Monday, October 22, 2007

Richard and Judy - and probability

Not withstanding Mike's suggestion that I drop this topic as its not up to usual standard, I'm going to stick with it for a bit.

Jeremy says:

...the R&J show was basically scamming a significant proportion of the population into believing that they had a chance of winning... when they had NO chance at all at the time they decided to commit their money. It is in the withholding of this information that I think the unfairness lies.

I'm not sure that's it, as, in my example where stage 1 of whittling down is that half the time entries are received they go straight in the bin, a significant proportion of the population have no chance of winning at the time they decide to commit their money (because their entry is immediately binned). Yet there is no unfairness in withholding this info from them.

I didn't make my point about determinism clear, as some of you think it irrelevant. Well, may be it is irrelevant, but let me try again (as the point I was trying to illustrate is the crux of my question)!

I was trying to illustrate the point that what we consider "random" and "non-random" depends on how we set the parameters. Given the parameters include the laws of nature and all antecedent physical conditions, that this dice should now roll 6 is not random. There's "no chance" of it doing anything else. On the other hand, given no more info than that the unloaded dice is given a vigorous unseen shake before being rolled in the normal way, it's getting a 6 is random (there's "just as good a chance it'll roll a 5 or 4 etc.").

We say the Richard and Judy case was unfair, and we say this because certain contestants - those who phoned after a certain point - were not excluded at random, but were ruled out from the start (so they had "no chance" of winning), whereas in my example where half the entries go in the bin, and the rest go into a random lottery, we say that those that went into the bin were fairly excluded at random: they did still have the "same chance of winning" as everyone else.

There's a question about what we consider "random" and "non-random", and thus fair (a "chance" of winning [even if very, very low, Mike!]) and unfair ("no chance" of winning at all), to which I don't know the answer.

Why, in the RandJ competion, is it right to say certain entrants have "no chance of winning" whereas in my hypothetical competition those who phone in during times when entries go in the bin have "the same chance of winning as everyone else"?

[[N.B. Note this problem is not, Mike, that I cannot tell difference between very low, and no, probability. I do know about that (in methodology I teach the Bayesian response to the paradox of the ravens, which hinges on precisely that point).]]

11 comments:

Joe Otten said...

A randomly selected entrant has the same chance of winning whether the winner is selected early or not.

But this of course is a very low chance bearing in mind the cost of entry and the value of the prize. This seems to be the real unfairness.

Perhaps, in the spirit of consumer protection, like on gaming slot machines, there should be some indication of the percentage payout for phone-in competitions.

Mithrin said...

I think the whittling down problem is subtly different from the original problem.

Take a national lottery, say that players are only allowed to buy tickets on the Saturday but only those bought before 6pm would be included in the 8pm draw.

Now if this information is not shared with the players some may buy tickets between 6pm and 8pm completely ignorant of the fact that their tickets won't participate in the draw. In this model there seem to be 3 sets of players:
a) those with *fair* potential tickets bought before 6pm.
b) those with *unfair* potential tickets bought between 6pm and 8pm.
c) those with *fair* impossible tickets bought after 8pm
[I think everyone would agree that (c) deserve no sympathy but (b) seem to be in a grey area.]

Here I have used fair to mean that the purchaser had complete awareness of the chance of his ticket winning.

Now if we change the conditions and all players are made aware that tickets bought between 6pm and 8pm will not be included can we not ask how many would buy tickets after 6pm?

I think the above is similar to the original phone in problem. If we take the whittling down example we would have the following model:

Players are allowed to buy national lottery tickets up until the draw. Of the tickets purchased only 50% of them will be included in the draw.

The question has to be asked how is that 50% determined?

If you say only those who buy tickets on even minutes are included; how many people would buy tickets on an odd minute? [This would not be random and would be no different to above]

However if you say that a coin is flipped for each ticket and only those that get heads would be included how does that change the situation? [This would be random]

Too me it therefore comes down to three factors:
1] Knowledge of the participants.
2] Knowledge of the organizers.
3] Differences in knowledge.

So for it to be random and fair I would define that in the case of a competition both organizers and participants can not know who has a chance to win i.e. they have exactly the same knowledge. In this model cheating would be when a participant has more knowledge than an organizer OR when an organizer has more knowledge than a participant.

Not sure if that makes sense.

Psiomniac said...

