Questions tagged [choice-theory]

a conglomerate of models and results concerning the aggregation of individual choices into collective choices

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4 votes
1 answer
86 views

Choice Function and Empty Set [duplicate]

Can the choice function of a non-empty and finite set be the empty set? Or is this by definition of the choice function impossible? Does there need to be always at least one winner if we evaluate non-...
2 votes
0 answers
83 views

How to find the substitution and income effects?

The usual definition of Substitution Effect (pg. 30; also found in Varian) tells that the Slutsky SE is $x(p_x', I') - x(p_x,I) = x(2,15) - x(1,10) = \frac{15}{2 \cdot 2} - \frac{20}{2 \cdot 1} = -6....
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2 votes
0 answers
22 views

Uncompensated and compensated demand functions

I came across this lecture note online and some of the points below confuse me. I have added the part that confuses me as an image and here is the lecture note for further reference, if needed. ...
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3 votes
1 answer
230 views

Hick's and Slutsky's approaches lead to different income effects. Why?

Suppose a cup of coffee and a plate of beans are sold at € 1 and € 3 respectively during the winter. In summer, the government decides to remove the subsidy on coffee and its new price per cup goes up ...
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1 vote
0 answers
38 views

Lump sum income grant is better than a subsidy grant

There is an exercise in a Microeconomics book that says an income grant to a person provides more utility than does a subsidy on one good that costs the same amount to the government. Something like ...
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0 answers
19 views

Influence on the consumption of $Y$ when the price of $X$ falls

Consider the utility function $u(x,y)$ and two budget constraints $B_1: px + qy = 1$ and $B_2 : p_1x + qy = 1$ where $p_1 < p$. If $(x^*, y^*)$ and $(x_1^*, y_1^*)$ maximize $u$ subject to $B_1$ ...
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1 vote
1 answer
54 views

If the income increases by $\$d$, will it mean the utility at the optimal point increases by $\lambda d$?

I read that if the income increases by $\\\$d$, then the utility at the optimal point will increase by $\lambda d$. How do I get a sense of this, both mathematically and intuitively? Can we write that ...
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2 votes
1 answer
107 views

Proof of the tangency condition in UMP

When an indifference curve is tangent to the budget line such that the preferences are convex and monotone, why is the point of tangency an optimal for an UMP? Given the budget line $p_1 x + p_2 y = I$...
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0 votes
0 answers
13 views

Intuitively, how is a nested logit model identified?

This is an informal question. I understand that nested logit models allow flexible substitution patterns that so that substitution would be more likely within a nest than across nests. But if you only ...
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3 votes
1 answer
259 views

Convexity of preferences (dissimilar definitions)

Varian's Intermediate Microeconomics describes convexity as $$\text{Given } x, y \in X: x \sim y \implies \forall t \in [0,1], tx + (1-t)y \succeq x,y$$ The other definition I read everywhere is: $$\...
2 votes
0 answers
26 views

What social aggregators are complete?

I am reading Eliaz (2004) for a general treatment of social-choice impossibilities. I am confused by the fact that it seems that we should get completeness even in situations where we should not have ...
2 votes
0 answers
220 views

Indifference curves representable by real-valued functions on $\mathbb{R}$ of a continuous preference relation

Let $X = \mathbb{R}^2$. Suppose $\succeq$ denotes a continuous preference relation. If every indifference curve can be represented by functions from $\mathbb{R}$ to $\mathbb{R}$, will it mean the ICs ...
2 votes
2 answers
183 views

Discontinuous function $U$ with continuous preferences can be written as a composition of discontinuous & monotone function and a continuous function

Conjecture: Every discontinuous utility function $U$ representing continuous preferences can be written as $U = f \circ g$ for some continuous $g$ and discontinuous strictly monotone $f$. The goal is ...
1 vote
0 answers
57 views

Perfect substitutes mathematical definitions not equivalent

Statement: Consider goods $X$ and $Y$ (and we denote the quantities of by the same notation) such that they are perfect substitutes with the substitution ratio $1:n$. Assume the basic axioms ...
0 votes
1 answer
51 views

Marginal utility meaning and properties

Consider goods $X$ and $Y$ such that the marginal utility of a unit of good $X$ is always that of $n$ units of good $Y$. $X$ and $Y$ are perfect substitutes. Question 1: What does the above mean ...
0 votes
0 answers
24 views

In consumer theory, shouldn't necessity good and neutral good be different ? What will be the IC and utility function for both?

