Its A Question Of Science

Originally posted by Tyrannosauruschex
Could human behavior be modelled mathematically?

This boils down to the question of whether or not human behaviour is deterministic. It is not since the universe is not deterministic, therefore the maximum one could achieve is a statistical description of the likeliness of certain actions.

Originally posted by aethsilgne
I'm not blaming economics, I'm blaming the lecturer! 🙂
To get back to the original post of this thread:

Given we have perfect information (preferences and indifference functions) on an individual, we can predict all his rational (minimal cost) behaviour. Given that no man is equal, everyone has different preferences. This may lead to some actions bei ...[text shortened]... n and set budgets).

I'll leave you to do the numbers 🙂.Thanks for the interesting thread!

This economic approach is not really relevant to the issue at hand IMO, not only because we know that humans do not act rationally, but they also simply don't have the storage capacity for the information needed to act rationally.

Originally posted by KazetNagorra
This economic approach is not really relevant to the issue at hand IMO, not only because we know that humans do not act rationally, but they also simply don't have the storage capacity for the information needed to act rationally.

Do you even know what was assumed so far?

Originally posted by Palynka
Do you even know what was assumed so far?

Haven't read the whole thread, so no.

Originally posted by Palynka
What are you waiting for?? 😠

I'm on it. There will in fact be chess.

Originally posted by Palynka
What are you waiting for?? 😠

Okay, Palynkinstein. I do not have your gift for writing things that are both clear and short, so here's some stuff and then over to you...

Introduction

What I will do here is discuss a few situations, which have been dealt with abstractly earlier in this thread, in more concrete ways. The goal is to fix intuition about different types of preference-orders and to illustrate how the proof above comes together. There will be some mathematical arguments, but they won't involve any of the sort of heavy machinery used in the earlier posts.

Finite Preference Orders: A Knight's Tour

Begin with a standard 8x8 chessboard with only a knight at d4. We define a preference relation on the chessboard by stipulating that we prefer some square T to a square S exactly when S can be reached in fewer knight moves than T, starting from d4. In other words, move the knight (legally) from a1 to S in the fewest moves possible. Then put the knight back at a1 and do the same for T. We prefer whichever of S and T required more moves. S and T are indifferent if the same number of moves are required.

I'm pretty sure, although the exact numbers are not important, that any square can be reached in at most six knight moves and at least zero, and that for each nonnegative integer up to six, there are squares requiring exactly that many moves. Thus we have seven indifference-classes of squares, in order according to how many moves they require. We then assign a number to a square in the obvious way: a square's number is just the number of knight-moves required, so that the number assigned to square T is less than that assigned to S exactly when we prefer S to T.

In fact this is essentially the only way to make such an assignment -- this is what Palynka said in his first post: "up to an affine map".

First Intuition In The Countable Case: Infinite Knight Tours

This case is exactly the same as before, except now the chessboard extends infinitely in all directions. It's clear that for any positive integer n, there is a square which requires at least n moves to reach it, and that any square is reachable in a finite number of moves, so that (using the same preference relation as before), we now have a countably infinite set of indifference classes (recall that an indifference class consists of the set of all squares which are reachable in a given number of moves; a square in an indifference class is called a "representative"😉, and this set is ordered according to how many moves it takes to reach a representative of each indifference class.

The obvious thing to do is to construct a function exactly as we did in the finite case, by simply assigning to each indifference class the number of moves it needs. Instead of only assigning the numbers 0-6, we'll be using an infinite set of nonnegative integers (in fact I'm certain we'll use all of them but a proof would be a heinous digression). The procedure is exactly the same, though. It's pretty much called "counting".

Suppose we complicate things slightly. Note that the starting square, with the knight on it, determines two infinite lines of squares: imagine the knight were instead a rook, and look at the infinite cross formed by all possible legal rook-moves. Now pick just one line of this cross. Now play the knight-moving game, except that whenever the knight ends up to the left of the line you picked, declare the number of required moves to be minus the number of moves you made.

We can still make the preference order exactly as before, but now we have an indifference class for each integer, positive, negative, or zero. Still, however, this number-assignment has Palynka's Order Property.

One might wonder why the method for finding a POP function on a countable ordered set given in my other post is more complicated than counting. It turns out that the ordering imposes some extra structure on the set, as, say, the rational numbers illustrate.

