Cardano: The Book on Games of Chance (1961/1663)

Cardano; from Wikimedia.

My library has refused to buy the new translation of Jacob Bernoulli's Ars Conjectandi (1713), possibly because the price tag is on the order of $3000. Consequently, I don't have an English translation of the complete book, arguably the most influential one in the history of probability (perhaps with the exception of Kolmogorov's little pamphlet).

Meanwhile, they hold two copies of Cardano's weird, chatty, and uneven book on gambling. I checked out both of them, only two find out that one (the 1961 version) was a reprint of the other, the translation by Øystein Ore appended to his own 1953 commentary on the book (which, incidentally, also included an interpretation of Cardano's erroneous calculus of probabilities).

I'll deal with Ore's book later. Now a few comments on quotes on Cardano.

Contents

Cardano's book can roughly be divided up into sections as follows. If it looks jumbled up, it's because it is.

• Chapters 1–10: Preliminaries, survey of games, moralistic preaching, etc.
• Chapters 9–11: Combinatorics of dice games.
• Chapters 16–19: On card games (mostly non-mathematical discussions of cheating, moral issues, and the like).
• Chapters 20–21: On luck.
• Chapters 22–25: More game taxonomies and definitions.
• Chapter 26: On theoretical and practical knowledge.
• Chapter 27: More on luck
• Chapter 28: Recommendations on playing styles in backgammon.
• Chapter 29: On why you shouldn't play against hotheads.
• Chapters 30–31: Games in ancient Greece.
• Chapter 32: Computing expectations for a die.

From the perspective of the history of mathematics proper, the most directly relevant parts are those on combinatorics and those on luck.

"If Fortune Be Equal"

One of the more remarkable features of Cardano's book is the prominent place it gives to "luck." As Lorraine Daston has noted, this concept seemed to fill out any gap his calculus didn't account for, including the gap between frequencies and probabilities, or between expectations and actual outcomes.

Mean playing dice; illustration from a 1531 print of Cicero's On Duties.

The first time the word occurs is during Cardano's dicussion of fair bets. Apparently, the most common dice game at the time was a bet on whether a specific throw (e.g., double six) would show up or not show up within n throws of a number of dice.

Cardano was therefore interested in computing the number n for which this bet would be fair — or in his words, for which "there is equality." He explains in Chapter 11:

This gives eighteen casts for equality for a throw (1,1); for in that number of casts this throw can appear and not appear with equal probability; and similarly for the throws (2,2) and (3,3).

But the throw (1,2) can turn up in two ways, so that for it there is equality in nine casts; and if it turns out up more frequently or more rarely, that is a matter of luck. (p. 11; my emphasis)

Later, in Chapter 14:

If, therefore, someone should say, "I want an ace, a deuce, or a trey," you know that there are 27 favorable throws, and since the [size of the] circuit [= sample space] is 36, the rest of the throws in which these points will not turn up will be 9; the odds will therefore 3 to 1.

• Therefore in 4 throws, if fortune be equal, an ace, deuce, or trey will turn up 3 times and only one throw will be without any of them;
• if therefore, the player who wants an ace, deuce, or trey were to wager three ducats and the other player one, then the former would win three times and would gain three ducats; and the other once and would win three ducats;
• therefore in the circuit [= observation] of 4 throws they would always be equal. (p. 16; my epmhasis; my itemization)

His language use here suggests that, on some level, he believes that the expected values must somehow be realized — they are what the game should pay off if it weren't for all these kinks and imperfections in the universe.

"The Length of Time … shows Forth All Forms"

Apparently, he is completely serious about this. In a later chapter on skill (ch. 27), he lists two "methods" by which fortunes can change, the second being the more occult one:

But of the other method there is also some secret principle. To these matters belong amulets, witchcraft, and the like, and just as in each case (as they say) the sword fits its own sheath and the foot its own show, so the hour, the day, the year, and the place must fit; so also in this question, what will make one man happy will make another wretched. (p. 44)

Austrian 16th century woodcut of two soldiers playing dice.

