## Bets and groups

I’ve been thinking about bets recently. As you do. I’m interested in the Dutch book theorem, and central to the proof is having a method to formally represent what a bet is. Borrowing from this paper by Frank Döring, we can think of a bet as a list of ordered pairs of an event and a number: . The are the events and the are the stakes. So if Alice and Bob are betting on whether a coin will land heads or tails, with the winner gaining €1 from the loser the bets they are accepting are as follows: (let’s say Alice bets the coin will land heads) . Bob is taking the other side of the bet, so he is accepting the bet: . Flipping the sign on each stake turns one bet into its “complement” if you like. Taking both bets (i.e. taking Alice’s bet and Bob’s bet) would mean a net profit of 0, however the world turns out. This is a kind of “neutral bet”. We can stipulate that the form a partition of the space of possible outcomes. That is, they are mutually exclusive and exhaustive. A bet has to say what happens in all eventualities. Say you were betting on the roll of a die, and the deal was that you won if a 1,2, or 3 came up and that your opponent won if a 5 or 6 came up. What would happen if a 4 was rolled? The bet as it stands doesn’t say… So for simplicity let’s say bets specify what happens in all eventualities.

Perhaps a better formulation of Alice’s and Bob’s bets is as follows: for Alice and the “signs reversed” version for Bob. The here is supposed to cover all remaining possibilities, and stipulates that in the event that the coin lands on its edge, say, then “all bets are off”.

I like Döring’s framework for bets. It’s wonderfully general. J.Y. Halpern’s book *Reasoning about uncertainty* has a different framework which is less general, but still sufficient for proving the Dutch book theorem. Döring’s framework allows him to show that no (nontrivial) kinds of conditional probability measure can be justified by a Dutch book argument.A Dutch book in Döring’s framework is a bet where all the stakes are negative.

I was thinking about this framework for talking about bets, and I realised a couple of things. Döring discusses combining bets by taking pairwise intersection of the events, and summing the corresponding stakes, to get you a new bet. The events will still partition the event space, and the net profit will be the same for all outcomes, whether you take each bet individually or take the combined bet. We’ve already seen how there’s a neutral bet, and it should be obvious that combining any bet with the neutral bet doesn’t change anything. And for each bet, flipping the signs on all the stakes gives you a bet which, when combined with the original bet, gives you the neutral bet. Basically, the set of all bets with the rule of combination form a group. (Associativity of the group action follows from associativity of taking intersections and addition of numbers). An abelian group, in fact.

For each event we can define a function that maps a bet to the net gain of the bet if that event occurs is a homomorphism from the group of bets to the additive reals. For each probability function over the events, there is a homomorphism from bets to additive reals that returns the bet’s expected value given that probability distribution.

I don’t know if there is any actual point to this, but I found it neat that you can do this stuff. I wonder if you can put a Dutch book in terms of the existence or otherwise of some group theoretic stucture. A Dutch book in this framework is a bet with all stakes being the same sign (that is, whatever happens, it’s always the same party that benefits). So perhaps one can exploit the fact that this is the same as the maximum stake being negative or the minimum stake being positive, and neither of these functions from bets to additive reals are homomorphisms.

## PhilTeX

Just a quick update to say that I am a contributor to the PhilTeX group blog for philosophers who use LaTeX. If you fit into that (rather niche) category, chances are you’ve already heard of PhilTeX, so this update is almost certainly completely superfluous.

That is all.

## White’s coin puzzle for imprecise probabilities

[Caveat lector: I use a whole bunch of different labels for people who prefer sharp credences versus people who prefer imprecise credences. I hope the context makes it obvious which group I’m referring to in each instance. Also, this was all written rather quickly as a way for me to get my ideas straight. So I might well have overlooked something that defuses the problems I discuss. Please do tell me if this is the case.]

On two occasions now people have told me that there’s this paper by Roger White that gives a pretty strong argument against having imprecise degrees of belief. Now, I like imprecise credence, so I felt I needed to read and refute this paper. So I sat out on my tiny balcony in a rare spell of London sunshine and I read the paper. I feel slightly uneasy about it for two different reasons. Reasons that seem to pull in different directions. First, I do think the argument is pretty good, but I don’t like the conclusion. So that’s one reason to be uneasy. The other reason is that it feels like this argument can be turned against sharp probabilists as well…

The puzzle goes like this. You don’t know whether or not the proposition “P” is true or false. Indeed, you don’t even know what proposition “P” is, but you know that either P is true or ¬P is true. I write whichever of those propositions is true on the “Heads” side of a coin, after having painted over the coin such that you can’t tell which side is heads. I write the false proposition on the tails side. I am going to flip the coin, and show you whichever side lands upwards. You know the coin is fair. Now we want to know what sort of degrees of belief it is reasonable to have in various propositions.

