s)\) for all s∈E (i.e., it is defined pointwise for the states in E).
A function V is strictly monotonic if V(p)≥V(q) whenever p dominates q according to first-order stochastic dominance, with strict inequality in the case of strict dominance, and V is mixture continuous if \(V(\alpha p+\left ( 1-\alpha \right ) q)\) is continuous in α for all p and q.
Suppose that ∣F∣=r and \(\mid F^{^{\prime } }\mid =k>r.\) Let \(s=\left ( c_{1},...,c_{k}\right ) \in C^{F^{\prime }}\), then \(\boldsymbol {P}_{C^{F}}\left ( s\right ) =\left ( c_{1},...,c_{r}\right ) \in C^{F}.\)
Gul (1991) axiomatized a model of disappointment aversion under risk. The argument here is analogous to that of Gul except that here it is cast in terms of uncertainty.
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We thank Graeme Doole, Asen Kochov, and Jacob Sagi for comments.
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Karni, E., Vierø, ML. Probabilistic sophistication and reverse Bayesianism. J Risk Uncertain 50, 189–208 (2015). https://doi.org/10.1007/s11166-015-9216-5
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DOI: https://doi.org/10.1007/s11166-015-9216-5