Countable preference-orderings like the infinite chessboard knight-move ordering have a property which makes them very nice, in the sense just described. We saw that they may as well be finite, and this has to do with a property called "well-ordering".

We'e considering sets with an ordering such that among any two different elements, one comes before the other (we are considering sets of indifference-classes, which amounts to collapsing collections of elements which do not have this property down to single elements). We say that such an ordering is a well-ordering if any subset has an element which comes first (is least) in the ordering. A set is also well-ordered if the same condition holds, with "least" replaced by "greatest".

For example, the nonnegative integers are well-ordered, while all of the integers are not.

The first version of the infinite knight-move ordering had this property, and this allowed the construction of our POP function simply by counting. However, the second version, which used infinitely many positive and negative integers, did not have this property, yet we were still able to do the counting thing with no trouble. The reason is that the integers are not far from being well-ordered: we can break them into two sets, (negative and nonnegative, say) such that each of the two sets is well-ordered (every set of positive integers has a least element, and every set of negative integers has a greatest element).

We thus say that an ordered set is "weakly well-ordered" if we can break it up into two disjoint sets, each of which is well-ordered. Whenever we have a weakly-well-ordered set, we can construct a function with the POP simply by counting. Equivalently, a set is WWO if every element has an immediate successor in the ordering (except the greatest, if one exists) or a predecessor (except the least, if one exists).

EPIC FAIL: The Tale Of The Trail Of The Square King

Unfortunately, countable ordered sets do not have to be weakly well-ordered. Rather than do something artificial with the rationals, which are the most obvious non-WWO countable set, I want to do something a little different.

Hopefully, you wondered why I brought the chess into the last section, since it's not used in an essential way; knight moves are just something to count. If you wondered this, then you understand intuitively that in some sense, WWO sets are pretty much covered by the example of the integers. Now we are going to make use of chess(boards) more essentially.

Begin again with the same infinite chessboard. There is one piece, the Square King* (SK). Pick a starting square and orient yourself so that you can see 4 directions on the board, north, south, east and west, reckoned from the starting square.

The square king starts at the starting square, and, as his name implies, he can move like a normal king, except not diagonally. He moves one square north, and then moves according to the following rules:

1. He starts by moving west one square.
2. At all times, he moves in such a way that if he had diagonal capabilities, a single diagonal move would put him somewhere he has already been.
3. As long as rule 2. is met, he moves in whatever direction he was already moving.
4. He never goes somewhere where he has already been.
5. When the other rules force him to make a turn, he turns counterclockwise with respect to the compass.

You'll find that the Square King has one choice of path, and it describes a spiral which goes through every square, eventually, exactly once. He also counts as he moves. The Square King proves that this chessboard has countably many squares.

Now we define a preference ordering on all the squares: if S and T are different squares, we prefer whichever is more east with respect to the starting square. If S and T are equally far east, we prefer whichever is further north**. If S and T are both equally east and equally north, then they are of course the same 😵.

When we attempt to give each square a number in such a way that S gets a bigger number than T exactly when it is preferred to T, we see that nothing like the counting procedure above we used earlier will work. Why? Well, such a counting procedure would have to look something like the following:

Pick an enumeration of the squares, ie give each square an integer, by countability. If the ordering given by this enumeration is inconsistent with the east-west, north-south ordering, we might hope to be able to do some swapping of the numbers in the enumeration to fix things. Unfortunately, this is impossible, and our chessboard gives us visual intuition about why.

The Square King provides an enumeration -- a numerical ordering -- of the squares, and the choice of preferences we stipulated gives an ordering. Now pick two different squares, S, west of T (which are not directly north and south of each other). Then the square king can get from T to S in finitely many steps. However, in the preference-order, there are infinitely many squares between S and T: everything due south of S, everything west of T and east of S, and everything due north of T. (This says that the Square King gives a WWO structure to the squares, while the preference order doesn't, so there is no way to map one to the other consistently.)

Therefore, the counting method, which tries to treat countable sets essentially the same way as finite ones, is a kind of misintuition. It is an EPIC FAIL.

