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Category: Philosophy

The philosophy of science, empiricism, a priori reasoning, epistemology, and so on.

December 27, 2018 | 29 Comments

Dutch Book: Or, How to Gamble

This post first appeared sometime before 2012, but I lost the original date due to the hacking.

If you are a gambler, you’ll be delighted to learn that there is a way to create a system of wagers such that you are guaranteed to win money no matter what happens. The system of sure profit is called a Dutch Book. Unfortunately, it only works if the bookie with whom you lay the bet has set his odds incoherently.

That doesn’t happen often; but what does is that the bookie (or stock broker, or insurance agent, etc.) usually and purposely makes Dutch Book against his gamblers (or investors, or customers, etc.). That is, the bookie is usually certain of making a profit no matter what happens. No surprise there, right?

There are few on-line sources which explain a Dutch Book. The Stanford Encyclopedia of Philosophy and Wikipedia entries are fine, but short on details and examples. This is because, as you will soon see, it is a subject difficult to understand.

Bets

A bookie sets odds for events, and then gamblers place bets based on those odds. If the event the gambler selected occurs, the bookie must pay the gambler the original amount of the bet plus the bet times the odds; this amount is called the payout. For all the other events (which do not occur) the bookie keeps the bet and there is no payout. The classic example is a horse race. One horse must, and only one horse can, win. A win by one preselected horse is the event of interest to the gambler. The bookie must however set odds on all horses.

If the odds for an event are “even”, i.e. 1 to 1, and a gambler places a bet of (say) 10, then if the event occurs the bookie must pay the gambler the original 10 plus the bet times the odds, i.e. 10 + 10 x 1 = 20. If the event does not occur the bookie pockets the 10. If the odds are 3 to 1 with the same bet, the bookie pays 10 x 3 = 30 for the bet plus the original 10 for a payout of 40. Again, if the event does not occur, the bookie keeps the 10.

The bookie doesn’t just set odds for one event, of course. He always, at least tacitly, sets them for at least two, with the second always being the “event” that the first event doesn’t happen. Make sense? For example, if the event is that the Detroit Tigers win tomorrow’s game, the second “event” is that the Tigers lose. The odds for the event and non-event, or for all events as in a horse race, should always be such that the probabilities add to 1. If they do not, then they are not coherent and a Dutch book can be made. The explanation follows.

Odds

Odds are a one-to-one function of probabilities. The function is

   probability = odds / ( 1 + odds).

For even odds, probability = 1 / (1 + 1) = 0.5; for 3 to 1 odds, probability = 1 / (1 + 3) = 0.25; for 1 to 2 odds, probability = 1/(1 + 0.5) = 0.67. The odds are, of course, a fraction, which is why “1 to 2” = 1 / 2 = 0.5, and “3 to 1” = 3 / 1 = 3. Bookies often state odds like “3 to 2”, but here to keep a common denominator of 1, this is written “1.5 to 1” (since 3 / 2 = 1.5), etc.

Dutch Book for the Gambler

Suppose the bookie has been taking too much Dutch courage (are we still allowed to say that?) before setting his odds and comes up with the following system of odds: the odds for the event are even, i.e. 1 to 1, and the odds for the non-event are 3 to 1. The probability implied by these odds sums to 0.75 (as shown in the table). Something has gone wrong. It is now possible for a gambler to make Dutch book against the bookie.

Book    Odds    Probability    Bet    Payout
Event    Even    1 / (1 + 1) = 0.50    20    20 + 20 = 40
Non-event    3 to 1    1 / (1 + 3) = 0.25    10    30 + 10 = 40
      0.5+ 0.25 = 0.75    30    40

 

One way is that the gambler makes a bet of 20 on the event, which is at even odds. If the event occurs, the gambler takes 40 (the original 20 plus 20 more). The gambler, or a confederate, also makes a bet of 10 on the non-event. If this non-event occurs, the gambler also takes 40 (the original 10 plus 30 more). The gambler paid 20 + 10 = 30 to play, but no matter happens what he wins 40, which is a sure profit of 10 regardless whether the event occurs or not. Dutch book!

