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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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 ...
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8 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, ...
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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 ...
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12 views

Combinations OR Decision tree ? Six Spices - Total flavors

this is a simple question for which I'm trying to reason. Suppose you have 6 spices, what is the possible number of flavors you can make ? You may assume that you can only combine one spice once to ...
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14 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 ...
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20 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 ...
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11 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}), ...
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6 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 ...
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1answer
15 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 ...
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17 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 ...
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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 ...
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22 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 ...
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1answer
22 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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1answer
32 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 ...
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0answers
15 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 ...
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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 ...
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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 ...
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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 ...
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15 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
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1answer
28 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 ...
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1answer
30 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 ...
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1answer
28 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 ...
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12 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 ...
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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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47 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 ...
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2answers
41 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", ...
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1answer
142 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 ...
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1answer
26 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) ...
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23 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 ...
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1answer
326 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 ...
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0answers
63 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 ...
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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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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 ...
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2answers
45 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 ...
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0answers
56 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 ...
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2answers
53 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 ...
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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 ...
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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 ...
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1answer
39 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 ...
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2answers
56 views

AS Level Decision mathematics (D1) problem [closed]

I've been doing the past papers of my D1 decision mathematics module and I came across a question I couldn't do. The question is part D of the below. I'm not sure how to tackle it as usually you turn ...
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8 views

Weighted Decision Making on the basis of Two Significance Indexes

I'm trying to make the best decision for assignment of books(some of the books have multiple authors that belong to different organizations).Only one author can be represented by a single book. I have ...
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0answers
46 views

Monotonocity of ratios of normal CDFs

I am solving a problem in decision theory under uncertainty and need to establish whether $\frac{\Phi(x)-\Phi(x-\varepsilon)}{\Phi(x+\varepsilon)-\Phi(x-\varepsilon)}$ $(\ast)$ is monotonically ...
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28 views

Adjusting probabilities in a Decision Tree

I've got a bit confused with adjusting the parameters to change the decision from a Decision Tree, and consequently with the sequence in getting optimal value. I'm thinking about a rather simple and ...
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1answer
43 views

How much is it worth to participate in a second price auction?

You have a valuation for an object (say $v_a$), which you don't know yet but you know is distributed U[0,1]. You will be competing in a second price auction against a completely identical guy as you, ...
0
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2answers
135 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 constraints. To ...
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51 views

Have there been any attempts to unify statistics and decision theory into a single framework that refrains from estimating probabilities?

If I understand correctly: statistics, narrowly construed, is all about using data to estimate probabilities. decision theory can then be applied to those probabilities in order to predict which ...
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99 views

Explaining daily life event using prospect theory

I am new to decision theory and currently I am reading the book 'Making Better Decisions: Decision Theory in Practice' by Itzhak Gilboa. I am fascinated by the discussion of utility function and risk ...
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32 views

Rao-Blackwell improvement for a nonrandomized estimator

Context: please consider a parametric statistical model $(\mathcal{Y},\{P_\theta:\theta\in\Theta\})$ and suppose that we are estimating $g(\theta)$. Associated with this is the set of decisions ...
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1answer
40 views

Reducing an I-optimal problem to a Pareto-optimal problem

Given a set $\textbf y\subset\mathbb R^2$, let $y = (y_1,y_2), y'=(y'_1,y'_2)\in\textbf y$ be elements of that set, let $\alpha_{min}\in\mathbb R$, $\alpha_{min}<1$, $\alpha_{max}\in\mathbb R$, ...
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1answer
39 views

Biobjective optimisation, pareto non-domination

Ok, so, I have a function $f_I(y_1, y_2) = \max\{\alpha y_1 + (1-\alpha)y_2:\alpha\in[\alpha_{min},\alpha_{max}]\}$ that I'm trying to minimise, and I'm asked to find, amongst a set of vectors $y$, ...