FIXED: The Shout Of The Square King

EPIC FAIL or not, Palynka said that we can always assign numbers to countable preference-orders consistently, so we'd better be able to do it; we'll just have to do it in a way that doesn't assign numbers which live inside a WWO set. For any square S of the infinite chessboard, there is an infinite set of squares to which it is preferred: everything due south, and everything west. The idea is, for each T to which S is preferred, to look at the number which the Square King would have counted when he got to T, appropriately add these numbers up, and assign the result to S.

To do this, we make the Square King move in exactly the same way as before, but instead of shouting out "N" when he gets to the Nth square, we have him shout out "(1/2)^N" (so he shouts out "one-eighth" on the third square he reaches etc). We keep a tally, forever, as follows. Each time the SK shouts a number, we check to see if he's on a square south or west of S. If he is, we add the number to his total. If not, we ignore it. At the end of time, we look at our total. It's going to be less than 2***. Assign this total to S.

If S is preferred to T, there will be more ignored shouts when we do this for T than when we do it for S, so our total for T will be lower. Conversely, if our total is higher for S, it means we ignored fewer shouts, so S is north or east of T.

Thus this procedure assigns a real number between 0 and 2 to each square, respecting the preference-order****. FIXED!

Next is to deal with the uncountable, separable case and the counterexample. I have some concrete examples in mind, but I'm sick of writing for tonight. Tomorrow there will be the rest of it, unless anyone with a serious interest in this thread explicitly asks me to stop*****. In the meantime, there's homework in the endnotes 😛!

Notes:

*Originally, this used some bishops, too -- the Hip Priests, but the same thing can be accomplished without them.

**There is a different, beautiful lexicography that almost works, involving a Hip Priest and a Normal Knight, but I couldn't manage to smooth out a detail, so I'm doing this stupid thing instead -- I am disappointed. Maybe someone over in Posers and Puzzles can do a repair job on the idea, but I won't hijack with it here 😳.

***So there are these things called "geometric series" and Wikipedia has an article. Actually, if you're reading this, I bet you already know about them.

****Homework: in fact, this number is always rational. Can you see the connection between these sums and the binary representation of the assigned numbers? Can you use this to show why they must be rational?

*****I have a bad feeling that I'm doing evil thread-hijacking right now, and I'm aware that this could be a massive distraction from the discussion I hope Palynka will continue.

This is the second-last of these. The last is almost done, but I need to think about it a little more. I want to emphasise that none of these posts is essential to following Palynka's discussion; they are merely useful for understanding why a few of his statements are true.

The Uncountable Case: Very Irregular Language

The Square King has more tricks up his sleeve. Suppose he plays a new game: beginning at the starting square, he starts to make legal moves: at each stage, he picks one of the four cardinal directions, and moves one square in this direction. Each time he moves, he shouts out the first letter of the direction in which he intends to travel -- N,S,E or W. In each instance of the game, he makes a countably infinite sequence of moves, and we record the letters in the order that he shouts them. Each such infinite sequence of moves is a "game". Let X be the set of all games which the Square King could possibly play.

First we will establish that X is uncountable. An uncountable set is an infinite set with the property that there is no way of assigning a unique nonnegative integer to each element (the spiral done by the SK yesterday showed that the squares of the chessboard are countable, recall).

Suppose that X were countable. Then we could write a list of the possible games, which would give us a corresponding list of shouts. Maybe it looks like:

1.NESSEWEEEEEEEESSSWWSSWS.....
2.WWWWNNNNNWWWWNNNNWWWWNN.....
3.ESWNESWNESWNESWNESWNESW.....
3.SEWNSEWNSEWNSEWNSEWNSEW.....
......

Since X is assumed to be countable, we'll suppose that EVERY POSSIBLE GAME is represented by a sequence on the list. Now, with this infinite list of infinite sequences in our (very large) hand, we put the Square King on the starting square, and we tell him how to play his game: at the Nth move, we look at the Nth letter of the Nth sequence in the list, and tell the king to go in the direction 90 degrees counterclockwise from the direction represented by that letter (so, for instance, sequence 3 has a W in the third place, so we tell the SK to go South -- 90 degrees CCW from West -- on the third move). Following these rules gives us a game like the others. However, for any N, the SK moves in a different direction on the Nth move than he did in the Nth game on the list, so this game is different from the Nth game, for each N. Thus our new game does not appear on the list, which contradicts our assumption that the list was exhaustive. Thus it is impossible to make an exhaustive list: X is uncountable*.