The example is contrived, but it is easy to show that the gambler can always find a way to take money from a bookie if the bookie miss-estimates the odds such that the probabilities implied by the odds sum to a number less than 1. That is, whenever you see a set of odds for a set of events which sum to a probability less than 1, you can be certain of making a profit. Of course, this works for more than two events in a set as well (such as horse races, stocks, and so forth).

Dutch Book for the Bookie

Now turn everything around and look at it from the bookie’s perspective. Is there a way the bookie can find a Dutch book against the gamblers? Yes, and they routinely do it. Dutch book is the very means Las Vegas, racetracks, and brokerage houses make their dough. The parimutuel system of betting is a perfect example of the house creating a Dutch book.

The exact details of how bets are placed are not interesting to us here. Suffice to say that bets are made by gamblers who each estimate their own odds, perhaps using their guts or a pre-event estimate of odds given by the bookie. Suppose the end results are in this table.

Book    Bet    Probability    Odds    Payout
Event    30    30/100 = 0.3    2.33 to 1    30 + 70 = 100
Non-event    70    70/100 = 0.7    1 to 2.33    70 + 30 = 100
   100    0.3+ 0.7 = 1       100

 

The pool of money bet is 100, which implies that the pre-event probabilities as judged by the gamblers are 30 / 100 = 0.3 and 70 / 100 = 0.7. This gives the odds 2.33 to 1 and 1 to 2.33, with the payouts being 100 no matter if the event occurs or not. Since the bookie only took in 100 and has to pay out 100 no matter what, he breaks even—and of course makes no money. To fix this, the bookie or broker or racetrack takes a cut off the top. He makes it so he pays out no more than the total amount bet minus some percentage.

Suppose that percentage is 10%. Then there is only 100 * (1 – 0.1) = 90 left for payouts. The table is then adjusted:

Book    Bet    Probability    Odds    Payout
Event    30    30/90 = 0.333    2 to 1    30 + 60 = 90
Non-event    70    70/90 = 0.777    1 to 3.5    70 + 20 = 90
   100 – 10 = 90    0.333+ 0.777 = 1.111       90

 

The implied probabilities have both shifted higher and now sum to 1.111, which is greater than 1, thus the system of bets is incoherent. The odds have also shifted, though they do not immediately appear strange, which is a disadvantage of working with odds for the uninitiated. The payout is always 90 no matter if the event occurs or not, and since the amount bet is 100, the bookie makes a profit of 10 no matter what happens. The bookie has made a Dutch book against the gamblers. Not necessarily against any individual gambler, you understand—some gamblers will still make money—but as a whole the gamblers are taking less than they should.

The bookie can put himself into deep kimchee if he cuts too much, however. Suppose he wants to skim 40% off the top, leaving only 60 in the payout pool. If the non-event occurs, the gambler who bet 70 is going to want not just his 70 back but will demand some kind of profit, however minimal. But with only 60 in the payout pool, he cannot even get his original 70 back. So the bookie must be content with taking less than 40%. Actually, for this system of bets, he must take less than 30% (the maximum skimmable amount is a simple function of the sizes of the bets for the various events and which event occurred).

The amount skimmed is called the “juice”, “vig”, “transaction costs”, among other things. Las Vegas sports bookies usually set the Dutch book so that the odds sum to a probability of about 1.05, which means they skim about 5% from the pool of bets.

Any sum of probabilities greater than 1 also guarantees a Dutch Book for the bookies, just as any sum of probabilities less than 1 guarantees a Dutch Book for the gamblers. The only “fair” bet is where the sum of probabilities equals 1.

It can happen that no gambler picks the event that occurred. Some racetracks fix this situation by refunding all tickets. Brokers manage it by not returning your calls.

