For questions regarding formal decision problems. In contrast, questions involving strategic aspects (where the solution depends on the behavior of others) are discussed in game theory.

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+50

Reducing an I-optimal problem to a Pareto-optimal problem

Given a set $\textbf y\subset\mathbb R^2$, let $y = (y_1,y_2), y'=(y'_1,y'_2)\in\textbf y$ be elements of that set, let $\alpha_{min}\in\mathbb R$, $\alpha_{min}<1$, $\alpha_{max}\in\mathbb R$, ...
0
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1answer
24 views

Biobjective optimisation, pareto non-domination

Ok, so, I have a function $f_I(y_1, y_2) = \max\{\alpha y_1 + (1-\alpha)y_2:\alpha\in[\alpha_{min},\alpha_{max}]\}$ that I'm trying to minimise, and I'm asked to find, amongst a set of vectors $y$, ...
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0answers
15 views

Is summing posterior probabilities to obtain total counts valid?

I'm using a Bayesian approach to classify my data set into two mutually exclusive groups. From what I read typically one would apply a decision rule to add a count into group 1 if $P_{g1}>P_{g2}$ ...
2
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1answer
19 views

Clarification - DeGroot Proof on Transitivity Property of Subjective Probability

In developing axiomatic foundation for subjective probability DeGroot (Optimal Statistical Decision, 2004, p71) gives two axioms/assumptions: SP1: For any two events A and B, exactly one of the ...
0
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1answer
31 views

Integration over a conditional cumulative density function?

In his text "Mathematical Statistics: A Decision Theoretic Approach," Ferguson describes a way to define the expected value as a Rieman-Stieltjes integral. Roughly, he says that we can define the ...
3
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2answers
49 views

What does “decidability” of a Model mean exactly?

I'm looking at the theorem concerning the Model of Arithmetic: M arith = (Integers, +, *, <) is undecidable. What does the "decidability" of a model mean exactly? Does that mean that "the ...
2
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2answers
55 views

Optimal solution to a statistical decision problem

Setup I'm trying to find condition(s) that characterize the solution to a statistical decision problem. The environment is as follows. $\Omega$ is a finite set of states of the world. A decision ...
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0answers
14 views

Introductory study of Survival Analysis and Decision Theory

I'm pursuing a compact Masters degree in Mathematics, a 4 year program at BITS Pilani, India. Except for a couple of introductory courses on statistical and probabilistic analysis, and operations ...
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0answers
23 views

Coming up with a condition for index selection from a set for a specific problem

In my research I have come across the following problem. I have two sets of real numbers, say $\{a_i\},\ \{b_i\},\ i=1,2,\cdots,\ n$. Let $S$ be a given set such that $S\subset \{1,2,\cdots,\ n\}$. ...
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0answers
37 views

Solving simple decision-making model over multiple periods

Consider the following model. Each period t=0,1,..., an agent makes an effort $x\in R_+$ to solve a problem. The value from solving the problem is $V>0$. The relationship between effort and ...
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1answer
10 views

Decision theory

Hi everyone I am new to the topic of decision theory and need some help answering some questions with the data below. $$ \begin{array}{c|lcr} Choice & \text{.4} & \text{.5} & \text{.1} \\ ...
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0answers
18 views

Max Expected Utility

I need help calculating the max expected utility. I want to LEARN this stuff, so if I can be so picky please give some explanation with answers (just an answer won't do me any good). Let P(x) = ...
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0answers
20 views

Can a randomized rule induce a random measure on the action space?

$D = \{d_i: X\to Y, i=1,\dots,n\}$ is a finite set of mappings from $X$ to $Y$, $(\Omega, \mathcal F, P)$ is a probability space, and $\delta: \Omega \to D$ is a measurable mapping. Can $\delta$ ...
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0answers
32 views

Choosing the best framework (expanded secretary problem)

I am a software developer and every year I am confronted with the same problem: The software frameworks evolve and change rapidly. For each new project I have to make a decision, which framework to ...
2
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1answer
36 views

probability matching strategy for coin flips

imagine a betting game where we observe $N$ independent coin flips $x_1,...,x_n$ (where each $x_i \in {H,T}$) from the same coin, whose true weight is $\theta$. the task is to predict how many Heads ...
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2answers
28 views

resolving expected utility of st. petersburg paradox with logarithmic utility

St. Petersburg paradox is a game where you toss a fair coin repeatedly and if it lands heads on the $k$th trial you get $2^n$ dollars. Expected utility of game is: $E(U) = \sum_{k=1}^{\infty}[0.5*0 + ...
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2answers
26 views

