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).
Negative transitivity means that if x~Py & y~Pz => x~Pz. Translating that means that if x not preferred to y and y not preferred to z, then x is not preferred to z.
These sound like reasonable axioms, don't you think?
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.
I'll continue as soon as someone proves this. The next step will be to prove that there is a unique (up to affine transformations) function u for which xPy => u(x) > u(y), for all pairs (x,y) in X.
this sounds a lot like microeconomics (consumer preferences).
Given people act as rational beings and strive to achieve maximum utility I'd say that human behavior could indeed be modelled mathematically. It would just take a LOT of variables 😀.
The only human behaviour that can not be modelled mathematically is irrational behaviour.
Originally posted by Palynka 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.
Our behaviour is dependent on a 'cost' (i.e. time, money, opportunity cost etc.). People try to minimize this cost. It's rational to minimize this cost.
Of course this is all under the assumption of perfect information, which we do not have! So maybe in the future when communication is even more efficient!
Originally posted by aethsilgne Our behaviour is dependent on a 'cost' (i.e. time, money, opportunity cost etc.). People try to minimize this cost. It's rational to minimize this cost.
Let me rephrase, minimal costs maximum utility. Me being on this website now gives me more utility than for eg. sleeping.
Taking bets may give you more utility than the opportunity cost for the given sum of money. Say you gain two utils of betting 1$ and only 1 util of saving that 1$ on the bank than you've acted rationally given your set of consumer preferences imo.
Originally posted by aethsilgne Let me rephrase, minimal costs maximum utility. Me being on this website now gives me more utility than for eg. sleeping.
Taking bets may give you more utility than the opportunity cost for the given sum of money. Say you gain two utils of betting 1$ and only 1 util of saving that 1$ on the bank than you've acted rationally given your set of consumer preferences imo.
Have a look at Game Theory bud, it's very interesting and while it doesn't answer your questions it shows a train of thought.
I don't understand where your contribution talks of betting and saving, and the payoffs for that behaviour. They are the same thing just dressed up nicely in 1 respect as opposed to the other
27 Jan '09 22:04
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