Necessity good for example salt, which regardless of income has to be consumed at certain quantity. But neutral good for example is Suppliments for a healthy person which regardless of income he/she ...
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1 vote
1 answer
174 views

Weak preferences and negative transitivity

Let $ \succ $ be a binary relationship on the set $X$ such that, given any $ x, y, z\in X $, if $x\succ y$: (Asymmetry): $\neg(y\succ x)$, (Negative transitivity): $(x\succ z) \vee (z\succ y)$. ...
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1 vote
1 answer
32 views

In a setting with N goods how many combinatorial bits do we need to construct a preference map

I am reading this paper: https://www.researchgate.net/publication/5208445_The_market_for_preferences By P.E Earl and J.Potts On page 3 the following is written: "If we think of individual ...
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2 votes
1 answer
61 views

Part of proof of Gibbard-Satterthwaite Theorem

I'm currently working through Nisan's Algorithmic Game Theory, Chapter 9 (Introduction to Mechanism Design). A part of the proof for the Gibbard-Satterthwaite Theorem is given as "obvious," ...
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20 views

How to empirically measure the underlying utility function for a 'max EU" SWF? Stated/revealed preferences over uncertain lotteries, or?

Direct answers, as well as pointers to the best literature and textbook treatments, as well as the names of key researchers, are appreciated. Suppose we are considering policies and transfers that ...
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5 votes
0 answers
37 views

Comparing voting methods when there are only two voters

Consider the Schulze, Kemeny-Young, Ranked Pairs and Borda count voting methods. (The last is obviously the odd one out in this list!) Suppose that there are only two voters. Each voter gives a ...
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3 votes
2 answers
142 views

Convex rationalization when the budget sets are segments?

Backgroud: SARP can be defined on general budget set. SARP: Assume for all $B$ the choice $c(B)$ is only one element. If $x_i,x_{i+1}\in B_i$, and $x_i = c(B_i)$, for all $i\in \{1,N-1\}$, then $x_1=...
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2 votes
0 answers
53 views

Market shares of Nested Logit demand model

Consider a Nested Logit demand model with two nests, $N_1, N_2$: $N_1$ contains the outside option only (labelled "0"), $N_2$ contains all the remaining alternatives (labelled "$j=1,...,...
0 votes
1 answer
148 views

max{x1,x2} where P1not=p2

I have seen min{x1,x2} functions representing perfect compliments but have never seen a max{x1,x2} function anywhere in my book or lectures, I also have never seen anything about p1 not equaling p2. ...
2 votes
1 answer
93 views

Market with changing number of goods and services

In the General Equilibrium framework of Arrow, Debreau and others, there are a fixed number of commodities, which I feel is a valid assumption in the short run but maybe not in the long run. Over time,...
3 votes
1 answer
122 views

Understanding the Choice Rule in MWG

I am reading the Microeconomics Theory book by MWG, and I am having a tough time interpreting what things mean to a real life example, so any help would be appreciated. For example, it gave this. ...
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2 votes
1 answer
649 views

Can the Certainty Equivalent be negative?

I am questioning if the CE of a lottery can be negative? For me it doesn't make much sense by definition. I encountered this problem on the following exercise: Imagine a case where we have a lottery(...
4 votes
1 answer
101 views

Utility representation of single peaked preferences

Is it true that a single-peaked preference (with the peak at some finite point) over the set of real numbers, always has a utility representation ?? If yes, can you please hint towards the proof or ...
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0 votes
0 answers
55 views

Binary relation on the set $X = \{v, w, x, y , z\}$ that is asymmetric and transitive but not negatively transitive

So I am trying to find a binary relation on the set $X = \{v, w, x, y, z\}$ that is asymmetric and transitive but not negatively transitive, and is quite tricky. Will $R = (v, w)$ be asymmetric and ...
0 votes
1 answer
72 views

Prove that Choice Coherence Implies IIA

Prove that Choice Coherence implies Independence of Irrelevant Alternatives (IIA). From the definition of choice coherence, we have this: A choice function c satisfies choice coherence if, for every ...
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4 votes
1 answer
112 views

Doesn't the Pareto-extension rule invalidate Eliaz's (2004) unified theorem of social choice?

Eliaz (2004) uses social aggregators to provide a unique "meta-theorem" from which Arrow and Gibbard-Satterthwaite follow as corollaries. He defines social aggregators as follows. Let $\...
4 votes
1 answer
271 views

Difference between social choice functions and social decision functions?

A social decision function (SDF) à la Sen (1970) is defined as a collective choice rule whose range is restricted to social preference relations which generate a choice function. From Gaertner (2009), ...
3 votes
2 answers
88 views

Exact definition of one-player Bayesian Correlated Equilibrium

Consider a game where a decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world $V$. The state of the world has support $\mathcal{V}$. ...
2 votes
1 answer
93 views

Why does Figure 2.F.1(b) (MWG page 30) satisfy the WARP (Definition 2.F.1)?