Consider the following subset Y of X: a game G is in Y exactly when the following happens. There is a finite number and a finite list of cardinal directions such that, after that finite number of moves have been made, the square king just continually follows that finite list of directions over and over. For example, he might do some crazy stuff and then just head northwest (...NWNWNWNWNWNW...) until the end of time. He might do crazy stuff and then go north-west-south-east until the end of time, thus staying on the same four squares. Such a game is called a "eventually cyclic game" -- an ECG -- and Y is the set of ECGs. We will need that Y is in fact countable.

To see this, note that each ECG is uniquely determined by a pair of finite strings of directions, namely the "crazy stuff" at the beginning, and the string of directions that gets repeated forever, the "cycle". Thus each ECG corresponds to a pair of FINITE games. How do we count these pairs? We pick an enumeration of the set of "crazy stuff" strings, and an enumeration of the set of cycle strings, and use these enumerations to associate each pair to a square on the infinite chessboard, using the north-south and east-west distances as coordinates corresponding to the two enumerations. Now we let the Square King do his spiral, calling out the crazy stuff and cycle corresponding to each square he hits! Thus the set of eventually cyclic games is countable**.

Now we'll put a preference-order on the set X of games. To do this, we put a simple preference-order on the four cardinal directions: we prefer E to N to W to S. Now take two games from X, and their corresponding direction-strings, say G and H. Reading from the beginning in each, find the first letter in which they differ. Whichever of G and H has the preferred letter (in the above sense) in this place is preferred. This gives a preference-order on X with no indifference.

Now we wish to find a function from X to the reals with the POP property. We start by doing this for Y.

Recall that Y is countable, and it inherits the preference-order we just described from X, since it is a subset. We also saw how to use the Square King Spiral to enumerate the elements of Y, so we use the same geometric series trick we used for the squares of the chessboard to assign a unique (rational) number between 0 and 2 to each eventually cyclic game in Y. For each G in Y, call this number u(G).

We are going to purchase a pair of bootstraps from Y****, then pull our function u up to X with said bootstraps. Suppose K and L are distinct games in X, with L before K in the preference-order. Then they look something like this:

L = [Some finite string], N, W ....
K = [SAME finite string], E, S ....

(So K is preferred since E is preferred to N.) Now consider a string:

M = [SAME finite string], N, N, [Ns forever!]

Then M is between K and L in the preference-order and M is an eventually cyclic game, since it eventually has Ns forever. In general, this is always possible: between any two games in the preference order lies an eventually cyclic game.

Now pick any game G in X, and pick an eventually cyclic game H(1) which comes before G in the preference-order, say. Now, proceed forever in doing the following: whenever we have H(n) for some n>1, choose H(n+1) to be an eventually cyclic game between H(n) and K. This is always possible by what we just did. Since we now how to construct u for eventually cyclic games, we know u(H(n)) for each n, and we know that since H(n+1) is preferred to H(n) for each n, u(H(n)) < u(H(n+1)). Thus our sequence of u's associated to this sequence of ECGs increases, and is bounded above. Define u(G) to be the smallest real number which is greater than all of the u(H(n))s. There are a few little things that need to be checked that would be out of place to check here, but this is the full construction: it gives a POP function for our uncountable ordered set X.

It turns out that X has a vital property -- the "always-betweenness" of some countable set described above -- not shared by all uncountable orderings, and the next section will deal with how and why this property is vital.

Notes:

*Is this lovely argument mine? I frickin' wish. This strategy of proof -- called a "diagonal argument" -- was invented by Georg Cantor to prove that the real numbers are uncountable (which is exactly what we have done, actually). All I did was dress it up like a chess piece with a diagonal (movement!) disability. Diagonal arguments have been showing up ever since Cantor. Kurt Goedel uses one in his paper on the famous incompleteness theorems, and Alan Turing uses one in his solution of the almost-as-famous Halting Problem.

**This corresponds to the proof that the rationals are countable, since there are fewer rationals than there are squares on an infinite chessboard.

***This is sometimes called the "Manhattan metric", for city-grid reasons which are clear. It's also called the "L1 metric" if you have fancy pants.

****Whatever those are.

I find all of this fascinating, I really do, and I'm very impressed with Chronicleaky's and Palynka's logical and mathematical skills, but I'm getting a little lost regarding the connection to the question at hand. In establishing that, given any list of alternatives, a well-ordered set of preferences can be constructed for the entire list, are you saying this is equivalent to human behaviour being predictable (at least theoretically)?

