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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Hamiltonian Path to Hamiltonian Cycle reduction

I have to show that HP polynomially transforms HC by following steps: $(1)$ Construct a polynomial transformation $f$ from HP to HC. $(2)$ Show for all graphs $G$ that $G ∈ YHP ⇒ f(G) ∈ YHC$. $(3)$ ...
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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 ...
120 views

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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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 ...
1k views

Pairwise majority voting and Arrow's axioms

The following is a question on Arrow's theorem with a pairwise majority decision. The bits I was unsure about was (bi) (is the 4th condition satisfied?) and also is (bii) correct? Thanks for your help ...
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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 ...
6k views

Basic concept of utility: utility of expected value vs expected utility

I'm new to the concept of utility and I'm struggling to understand an important idea. Say we have some bet. I don't understand how the utility of the expected value of the bet differs from the ...
1 vote
236 views

Is there a known example of a voting system that does not satify the dictator fairness criterion but does satisfy the others?

As Wikipedia says, Arrow's impossibility theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria: If every voter prefers ...
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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 ...
532 views

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 ...
134 views

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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Would a risk-averse agent ever accept gambles with negative expected value?

Consider a risk-averse agent (whose 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 ...
1 vote
305 views

Far-too-simple proof of (limited) Debreu representation theorem; where's the error?

Debreu's theorem: Let $X$ be a topological space that is connected separable or second countable. A binary relation relation $\succsim$ on $X$ is complete, transtitive, and continuous if and only if ...
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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 ...
647 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 ...
97 views

Dutch Book for VNM Axiom 4

I'm unconvinced that my refusal to accept the fourth of the von Neumann-Morgenstern axioms is irrational. Wikipedia claims that there is a Dutch book argument against me, but I do not see how that can ...
1 vote
71 views

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
148 views

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 ...
1 vote
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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 ...
157 views

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 ...
1 vote
2k 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 ...
1 vote
161 views

Secretary problem

I have a problem with the secretary problem, I wanted to prove that maximum value of the probability function of choosing the best applicant is decreasing as n gets bigger. So, in other words: F(n):=...
132 views

Does an ergodic Markov Decision Process have a unique optimal gain?

It is known from chapter 5 of Dynamic Programming and Optimal Control Vol II that a uni-chain Markov Decision Process (MDP) has a unique gain-bias solution $(J,\vec{h})$ to the following infinite-...
Do there exist a family of measurable functions $(f_t^\delta)_{t \in \mathbb{N}, \delta \in (0,1)}$ and constants $C,c>0$ such that, for each $t \in \mathbb{N}$ and $\delta \in (0,1)$ we have that \$...