I think this is an area that is very difficult to think about for most people and I certainly include myself.
I wonder whether the problem at the core of this is that epistemic probability versus physical probability is one of those dualisms that hinders rather than helps.
I think that under determinism, there is only epistemic probability because, from a physical perspective, there is no 'likelihood'. But from a human point of view, even if we think determinism is false, it is not clear what indeterminacy adds to our understanding of the probability of events.
Suppose we are playing a dice game and sometimes we use an electronic version connected to a source of radioactive decay (suitably calibrated to give a 1/6 chance of any given number) and sometimes we use a physical dice that we assume is essentially deterministic in its operation. In the first case, if we interpret quantum mechanics as having irreducible randomness, then there is a physical probability, in the second only epistemic. I cannot see how the cases differ in terms of the fairness or conduct of the game. From the human end-user perspective, there is only epistemic probability.

I think we should distinguish between different types of epistemic probability. For want of better terms, let's call subjective epistemic probability (SEP) the likelihood of an event X given the evidence available at time t, to some agent S. Let's call objective epistemic probability (OEP) the likelihood of an event X given all the evidence that could in principle be available to S if S had the time and means to gather it.

Now consider the dice example given earlier:
Six people each choose a number on a dice. The dice is rolled and one person wins. Does the fact that the dice was, in fact, loaded make this competition unfair? No. Even if, because of the loading, it was a dead cert that the number 5 would win and the number 3 wouldn't, there's no unfairness - not even to the person who chose number 3. What matters is that everyone who enters enters with an equal epistemic (for-all-they-know) probability of winning.]

Here the SEP for each person is 1/6 but the OEP is 1 for 5 and zero for all other numbers. What makes this fair on the face of it is that when people are choosing the numbers they all have the same SEP and so nobody is advantaged. Also, the fact that the organiser has a different pool of evidence does not seem to influence the likelihood of who wins. I would still say that this game is unethical though, because the organiser of the game does not have the same SEP as the participants in such a way that the organiser's SEP better approximates the OEP and yet the participants are not informed of this.
The game can be fair and yet unethical. But suppose the organiser knows that her nephew will participate and that his favourite number is 5. Is the game still fair? Or does our view of what people know when, influence our judgement of fairness?

Was the R£J case unfair as well as being unethical? I would say best practice would mean that the protocol for how the competition is actually run should be available to all in advance. So it was certainly unethical. The same is true of the winnowing case but now we have a problem, because it seems that the winnowing game is fair. But why? Is it unfair if calls are screened on the basis of odd versus even minutes yet fair if the organisers toss a coin on receipt of each call? I think it is fair in both cases. There is disparity between the SEP and OEP in the first but not the second case. But if we assume that people phone in at random with respect to odd and even minutes then at time t just before the game begins, even if the organisers know the objective epistemic probability for a participant selected at random, it is the same as their subjective epistemic probability. This is also true for any participant whilst the game is running prior to them initiating the call process. This is not true in the actual Richard and Judy quiz and so it was unfair.

Mithrin said...

My background is not philosophy so forgive me if I make any beginners mistakes.

I actually don't see how a dice can be deterministic? To me that suggests you know what you will get when you roll the die. I know there are a set of possible outcomes and that when I roll the die I will get one of them, but not which one? Is it deterministic because the set of outcomes are finite and known?

I'm not sure I see the difference between physical and epistemic probability. Or perhaps I should say the difference between them in pragmatic terms - because as you say from a human perspective they are effectively the same thing?

The idea of Subjective Probability and Objective Probability fits with my thinking; I would imaging there could be any number of Subjective Probabilities depending on number of observers and their perspective? While Objective Probability would be *absolute*? So would Subjective Probabilities be a function of the Objective?

Would a fair game then be defined as when all participants have the same subjective probabilty?

"Six people each choose a number on a dice. The dice is rolled and one person wins. Does the fact that the dice was, in fact, loaded make this competition unfair? No."

I don't think the answer is as simple as no, I think it depends. Say there is a 7th person who owns the die and he does not participate in the game, but he knows number 5 will always win. Could you say this game was "subjectively fair"? But only as long as the owner of the die does not participate? But it would not be objectively fair? Or is this where physical and epistemic probabilities separate? The die has the physical probability of rolling a 5 100% all other 0?

Although if you take {one of the} the definition{s} of fair - free from bias - then the game is never fair as the die is never free from bias. Although this equates game fairness to die fairness...