PBE6 has a point. Chronic's examples are excellent for people who are not understanding the proofs (I was thinking of giving even less formal examples), but one needs to keep in mind that the goal here is formalizing choices.

Why? The thing is that if we are to construct a mathematical model of behaviour, we will need a function that represents the preferences underlying an individual's choices/decisions. But does such a construct (function) exist? Under which conditions? This is what we're trying to answer here.

You see, what sonhouse said seems a more intuitive path, but it actually requires our little thought experiment here. To go from ex-post choices to modelling we first need to see under what conditions revealed preferences can be represented by a function.

Besides, I find it extremely cool to think about it in these terms. So why not? 😵

Edit - I'm not making this up as I go, so there's really no reason to be impressed with what I'm doing. There is an old and vast literature on this from which I'm just presenting the main building blocks... I'm sorry if it came across as MY theory. 😳

Originally posted by Palynka
PBE6 has a point. Chronic's examples are excellent for people who are not understanding the proofs (I was thinking of giving even less formal examples), but one needs to keep in mind that the goal here is formalizing choices.

Why? The thing is that if we are to construct a mathematical model of behaviour, we will need a function that represents the prefe just presenting the main building blocks... I'm sorry if it came across as MY theory. 😳

I agree with you and PBE6; those posts are out of place. I'm worried that my examples, in their Nemesian length, are more off-putting than helpful, too. The last example will be fun for the sort of person who finds that sort of thing fun, so I'm still going to post it, but I will do so in a new, appropriately labeled thread. Once this is done, I will move the others to that thread and get the mods to delete them from here, to keep the flow. That way, if we don't understand something you post in here, say, but a detailed explanation would interrupt the flow, we can ask you to put it over there.

EDIT As for whether or not human behaviour is predictable, we have to take an approach at least a bit like the one Palynka is presenting, because "It's a Question of Science" 😛.

Originally posted by Palynka
But does such a construct (function) exist? Under which conditions? This is what we're trying to answer here.

It does not exist. The universe is undeterministic, which prohibits such a function from existing. Even if human thinking is deterministic intrinsically, it's still influenced by things that are stochastic in nature (e.g. getting cancer).

Originally posted by KazetNagorra
It does not exist. The universe is undeterministic, which prohibits such a function from existing. Even if human thinking is deterministic intrinsically, it's still influenced by things that are stochastic in nature (e.g. getting cancer).

It's good that you make assertions without knowing anything about them. So stochastic processes cannot be modeled? We'll prove in a while that the function exists even in the presence of stochasticity.

But I doubt you have even a small iota of the intellectual capacity required to follow so you're content to send potshots from your armchair. I bet it's a comfortable one.

Originally posted by Palynka
It's good that you make assertions without knowing anything about them. So stochastic processes cannot be modeled? We'll prove in a while that the function exists even in the presence of stochasticity.

But I doubt you have even a small iota of the intellectual capacity required to follow so you're content to send potshots from your armchair. I bet it's a comfortable one.

The thread states a simple question, and those educated enough are doing there best to answer in a mathematical way. Though it may be possible, its not probable that we will figure it out. If we look at this question on a surface level it may be solvable mathematically (although extreamly difficult)? One problem is that while this is all quite logical, one must examine logic. I noticed that people are working with decisions as a base, but if we chose to look at decisions as ideas, and ideas are nothing more that collisions between subatomic particles, perhaps even wiggling of strings, the whole idea of predicting human behavior becomes quite convaluted and unfathomable...just my opinion, spouted from my comfortable armchair...lol

Originally posted by Palynka
It's good that you make assertions without knowing anything about them. So stochastic processes cannot be modeled? We'll prove in a while that the function exists even in the presence of stochasticity.

But I doubt you have even a small iota of the intellectual capacity required to follow so you're content to send potshots from your armchair. I bet it's a comfortable one.

Actually, what I said is that the maximum achievable model is a stochastic one; one important thing to realize however is that the stochasticity is not some artifact of a deterministic system, but fundamental in nature. At least, this is what I learned when I applied my iota of intellectual capacity to studying quantum physics and Bell's inequality from my armchair.