What is the density of $y|z$ in the following problem

I have three random variables: $x$, $y$, $z$ in $\mathbf{R}$. I know the following about their distributions: $x \sim \text{unif}[-\infty, \infty]$, $y \sim \mathcal{N}(x, \sigma)$, $z \sim ...
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1answer
54 views

Converting a Summation to an integral

Please how do I convert this summation $$ \frac{r-1}{n} \sum_{i=r}^n \frac{1}{i-1} $$ to the integral $$ x \int_x^1 \frac{1}{t} dt = -x \ln x? $$ by substituting $x = r/n$, $t=1/n$ and $dt =1/n$. ...
1
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1answer
42 views

Stopping rule for house selling problem

We have a house to sell. Each day an offer of $X_n$ comes for the house. Each offer costs an amount $k$ to observe. You may think of $k$ as advertisement costs. When you receive an offer you must ...
2
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0answers
20 views

Minimax decision

Suppose an observation $x$ comes from a Bernoulli distribution: $$p_{\theta}(x)= \theta^{x}(1-\theta)^{1-x}, x=0,1 \ \ 0 \leq \theta \leq 1$$ Let's say that a future one-step observation $y$ also ...
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0answers
45 views

Secretary Problem with rank based selection and cardinal payoff

Background: The cardinal payoff variant of the Secretary problem aims to maximize the expected value of the selected applicant, assuming values of applicants are random variables X drawn i.i.d. from a ...
5
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1answer
116 views

Probability/Decision- infimum over set of expectations (can be interpreted as decision problem)

Let $X$ be a random variable over $\mathbb{R}$ with finite first moment (mean). Let $H$ be a piecewise function defined such that $H_a=c_1(x-a)$ for $x>a$, and $H_a=c_2(a-x)$ for $x<a$, with ...
0
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1answer
27 views

Identifying random variables with their generated distribution function - Necessity of countable additivity?

Let the state space $\Omega=[0,1]$ and $\lambda$ be the Lebesgue measure defined on the Borel $\sigma$-algebra on $[0,1]$. Consider measurable functions (random variables) $f:\Omega\to\mathbb{R}$ and ...
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0answers
33 views

Optimal Number of Entries for a Contest of Skill…

Objective: I'm looking for the optimal number of unique entries in a contest of skill with monetary prizes. Description: The contests vary from as little as 20 entries up to 10,000+ entries. You ...
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0answers
23 views

A Markov Decision Process problem

Consider an Stochastic shortest path problem where all stationary policies are proper. A stationary policy is said to be proper if, when using this policy, there is positive probability that the ...
0
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1answer
29 views

the intuitive difference between expected utility and utility of expected profit in a gambling game

What is the intuitive difference between expected utility and utility of expected profit in a gambling game ? Which one is the "usefulness of the game" to a player ?
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1answer
68 views

Shape of utility function

I have read in a paper (http://www.public.asu.edu/~kirkwood/DAStuff/refs/risk.pdf) that the shape of the utility function depends on the attitude towards risk. My question is does not it also depend ...
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0answers
22 views

Parallel lines in 2-simplex

I do have a problem in understanding a statement in the following argumentation. Consider a 2-simplex $\Delta := \{ (x_1, x_2) : x_1, x_2 \geq 0, x_1+x_2\leq 1 \}$. Assume that for every $P,Q,R \in ...
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0answers
49 views

How to interpret the indicator function?

I am reviewing a paper titled " Bayesian Sampling Approach to Decision Fusion" by Biao Chen and Pramod K Varshney. This paper uses an indicator function that I am not being able understand. The ...
1
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1answer
37 views

How to solve Bellman's optimal equation from the first principle

How to solve the following set (finite) of equations $$ v_*(s) = \max_{a\in A(s)} \sum_{s'} p(s'|s,a) [r(s,a,s') + \gamma v_*(s')]$$ $p$ and $r$ functions are given.
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2answers
135 views

How to use the 1/e law of best choice?