"Is it unfair if calls are screened on the basis of odd versus even minutes yet fair if the organisers toss a coin on receipt of each call? I think it is fair in both cases."

I agree they would be both fair, but as I said only one of them is random. The coin technique would work regardless of whether the participants knew how it was being done and would not impact the behaviour of the participants; the other technqiue would only work if participants had no knowledge of it. However I think the important point is that all participants know that only 50% of calls are allowed through, if they don't then it ceases to be fair because while all participants think they have the same subjective probability they actually do not.

So with competitions could you define fair in human terms - if the behaviour of a participant would change given full knowledge a competition is defined as unfair? Effectively the organizers have to ensure that all participants have the same subjective probability?

Mike Nicholson said...

"in methodology I teach the Bayesian response to the paradox of the ravens, which hinges on precisely that point"

Fair enough ;) I have seen a number of extremely intelligent people fall down on the most basic of mathematical concepts though, so it was always a possibility. Certainly no offence intended.

If we are talking about a purely deterministic point of view then surely the winner would have been determined long before he or she was picked (although the calculations to work out who it was would unworkable to anybody except Laplace's demon). By that argument any competition is determined in advance - the national lottery springs to mind - and it's really not worth anybody's time entering unless they already have the winning numbers.

For this argument I confess the Richard and Judy show seems an odd example.

But (being in an argumentative mood today ;) ) it is possible that the conditions under which the winner is picked could be altered by the number of entrants.

e.g. In the case of the national lottery it is just balls extracted at random. This is the same regarless of how many people buy tickets.

Let's assume though that they have some kind of mechanism for picking a winner from a list, be it a person picking a bit of paper out of a basket or an algorithm choosing a phone number from a list.

Now it is entirely possible (in fact I would go so far as to say probable) that this selection process would produce a different result if there are 10,001 entries instead of 10,000, as the extra number in the list causes the algorithm to binary chop in a different place or the extra bit of paper in the basket means all the other bits of paper move in a slightly different way while somebody is rummaging around for the winning bit of paper.

This means that the winner was not determined in advance of the competition, merely in advance of the choosing once all the entries are in.

Which in turn means that your example of discarding entries would be fair, as they had received consideration and had been selected for discarding based on the full list passed into the algorithm (or equivalent).

the RandJ selection, on the other hand, did not have half the entries which could have meant a different result.

So, yes, I would say that it is unfair.

Psiomniac said...

mithrin,
I will try to reply in full later but for now let me address the deterministic dice question. There are reasons why determinism does not imply predictability that I won't go into here but we could consider a dice as a deterministic system if it is the case that the outcome is dependent only on the operation of physical laws on the macro scale. By this I mean that, regardless of whether the calculation of what the outcome will be could be done, (even in principle), the outcome itself is not affected by quantum indeterminacy.

Steve said...

From what I can gather, the draw as advertised was essentially a raffle. Analogically, if I go door to door selling tickets for a raffle that was drawn last week and lie about the draw date, I'd be defrauding anyone and everyone who bought one. The subterfuge displayed on the show similarly exploited those encouraged to call after the deadline by taking their money and adding it to the prize pot without giving them a valid entry.

Hugo Hadlow said...

I think I see what you're getting at now, Stephen, and I agree with you.

First, some maths. Say we have a 1000 contestants. We could roll a die with 1000 sides. Or, we could flip a coin for every contestant, exclude all heads. Say that leaves us with 512 people. We then roll a 512 sided die to and pick the winner. Mathematically, these systems for picking the winner are equivalent. You have the same odds of winning under each: 1 in 1000.

It would not be less fair to use the latter system, even if the punters didn't know about it (say, if you told them you were going to use a 1000-sided die, but actually used the the latter system).

What then, if we do the same with phone-ins. Instead of adding everyone who calls in to a list from which the winner is picked, we flip a coin and only add the called to the list if they get tails. The punter has the same chance of winning: no unfairness.

What then, if we flip a coin at the beginning of each minute, and if it's heads, all callers during that minute are not added to the list. Mathematically equivalent: no unfairness.

What then, if instead of flipping a coin, we decide a week in advance that callers will only be added to the list if they call in an even minute. If none of the punters know this, punters calling in even minutes think they have same chance of winning as punters calling in odd minutes, and vice versa. No unfairness.

