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I strongly suspect that an emerging important area for applications of measure theory will be in approximate dynamic programming techniques. Approximate dynamic programming (aka "reinforcement learning" in the computer science literature) has been the direction of research work in the last ~10-20 years of the dynamic programming literature. Economics is only ...
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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 $\...
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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 ...
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I think CompEcon covered most of the points that I was going to mention. Just a few last thoughts:
1) Why are Epstein-Zin preferences important?
The preferences are important because they allow you to separate two of the dimensions along which people care about their allocations; namely, risk aversion and intertemporal substitution.
Additionally, one ...
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This is only a quick answer, unfortunately. The key intuitive insight for Epstein-Zin is that they separate two distinct properties of preferences: risk aversion ("I'd prefer less uncertainty to more uncertainty*") and intertemporal substitution ("I may want to shift consumption forward or backwards in time**").
In the very popular Constant Relative Risk ...
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There are definitely economic justifications for playing the lottery, even if all (I hope) players understand that it is unlikely to pay off.
One such justification is that what you actually buy when purchasing a lottery ticket is the fantasy of winning.
Here are a few sources. Lotterys are relatively well understood in economics.
The Economics of ...
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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 ...
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In the below figure, CDF $F(\cdot)$ is first-order stochastically dominated by $G(\cdot)$. But $X_1$ and $X_2$ fall within the support of both distributions. So it would be possible to draw $X_1$ from $F$ and $X_2$ from $G$, or to draw $X_2$ from $F$ and $X_1$ from $G$.
More generally, if $X_G$ is a draw from $G$ and $X_F$ is a draw from $F$ then $X_F-X_G$ ...
8
As often with models embodying some form of "irrationality" (whatever that means), HD does a great job at matching a whole lot of behaviors, but leaves room for rather annoying Dutch Book situations (also know as "money pump" situations). These suggest that HD might generate some inaccurate predictions, and induce undesirable behaviors when included in ...
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This was too long for comment. "Post 1960" seems an arbitrary and very high bar for an applied field, including micro theory. Most of the topics you name would not be considered contemporary mathematics. For example, measure theory started with Lebesgue's thesis and is over a century old. Topology is even older and started with Poincare, who introduced ...
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To understand why $\alpha$ must be constrained in $(0,1)$, one has to contemplate the meaning of the expression
$$\alpha L$$
when $L$ is a "lottery". How is a lottery denoted mathematically? Authors do not agree on that: for example, the way Jahle and Reny define a lottery (a "gamble" in their terminology), a lottery can be written as a vector whose ...
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As a first remark: the Anscombe-Aumann axioms, in particular Independence, are defined over acts taking the state space to a linear space (generally simple lotteries over consumption objects). Even when we consider the restriction of the model to purely subjectively uncertain acts, we still need to employ the full model or we will lose information.
That ...
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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 objective) expected utility maximizer, randomizing can only be optimal if it is over pure optimal choices. This is a direct consequence of expected ...
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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, ...
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I want to add some other justifications for buying lottery tickets:
general risk-seeking behavior (which is probably pretty rare)
risk-seeking when it comes to low monetary values (Cumulative prospect theory)
Cognitive biases, e.g.,
with respect to probabilities (over-weighting low probabilities, Knightian unicertainty)
the money (under-weighting small ...
7
I do not know about social choice, but for utility representations I think the two most cited books are "Convex analysis" by Rockafellar and "Infinite Dimensional Analysis: A Hitchhiker's Guide" by Alliprantis and Border. They contain most (if not all) results on convex analysis and functional analysis used by economists.
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The first order stochastic dominance relation is convex.
An easy way to prove this is to use the property that a cdf $F$ FOSD another cdf $G$ if and only if $F(x)\le G(x)$ for all $x$. That is, $F$ FOSD $G$ if and only if the graph of $F$ is never above the graph of $G$. It is then easy to show that $F$ is never above any convex combination $H(x)=\alpha F(...
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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, ...
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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. ...
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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
...
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A utility function is additively separable if it can be written as:
$U(x,y) = u(x) + v(y)$
Examples:
*$U(x,y) =ax + by$ is additively separable by inspection.
*$U(x,y) = ax + bx2 + cy$ is also.
*$U(x,y) = x^a y^b$ is additively separable, because you can write it as $U(x,y) = log(x^a)+log(y^b)=alog(x)+b log(y)= u(x) + v(y)$
*$U(x,y) = \frac{xy}{x+y}$ ...
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It's well known that if $\succsim$ satisfies independence, then it is also convex.
Since $\succsim$ satisfies independence,
$L\succsim L^{'} \iff \alpha L+(1-\alpha)L^{''}\succsim \alpha L^{'}+(1-\alpha)L^{''}$ for all $\alpha \in [0,1]$ and $ L, L^{'}, L^{''}\in \mathfrak{L} $
Convexity requires:
$L\succsim L^{''}$ and $L^{'}\succsim L^{''} \...
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Measure theory is widely used in the problem of fair division (aka "cake-cutting"). See the many papers about fairness in economics journals.
For a particular example, see Tatsuro Ichiishi and Adam Idzik, "Equitable allocation of divisible goods", JME 1999.
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Loeb spaces have been used to model situations with a continuum of agents. See http://eml.berkeley.edu/~anderson/Book.pdf and the chapters by Sun on economic applications in the book Nonstandard Analysis for the Working Mathematician.
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The term bounded rationality was introduced by Herbert Simon. He wrote
"The term, bounded rationality, is used to designate rational choice
that takes into account the cognitive limitations of both knowledge
and cognitive capacity. Bounded rationality is a central theme in
behavioral economics. It is concerned with the ways in which the
actual ...
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Yes this is a blossoming area of research in economics and it spans approaches from psychology to game theory, social theory, and cultural biases.
A lot of these approaches fall under the rubric of "Behavioral Economics". Unlike classical market economics which often begins with assumptions around individual and collective rationality, behavioral economics ...
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After the exchange with the OP in my other answer, let's work a bit with his approach.
We have a discrete random variable $X$ with finite support, $X = \{x_1,...,x_k\}$, and probability mass function (PMF), $\Pr(X=x_i)=p_i, i=1,...,k$
The values in the support of $X$ are also inputs in a real-valued cardinal utility function, $u(x_i) > 0\; \forall i$....
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A function is additively separable in its arguments if it has the form
$$f(x,y) = g(x) + h(y)$$
This means that the cross partials are zero, and so there is no "cross" effect of the one argument over the marginal effect that the other has on the value of the function. Since marginal effects are at the very heart of Economics (see here), assuming additive ...
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There is a series of papers that address precisely this question. The most famous ones are probably Walker and Wooders (2001) and Chiappori, Levitt, and Groseclose (2002) that deal with penalty kicks and tennis serves. Both papers conclude that the behavior of professional athletes is consistent with them playing g a mixed strategy equilibrium. A more recent ...
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The real-life question is "how do you persuade people to use mixed strategies"?
To stick with your example, Consider a person that has to make a binary choice $(H, T)$, and, after contemplation, they conclude that the optimal strategy is the mixed strategy $(2/3, 1/3)$.
I have never know of anyone putting two red and one blue ball in a vase and ...
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