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Questions tagged [decision-theory]

the mathematical study of strategies for optimal decision-making between options involving different risks or expectations of gain or loss depending on the outcome.

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How could define the certainty equivalent in a Bayesian Persuasion model?

For once again I will start describing the Kamenica and Gentzkow Bayesian persuasion model. Suppose that $\Theta$ is a finite set of states and $\theta$ is the element of the state set. To simplify ...
Oliver Queen's user avatar
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1 answer
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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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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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Kohlberg-Mertens theorem

I am trying to understand the Kohlberg-Mertens theorem. Here is the context and the theorem. We fix $N$ a set of players and $S_i$ the finite set of actions of each players with $\lvert S_i\rvert =m_i$...
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Independence Axiom and Expected Utility Theorem Proof

In my micro class we covered the proof of the existence of a Von Neumann–Morgenstern utility representation of preferences $\succeq$ over a set of lotteries $\Delta(Z)$ - where $Z$ is some finite ...
Joseph Basford's user avatar
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1 answer
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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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Trying to find a proof for Strong Axiom of Revealed Preference with general choice set

Note this is question is not about consumer demand with price and income data. This is a question about general choice theory. For reference, see: https://www.jstor.org/stable/2550390 See Debreu's ...
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Equivalence of two definitions of revealed preference

Given a choice structure $(\mathscr{B},C(.))$ we can construct a preference align with this structure, write it as $\succcurlyeq^C$ defined as $$x\succcurlyeq^C y\Leftrightarrow \exists B \in \mathscr{...
Nonenicht's user avatar
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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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What economic theory models describe a financial regulator's decision to monitor a company?

Let's consider an example (that may not describe how these entities actually work, but humor me): You are the Financial Stability Board (FSB), you need to designate systemically important financial ...
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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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Strict monotonicity implies FOSD under EU?

I think this result is very simple and very useful. However, I find no paper cover it. Is it on some textbook like MWG? Strict monotonicity means: $f(s)\succ g(s)$ for all $s$ implies $f\succ g$ and $...
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Arrow’s Impossibility Theorem Proof - Unicity of "dictator"

I have a hard time understanding completely Arrow’s Impossibility Theorem proof, even the very pedagogical one by Geanakoplos, that can be found for instance here : https://users.ssc.wisc.edu/~dquint/...
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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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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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MWG Exercise 2.E.5

Exercise Suppose that $x(\mathbf{p},w)$ is a demand function which is homogeneous of degree one with respect to $w$ and satisfies Walras' law and homogeneity of degree zero. Suppose also that all the ...
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Determining Perfect vs. Imperfect Information in Calculating Expected Value

In this scenario, you are presented with an opportunity to engage in a game for a fee of $50. On a table, there are two boxes: a large box and a small box. The large box contains a total of 40 balls, ...
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Estimating willingness-to-pay for a risk-averse person who can 'select' lotteries

I'm studying how the willingness-to-pay differs for individuals who can 'select' lotteries. Individuals are presented with L1 first and can pay some amount to get lottery L2. Assume these are my ...
comparing-lotteries-help's user avatar
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1 answer
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Decision theory: elicitation method

I'm stuck with the following question: Let's say that C1, C2 and C3 represent the certainty equivalents and (x,p,y) the prospects. C1 ~ (x, p, 0) C2 ~ (x, p, C1) C3 ~ (C1, p, 0) What is C3 such that ...
EcoSTUD233's user avatar
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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 ...
homo-economitux's user avatar
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1 answer
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Can Debreu's axiomatization of cardinal utility use equivalent relation instead of preference relation?

Theorem ([Debreu 1959][1] page 9, 10) Let $X_i$ be space of real numbers. If $\succsim$ is continuous, rational, independent and at least three factors are essential, then there exist functions $u_i:...
High GPA's user avatar
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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 ...
High GPA's user avatar
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2 votes
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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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Is my understanding of Arrow's dictatorship correct? The dictator is free to update her preference and the social choice will always follow her taste

Suppose $R$ is a social ordering, $f$ is the social choice function, and $R_i$ is an individual preference. A profile of individual preference is $<R_i>$. $f(<R_i>)=R$ Suppose $i=1$ is ...
dodo's user avatar
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Debreu's ordinal representation theorem is unique up to a positive monotonic transformation, what is the source?

In Debreu's 1954 ordinal utility representation theorem, the utility is unique up to a positive monotonic transformation. While the uniqueness result is well-known, I fail to find a proper reference. ...
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Question About Stochastic Choice - MWG Exercise 1.D.5

I am studying microeconomic theory using MWG. I got stuck on Exercise 1.D.5, specifically part (c), but I would also like to have my part (a) and (b) checked by someone. Here is the exercise and my ...
Beerus's user avatar
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1 answer
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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 ...
Beerus's user avatar
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2 votes
1 answer
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About The Bayesian Conditional-Probability Systems in Myerson's Game Theory: Analysis of Conflict

I am self-studying game theory using Myerson's Game Theory: Analysis of Conflict. I got some trouble understanding his Bayesian conditional-probability system. The Bayesian conditional-probability ...
Beerus's user avatar
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0 answers
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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 ...
High GPA's user avatar
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1 vote
1 answer
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Proving duality of UMP and EMP arguing with continuity of utility

