Questions tagged [preferences]
Binary relations that reflect which states of the world an agent considers to be most desirable. Preferences are a fundamental ingredient in the axiomatic study of consumer choice decision theory.
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How can we measure bias in content recomendation systems?
Sorry if this is the wrong place to ask this (maybe I should ask a statistics forum?).
I want to know whether there has been any good work for econometrics people about measuring bias in content ...
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Anscombe–Aumann model presentation for more general spaces
I am taking a look in the book of Gilboa . In chapter $14$ there is a presentetion on Anscombe–Aumann’s Theorem. Although he does not present the assumptions about the sets of outcomes and states, and ...
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Lexicographic Preferences and The Axiom of Choice
I am once again thinking about the proof that lexicographic preferences over $\mathbb{R}^2$ have no utility representation. The proof I have seen is the following.
Suppose that there does exists $u:\...
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Are weakly monotonic preference and strictly monotomic preference the same?
Recently I am reviewing preference theory and I found that I am confused about the difference between weakly monotonic preference and strictly monotonic preference.
Now, let us suppose $X$ is the ...
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Normalizing one price with non-homothetic utility
In an $n$-good model with prices $p_i \in \{ p_1,...,p_n \}$ can you normalize one price to unity if the underlying preferences are non-homothetic? Or is this price normalization property only a ...
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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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Convexity and Strict Preferences
I was looking into convexity while reading Varian's intermediate micro textbook, and an article by Richter and Rubinstein said, "The canonical definition of convex preferences requires that if a ...
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How to formally define the preference profile in which the top alternative of one player becomes the bottom alternative of the next player?
I need to generically define a strict preference profile with Condorcet cicles when the number of players and alternatives coincide. To illustrate my problem, consider the following four-player & ...
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If both goods are considered economic goods in the economy, then how can their demand be negatively sloped when in a state of satiation or bliss?
if both goods are economic bad, then we can say that they have a negatively sloped demand curve. But how can we say the same for economic goods in a state of satiation or bliss?
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Non-nullity assumption in vNM theorem of cardinal utility
The vNM theorem suggests that weak-ordering, continuity, and independence is equivalent to the existence of expected utility, unique up to an affine transformation.
In Savage's axioms of expected ...
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Mapping of Utility function to a single vector
So I was watching this YouTube video and during it, he said that it is possible to create a bijection between a utility function and vector, which is then used to understand the ordinal ranking. I was ...
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Debreu's cardinal representation theorem for finite outcome set
Suppose there are three dimensions. $x,y,z\in X^3$.
Independent: $z_ix\succsim z_iy\iff z'_ix\succsim z'_iy$.
When $X$ is connected topological space, Debreu proved that weak order, independent, and ...
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Reconciling Continuity of Binary/Preference Relations with Continuity of Functions/Correspondences
There are various ways to express the concept of continuity of a binary relation, but one I've come across seems to imply the closed-graph property is sufficient. That is to say:
For a nonempty set $X$...
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Equivalence of two definitions of monotone preference
In MWG, the definition of weak preference is for all $x,y \in X$, $y>>x$ implies $y\succ x$ . But I have read some other articles that define weak preference as $y\geq x\implies y\succeq x$.
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What are the most common axioms to define a strict preference relation?
I am wondering which are the standard axioms with which indifference and (particularly) strict preference are usually defined. Normally, a weak preference relation for agent $i$ on a set $X$, denoted $...
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Question About Proving $\alpha(\cdot)$ is Continuous in the Proof of Proposition 3.C.1 from MWG
See Proposition 3.C.1 from MWG
Continue from this post, Microeconomic Theory by Mas-Colell et al. book said the following:
What remains is to establish that all convergent subsequences of $\{\alpha(x^...
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Proof: Let $\epsilon>0$ and $x'\in\mathbb{R}^L_+$ be such that $\|x'-x\|\geq\epsilon$. Then $\alpha(x')$ belongs to some $[\alpha_0,\alpha_1]$
See Proposition 3.C.1 from MWG
Continue from this post, the book (MWG) then started the proof that $\alpha(x)$ is a continuous function:
We now argue that $\alpha(x)$ is a continuous function at all $...
