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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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
5 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
13 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
17 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
29 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
30 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
24 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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0answers
9 views

Decision Analysis, Utility Function

Tom planning to invest in the stock market. Utility function U(m) = m^2 for what values of x & y maximises utility when action a1 is applied?
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2answers
25 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 $dt =1/n$. ...
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1answer
40 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
19 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
32 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
112 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
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
32 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
18 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
28 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
62 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
20 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
48 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
35 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
128 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
32 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
44 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
9 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
17 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
25 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
66 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
127 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
251 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
30 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
23 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, ...
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0answers
24 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 ...
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1answer
35 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 ...
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1answer
83 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
98 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
193 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
33 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
67 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
85 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, ...
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0answers
48 views

Decision Making Algorithms - Chaos Theory

I'm doing research on decision making algorithms on robotics. And recently I've read a lot of about Chaos Theory. I've searched all over the web, in IEEEXplore, ACM Digital Library, but can't find any ...
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0answers
61 views

Classification problem: admissible rule is a Bayes rule for some prior $\pi$

I have a classification problem where I want to place an observation $X$ into a population described by a pdf equal to either $f_1$ or $f_2$. Given $P_{f_i}(\frac{f_1(X)}{f_2(X)}=j)=0$ for all $j\in ...
4
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1answer
128 views

Maximising probability for financial advice

I have the following problem: A financial advisor tries to impress his clients if immediately following a week in which the ftse index moves by more than $5\%$ in some direction he correctly ...
3
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1answer
149 views

Mistake in Wikipedia article on St Petersburg paradox?

I suspect that there is a mistake in the Wikipedia article on the St Petersburg paradox, and I would like to see if I am right before modifying the article. In the section "Solving the paradox", the ...
2
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1answer
160 views

Question on the equivalence of behaviour strategy and mixed strategy for a player with a single information set

Prove that if a player in an extensive-form game has only one information set, then his set of mixed strategies equals his set of behavior strategies. This is the exercise $6.4$ on page $246$ in ...
3
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6answers
161 views

When to stop doubling down?

My question is similar to this one but very specificly different When to stop in this coin toss game? Imagine a game where you would start with $100. Every time you can roll a die (d6), if it is 1-5 ...
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1answer
65 views

simultaneous probability elicitation from multiple agents without an exogenous banker

A proper scoring rule is a function $f:[0,1]\times\{0,1\}\to \Bbb R$ such that, if a subject will receive a reward of $f(x,0)$ for reporting his estimate of the likelihood of an event as $x$ if the ...
0
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1answer
78 views

Decision theory question about selling a house

I have a real world problem and I was wondering if you guys have any nice insight on the best way to solve it mathematically. I'm not sure there is a decisive solution, but it would be nice to have a ...