Casinos

Update A roulette wheel is a Dutch Book for the house. For American roulette, there is a 1 in 38 chance, i.e. a probability of 2.63% of hitting any number on the wheel. The odds, however, are 34 or 35 to 1, depending on the casino (35 to 1 is the most common). That implies a probability of 2.86% or 2.78% for the single number/slot.

Here is the tricky part: since the odds are 34 or 35 to 1 for the single number, they are that way for each number. That means the implied probability of hitting any number is 38 * 0.0286 = 1.086 or 38 * 0.0278 = 1.056. Since both of these are greater than 1, you can see that the house has guaranteed itself a Dutch Book in its favor.

Almost. The Dutch Book applies only if all numbers on the wheel are bet by gamblers on each roll. If only one gambler bets on one number, then a Dutch Book isn’t technically present, but the odds are still against the gambler. Technically, the Dutch Book calculation is for large numbers of spins of the wheel. That is, it is one of the calculations the casino uses to figure their take.

Philosophy

Dutch Books figure prominently in the foundations of subjective Bayesian philosophy. There, axioms and theorems say that degrees of belief/probabilities should be coherent in the same way that bets are. Objective Bayesian foundations don’t start with Dutch Books, but say they arise from even more basic axioms. However, these are subjects for another day.

December 18, 2018 | 14 Comments

Probability Of Abiogenesis? There Isn’t One

Remember before when we agreed nothing has a probability? Bully. So if no thing has a probability, neither does abiogenesis, since abiogenesis is a thing.

Since abiogenesis is a thing and things have no probabilities, it makes no sense to speak of “the probability of abiogenesis.”

James Chastek hit upon this commonsense idea in lovely article at Just Thomism (a site to be bookmarked): Creationists vs. evolutionists on abiogenesis.

Given the level of passion on this topic from fire eaters on the sides named in the title—and there are more than two—Chastek wisely turned off comments to his post. I am reckless and leave them on. I hope the gentleman won’t mind my quoting him extensively. In any case, I am going to assume below that everybody (which includes you; yes, even you) have read his original.

To sum up, the sides are fighting over there is a 10390th [creationists] or 1040th [evolutionists] chance of some process terminating in life before either side specifies what the process would be. It’s hard to understand what this means. Compare the debate to a time when we can actually figure out the chances of something happening, like Bingo. A clever seventh-grader can tell you your odds of selecting B5, but only after he knows the process by which bingo balls get selected. Without this, what sense does the question have? What are your odds of choosing B5 right now? To be clear, I don’t mean (your odds of leading a bingo game)(your odds of picking B5). That sees like a question with a possible answer. I mean the much more incoherent question that asks without specifying any process by which a given number gets picked, what is your chance of picking it? The contradiction practically has a siren on top: you can’t know how likely you are to pick something without knowing how you could pick it at all.

I assume that both sides are assuming that all there is to forming life is having molecules bang around and form stuff, but the problem with this theory is not whether its parameters of probability are acceptable or not, but that it is not a theory at all. Saying “things just bang around and stuff happens” is not a process, since a process is a specified set of steps toward a definite terminus while “things just bang around” is not a specified set of steps and “stuff” is not a definite terminus. If we could actually specify the bingo-machine that led to the number of life, neither side would need to figure out the probability of the number popping up, any more than any winner at a game of chance cares what his odds were.

The problem is that we don’t have an acceptable mechanism for abiogenesis and so we have no idea how probable it is. Creationists and evolutionists are fighting over something that neither one of them has.

Putting it in my language, we don’t have a probability for abiogenesis because we haven’t specified the premises. Probability must needs have premises, for all probability is conditional.

We can say this more strongly. If we knew the process by which abiogenesis occurred, again it would not have a probability (except for 1), nor would anybody think to ask for one, since we would know the cause or causes of it. Since we do not know the cause, it’s appropriate to speak of probability, but because we cannot set up the premises, i.e. specify the process, we can’t even speak of probability.