What then, if instead of flipping a coin, or excluding half the minutes, we exclude five out of every six minutes? No unfairness. Say with 600 people. Either 1/2 (coin flip, or every other minute) * 1/300 (remaining people after first elimation round) = 1/600 (correct odds). Or 1/6 (only including every sixth minute) * 1/100 (remain people) = 1/600 (still correct odds).

Say the phones are open for an hour. What then if instead of including every sixth minute, we include the first 10 minutes, and exclude the remaining 50?
We could then pick the winner from the list after 10 minutes, safe in the knowledge that no one will be added to the list later. Why would this be any different from waiting until the advertised hour was up?

Ron Murphy said...

Fair? Are we talking (A) fair in terms of agreed and understood conditions, or (B) fair as in even chances?

What is clearly unfair(A) in the R&J case is that players paid to enter, thinking they would be entered.

In Stephen's case of binning alternate entries - bin-or-in - this would be fair(A) if:
1) All participants knew that was happening
2) Any participant knew that not only he, but all other participants, could not influence his bin/in slot.

Even in the actual R&J case, it would be fair(A) if participants were told that:
1) If you call towards the end of the competition deadline, you may pay and not be entered.
2) A winner 'may be pre-picked before you call', meaning your paid entry will have no chance of winning.

The reasoning is as follows:

Suppose that the universe was completely deterministic, and that physicalism was accepted, so that not only the outcomes of rolled dice were in principle predictable, but also the choice to enter and when to enter a competition (no free will). In this case fairness would have no meaning - everything would be simply a sequence of deterministic events. However, even in this case any particular participant, not having the capacity to calculate and predict, would not know the outcome. As long as on the macro scale outcomes such as rolled dice 'appear' to be random, and follow expected statistical laws, then the participants could 'feel' that the competition is fair (assuming knowledge of this universes determinism didn't put them off entering - but then they couldn't 'choose' not to enter could they).

Being more pragmatic, as long as the organisers and participants agree on the rules before hand, and both accept the risks according to their understanding of the universe, be it determinism, faith, fate, mind-over-matter, astrological signs, or whatever understanding they bring to the competition, then it is fair(A).

In some competitions, such as card games, you can calculate the odds of your next move pretty accurately, if you have the skill and capacity count cards. I presume professional gamblers would agree with that. Other competitions contain more complex variables, such as in horse racing, where participants don't feel it unfair to bet on long odds for chance of a big win. Fans of a lower league football team might place bets on their team against the Premiership champions, with little chance of winning, without feeling it to be unfair. So, the fairness according to epistemic or objective probabilities has little to do with whether the player considers the competition fair(A), though it may not be fair(B).

What would make the above bets unfair(A), is if the cards were marked, the horse nobbled, or the referee bribed, all without the punters' knowledge. Note that if you're on the side that has fixed the event and stand to win, it's not merely unfair, it's better than fair(A and B). And this is the case with the R&J organisers. Whereas the punters are betting on what they think is a fair (A and B) competition, the organisers are making more money than they otherwise would by making it unfair (A and B) - taking money for bets after the competition has effectively 'closed'.

Mike Nicholson said...

Hugo, there is a difference between your first examples and your final conculsion.

The difference is if you exclude people from the full list based on some criteria (they called in an even minute, you tossed a coin etc.) then it makes no difference whether the people know or not - they would likely still enter. Even with the even minute they'd need to be sure their clock matched yours.

With the conclusion it certainly makes a difference for who would then call after the first 10 minutes?

Add to that the fact that these calls are being charged and the gulf between exclusion based on a coin toss (fair) and exclusion based on a time limit that is different to the one advertised widens from merely unfair to fraudulent.

Cheers

Mike

Sally_bm said...

I'm sorry to go back in time, especially if I'm repeating already constucted and defeated points!

I think this depends on how the random caller is chosen. If a random time is selected, then I don't see why it shouldn't be selected before the dialing begins rather than afterwords. If, however, it's done by first names r something, closing the phone lines early could change the outcome of the competition. Like you said, there's always only one person who really can win- the person who does. But, who that person is might change if the pool is left open for longer. The randomness machine is selecting from a different data pool. The person it selects might be one of the later callers. If the lines are closed early, however, that person must be one of the earlier callers. So it might deny someone the chance of winning who would otherwise have won, which is, I think, where the unfairness lies.

However, as Richard and judy don't say how the random caller is chosen, I don't see why there woudl be a problem with using the first method of selecting a random time and calling the person who calls then the winner.