Caution: I'm not a mathematician, but I remember some of what I learned in college. I was reading about the Secretary Problem on Wikipedia, essentially about determining the optimal moment to stop ...
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0answers
36 views

Prospect of research in some stochastic optimization/approximation field

This question is a not a technical one. Sorry for that. As I am new to the area of stochastic optimization/control, I want to know the active prospect of research in the following areas 1) ...
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0answers
45 views

Simple Math Problem on Interval

It's not clear for me. I see this wikipedia page for a difference of half interval on $\mathbb{R}$ and interval on $\mathbb{R}$? For example $$ \{ (-\infty \le x \le a) \, \left|\, a \in \mathbb{R} ...
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0answers
10 views

A confusion on adaptive algorithm

Consider flight trajectory control problem i.e. find out the control parameter for which the average error of the actual output and desired output is minimized. Can we call any algorithm for solving ...
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0answers
20 views

A doubt on markov decision process

Given that a policy is a function from a state action pair to probabilities, the set of policies for a MDP forms a POSET (the partial order is due to value function for a policy). Why there should be ...
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0answers
26 views

Decision function problem based on the logistic function

Suppose we have a bunch of a sampled pairs $(x_1,y_1)...(x_n,y_n)$ with the $y_i =\pm1$. Then consider the decision function $h(x) = -1$ if $p(x)=\frac{1}{1+e^{-x}}\leq0.5$, and $h(x) = 1$ if $p(x) ...
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0answers
67 views

Decision Tree and rank?

Consider all strictly decreasing functions from $\{1,2,3,4\}$ to $\{1,2,3,4,5,6\}$, or in other words, all functions defined on $\{1,2,3,4\}$ such that $f(1)>f(2)>f(3)>f(4)$. Draw a decision ...
0
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1answer
137 views

Von Neumann–Morgenstern independence axiom vs. Savage independence theorm

Von Neumann–Morgenstern independence axiom: Savage independence theorem: What is the difference between the two? I'm think Von Neumann is talking about the prizes (outcomes) and Savage is talking ...
4
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3answers
284 views

How practically relevant is game theory?

I usually don't care too much about the practical relevance of nice mathematics :-) But this time, as I am looking to find some areas where I can apply maths and possibly collaborate with ...
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0answers
31 views

Modeling a multiple criteria decision problem

Sorry for the generic-sounding title. I am currently tasked to solve a decision problem. Currently there are two factors that I must consider before making the decision. Suppose a customer walks ...
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0answers
24 views

Name of decision method in which probability of taking an action is exactly past successes / past attempts, while alternative actions normalize

The probability of choosing among options $X_1$, $X_2$, $X_3$, $...X_n$ is initially uniform, i.e. $P(X_j)=1/n$. On choosing $X_j$, either success or failure will occur (with unknown probabilities, ...
0
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0answers
25 views

Characterizing relation “ has no less information than” between information systems represented by Markovian matrices

I'm reading this note on Blackwell's theorem which establishes the monotone relationship between the information in a information system a decision maker faces and his expected welfare. An ...
0
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1answer
37 views

Natural Decision Problem not in PTIME

Are there any natural decision problems which are guaranteed not to be in $\mathsf{PTIME}$? Preferably natural graph problems like $\mathsf{CLIQUE}, \mathsf{VERTEXCOVER}$ etc. (However, they would be ...
0
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1answer
95 views

Maximum-Value Secretary Problem

Background: The classic secretary problem has the simple solution of rejecting the first 1/e applicants and then selecting anyone who was better than the best in the rejected set. However, in the ...
5
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1answer
105 views

Is this a problem that has already been solved?

I have a question paper with $n$ True/False questions and I don't know the answer to any of those questions. My objective is to find the answer key of the question paper. All I have is a machine which ...
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3answers
3k views

Can I use the “Secretary Problem” to find the worst candidate, too?

As you know, we use the "Secretary Problem" to choose the single best candidate. Now I would like to know can we use this rule to find the worst candidate, too? If yes, how to accomplish this?
2
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1answer
198 views

Would a risk averse agent ever accept gambles with negative expected value? [closed]

Consider a risk-averse agent (his utility for money is strictly concave) that maximizes expected utility. Would such agent ever a accept a gamble whose expected value is negative? (e.g. think of state ...
2
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0answers
35 views

Decide whether a function has an elementary indefinite integral without determining it!

Risch, who developed the algorithm in 1968, called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral; and also, if ...
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0answers
68 views

How to test a flash light.

Let us say you have a flash light. At full charge, it can last for 2 hours (or $T$.) Right now, it charge is a random variable $C$ which has a uniform distribution (or distribution $D$.) You may turn ...
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0answers
94 views

MDP problem - How is the expected cost calculated?

I have been stuck with a problem for a while regarding Markov Decision Processes for a Policy improvement algorithm. Assume that I have probabilities for certain states to evolve the system into, ...