In short, both sides are bluffing.

By a process Chastek (and I) mean something like this: molecule A hooks to molecule B, and the joint AB becomes more than A+B; and this hooking can happen in the following specified ways. These specifications do not need be true: if they were true, then again we would know the cause of abiogenesis. They can be guesses based on knowledge of chemistry, biology, and so forth. They can be wrong guesses. They can invoke fictional mechanisms known to be false, even. As long as we have the process, we can derive the probability.

This last fact emphasizes probability cannot be used to prove theories true or false.

About actual abiogenesis, I share Chastek’s opinion—which you will remember, because you read his article.

December 17, 2018 | 9 Comments

Bishop Barron and Ben Shapiro Chat About Salvation

Atheists reject Christ in the sense that they deny Jesus was crucified and rose from the dead because he is God. Atheists also therefore reject the idea of eternal life and the necessity of salvation through Christ Jesus.

Everybody knows this. And nobody is shocked if a Christian tells an atheist, “Repent, for the kingdom of heaven is at hand.” And if he retold Jesus’s own words “I Am, The Way, The Truth, The Life, no man comes to the Father but through Me.” Or even if he spoke the words of John the Baptist, which Catholics the world over heard on Sunday:

I am baptizing you with water, but one mightier than I is coming. I am not worthy to loosen the thongs of his sandals. He will baptize you with the Holy Spirit and fire. His winnowing fan is in his hand to clear his threshing floor and to gather the wheat into his barn, but the chaff he will burn with unquenchable fire.

Harsh, that. But who are you to judge God?

Of course, you don’t have to believe any of that, the Christian might tell the Atheist, but you’ve been told and you’re on you’re own.

Just as naturally, atheists in general don’t like to be annoyed with this kind of talk. But they don’t put the Christian’s behavior down to animus. Except, these days, to the extent the atheist embraces progressive ideology and believes the Christian’s ulterior motive is to ban his favorite non-reproductive behaviors.

An atheist might say the Christian is expressing “anti-atheism”, or that the Christian is “anti-atheist”, but both sound absurd. They sound absurd because it is taken for granted by both sides in the debate that each believes the other is wrong.

Now if we were to swap “atheist” for “Hindu” in the above not much changes, except the Hindu will be suspicious to the extent he believes the Christian wants to change the Hindu’s very culture and way of life. This same suspicion would be felt if the Christian were swapped by an NGO worker pushing acceptance of one of those non-reproductive behaviors.

The Christian might be called “anti-Hindu”, but it would be understood in the sense that the Christian is trying to talk the Hindu out of an error; and vice-versa as the Hindu answers the Christian in debate.

Again, nothing is changed if we switch out “atheist” for “Jew”. Enter this video (Shapiro is one of the forces behind The Daily Wire; Barron is an auxiliary bishop in LA):

Privileged? As in it would be nice, but, really, don’t put yourself out? Well, people are not used to hearing Church leaders say “You are wrong, the Church is right”. Most recoil at religious certainty, especially Christian certainty. They are affronted and disbelieve because of the certainty. Strangely, this is not the case of contingent scientific statements. Anyway, perhaps Barron had this in mind.

Reaction, as they say, was swift. Here’s one. Another. Another. Radical Catholic said:

I’m so glad Bishop Barron didn’t say anything to offend Ben Shapiro.

That would have been horrible.

“No, you have to convert. Extra Ecclesiam Nulla Salus.”

“Why do you hate Jews so much?!?”

PR disaster averted, I say.

Extra Ecclesiam Nulla Salus = Outside Church No Salvation. This is dogma in the Catholic Church, and is therefore required to be believed. There is also the idea of invincible ignorance, which describes the possibility of salvation to those who never had the opportunity to hear of Jesus. And there is the admission that, of course, Jesus will judge each man individually, regardless what each man calls himself. An unformed and error-filled conscience will not be admissible as a plea at your judgement, however. Particularly if you have carefully and over a long period of time been told of your errors (this is my own concern for myself).

