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
40 views

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)...
3
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1answer
57 views

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 ...
1
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1answer
9 views

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 ...
0
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0answers
14 views

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?
1
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1answer
36 views

Decision theory references for advanced undergrad/early grad students?

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 ...
0
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1answer
27 views

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 ...
0
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0answers
13 views

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) - ...
0
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0answers
24 views

Decision tree probabilities

I am solving the following decision problem. I formulated the problem into the tree OK but am unsure of what the probabilities along the branches are. i.e. how do I go from, for example, $P(H_1 \cap ...
4
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1answer
59 views

Analyse the US presidential election from the perspective of Arrow's theory

I want to consider the voting system against the 4 Arrow's axioms. So first the voting system simplifies to first-past-the post system where the highest polling candidate wins the election. How does ...
0
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0answers
7 views

Normalize the output in decision theory

I have more inputs each with a weight. One user can have more inputs and based on sum of input * weight he will receive a score. The problem is that range of inputs are from 0 to 100, but the sum of ...
-2
votes
1answer
26 views

Expected Utility Decision Theory Problem [closed]

Consider a person who choose among lotteries. Each lottery is of the form (p1, p2, p3) where p1 is the probabilty of getting Rs.5, p2 is the probabilty of getting Rs.1 and p3 is the probabilty of ...
0
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1answer
24 views

Algorithm with undecidable input set?

I am interested in "Relative Decision Problems" in the following sense: Let $\mathbb{N} \supseteq U \supseteq S$. Is there an algorithm such that on a given input $u \in U$ decides whether $u \in S$? ...
0
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0answers
76 views

Advantages/disadvantages of a utiliarian social welfare function

If a planner obeys anonymity and strong Pareto principle (individual preferences carry over to the group), then the social utility function will be: $W(x)=\sum_{i=1, ..., n}U_i(x)$ i.e. summing ...
0
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0answers
13 views

Scotts axiom, representation theorems for Qualitative -Numerical Probability function relations

Scotts theorem/axiom and other representation theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, ...
0
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0answers
5 views

How can I decide, using a strictness parameter, whether a collection of values are “equal enough” to be trusted?

Suppose I have a set of N scalar data points which I don't entirely trust the measurement of but can't repeat the measurement myself, nor is there any way to generate further data. For example, I am ...
1
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0answers
16 views

What are some techniques of constructing a good utility matrix?

A utility matrix is considered to be subjective and arbitrarily defined. Therefore, we run the risk of over-emphasizing or under-emphasizing the possible alternatives. Are there ways to design an ...
0
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0answers
24 views

Travelling salesman - organising a tour of any European destination based on the cheapest flights available.

I apologise if this has only a tenuous link to a mathematics forum I'm sure everyone is familiar with the £10 one-way flights by Ryanair and similar airlines in Europe. I was wondering whether there ...
0
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0answers
13 views

Marginal probabilities

I am stuck on the following problem of calculating marginal probabilities, which I have highlighted in yellow: Given the information below, how do we calculate$ p(X=0|w=\frac{1}{4}), p(X=0|w=\...
1
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0answers
11 views

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 ...
0
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1answer
23 views

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 ...
0
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0answers
20 views

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 ...
0
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0answers
12 views

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 $£10$...
0
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0answers
32 views

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 ...
0
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1answer
30 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 ...
0
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1answer
36 views

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 ...
0
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0answers
19 views

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 ...
1
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0answers
32 views

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 ...
0
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0answers
6 views

Show that if $g_w(M_i)>g_w(M_j)$ and $g'(M_i)>g'_w(M_j)$ then $g_w^{\alpha}(M_i)>g_w^{\alpha}(M_j)$

Let be $w$ and $w'$ two weight vectors. Let be $g_w(M_i)$ the weighted sum of the evaluations of the $M_i$ model considering the weights $w$. Show that if the weighted sum leads to prefer $M_i$ to $...
-1
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1answer
22 views

Condorcet winner in an examen subject with typo

I took a course in decisional models this semester. It is delightful because it allows to understand why are the weaknesses of every decision making process. Yet, the teacher gave us an old exam ...
0
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0answers
17 views

An algorithm to best match a collection of products based on several variables

I am not sure how best to describe what I am thinking, but I will give it my best shot. Lets say your body needs: 50g of glucose, 20g of carbohydrates, 5g of fibre and 120g of protein. Your fridge ...
2
votes
1answer
33 views

Bayesian Expected loss integral

Thanks. I don't understand how to calculate the integral for a Bayesian Expected Loss. The problem is from Berger 1985 Stat Decision Theory and Bayesian Analysis page 8. Example 1. Assume no data is ...
1
vote
1answer
39 views

Emptiness and infiniteness decidable for recursive languages?

The problem of determining whether a recursively enumerable language is empty or infinite cannot be solved. The proof goes by reduction to the problem of decidability, which is known to be unfeasible ...
3
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1answer
30 views

Applying Markov Decision Processes to an arrival forecasting problem

I have the following problem and I'd like to know if it's something that was already studied in the literature or not. I'm not sure about the naming conventions either. I have a system $S$ that can ...
0
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0answers
15 views

Prospect Choice Problem

We're starting to work on stochastic dominance in my mathematical finance course, and I came across some issues related to three particular prospects. Consider two states $s_1,s_2,$ and the prospects $...
1
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0answers
18 views

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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0answers
49 views

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 ...
1
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2answers
42 views

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", ...
0
votes
1answer
186 views

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 ...
1
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1answer
29 views

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) ...
0
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0answers
24 views

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 ...
8
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1answer
368 views

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 ...
0
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0answers
66 views

.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-...
1
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0answers
39 views

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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0answers
31 views

Understanding the difference between Posterior Probability versus Likelihood

I am trying to understand the difference between posterior probability versus likelihood. Here is the source material I'm working off of: To say that $H$ has a high likelihood, given observation $...
4
votes
2answers
46 views

How to group people based on their choices? What algorithms are available?

For example I have eight kids, A,B,C,D,E,F,G,H If I ask them to go into groups of two, their choices are ...
6
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0answers
57 views

Approximate Dynamic Programing - Discount Factor for Very Long Horizons

I want an optimal strategy for a very long time horizon, say $K=100000$. I have dynamic decision making problem where next state $x_{k+1}$ is determined by the probability distribution $f(x_{k+1}|u_{k}...
0
votes
2answers
55 views

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 ...
3
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1answer
31 views

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 ...
1
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1answer
26 views

How to Solve Multiple Stopping Problem with a Known Payoff Distribution

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 ...
0
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1answer
41 views

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 ...