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There is a problem in how you translate completeness into behavior. Let $R$ be any binary relation, representing preferences, on a set $X$ of alternatives and $A\subseteq X$ be a nonempty set of alter …

Trivially, the decision maker can ignore all additional information and simply play the same action they would play without any information. The optimal strategy must be then at least as good, on aver …

Note first that for each nontrivial (more than one point) compact interval $I$, there exists a strictly increasing function from $\mathbb{R}$ to $I$. Let $\langle q_1,q_2, q_3,\ldots\rangle$ be an en …

Here is a partial answer: The two conditions are equivalent if preferences are strongly monotone. To see this, let $x_Iy\succ x_Iy'$. Let $e=(1,1,\ldots,1)\in\mathbb{R}^n$. By strong monotonicity and …

Not every probability distribution comes from a density. In particular, point masses have no density (with respect to Lebesgue measure). Here, being perfectly informed requires that the conditional di …

This is a special case of Theorem 1 of [Richter, Marcel K. "Revealed preference theory." Econometrica (1966): 635-645.]

Here is an example that shows that certainty equivalents need not exist: Let $f:\mathbb{R}\to (0,1)$ be an increasing bijection. Let $0<\alpha< f(1)-f(0)$ Define $u:\mathbb{R}^\mathbb{N}\to\mathbb{R}$ …

Here is the sketch of a proof. All we need is that every continuous weak order on each $X_i$ admits a continuous utility representation. One sufficient condition is that each $X_i$ is a connected sep …

The other answer explained why the result is trivial; here is why it is not true without the modifier "on the range" and under the most literal reading. Consider $[0,1]\cup(2,3]$ with the usual order. …

No. Let $X=[-1,1]$ and $u:X\to\mathbb{R}$ be given by $u(x)=-|x|$. Then, $u$ represents strictly convex preferences. Moreover, $0.5\succ -1$. Now, the only points that $0.5$ is revealed preferred to a …

Let $\succeq$ be a relation on $\mathbb{R}^l_+$ such that $x\gg y$ implies $x\succeq y$ for all $x,y\in\mathbb{R}^l_+$, and such that all upper contour sets are closed. Then $x\geq y$ implies $x\succe …

A preference such that $p\succeq q$ for all lotteries $p$ and $q$ clearly does (!) have a cardinal utility representation: The utility function over outcomes must be constant, and this is clearly also …

The paper [Reny, Philip J. "A characterization of rationalizable consumer behavior." Econometrica 83.1 (2015): 175-192.] answers the question. Here is the abstract: For an arbitrary data set $D =\{(p …

If in equilibrium, a player "chooses a mixed strategy" that plays $H$ and $T$ with positive probability, $H$, and $T$ must be both optimal choices. It is a standard result that for a (subjective or ob …

Many people accept the axiomatizations of expected utility as normatively appealing, especially in contexts of pure risk. For people with this view, rational decision-makers should behave in accordanc …