I can see that Figure 2.F.1(a) satisfies the WARP (Definition 2.F.1) in MWG (page 30). However, as the choice $x(p',w')$ is only feasible under the price-income level $(p',w')$ and $x(p'',w'')$ is ...
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1 vote
1 answer
59 views

Risk neutrality in single-agent choice problem under uncertainity

Consider the following static single-agent choice problem under uncertainty. Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{V})$. ...
3 votes
2 answers
247 views

Sen's property $\alpha$ holds when limited attention in choice?

Consider the limited attention choice framework by Matejka and McKay (2015). This framework can give rise to consideration sets, as roughly summarised below. Consideration sets in the limited ...
1 vote
1 answer
74 views

Is the set of optimal strategies convex in a single-agent decision choice problem?

EDITED with insights from the comment below. Consider a decision maker who has to choose an action among $\mathcal{Y}\equiv \{1,2,...,L\}$. The payoff from choosing action $y\in \mathcal{Y}$ depends ...
1 vote
1 answer
163 views

Boots' Theory by Pratchett

Consider the following quote from the book, $\textit{Men at Arms}$ by Terry Pratchett. “The reason that the rich were so rich, Vimes reasoned, was because they managed to spend less money. ...
0 votes
2 answers
787 views

The mathematical proof of a monotonic utility transformation does not restrict the use of strictly decreasing monotonic functions. Why bar them?

I understand from an intuitive sense that decreasing monotonic transformations will skew the choices and ordinality. But mathematically the $F'(U(x,y))$ just cancels out each other out in numerator ...
5 votes
0 answers
113 views

Afriat theorem for negative goods

GARP and Afrait theorem assume that the alternative $x\in\mathbb R_+$ is always positive. In some economic contexts, such as financial choices, the attribute can be negative. I wonder if we can ...
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1 vote
0 answers
155 views

Archimedean but not mixture continuous

In the context of preferences on a set of lotteries on a finite set $X$, what is an example of a preference that is independent, Archimedean but not mixture continuous? I know the mixture continuous ...
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1 answer
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What does irrationality mean under the notion of revealed preferences? An example with terrorism.

I am currently studying for an economics of conflict exam and one of the potential questions is likely to be in the vein of "Are Terrorists Rational - Discuss". However how would rationality be ...
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3 votes
2 answers
303 views

a risk lover agent preferences and the preference of risk natural agent may be the same

Consider two lotteries $N$ and $M$. Agent $i$ is risk-averse and prefers $N$. Agent $j$ is risk-neutral and prefers $M$. Would any risk-loving agent $k$ also prefer $M$? That is, would $j$ and $k$ ...
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2 votes
1 answer
565 views

Choice correspondence notation

My question is about the following notation: I have noticed in several places (for example here (page 15) and here (page 1)) that different authors use different notations for choice correspondences....
6 votes
1 answer
357 views

Envelope theorem for discrete choice sets?

If we have a function $$f(x)=\max_yg(x,y)$$ Then we can find the derivative $d/dx \ f(x)$ by realizing that $$(1): \quad \frac {\partial }{\partial y}g(x,y^*)=0$$ because of the first order ...
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3 votes
1 answer
439 views

What is the economic incentive to cheat? How does an experiment capture exogenous deviations?

In my undergraduate career an economic experiment was conducted on my class: one class was the control two classes were the experimental (I was in an experimental class) The basic premise was a ...
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1 vote
1 answer
65 views

Is this proof correct?

i'm new at this, so i´m really sorry sorry if i do something wrong. The problem is this: A choice function satisfies condition α if whenever $x = C(A)$ and $x ∈ B ⊂ A$, it follows that $x = C(B)$ as ...
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3 votes
0 answers
542 views

Choice rule and path independence

Question: A choice rule $C$ satisfies path independence if for all $A, B \in 2^X \setminus \emptyset$, $C(A \cup B) = C(C(A) \cup C(B))$. Prove that if $C$ is nonempty and rationalizable, then $C$ ...
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1 vote
1 answer
46 views

Vocabulary/Name for Utility of a Set of Choices

I am trying to find if there is some litterature or definition of the concept of the Utility of a set of choices. My google searches return nothing, and I expect it is because I do not know the right ...
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1 vote
1 answer
93 views

What is one dimensional, ordered type?

I am reading papers about moral hazard. What is one dimensional, ordered type $\theta\in\Theta$? What is one dimensional, not ordered type $\theta\in\Theta$? Could you please give an example? ...
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