11

It is. Prior to continuity, which is a property of the preference relation, the preference relation $\succsim$ itself has been defined to be a binary relation that is characterized by transitivity, and, to begin with, by completeness. Then if $S_1\cup S_2 \neq [0,1]$, it means that there exist some values of $\alpha$ somewhere in $[0,1]$, call them $\...


10

this paper http://else.econ.ucl.ac.uk/papers/uploaded/243.pdf (Choi 2007) has a nice state of the art experiment that deals with rationality and expected utility is a special case of it. In general only 17% of consumers are compatible with rationality ergo the remaining part cannot be expected utility maximizers. Quah has a nice paper on the revealed ...


9

No, not necessarily. Without the independence axiom (or something else to replace it) there is not much you can infer about preferences over (non-degenerate) lotteries from knowing preferences over outcomes only. For instance, let $p^L_n$ be the probability of outcomes $n \in \{1, 2, 3\}$. Then preferences over lotteries $\succeq^*$ represented by the ...


9

The name for the amount $56.25 is certainty equivalent. The expected utility for the individual from taking the bet is calculated as follows: $$E[U]=\frac12U(100+125)+\frac12U(100-100)=75$$ Suppose the individual can pay an amount of money $x$ so that she can avoid taking the bet (which leads to expected utility $75$). What's the maximum amount of money $x$ ...


9

In order to understand this problem, I will work through the generic case. Say that a user had generalized quadratic (Bernoulli) utility, similar to your problem: $$u(x) = \beta x^2 + \gamma x$$ and suppose that there is a distribution for the outcome of $x$, denoted $F(x)$. Thus, utility over this distribution is equal to $$\begin{align} \int u(x) \text{...


8

The primary literature concerned with this type of question (at least where classical results break down) is behavioral economics. There's a great general compilation of papers put together by the Russell Sage Foundation called the "Behavioral Economics Reading List" that includes, among other things, a General Introduction section with overview papers by ...


8

Consider the version of the paradox from Wikipedia: A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 2 dollars and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if a tail ...


8

\begin{eqnarray*} \displaystyle U(L) & = &\sum_{s=1}^{S}\pi_s U(Y_s) = \sum_{s=1}^{S} \left(-\frac{1}{2}\pi_s(\alpha - Y_s)^2\right) = -\frac{1}{2}\sum_{s=1}^{S} \left(\pi_s(\alpha^2 + Y_s^2-2\alpha Y_s)\right) \\ &=& -\frac{1}{2}\left(\alpha^2\sum_{s=1}^{S} \pi_s + \sum_{s=1}^{S} \pi_sY_s^2-2\alpha \sum_{s=1}^{S} \pi_sY_s\right) = -\frac{1}{...


7

Adding to the list of paradoxes, consider Machina's paradox. It is described in Mas-Colell, Whinston and Green's Microeconomic Theory. A person prefers a trip to Paris to watching a television program about Paris to nothing. Gamble 1: Win a trip to Paris 99% of the time, the television program 1% of the time. Gamble 2: Win a trip to Paris 99% of the time, ...


6

I'm somewhat surprised that no one has linked to this paper: Backus, Routledge, and Zin (2004) Exotic Preferences for Macroeconomists (this version has some fixed typos, vs the NBER print). Their abstract is concise and extremely descriptive: We provide a user's guide to 'exotic' preferences: nonlinear time aggregators, departures from expected utility, ...


6

This won't get at individual choice, but how about evolutionary approaches? Perhaps this isn't what you are looking for, but one way to model decisions is to wander from the rational paradigm entirely. All changes in behavior are driven by natural selection, and so an equilibrium is based on stability. In a symmetric normal form game, an evolutionarily ...


6

No, I would not say that this resolves the Machina paradox, because it is exactly the same as the Machina paradox: the paradox does indeed require from you to look at the three possible outcomes. The M-C/W/G book discuss only the $B$ and $C$ outcomes because it is there where the paradox focuses on whether a violation of the axiom of independence may happen. ...


6

Is there any (economic) rational for the first-order expansion of the RHS? And for its different neighborhood evaluation? As for your first question: This is a purely mathematical tactic in order to obtain an (approximate) equation for $R$. The expansion of first order on the RHS is motivated by this fact, i.e. to bring $R$ alone "in the surface". The ...


6

Not all cdf’s have a density function, (for example if $F$ is not differentiable). However, when they do have a density, the notation $dF(z)$ is equivalent to $f(z)dz$. When performing integrals. However, even if the density does not exists, you can still write the expectation using the notation $dF(z)$. The details of what it actually means and the subtle ...


6

Gains and losses presuppose a reference point, which is not a feature in standard expected utility theory. In this theory, the only argument in the utility over wealth is $w$, the absolute level of wealth. A common form of utility function is the constant relative risk aversion (CRRA) form: \begin{equation} u(w)=\frac{w^{1-\rho}}{1-\rho}, \end{equation} ...


