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Originally posted by aethsilgneHow about I break the mould and get your " basket" and ram it down your throat. You're dead, right, so your considerations and maths don't count anymore.
What were trying to show is that man is free to make his choices but that the choices man makes are always quite logic given his/ her preferences. If we would know what your preferences are, how many utils you would give to all your goods in your 'basket' than we would be fairly able to predict your behaviour.
NB. try to see goods not only materialistically, but also as 'going for a walk' , 'going on a date'
it's quite lame, have you seen " Who wants to be a @~{aire?
Would you guess when you have 50/50 for 1 mill or 50k. Maths says you guess, the pay off is greater than the loss, but it's not that easy. A person uses other things, what that money could do for them. It's not just maths and statistics, that's what makes it interesting.
Originally posted by PalynkaI can try this. But first just let me know what do you mean with imply completeness.
First homework, prove that the properties of asymmetry and negative transitivity over strict preferences imply completeness and transitivity of the weak preferences. Note: weak preference of x over y just means that x is strictly preferred or indifferent to y.
Originally posted by RoostySay you divide all actions you take into rational and irrational actions. To make my point clear; only rational actions can be explained mathematically.
OK 🙂 I was being sarcastical
Irrational actions (like most of the bets) are based on a wrong perception of cost and benefit analysis imo.
Originally posted by aethsilgneAdd in the coefficient of risk aversion while you're at it!
Say you divide all actions you take into rational and irrational actions. To make my point clear; only rational actions can be explained mathematically.
Irrational actions (like most of the bets) are based on a wrong perception of cost and benefit analysis imo.
Originally posted by PalynkaI'll write xWy for weak preference, and xIy for indifference, i.e. xIy iff x~Py and y~Px. Note that I is symmetric. The definition of weak preference is: xWy iff xPy or xIy.
Let P be a (strict) preference relation over a given set X. A preference relation is a binary relation (i.e. it relates two elements of a given set) that has the properties of asymmetry and negative transitivity.
More specifically, asymmetry means that if x strictly preferred to y (from here on xPy) then y is not preferred to x (from here on x~Py).
Ne ...[text shortened]... to affine transformations) function u for which xPy => u(x) > u(y), for all pairs (x,y) in X.
Also, indifference is transitive: if xIy and yIz, suppose x~Iz. Then wlog xPz. But by the hypothesis, x~Py and y~Pz, so by negative transitivity, x~Pz, a contradiction. Thus xIz, so I is transitive (it's an equivalence relation, which it should be, intuitively).
Suppose that x~Wy. Then x~Py and x~Iy. The latter implies that xPy or yPx, so by the former, yPx. In particular, yWx. Thus either xWy or (inclusive) yWx (note that both hold in the case of indifference). This is completeness of W.
Suppose that xWy and yWz and x~Wz. Then x~Pz and x~Iz. Since xWy, either xPy or xIy. In the first case, asymmetry of P implies y~Px. In the second case, the definition of I implies y~Px. Thus y~Px and x~Pz, so y~Pz by negative transitivity. Since yWz and y~Pz, yIz.
If xIy, then yIz implies xIz by transitivity of I; in particular xWz, a contradiction.
If xPy, then, since yIz and x~Wz, we have x~Pz and z~Py, so by negative transitivity, x~Py, a contradiction.
Thus xWy and yWz imply xWz, so W is transitive.
Now consider X/I, the set of equivalence classes such that x and y belong to the same class iff xIy. Denote the I-class of x by [x]. If x and y represent different I-classes, then xPy or yPx (but not both). Thus P totally orders X/I, so we can assign a real number u to each I-class in such a way that u([x]) < u([y]) when xPy. By assigning u([x]) to every element of [x], we have a function u:X-->R with the desired property.
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Originally posted by ChronicLeakyI knew it would be straightforward for you. 🙂
Now consider X/I, the set of equivalence classes such that x and y belong to the same class iff xIy. Denote the I-class of x by [x]. If x and y represent different I-classes, then xPy or yPx (but not both). Thus P totally orders X/I, so we can assign a real number u to each I-class in such a way that u([x]) < u([y]) when xPy. By assigning u([x]) to every element of [x], we have a function u:X-->R with the desired property.
Give me a method for assigning a real number to an infinite, but countable set. What happens if X is uncountable?
Edit - Adam, did his proof made it clear what I meant by that?
Originally posted by ChronicLeaky🙄
I'll write xWy for weak preference, and xIy for indifference, i.e. xIy iff x~Py and y~Px. Note that I is symmetric. The definition of weak preference is: xWy iff xPy or xIy.
Also, indifference is transitive: if xIy and yIz, suppose x~Iz. Then wlog xPz. But by the hypothesis, x~Py and y~Pz, so by negative transitivity, x~Pz, a contradiction. Th ...[text shortened]... ssigning u([x]) to every element of [x], we have a function u:X-->R with the desired property.