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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1answer
16 views

Emptiness and infiniteness decidable for recursive languages?

The problem of determining whether a recursively enumerable language is empty or infinite cannot be solved. The proof goes by reduction to the problem of decidability, which is known to be unfeasible ...
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0answers
15 views

Applying Markov Decision Processes to an arrival forecasting problem

I have the following problem and I'd like to know if it's something that was already studied in the literature or not. I'm not sure about the naming conventions either. I have a system $S$ that can ...
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0answers
8 views

Prospect Choice Problem

We're starting to work on stochastic dominance in my mathematical finance course, and I came across some issues related to three particular prospects. Consider two states $s_1,s_2,$ and the prospects ...
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0answers
16 views

Expected utility of action, given probability model

We record measurements of an appartus every day. If apparatus doesn't break (it has probability equal to $1-p_2$), it will measure zero with probability $p_1$. If apparatus breaks (probability $p2$), ...
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42 views

how to calculate expected utility for probability decision problem?

consider a decision problem: classifying $x$ as belonging to one of two classes $C_1, C_2$. there are prior probabilities for each class, $p(C_1), p(C_2)$ and likelihood probabilities for data given ...
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2answers
40 views

Connection between two notions of undecidability

As far as I understand, there are two common notions of "undecidability" that I will call "Mathematical undecidability":="the statement can neither be proved nor disproved from the axioms", ...
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1answer
111 views

How we decide for a given context free grammar generate an infinite number of strings?

Consider the following decision problems: (P1) Does a given finite state machine accept a given string? (P2) Does a given context free grammar generate an infinite number of strings? Which of the ...
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1answer
21 views

Value of experimentation on decision analysis problem?

The following table represents a decision analysis problem (in units of a thousand dollars) Suppose you can obtain information which will tell you, with certainty, whether or not state 1 (S1) ...
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0answers
22 views

AHP (Analytical Hierarchical Process) for individual selection decision with equally important objectives

How can I apply AHP for individual selection decision when all the objectives are important? For example, I want to select a car. The car needs to have all the characteristics (all the components). In ...
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1answer
291 views

Best strategy to find a parking lot

New Bounty Edit (2 days remaining on the Bounty): To point out that the only answer given at this time cannot be considered an answer, because it simply gives a hint on how to formally model the ...
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0answers
42 views

.Decision Theory and Bayesian Inference

An organization uses a spam filtering software to block email messages that may potentially be spam messages. The spam filter can be set to one of two security modes: High-Security-Mode (H) or ...
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0answers
35 views

Listing possible decision functions and their meanings

Question: A statistician has to decide on the basis of two observations whether the parameter ? of a binomial distribution is $\frac{1}{4}$ or $\frac{1}{2}$; his loss (a penalty that is deducted from ...
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0answers
24 views

Understanding the difference between Posterior Probability versus Likelihood

I am trying to understand the difference between posterior probability versus likelihood. Here is the source material I'm working off of: To say that $H$ has a high likelihood, given observation ...
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2answers
40 views

How to group people based on their choices? What algorithms are available?

For example I have eight kids, A,B,C,D,E,F,G,H If I ask them to go into groups of two, their choices are ...
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0answers
53 views

Approximate Dynamic Programing - Discount Factor for Very Long Horizons

I want an optimal strategy for a very long time horizon, say $K=100000$. I have dynamic decision making problem where next state $x_{k+1}$ is determined by the probability distribution ...
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2answers
48 views

Maximal clique problem

I understood what clique is all the nodes of the sub graph have to be connected to each other. In the following figure, it says that the maximal clique is {1,2,3,4,5}. But as per the definition of ...
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1answer
31 views

Proof that $\min_{b\in B} u(a,b)\leq \min_{b\in B}\max_{a\in A}u(a,b)$

So I have two finite sets $A,B$ and $u:A\times B\rightarrow \mathbb{R}$ a utility function. I am asked to give a certain proof but I don't need help with the whole thing, I just need help figuring ...
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1answer
25 views

How to Solve Multiple Stopping Problem with a Known Payoff Distribution

I'm interested in learning how to optimally solve a multiple stopping problem with a known payoff distribution, like the following: You are observing a sequence of forty $(40)$ opportunities, each ...
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1answer
36 views

Non-integer $n$ in sample size problem

Setup Consider a sample size determination problem with the maximization of expected utility approach (as in Lindley 1997). Let $\theta$ be the state, $x=(x_1,\dots,x_n)$ a sequence of $n$ iid ...
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2answers
52 views

AS Level Decision mathematics (D1) problem [closed]

I've been doing the past papers of my D1 decision mathematics module and I came across a question I couldn't do. The question is part D of the below. I'm not sure how to tackle it as usually you turn ...
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0answers
7 views

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
44 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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0answers
28 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
40 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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0answers
18 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
119 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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0answers
50 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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0answers
39 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
85 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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0answers
29 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 ...
0
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1answer
40 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 $\alpha_{min}\in\mathbb R$, $\alpha_{min}<1$, $\alpha_{max}\in\mathbb R$, ...
0
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1answer
38 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
40 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
86 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
19 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
25 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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47 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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42 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 $\delta$ ...
2
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1answer
83 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
85 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
61 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
164 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
26 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
163 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
161 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
49 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
39 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 ...
0
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1answer
41 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 ?