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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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: The law of demand always holds if the walrasian demand function $x(\mathbf{p},w)$ satisfies the weak axiom of revealed preference (WARP), Walras' law,...
3 votes
1 answer
88 views

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 ...
2 votes
1 answer
71 views

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 ...
1 vote
0 answers
49 views

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, ...
0 votes
0 answers
41 views

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 ...
0 votes
1 answer
37 views

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 ...
3 votes
1 answer
77 views

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 ...
1 vote
1 answer
52 views

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:...
4 votes
1 answer
76 views

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 ...
2 votes
0 answers
26 views

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, ...
1 vote
1 answer
121 views

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 ...
0 votes
1 answer
28 views

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 ...
0 votes
1 answer
67 views

What does the Arrow-Pratt risk aversion measure means 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 ...
3 votes
2 answers
445 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
116 views

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 ...
0 votes
2 answers
53 views

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. ...
0 votes
0 answers
46 views

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 ...
9 votes
2 answers
1k views

The Savage sure thing principle and Subjective utility representation

I have tried reading and understanding Savage's proof of the subjective utility representation, it is too complicated. Is anyone aware of a shorter/more elegant proof of this? It is not a problem if ...
7 votes
1 answer
952 views

Lexicographic Preference Relation on the QxR

I would like to ask for your help. I recently learned that the Lexicographic Preference relation can be represented by a utility function $u:X\to\mathbb{R}$ on $\mathbb{Q}\times\mathbb{R}$ (but not $\...
2 votes
1 answer
68 views

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 ...
13 votes
4 answers
2k views

Is there an economic analysis of the rationality of buying lottery tickets?

It's quite clear that the expected return on a lottery ticket is less than 1. However, I think it can still be argued that buying lottery tickets is still a economically rational decision by ...
3 votes
0 answers
61 views

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 ...
1 vote
1 answer
71 views

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 ...
2 votes
2 answers
141 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 ...
3 votes
0 answers
101 views

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: $\...
0 votes
1 answer
26 views

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 ...
0 votes
0 answers
24 views

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,...
2 votes
0 answers
100 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 $q$ ...
2 votes
1 answer
207 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 ...
4 votes
1 answer
83 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 ...
4 votes
1 answer
281 views

What is the difference between Impression Management and Signaling Theory?

I'm interested in theories on how organisations shape their stakeholders' (especially consumers' and investors') perceptions and decisions. I read about Impression Management and Signaling Theory. ...
5 votes
2 answers
122 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 ...
1 vote
1 answer
54 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 ...
1 vote
0 answers
74 views

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 ...
6 votes
1 answer
586 views

Why is the Marginal Utility of losses diminishing in Prospect Theory?

This is Kahneman's value-plot on prospect theory: QUESTION: Why is the Marginal Utility of losses deminishing? CONTEXT: I fully understand that the Marginal Utility of gains deminishes: 100 dollar ...
2 votes
1 answer
53 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}...
2 votes
1 answer
81 views

$a\geq 0$, $x\succsim y$ implies $x+a\succsim y+a$ so the preference is linear?

$\succsim$ is a continuous and local non-satiate weak order. $x,y,a$ are vectors in $\mathbb R^n$ We say $a\geq0$ if all directions of the vector $a$ is greater or equal to zero. We want to prove (or ...
8 votes
2 answers
185 views

Ordinally Separable Utility Representation

Let $X_i$ be a separable, compact, Banach space. Definition: A weak order $\succeq$ on $X=\prod_{i=1}^NX_i$ has an ordinally separable representation if there exists $u_i: X_i\rightarrow \mathbb{R}$ ...
1 vote
0 answers
65 views

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 ...
1 vote
0 answers
49 views

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 ...
4 votes
1 answer
270 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 ...
2 votes
2 answers
452 views

Demand correspondence is both upper and lower hemi-continuous; is the preference continuous?

$\succsim$ is a weak order over $\mathbb R^L$. For a closed budget set $B\subset\mathbb R^L$, define demand correspondence: $$D(B)=\{x\in B|x\succsim y\forall y\in B\}$$. We know that $D$ is always ...
4 votes
1 answer
104 views

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 ...
3 votes
0 answers
86 views

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 ...
2 votes
1 answer
157 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 ...
0 votes
0 answers
58 views

Auction Theory and Elections

Can we (is it reasonable to) apply auction theory and the various incentive compatible mechanisms to elections and guarantee cardinal voting? I am thinking specifically of VCG auctions and ...
0 votes
0 answers
25 views

Risk Aversion under Worst Case Utility Representation

The preference relations (≿A and ≿B) over lotteries is defined as: p ≿ q iff min{v(z) : p(z) > 0} ≥ min{v(z) : q(z) > 0} Under what conditions can you say that ≿A is more risk averse than ≿B?
11 votes
2 answers
1k views

Can the Machina Paradox be solved by expanding the choice set?

In another question, the Machina paradox is mentioned as a possible counterexample to the expected utility model: Adding to the list of paradoxes, consider Machina's paradox. It is described in Mas-...
3 votes
1 answer
114 views

How to determine if people behave optimally with a generic utility function?

I have a some real-world data and a real-world choice for which I know the optimum solution is something like $y^* = u(x)$, where $u(.)$ is some utility function. I have data on $y$ and $x$, so if I ...
10 votes
3 answers
259 views

Do a group of economic agents really act as if they are rational?

When questioning the rational choice hypothesis, I often get responses that are similar to the followings: "Individuals may sometimes make irrational decisions, but a large group of economic ...