# Questions tagged [decision-theory]

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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### Estimating the value of information for determining when to stop an experiment

Suppose that there are two parameters $p_1, p_2 \in [0, 1]$, and you begin with an uninformative prior $\textrm{Beta}(1, 1)$ for both of them. An experiment has been running for $d$ days. On each day ...
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### Optimal strategy in a number-picking game against a perfect logician?

I'm thinking about a game scenario involving three players: myself, an opponent, and a referee. Each player picks a real number between 0 and 1, and the referee will select a number randomly between 0 ...
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### How to solve for Nash Equilibrium?

Image: I am currently studying for a college exam next week in Games Theorie. Unfortunately the example questions are very different to the course material and im stuck on this one. I would solve the ...
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### Proving a (Representing Utility) Function is Continuous

I sincerely apologize for posting such a long question. The question involves a complicated proof of a theorem in mathematical economics. I feel it will be better for me to state my question first. I ...
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### Calculation of normalized values for cost-type criteria in the weighted sum model

According to pymcdm WSM is calculated as follows: $$A_i^{\text{score}} =\sum_{j=1}^n \bar{x}_{i j} w_j \quad \text{for } i=1,2,3,\ldots ,m$$ Where: $m$ is the number of alternatives $n$ is the ...
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### Minimax Estimation: What's the difference between Minimax, Sharp Minimax, First-Order Sharp Minimax and Second-Order Sharp Minimax Estimator?

I am currently working on my dissertation on Biased Data, and the second chapter focuses on distribution function estimation. Efromovich's work appears to be an outstanding reference on the topic, but ...
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### Every cut is the union of (edge-disjoint) minimal cuts

I am tasked with proving the following statement: "Every cut is the union of edge-disjoint minimal cuts" The only information given, is the existance of the cut-set subspace $W_S(G)$. It ...
1 vote
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### Infinite sum $\sum_{t=c}^{n-1}(\frac{1}{t^2-1})$ as part of cardinal payoff variant

What does this sum $\sum_{t=c}^{n-1}(\frac{1}{t^2-1})$ equal? For context, I am trying to digest the cardinal payoff variant of the secretary problem. There is an interview process with $n$ candidates ...
1 vote
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### Example where r.v. $X_2$ stochastically dominates $X_1$ but $P(X_1 > X_2) \geq 0.95$

The problem is from a textbook I'm reading, but even with the hint, I'm not being able to come up with a solution. Let $X_1$ and $X_2$ be two random variables with CDFs $F_1$ and $F_2$. We say $X_2$ ...
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### Expected Utility, decision theory

I’m slightly confused how to assign utility if there are 2 pieces of information given about its value. For example, there are 2 decisions: to go to the movies or to go fishing. Provided that you get ...
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### two persons roll dice and bid game: optimal strategy

Two persons $A, B$ roll a fair $n$-face dice separately and get $1 \le x,y \le n$ points. Then the third party will put $x + y$ dollars in a black box. $A$ and $B$ only know the point they roll and ...
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### Minimizing the expected loss when there is a general loss matrix

This question is related to question 1.23 of "Pattern Recognition and Machine Learning" by Bishop. The question asks "Derive the criterion for minimizing the expected loss when there is ...
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### Recommendations on Intermediate Level Probability/Applied Statistics Book

So I'm an Internal Medicine Resident with an interest in mathematics and I have a BS in physics and MS in math. Lately I've been getting more into the statistical interpretation of diagnostic test, ...
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### Will the duality gap be zero if the constraint is satisfied with equality?

Consider the following problem that aims to find an optimal policy $\pi$ mapping a state $s\in\mathcal{S}$ to an action $a\in\mathcal{A}$: \begin{array}{cl}\tag{1} \displaystyle \underset{\pi:\mathcal{...
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### What to do when in Coombs voting method there two equal weights for candidates to be elimenated?

I've read about Coombs method on Wikipedia. I understand that we eliminate candidate with the most last-place votes. But what do we do when, for example, two candidates A and B have equal number of ...
1 vote
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### Minimizing average depth of a decision tree: when is greedy optimal?

Fix parameters $m$ and $n$. Consider a finite set $A$ and a set of attributes $f_1,\ldots,f_m$ where $f_i: A \rightarrow [n]$. We want to construct a decision tree $T$ with minimum average depth. We ...
1 vote
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### Proof in a clique decision problem (karp reduction)

Considering the following decision problems: E_CLIQUE(G, k), where G = (V, E) is a simple graph and k >= 1 an integer. Does G have a clique of size 2 · k? and CLIQUE(G, l), where G = (V, E) is a ...
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### Why does the uniform law of large numbers hold with non-i.i.d. random variables in Bayesian experimental design?

This paper, Asymptotic theory of information-theoretic experimental design, studies Bayesian experimental design where in each round $n$, the experimenter selects a stimuli $X_n$ that maximizes mutual ...
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### An interesting game "The Truel"

There is an interesting paper called The Truel. It is about 3 players A , B and C shooting under some rules.The two snippets of the pages relevant to my Question are given below, the full paper is ...
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### Prove that the preferences follow the Von Neumann and Morgensten's axioms

I'm studying Decision Theory from the book 'An introduction to decision theory' by Martin Peterson, and there is a problem that I don't understand how to solve. The problem is: You prefer a fifty-...
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### How lower than P(S|R) P(S|~R) must be in order for the expected value of R to be higher than the expected value of ~R

I was asking myself how lower than P(S|R) P(S|~R) must be in order for the expected value of option R to be (strictly) higher than the expected value of ~R, given the following value assignments to ...
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