Questions tagged [choice-theory]
a conglomerate of models and results concerning the aggregation of individual choices into collective choices
90 questions
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Choice function and IIA
I'm new to microeconomics and struggle to understand choice function in practice. E.g. lets say I have a set X=(a,b,c). How many possible choice functions I could in theory have? Then other concept ...
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Non decreasing function for utility transformation
I was going through the Theorem 2 :
Suppose $u(x)$ represents agents preferences, $\succsim$ and $f : \mathbb{R} \rightarrow \mathbb{R} $ is a strictly increasing function. Then the new new utility ...
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Question About Proof of Proposition 3.C.1 in MWG - Step 1
I have difficulties understanding the first step of the proof of Proposition 3.C.1 in MWG.
Proposition 3.C.1$\quad$ Suppose that the rational preference relation $\succsim$ on $X$ is continuous. Then ...
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Understanding the definition of monotone
In Microeconomic Theory by Mas-Colell, Whinston, and Green, the definition of monotone preference relations is given as follows:
Definition 3.B.2$\quad$ The preference relation $\succsim$ on $X$ is ...
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Prove: The law of demand holds if WA, Walras' law, homogeneity of degree 0, and homogeneity of degree 1 in wealth hold for Walrasian demand functions
Problem
I am asked to prove the following result (MWG Exercise 2.F.5):
The law of demand always holds if the walrasian demand function $x(\mathbf{p},w)$ satisfies the weak axiom of revealed ...
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Proof for Marshallian Demand function
If you have a Marshallian demand function that is strictly convex, then it satisfies WARP.
How to prove this?
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Proving the Choice with Recommendations
Suppose that there are two types of outcomes, i.e. $X=X_1 \cup X_2$ with $X_1 \cap X_2=∅$. All outcomes in $X_2$ are the same to the decision maker (he doesn't understand these kind of products). He ...
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In revealed preference (RP), is any two points $x,y$ related by the indirect revealed preference relation?
Let $X$ be the closed compact convex set of alternative and $B$ be a closed compact convex subset of $X$. $C$ is defined on all closed compact convex set $B\subseteq X$.
$X$ is ordered by a strictly ...
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Find a choice function such that WARP or SARP is violated
WARP implies choice function is raionalizable.
Say we have a choice function $C(B)$, $B$ is a closed convex compact set. I am looking for a intuitive example of $C$. The $C$ is economic meaningful, ...
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In Debreu's representation theorem of ordinal utility, is the assumption of "second countability" necessary?
Debreu's representation theorems
Debreu 1959 states that: second countability, continuity, and weak ordering sufficiently implies the existence of real (continuous) utility function. The second and ...
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Minimal assumption for a “certainty equivalence” exists
Let $R$ be the set of real number. Let $N$ be an infinite set. Let utility $u:R^N\to R$. The utility function is strictly monotonic.
My question is, does the certainty equivalence $CE$ exist? Do we ...
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Is the relation $\mathcal{R}=\{(1,2),(2,3),(1,3)\}$ on $X=\{1,2,3\}$ complete?
Is the relation $\mathcal{R}=\{(1,2),(2,3),(1,3)\}$ on $X=\{1,2,3\}$ complete?
By looking at the completeness definition in preference: Definition 1.1(c), this is same as the connected relation in the ...
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Choice theory vs decision theory
I always thought that Decision Theory and Choice Theory are the same fields. But when reading the Wikipedia entry for Decision Theory recently, I read the explicit clarification: "not to be ...
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Can the following statement be rationalized if it yields a choice function?
A person choose an alternative to maximize another person's suffering.
I thought we could define a sort of relation where the person suffers more from x than y. And if we can always do this, we can ...
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Can the following behavior be rationalized if it yields a choice function?
The decision maker has an ideal point in mind and chooses the alternative closest to it.
I am not sure if I am right, but in order to rationalize it, we first have to construct a choice function. So, ...
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Example of consumer preferences that switches from being concave to being convex
Question
Is there an example of consumer preferences over consumption bundles $(x,y)\in \Bbb R^2$ that would be concave when $x$ is abundant relative to $y$ and convex otherwise?
Are there known ...
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State dependent preferences vs state independent preferences in utility theory
I am working on changes in preferences and found papers on state-independent preference. What is the difference between state-dependent and state-independent preferences and utility functions? What ...
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Consumer surplus in Logit model should minus actural payment?
In Logit model, as Train(2003) said in his book(page 55) said,
"By
definition, a person’s consumer surplus is the utility, in dollar terms,
that the person receives in the choice situation. The ...
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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-...
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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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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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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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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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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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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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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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417
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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: $$\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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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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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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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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,...,...
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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. ...
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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,...
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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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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(...
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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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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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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 $\...
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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), ...
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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 $\mathcal{V}$. ...
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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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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 $P_V\in \Delta(\mathcal{V})$. ...
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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 ...
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