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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Weighted Decision Making on the basis of Two Significance Indexes

I'm trying to make the best decision for assignment of books(some of the books have multiple authors that belong to different organizations).Only one author can be represented by a single book. I have ...
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0answers
30 views

Monotonocity of ratios of normal CDFs

I am solving a problem in decision theory under uncertainty and need to establish whether $\frac{\Phi(x)-\Phi(x-\varepsilon)}{\Phi(x+\varepsilon)-\Phi(x-\varepsilon)}$ $(\ast)$ is monotonically ...
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13 views

Extension of Tarski's result on the decidability of reals

Due to Tarski's result, it is well-known that the theory of reals $(\mathbb{R},+,\cdot,<,=,0,1)$ is decidable. I am working on a paper where I need an extension of this result. More precisely, I ...
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10 views

How to get attribute weights from a tradeoff? [closed]

Assume that we are getting oranges and apples from a fruit basket, getting apples is seen more important than oranges, and the outcome of x=(0 apple, 25 oranges) is equally preferred as x=(10 apples,0 ...
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0answers
15 views

Adjusting probabilities in a Decision Tree

I've got a bit confused with adjusting the parameters to change the decision from a Decision Tree, and consequently with the sequence in getting optimal value. I'm thinking about a rather simple and ...
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1answer
28 views

How much is it worth to participate in a second price auction?

You have a valuation for an object (say $v_a$), which you don't know yet but you know is distributed U[0,1]. You will be competing in a second price auction against a completely identical guy as you, ...
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1answer
21 views

Binary string that contains all substrings of length k exactly once.

Here is the problem I'm working on (Satan's apple with bounded memory decision problem): Suppose I offer you slices of an apple, first $1/2$, then $1/4$, ... then $1/2^k$, etc. You want to eat as much ...
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17 views

Decision Theory - Utility for Health States (QALY)

The following comes from the book 'Decision Theory: Principles and Approaches' by Giovanni Parmigiani and Lurdes Inoue. In this example, we make use of the concept of QALY - quality-adjusted life year ...
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2answers
49 views

Methods to translate global constraints to local constraints

Are there any general methods for (global) optimisation which can translate a global optimisation problem to a "local" one? Or in other words, translate global constraints to local constraints. To ...
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45 views

Have there been any attempts to unify statistics and decision theory into a single framework that refrains from estimating probabilities?

If I understand correctly: statistics, narrowly construed, is all about using data to estimate probabilities. decision theory can then be applied to those probabilities in order to predict which ...
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36 views

Optimal choice of job based on multiple ranks

First of all I should state that I am a non-mathematics student but am pretty mathematically-inclined. I have a problem that I can't find a solution to on Google. Here is the hypothetical: I have ...
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0answers
46 views

Explaining daily life event using prospect theory

I am new to decision theory and currently I am reading the book 'Making Better Decisions: Decision Theory in Practice' by Itzhak Gilboa. I am fascinated by the discussion of utility function and risk ...
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1answer
65 views

some Graph and NP Theory Problems [closed]

my instructor solve the following problem, that which of the following is into a NP Class? ...
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31 views

Expected utility for cost

I want to compare two alternatives under uncertainty. I know if I had the values and the probability of their occurrence, I would be able to calculate the expected utility of each alternative. ...
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0answers
14 views

Rao-Blackwell improvement for a nonrandomized estimator

Context: please consider a parametric statistical model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ and suppose that we are estimating $g(\theta)$. Associated with this is the set of decisions ...
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1answer
37 views

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 $\alpha_{min}\in\mathbb R$, $\alpha_{min}<1$, $\alpha_{max}\in\mathbb R$, ...
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1answer
31 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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1answer
21 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 ...
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2answers
53 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 ...
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2answers
67 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
15 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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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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45 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
16 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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28 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
22 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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37 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 ...
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1answer
43 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
54 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
28 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
56 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 $dt =1/n$. ...
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1answer
56 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 ...
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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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93 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 ...
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1answer
148 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 ...
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1answer
34 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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34 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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29 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 ...
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1answer
35 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
99 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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25 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
52 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 ...
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1answer
49 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
184 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
43 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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1answer
39 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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68 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 ...
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1answer
176 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 ...
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3answers
428 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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33 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 ...