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
21 views

Does anybody know about an ebook version of Saaty The Analytic Hierarchy Process?

I'm looking for Thomas L. Saaty The Analytic Hierarchy Process in pdf, but I only found hardcover versions to order.
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0answers
25 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
8 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
22 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
42 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 ...
-5
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1answer
55 views

Attitude toward risk taking and the exponential utility function [closed]

I want to know some reference/book on the following topic: "Attitude toward risk taking and the exponential utility function". Thanks in advance.
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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
44 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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0answers
24 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
70 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
26 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
41 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
8 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
23 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
61 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
75 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
184 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
28 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
22 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 ...
0
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1answer
34 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
53 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 ...
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1answer
85 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?
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1answer
158 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 ...
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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
66 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
71 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
39 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
47 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
119 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
114 views

Mistake 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
138 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 ...
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6answers
158 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 ...
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1answer
68 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 ...
2
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4answers
121 views

death as a cost in decision theory

If I presented with an optional task for which I have an outcome independent investment $I$, a probability of success $P$ and a reward for success $R$, then I chose to undertake this task iff $PR > ...
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0answers
38 views

Inadmissibility of constant decision rule

My loss function is the squared error loss, $L(\theta,a)=(\theta-a)^2$. Suppose $X\sim U(0,\theta)$ given $\Theta=\theta$. I need to show that the constant decision rule $\delta(x)=c$ is inadmissible ...
3
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1answer
172 views

Formal Reduction: Pushdown Automata recognizing context free languages with bounded stack

I am studying for an exam in automata theory and I am having trouble solving the following: Consider pushdown automata and context free languages. Show that the following decision problem is ...
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1answer
2k views

How to compute the consistency index in Analytic Hierarchy Process?

By studying AHP (Analytic Hierarchy Process), I have a question about RI. The (CI) is the consistency index. The RI value is fixed and is based on the number of criteria evaluated. $CR = CI / RI$ ...
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2answers
84 views

Making a decision

Like the title "Making a decision", I am standing for a problem. I've calculated two probabilities. $$P(\text{Match}) = 0.24$$ $$P(¬\text{Match}) = 0.76$$ Now I've to make a decision. Does ...
2
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0answers
107 views

Lower bound on uncertainty reduction

Let $T$ be a set of tuples such that each score tuple $s(t_i)$, $t_i \in T$ is uncertain (i.e., not known deterministically). The score $s(t_i)$ can be represented as a uniform probability density ...
3
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0answers
96 views

Inadmissibility of Simpson's rule

Let $B_t$, $t\ge0$ be a standard Brownian motion and suppose $0<x_1<x_2<\cdots<x_n<1$. Then the conditional expectation $$ \mathbb E\left(\int_0^1 B_t\,dt \,\middle\vert\, B_0, ...
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1answer
65 views

Lower Expectation

Let $X$ be, for simplicity, a finite set (with the discrete topology). Denote with $M(X)$ the set of probability measures on $X$ endowed with the weak topology. For $\mu\in M(X)$ and a (necessarily ...
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2answers
130 views

What is the most common decision making math model?

I have been told that there are psychological math model to make decision. I want to know what is it and how it work.
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1answer
234 views

Coin Game, Probability and Fairness

The following game is being played : Player $\mathrm{B}$ pays to Player $\mathrm{A}$ an amount $\mathrm{X}$ and throws a coin at most $20$ times. If at the toss $k\space (k \leq 20)$ tail is thrown, ...
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1answer
128 views

question related to Bayes' rule and Bays' risk.

Let $X_1, X_2, X_3, \ldots, X_n$ be a random sample for $N(e,1)$. Let the prior p.d.f. of $e$ be $N(0,\sigma^2)$ under the square error loss function $L(e,d)={(d-e)}^2$. Find the Bayes' decision rule ...
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1answer
99 views

Deducing probability of an event, when opponent's type is uncertain

Suppose, two players I and II, given a state space of three states$\{a,b,c\}$ with a common prior, $p(a) = p(b) =p(c) =1/3$, are endowed with two partitions of state space, $\mathscr{P}_\text{I} = ...
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2answers
914 views

Sum of two stopping times is a stopping time?

Let $\sigma$ and $\tau$ be two stopping times in $\mathscr{F}_t$ and let this filtration satisfy all the usual conditions. Question: Is $\sigma + \tau$ a stopping time? Attempt at a solution: I ...