# Tagged Questions

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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### Arrow's impossibility theorem implications [closed]

If we knew that in an unfair voting system (ie: it is known that IIA does not hold) then if (1) holds, how would we go about inferring that (2) holds and then is it possible to show that if (1) and (2)...
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### How exactly is the St Petersburg Paradox giving bounded payoff in average-of-N-trials?

I understand why the expected value of the St Petersburg Paradox is algebraically infinite, but intuition tells me that in practice any given round of the game will not go on multiplying the pot for ...
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### Minimum Spanning Tree - Finding X in an interval (adjustment of an arc)

I am attempting a question on MST below, but I do not understand the answer of the question. I am trying to work backwards form the answer to find out how they got to the solution. The question I am ...
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### Sensitivity Analysis for Probabilities of States of Nature

QUESTION: SOL:Choose N if p < .78 Really usure how the solution has been derived. Can someone please explain how they came to this solution?
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I'm studying measure theoretic stochastic calculus, and I was hoping to pick up some knowledge of decision theory along the way. I'm very happy with Rudin or Karatzas in level of rigor, and I was ...
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### Counterexample to each Arrow's axiom

I am trying to find ways that combine group preferences which hold for 3 out of the 4 Arrow's axioms. Here is what I have come up with so far: Unrestricted Domain Not sure what scheme would ...
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### Which of these influence diagrams are not valid?

Which of these are valid influence diagrams? For those that are not, state the exact reasons why The squares, circle and diamonds represent decision, chance and value nodes respectively. (1) - ...
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### Bayes risk and Bayes decision

We are considering a sample of size $n$ from an exponential distribution, with parameter $w >0$. We wish to produce an estimate for $d$, for $w$ , with loss function: $L(w, d)=w(w-d)^2$ The prior ...
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### Doesn't “Almost Decisiveness” (ADC) imply “Decisiveness” (DC)?

The title may seem a bit counter-intuitive, but I can't get my head around this: Almost Decisiveness: If, when all individuals in a (finite or infinite) group G prefer an alternative x to another ...
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### Are these preferences rational according to utility theory?

I have this question about whether preferences satisfy the $6$ properties of utility, and am particularly stuck on the boundness, coherence and continuity conditions. Here is the problem: If one ...
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### Does there exist a utility function for these gambles and preferences?

Suppose that you prefer a gamble which pays $£100$ with probability $0.5$ and $£10$ with probability $0.5$ to a gamble which pays $£200$ with probability $0.25$, $£50$ with probability $0.25$ and $£10$...
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### Show, by example, that we can find social welfare functions which satisfy any three of the four Arrow's axioms.

I want to show, by example, that we can find social welfare functions which satisfy any three of the four Arrow's axioms. Given at least three rewards, and at least two individuals, there is no ...
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### 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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### Arrow's theorem, strategic thinking and utilitarianism

I have this problem as part of a course on Decision Theory, and was not sure about question a (4th condition of Arrow's theorem) and question dii (utilitarianism). I have provided the whole question ...
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### Sen's theorem on minimal liberalism

I want to solve the following question regarding Sen's theorem and preferences orderings. The bit I am having trouble with is 2b Question 1: Describe Sen's theorem ('paradox') There is no social ...
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### Undecidability of first-order satisfiability problem?

I need some clarification on understanding the "undecidability of" First-Order Logic (onwards, FOL). I understand that it means that the set of FOL theorems is undecidable (i.e. there is no effective ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### .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 Low-...
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### 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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### 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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### 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 ...
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 ...
### 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 ...