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Question About the Step Proving $\alpha(x)$ Represents Preferences in the Proof of Proposition 3.C.1 from MWG
Continue from this question, the book Microeconomic Theory by Mas-Colell et al. said
We now take $\alpha(x)$ as our utility function; that is, we assign a utility value $u(x)=\alpha(x)$ to every $x$. ...
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Question About the Step Involving the Connectedness of $\mathbb{R}_+$ in the Proof of Proposition 3.C.1 from MWG
This is a small issue, but it confuses me.
In the proof of Proposition 3.C.1 of Microeconomic Theory by Mas-Colell et al, the book said
... $\mathbb{R}_+\subset (A^+\bigcup A^-)$. The nonemptiness ...
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Revealed preferences - Commuter preferences
I'm working on a paper about commuter preferences and their level of satisfaction in a South American country. Do they prefer the metro or informal buses? That sort of thing. The data tells me that ...
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Prove a preference preserved under limits if and only if its upper and lower contour is closed
I'm concerned with the reverse direction, that upper and lower contour is closed implies the preference is continuous, that is for any sequence $x_n$ and $y_n$, $x_n\succcurlyeq y_n$ for all $n$, ...
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Positive monotonic transformation
I want to draw an indifference curve for this preference relation on $ℝ^2$
$$x ≿ y \leftrightarrow (x_1+x_2)^2 ≥ (y_1+y_2)^2$$
Can I just take the square root(and then drop the absolute value sign ...
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Indifference curves given preference relation
Let's say I have a preference relation on $ℝ^2$ given by
$$x≥y \leftrightarrow x_1+x_2 ≥ y_1+y_2$$
So, from my understanding, the bundle x is preferred at least as much as the bundle y if and only if ...
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Prove that any lexicographic preference $(u_1,u_2)$ must be complete and transitive
Let $\succsim$ be a lexicographic preference represented with $(u_1,u_2)$. $x\succsim y$ if $u_1(x)>u_1(y)$ OR $u_1(x)=u_1(y)$ and $u_2(x)\geq u_2(y)$.
Is it obvious that $\succsim$ must be both ...
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Equivalent assumptions in Revealed Preference and Classical Preference Theories
I have read that there are dualities between Revealed Preference Theory and Classical Preference Theory.
Suppose, I have $X$ as the set of all conceivable choices, and $\mathscr{B} \subseteq 2^X$ the ...
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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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Preference aggregation rules that satisfy the property that the most frequent best alternative is the collective best alternative?
Let $X$ be the set of alternatives, and $(R_i)_{i\in N}$ a profile of partial orders for $n= \#N$ individuals. I make the assumption that those partial orders are so that they all give rise to one ...
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What is a utility function rationalizes these preferences?
For a consumer deciding between goods A and B, with a budget of w:
If A and B are the same price (or if A is cheaper), the consumer will spend their entire budget on A.
As the relative price of A ...
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How to force two utility functions representing the same preference to generate expected utility functions representing the same order on lotteries?
Let $i$ be an agent, and let $A=\{x,y,z\}$ be a set of three alternatives. Then, suppose that player $i$’s linear order (i.e., complete, transitive, antisymmetric and reflexive binary relation) on $A$,...
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Cobb-Douglas utility function
Why in Cobb-Douglas utility function, the exponents have to sum to one? Can they not be equal to 1 and why?
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Convex preference and convex utility
What are the differences between convex preferences and covex utility function? Why are convexity preferences usually represented by the quasi-concave function and not the convex function?
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Question on Isolating The Wealth Effect in Analysis of Changes in Price-Wealth Combinations - MWG Exercise 2.F.3 Parts (e) and (f)
I am doing exercises in Chapter 2 of MWG. I feel I got completely lost in exercise 2.F.3 parts (e) and (f).
$\textbf{Exercise}$
Here is the question:
I have solved parts (a) to (d). In particular,
I ...