Unwoke Duffy said:

He can be saved–by repenting and rejecting his Jewish delusions. The status of indigenous tribes who never heard the gospel is open to possibility. This is not true of open, explicit Christ-deniers. On that there is copious biblical data to the contrary

Such delusions include relying on or believing in religious texts which have Jesus himself boiling in excrement for the “crime” of calling himself God. This is not a minor point of difference in theology. This is a whopper; it creates an unbridgeable theological chasm. Both sides can be simultaneously right. Eucmenicism is a limited tool.

Taylor Marshall said:

Bp Barron said that Our Lord Jesus Christ is the “privileged route” of salvation? Is this Catholic soteriology? Privileged? St Peter said (to Jews): “There is salvation in no one else, for there is no other name under heaven given among men by which we must be saved.” – Act 4:12

Replying to Marshall John Zmirak said, “It’s one thing to hope that the good will of the invincibly ignorant will be counted for implicit baptism of desire. It’s quite another to treat actual faith in Christ as if it’s just a first-class seat on the bus that has only one stop, Heaven.”

This is a sobering reminder that belief, even for Christians, isn’t enough. Every man must approach his own salvation with “fear and trembling”. But an explicit, knowing rejection of Jesus, the Church insists, puts you among the chaff.

Again, harsh, that. But easily amendable.

December 16, 2018 | 1 Comment

Summary Against Modern Thought: Eternity And The Mind of God

Previous post.

Eternity is a very long time, so settle back.

THAT THROUGH THE VISION OF GOD ONE BECOMES A PARTAKER OF ETERNAL LIFE

1 From this consideration it is apparent that the created intellect becomes a partaker in the eternal life through this vision.

2 For, eternity differs from time in this way: time has its being in a sort of succession, whereas the being of eternity is entirely simultaneous. But we have shown that there is no succession in the aforesaid vision; instead, all things that are seen through it are seen at once, and in one view. So, this vision is perfected in a sort of participation in eternity. Moreover, this vision is a kind of life, for the action of the intellect is a kind of life. Therefore, the created intellect becomes a partaker in eternal life through this vision.

Notes Being beings stuck in time, and noticing nothing but successions, such as lessons, this is extremely difficult to see.

3 Again, acts are specified by their objects. But the object of the aforementioned vision is the divine substance in itself, and not in a created likeness of it, as we showed above. Now, the being of the divine substance is in eternity, or, rather, is eternity itself. Therefore, this vision also consists in a participation in eternity.

4 Besides, if a given action is done in time, this will be either because the principle of the action is in time—in this sense the actions of temporal things are temporal; or because of the terminus of the operation, as in the case of spiritual substances which are above time but perform their actions on things subject to time. Now, the aforementioned vision is not in time by virtue of what is seen, for this is the eternal substance; nor by virtue of that whereby the seeing is accomplished, for this also is the eternal substance; nor even by virtue of the agent who sees, that is the intellect, whose being does not come under time, since it is incorruptible, as we proved above. Therefore, this vision consists in a participation in eternity, as completely transcending time.

5 Furthermore, the intellective soul is created “on the border line between eternity and time,” as is stated in the Book on Causes, and as can be shown from our earlier statements. In fact, it is the lowest in the order of intellects, yet its substance is raised above corporeal matter, not depending on it. But its action, as joined to lower things which exist in time, is temporal. Therefore, its action, as joined to higher things which exist above time, participates in eternity. Especially so is the vision by which it sees the divine substance. And so, by this kind of vision it comes into the participation of eternity; and for the same reason, so does any other created intellect that sees God.

6 Hence, the Lord says: “This is eternal life, that they may know you, the only true God” (John 17:3).

Notes Even after forever, you won’t know all of God.