5

I think I've found an answer to my question, in this excerpt from Nobel laureate John C. Harsanyi's 1994 paper "Normative validity and meaning of von neumann-morgenstern utilities", presented at the Ninth International Congress of Logic, Methodology and Philosophy of Science. Harsanyi starts by proving the same lemma that Alecos proved in his answer, namely ...


5

The utility function is a representation of preferences, which are traditionally inferred from choices. Preferences come before utility. I would not call the connection between utility and preferences causality, just a mathematical relationship. Risk aversion (risk preference) is not connected to discounting, which measures time preference. It does not make ...


5

This is perhaps a good opportunity to point out that the "certainty equivalence" concept means one thing in microeconomics/choice under uncertainty theory, while it means something different in macroeconomics. Microeconomics/choice under uncertainty The Certainty Equivalent of a lottery/gamble, is the amount of wealth which, if given with certainty, ...


5

Here is an "expected utility maximization/ game theoretic" approach to the matter (with a dash of set-theoretic probability). In such a framework, the answers appear clear. PREMISES We are told in absolute honesty that, for $x$ a strictly positive monetary amount, the following two tickets were placed in a box : $\{A=x, B= 2x\}$ with assigned ...


4

I think you are correct that this solves the Machina Paradox but I am not sure I would associate your reformulation of the model with the idea of state-dependent utility. State-dependent utility is more than a mere extension or modification of the set of outcomes of the expected utility model. To make sense of state dependent utility, you need to have a ...


4

A first price standard and reverse auction are formally equivalent to each other, and the same method can be used to solve both: First Price Auction In a first price auction, $n$ bidders choose their bid, $b_i$, as a function of their value $v_i$ (distributed according to $F$. They seek to maximise their expected payoff: $$[v_i-b_i(v_i)]\Pr(b_i\geq\max_j ...


4

Your notation is a bit misleading: it would be better to write $\mathbb{E}u(p)$ or $U(p)$ for the expected-utility associated with $p$ instead of $u(p)$, and $u(\mathbb{E}p)$ for the utility of the expected value of $p$. Formally $u$ is defined on $X$ and not on $\Delta(X)$. Regarding your proof, it seems to me that: $(i)$ you don't explain how to find $s$; ...


4

Have there been any empirical attempts to estimate the value of being taught specific skills - for example, phonics or solving algebraic equations? If I may be brazen enough to challenge the basis of this question. We first need to have an economic criteria to evaluate a subject in a curriculum before we start analysing its relevance to the labour market . ...


4

One must first distinguish two different senses in which the Allais Paradox can be seen as a "contradiction" of vNM independence; these correspond to the two different interpretations which can be given to any model in Decision Theory. (von Neumann-Morgenstern (vNM) utility theory is one such model.) According to the descriptive interpretation, a model in ...


4

Yes, there is such an interpretation in Section 3 of the original paper by Pratt: Pratt, J. (1964). Risk Aversion in the Small and in the Large. Econometrica, 32(1/2), 122-136. Under some regularity conditions, the coefficient of absolute risk aversion approximates the risk premium divided by half the variance for a small actuarily fair gamble. In ...


4

There are many such utility functions. Most commonly: \begin{equation} u(x)=L-\mathrm e^{-ax},\tag{1} \end{equation} where typically $a>0$. Or \begin{equation} u(x)=L-(x-a)^2.\tag{2} \end{equation} Both of these are uniformly bounded below $L\in\mathbb R$ and both tend to $L$ as the functions approach their respective maximum.


4

The von Neumann-Morgenstern (vNM) utility function takes the form \begin{equation} U(p,x)=\sum_{i=1}^np_iu(x_i) \end{equation} where $x=(x_1,\dots,x_n)$ with $x_i$ being the (monetary) payoff associated with outcome $i$ and $p=(p_1,\dots,p_n)$ with $p_i$ being the probability that $i$ occurs. In behavioral economics, generalizations of the vNM utility ...


4

Pick any set of non-negative $p_1,\dots,p_n$ such that $p_1+\cdots+p_n=1$. The convex combination of degenerate lotteries $L^1,\dots,L^n$ with the $p_i$'s can be written as \begin{align} p_1L^1+\cdots+p_nL^n&=p_1(1,0,\dots,0)+\cdots+p_n(0,\dots,0,1)\\ &=\bigl(p_1(1)+p_2(0)+\cdots+p_n(0),\;\dots,\;p_1(0)+\cdots+p_{n-1}(0)+p_n(1)\bigr)\\ &=(p_1,\...


3

The short answer seems to be yes your example violates expected utility... It mostly seems to me like a simple transformation of the first example you gave (but you got rid of the red balls). As mentioned in other answers expected utility is not equipped to handle uncertainty because it deals with taking expectations and expectations cannot be computed when ...


3

There is a typo in the figure that introduces some confusion in the previous answer, which is basically wrong. Based on the numbers and the figure, the utility is such that $$u=\sqrt{x},$$ so $$E[u]=\frac{1}{2} u(100+125) + \frac{1}{2} u(100−100)= \frac{1}{2} u(225) =\frac{1}{2} \sqrt{225} = 7.5$$. By definition, the risk premium (R) must satisfy the ...


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