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Question on The Weak Axiom of Revealed Preference and The Definition of Revealed Preference Relation
I am solving the following problem (from Exercise 2.F.3 (b) in MWG) and I got confused by the weak axiom of revealed preference and the definition of the revealed preference relation. Here is the ...
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Considering there are four different goods {w,x,y,z}, how many different relations can be defined using all goods
In this problem, you can either prefer a good to another, disprefer a good to another or be indifferent between the two goods. While I have been able to do it mechanically, I am very confused as to ...
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What does GARP in revealed preferences mean and how to regocnize it graphically?
I have a froblem understanding GARP in revealed preferences. I know what WARP means, and that SARP adds transivity such that it allows for indirect preferences, but what about GARP?
How can you ...
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How do I prove 2 indifference curves have the same properties?
I understand questions a) but I'm completely stumped at c).
What do they mean by U'=1 having same properties as U=10??
And w,for questions b and c, what are they asking by "general expression for ...
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Stock market existence and saving money
I have a generic question that perhaps has more to do with psychology than with anything else.
The question is, how effective is the stock market's existence in being effective in making people save ...
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Proof for convex preference relation
⪰ is a strictly convex preference relation on a set X which is the set of all N-tuples of nonnegative real numbers. Further are x,y,z element of this set X and i have the preferences x ≻ y ≻ z given.
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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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Why is monotonic preference and monotonic utility function non-decreasing?
Obviously a monotonic function can be either nondecreasing or nonincreasing:
However, in Economics, a quick google search gives:
I am interested in the history or the motivation behind the econ ...
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What was the study that shows peoples' tax preferences shift when they get more information about their income?
I am trying to find a study I read a few years ago, but I forgot the names of the authors and the title of the study. I have already tried to search for the study on Google Scholar without much luck. ...
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Determine Whether A Preference Relation Satisfies The Continuity Axiom - from Exercise 1.1 in Game Theory: Analysis of Conflict by Roger Myerson
I am self-studying game theory using Game Theory: Analysis of Conflict by Roger Myerson. Here is an exercise from the textbook. I tried it myself, but I am not sure if it is correct. I would really ...
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y is weakly preferred over x if and only if x+y ≤ 4 defines a preference relation on {0,1,2,3} why is this incomplete?
y is weakly preferred over x if and only if x+y ≤ 4 defines a preference relation on {0,1,2,3}. True or False?. I can see why it's not transitive but I was told it was incomplete if we take 2 and 3. ...
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Transitivity of Preferences paper
I am going through some of my old grad school notes, and in my microeconomics notes on transitive preferences, the teacher made a note of a behavioral economics result where when presented with two ...
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Give bundles $x,y\in \mathbb R^n$, there must exist a budget $B\supset\{x,y\}$ and a demand $D(B)\in[x,y]$?
For a problem in revealed preference. Give bundles $x,y\in \mathbb R^n$, must there exist a budget $B\supset\{x,y\}$ and a demand $D(B)\in[x,y]$?
Intuitively, this mean that we have two bundles, and ...
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Lexicographic preference relation not continuous
I’m having trouble understanding why lexicographic preference relations aren’t continuous. I need inspiration on how this proof would work.
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Quasiconvex and quasiconcave utility function
I saw that the model of the quasi-convex utility function is similar to the concave utility function and also the quasi-concave utility function is similar to the convex utility function. How can ...
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Continuity of preference
"A preference is continuous if for any $a,b\in X$ with $a\succsim b$ there are some neighborhoods $N_{\varepsilon}(a)$, $N_{\delta}(b)$ around $a$ and $b$ such that for every $x \in N_{\...
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Generalization of Debreu's additive utility function $\sum_nu_n(x_n)$ with infinite number of commodities
I want to generalize: $\sum_nu_n(x_n)$.
Here $x_1,x_2,..,x_n,...$ are commodities. There are infinite number of commodities: $n\in\mathbb N$ or $n\in \mathbb R_+$
The following not a candidate: $\...
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