12

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 ...


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 $\...


11

Well there is no opinion poll among economists on specifically this problem, but what can be judged from reaction of economists the consensus is that the Ole Peters paper is misguided and irrelevant at best. I think the economists' consensus was already very well and succinctly summed up by Doctor, Wakker and Wang you cite (and is an example of this ...


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

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, ...


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

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

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

\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

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 ...


7

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

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

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

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

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 ...


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

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} ...


6

Junior econ professor here. I saw Ole Peter's work and I was intrigued, so I actually looked into it to see if there was something original/insightful for me to learn. I even run a little simulation of what he calls the "St. Petersburg Paradox", to make sure I understood what the guy is actually talking about. Just to be clear, I didn't spend days ...


6

I’am the guy who wrote the short note that was mentioned in the original question. You can reach the note here. https://osf.io/preprints/socarxiv/axkfg/ I got interested in Peter's paper because of my interest in evolutionary games, where the question of whether the equilibrium selection process is or is not ergodic plays a crucial role. I agree that most ...


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

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 ...


5

Following @Pburg answer and the subsequent discussion in the comments, I wanted to post an alternative Machina Paradox I thought of. Although it might be less pervasive in real life, it seems stronger to me in the sense that it does not rely on some kind of complementarity between the "different" components of each outcome. Consider the following alternative ...


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

Suppose that the vector $W=\left(w_1,w_2,\dots,w_n\right)$ represents wealth in $n$ possible states. In addition, assume the probability of each state occurring is represented by the vector $\pi=\left(\pi_1,\pi_2,\dots,\pi_n\right)$. We can express this as the simple gamble: $$g = \left(\pi_1\circ w_1,\pi_2\circ w_2, \dots, \pi_n\circ w_n\right)$$ The ...


5

As pointed in the comments this was done by Ragnar Frisch. At least Barten and Böhm. (1982) as well as Johansen (1969) attribute these axioms to one of these two publications: Frisch, Ragnar (1926). "Sur un problème d'économie pure [On a problem in pure economics]". Norsk Matematisk Forenings Skrifter, Oslo. 1 (16): 1–40 Frisch,(1926). "...


5

$$ \left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right) $$ is the lottery where you get $1000$ with probability $2/100$ and $0$ with probability $98/100$. The expression $$ 20 \sim \left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right) $$ usually says that the decision maker is indifferent (in terms of preferences) between taking ...


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

Picking up my comment under this answer. One striking issue relevant to decisions not captured by expected utility is the framing effect discussed by Tversky and Kahneman (1981) and others. In their experimental study, they let two different (but with the same characteristics) groups choose between two options. Both groups actually face the same choices, ...


4

Let me mention another quite well-known one: The calibration theorem by Rabin (2000) and Rabin and Thaler (2002). The idea is that over small stakes individuals must be essentially risk-averse, but in reality they are not. Only assuming a weakly concave and strictly increasing utility function, Rabin shows that risk aversion on small stakes implies ...


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