It seems clear that your degree of belief in the proposition “The coin will land heads” should be a half. I’m not in the business of arguing why this is so. If you disagree with that, I take that to be a reductio of your view of probability. Whatever else your degrees of belief ought to do, they ought (ceteris paribus) to make your credence in a fair coin’s landing heads 1/2.

What ought you believe about P? Well, the set up is such that you have no idea whether P. So your belief regarding P should be maximally non-committal. That is, your representor should be such that C(P)=[0,1], the whole interval. This is, I think, the strength of imprecise probabilities over point probabilities: they do better at representing total ignorance. Your information regarding P and regarding ¬P is identical, symmetric. So, if you wanted sharp probabilities, the Principle of Indifference (PI, sometimes called the Principle of Insufficient Reason) suggests that you ought to consider those propositions equally likely. That is, if you have no more reason to favour one outcome over any other, all the outcomes ought to be considered equally likely. In this case C(P)=1/2=C(¬P). In sharp probabilities, you can’t distinguish total ignorance from strong statistical evidence that the two propositions are equally likely. Consider proposition M: “the 1000th child born in the UK since 2000 is male”. We have strong statistical evidence that supports assigning this proposition equal weight to the proposition F (that that child is female). I’ll come back to that later.

So what’s the problem with imprecise probabilities according to White? Imagine that I flip the coin and the “P” side is facing upward. What degrees of belief ought you have now in the coin’s being heads up? You can’t tell whether the heads or tails face is face up, so it seems like your degree of belief should remain unchanged: 1/2. Given that you can’t see whether it’s heads or tails, you’ve learned nothing that bears on whether P is the true proposition. So it seems that your degree of belief in P should remain the same full unit interval: [0,1].

But: you know that the coin landed heads IF AND ONLY IF P is true. This suggests that your degree of belief in heads should be *the same* as your belief in P. But they are thoroughly different: 1/2 and [0,1]. So what should you do? Dilate your degree of belief in heads to [0,1]? Squish your degree of belief in P to 1/2? Neither proposal seems particularly appetising. So this is a major problem, right?*

What I want to do now is modify the problem, and try and explore intuitions about what sharp credencers should do in similar situations. First I should note that the original problem is no problem for them, since PI tells them to have C(P)=1/2 anyway, so the credences match up. But I worry about this escape clause for sharp people, since it is still the case that *the reasons* for their having 1/2 in each case are quite different, and it seems almost an accident or a coincidence that they escape…

Consider a more general game. I have N cards that have a number 1..N on one side of each. The reverse sides are identical. Now, on those reverse sides I write a proposition as before. On card number 1 I write the true proposition out of P and ¬P. On the remaining 2..N I write the false one. Again you don’t know which is which etc. Now I shuffle the cards well, pick one and place it “proposition side up” on the table. For simplicity, let’s say it says “P”. I take it as obvious that your credence in the proposition “This is card number 1” should be 1/N. What credence should you have in P? Well, PI says it should be 1/2. But it should also be 1/N, since we know that P is true IF AND ONLY IF this is card number 1.

Imagine the case where N is large, 1 million, say. I get the feeling that in this case, you would want to say that it is overwhelmingly likely that P is false: 999,999 cards have the false proposition on, so it’s really likely that one of them has been picked. So my credence in P being true should be something like 1/1,000,000. Put it the other way round. Say there’s only 1 card. Then if you see the card says “P”, that as good as tells you that P is true, so your credence should move to 1.

On the other hand, if we’re thinking about a proposition we are very confident in, say A: “Audrey Hepburn was born in Belgium” (it’s true, look it up.). Let’s say C(A)=0.999 (not 1 because of residual uncertainty regarding wikipedia’s accuracy.). Now, if we have 2 cards and the one drawn says A, that’s good reason to believe that that card is the number 1 card. So in this case, it’s the belief regarding the card that moves, not the belief regarding the proposition.

What about the same game but with a million cards? Despite my strong conviction that Audrey Heburn was, if briefly an Ixelloise (?), the chance of the card drawn being number 1 is so small that maybe that should trump my original confidence and cause me to revise down my belief in A.