In Mas-Colell et al.'s Microeconomic Theory Proposition 3.E.1(ii) (p. 58) states that if $\succsim$ is a rational (i.e. complete and transitive), continuous, and locally nonsatiated preference ...
manifold's user avatar
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2 votes
2 answers
165 views

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 ...
High GPA's user avatar
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2 votes
1 answer
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About Theorem 1.1 (the Expected-Utility Maximization Theorem) in Game Theory: Analysis of Conflict by Roger Myerson

I am self-studying game theory using Myerson's Game Theory: Analysis of Conflict. I got some trouble understanding his proof of Theorem 1.1, the Expected-Utility Maximization Theorem. The Theorem goes ...
Beerus's user avatar
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0 answers
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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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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 ...
LJNG's user avatar
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Can a business's search for profits be considered as movement in state space?

When businesses make decisions to increase profits, they have to adjust several "parameters" of the business such as quantity to output, pricing, choosing a production mix, brand positioning,...
Joebevo's user avatar
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1 answer
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What does the Arrow-Pratt risk aversion measure mean in the deterministic case?

What type of preferences that are not related to risk aversion can the Arrow-Pratt measure of absolute (or relative) risk aversion model? So far, it seems to me that low RRA/ARA preferences imply that ...
ju_pi_car's user avatar
2 votes
0 answers
104 views

Proof of First-Order Stochastic Dominance with Riemann Sums

Let $A = \mathbb{R}$. For $p,q\in \mathcal{L}(A)$, $p$ first-order stochastically dominates (FOSD) $q$ if $F_p(a)-F_q(a) \leq 0, \forall a\in A$. Show that $p$ first-order stochastically dominates $q$ ...
homo-economitux's user avatar
2 votes
1 answer
343 views

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 ...
Ishan Kashyap Hazarika's user avatar
4 votes
1 answer
120 views

What is the economic intuition of prudence in the static case?

How can we interpret a "prudent" agent in the static case (i.e., someone with $u'''(\cdot)>0$)? I understand that in a dynamic setting, someone exhibiting prudence would do precautionary ...
ju_pi_car's user avatar
1 vote
1 answer
69 views

What is the risk aversion domain and how this could change in a dynamic market game?

Most of the market microstructure theory models assume a risk aversion coefficient, say $\gamma$ that is indexed with $i$ since any individual $i$ has her own $\gamma_i$ coefficient. Also, the inverse ...
Oliver Queen's user avatar
2 votes
1 answer
54 views

Lower bound for the utility in a decision problem with uncertainty

Model Consider a single-agent decision problem with uncertainty. A decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world. $\mathcal{Y}...
Star's user avatar
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5 votes
2 answers
128 views

Information structure for complete information

Model A decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world. $\mathcal{Y}$ is a finite set. The state of the world is a random ...
Star's user avatar
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1 vote
0 answers
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Representation theorem for $\succsim\supset>\cup\sim$

On $\mathbb R^2$, define $x=(x_1,x_2)>(y_1,y_2)=y$ if $x_i\geq y_i$ for all $i$ and $x_j>y_j$ for some $j$. Let $\sim $ be an equivalence relation that $x\sim y$ implies $x\not> y$. Define ...
dodo's user avatar
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Say a preference is "constant“, by analogy with a constant function

Let $f$ be a function with range of $\{-1,1\}$ and $f(x,y)=-f(y,x)$. Let the preference $\succ\subset X\times X$ where $x\succ y \iff f(x,y)=1$. In math we can define $f$ to be a constant function on ...
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Can the topological assumption in Debreu's representation theorem of cardinal utility be altered from "connected separable" to "second countable"?

Theorem (Debreu 1959 page 9, 10) Let $X$ be connected separable topological space endowed with product topology. If $\succsim$ is independent and at least three factors are essential, then there exist ...
High GPA's user avatar
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4 votes
1 answer
280 views

Does Debreu's representation theorem of ordinal utility require Hausdorff topology?

By Debreu's theorem of ordinal utility, any continuous weak order on $X$ is represented with a continuous utility function, if $X$ is a second countable or connected separable topological space. My ...
High GPA's user avatar
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3 votes
0 answers
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How did econometricians justify the use of $EU$ instead of $EU^2$?

Consider the following two utility functions: $EU(p)=\sum_i u_ip_i$ $EU^2(p)=(\sum_i u_ip_i)^2$. In preference theory, $EU$ and $EU^2$ are equivalent because they represent the same preference. A ...
dodo's user avatar
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2 votes
1 answer
187 views

Is Epstein-Zin utility a generalization of dynamic expected utility (DEU)?

Epstein-Zin (EZ) utility is the solution to: DEU is relatively simple: $\sum_t \delta ^t\mathbb E[u(c_t)]$. Is DEU a special case of EZ? How are those two models compared? Since EZ is a solution of a ...
High GPA's user avatar
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4 votes
1 answer
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Most utility functions under risk and uncertainty generalizes expected utility. What is deadly wrong if a model does not include EU as special case?

Why do people generalize EU instead of making an entirely new model, or create a model that is neither a special case nor an extension of EU? To my knowledge, most utility functions under risk and ...
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