Here’s another trickier case. Now imagine playing the same game with some proposition I have strong reason to believe should have credence 1/2, like M defined above. And let’s say we’re playing with only 3 cards. For simplicity, let’s imagine that M is shown. How should your credences change in this situation? Again, it seems that C(M)=C(1) is required by the set up of the game. But I’m less sure which credence should move.

In any case, is there a principled way to decide whether it’s your belief about the card or your belief about the proposition that should change. And if there isn’t, doesn’t this tell against the sharp credentist as much as against the imprecise one?

On objection you might have is that this is all going to be cleared up by applying some Bayes’ theorem in these circumstances, since in these cases (as opposed to White’s original one) see which proposition is drawn really does count as learning something. I don’t buy this, since the set up requires that your degrees of belief be *identical* in the two propositions. Updating on one given the other is going to shift the two closer together, but I don’t think that’s going to solve the problem.

________________

* The third option, a little squish and a little dilate to make them match seems unappealing, and I ignore it for now, since it seems to have BOTH problems that the above approaches do…

## World Cup gives insight into financial crisis

It has been reported today that JP Morgan analysts have predicted that England will win the world cup. Their analysis suggests that the schedule favours England over the tournament favourites Brazil and Spain. But there’s a flaw in their reasoning. One of the factors looked included in their model is the odds offered on various teams winning on Betfair.com. Since betfair is an English website, it seems there is a danger of “the market’s” assessment of relative likelihoods of various teams winning being skewed in favour of the home country. My guess is that this analysis’ outcome wouldn’t be robust if you substituted in a different betting exchange’s odds, say from a different country. Imagine taking, say, Italy’s biggest betting exchange and using their odds. I doubt England would prevail there. So either they should pick the odds offered by a betting exchange from a country with no realistic chance of winning (so the team with a chance don’t have their odds skewed), or aggregate the odds of various markets from various countries.

And how does this give an insight into the financial crisis? The mistake made in both cases is the same. It is to assume that the market price reflects the value of the asset. Economists call this the Efficient Market Hypothesis. In the world cup prediction case, the assumption is that odds offered reflect the real chance of the outcome occurring.

Odds offered by proper bookies obviously don’t straightforwardly reflect their expert opinion of the chance of the event: bookies shorten odds in order to make a profit. (Consider roulette: betting on red doubles your money, but the chance of red is slightly less than a half, and therein lies the house advantage; the profit.) Another confounding factor here is that odds on England offered by English bookies are shortened a lot, since many more people will bet on England here than in other countries (regardless of odds), so if England did win bookies would have to pay out a lot. So bookies make the odds shorter to limit their exposure to huge payouts. But betfair isn’t like a normal bookies. It’s a betting exchange. It’s much more like a stock exchange. The thing that is being bought and sold are bets. This should counteract some of the distorting effects inherent in standard bookie’s odds. But there is still a bias in favour of England, I think. People on betfair aren’t all betting as disinterested fully informed rational agents. So there is no reason to think the odds offered on England really do reflect the best estimate of England’s chance of winning. (I mean, come on. Emile Heskey is in the squad. Compared with Spain, who will probably leave Dani Guiza on the bench…)

To be clear: the problem is not with the idea of using the market in general, but the problem is that the betting exchange they used has a clear bias that JP Morgan don’t seem to have acknowledged. (From what I’ve read in the papers. Maybe they did, but I expect not). The insight into the financial crisis is this: if JP Morgan didn’t spot this flaw and they SURVIVED, imagine how dumb the financial companies that folded must have been!

Here’s another take on analysing the world cup which won’t be as popular in the England, since it doesn’t have England winning…

Important caveats: I know very little about economics and even less about football. But this is the internet, so my opinion is just as important as all those so called experts.

## Reverse LaTeX?

I know LaTeX better than I know how to do accents in word or equivalent. What I’d find useful is a way to type TeX commands and have something automagically replace that command with the unicode character.

For example: I’d type \’a and it would transformify into á. That would be cool. I can’t imagine many people would use it…

I actually quite like how my phone handles it.* You hold down the letter in question and a little menu appear containing a useful symbol, and then various accents you can put on the letter. Could that be implemented on laptops? Could you hold down the “a” key until a menu of accents popped up? I think that might have more appeal than my TeX geekery idea…

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*Yeah. I typed this blog post on my phone. Welcome to the twenty first century, baby!