Cardano does seem to think, however, that time tends to cancel out luck. In the chapter on sequential successes, he computes the probability of observing an unbroken string of 20 repetitions of a probability 1/2 event, wrongly getting the answer to be 1/8000. He comments:

Yet it can scarcely be believed that in 7,999 throws a player can cast an even number twenty times in a row. But we have demonstrated this; because, although in 8,000 performed only once, this reasoning can deceive us, still in an infinite number of throws, it is almost necessary for it to happen; for the magnitude of the circuit is the length of time which shows forth all forms. (pp. 19–20).

Here, "luck" almost seems to be synonymous with "noise."

Squares and Cubes

Since I'm talking about this error anyway, and since this is essentially the sole remaining mathematical component of the book, let me just quickly summarize what Cardano seems to be doing in the rest of Chapters 14 and 15.

At the end of Chapter 14, he claims that if the odds in favor of a single success is p : 1 – p, then odds in favor of k successes in a row are p^k : (1 – p)^k. This is not correct, since (1 – p)^k is not in general equal to 1 – p^k. He recognizes this in the opening of Chapter 15, noting that if it were really true, any run of consecutive probability 1/2 events would also have probability 1/2:

But this reasoning seems to be false, even in the case of equality, as, for example, the chance of getting one of any three chosen faces in one cast of one die is equal to the chance of getting one of the other three, but according to this reasoning there would be an even chance of getting a chosen face each time in two casts, and thus in three, and four, which is most absurd. For if a player with two dice can with equal chances throw an even and an odd number, it does not follow that he can with equal fortune throw an even number in each of three successive casts. (p. 19)

 Cardano 1 

 Cardano 2  

Correct 

1 : 1

1 : 1

1 : 1

1 : 1

3 : 1

3 : 1

1 : 1

8 : 1

7 : 1

1 : 1

15 : 1

15 : 1

1 : 1

24 : 1

31 : 1

1 : 1

35 : 1

63 : 1

1 : 1

48 : 1

123 : 1

So now Cardano owes us a different argument. He therefore goes on to claim that the correct answer for p = 1/2 in fact is k² – 1 : 1. This coincides with the correct answer for a couple of small values, but then diverges exponentially from it. This leads him to make the "infinity" remark quoted above.

Parenthetically, I'm not sure why he would cube rather than square the number 20 in that example. Perhaps Ore has something intelligent to say about this.

Late 15th century book illustration showing a dice game.

How To Gamble If You Must

In Chapter 20, Cardano tells a little autobiographical anecdote as an illustration of his points about fortune, luck, etc. This story is not strictly relevant to my topic here, but it is simply to bizarre not to quote. Hence, Ladies and Gentlemen, Cardano without filter:

Yet I have decided to submit to the judgment of my readers what happened to me in the year 1526 in the company of Thomas Lezius, the patrician of Venice, leaving it to each reader to form his own opinion. I had just duly resigned from the office of rector of the scholars in the University of Padua on the third of August, and now I was journeying with Hieronymus Rivola, a scholar from Bergamo, on a certain night of the same month toward Venice. We were playing a game (called Bassette) and I won all the money he had. Then he asked me to play with him on credit, if I am not mistaken up to two or three aurei, and I won again. Then, finally, he wanted to carry it on endlessly, but I refused. He promised to pay what he owed me within three ways; he did not come.

Then he chanced to meet me and said that he would come to pay the money on Saturday (which was the day of the Nativity of the Virgin) and promised to take me to a beautiful prostitute. At that time I was just completing my twenty-fifth year, but I was impotent. Nevertheless, I accepted the condition; there was not a word about the game. He came on the day agreed; and in that year the festival of the Blessed Virgin was on Saturday. He took me to the home of Thomas Lezius; there was no Thais there, but a bearded man with a young servant. No money was paid but we played with marked cards. I lost to him all the money which he owed me, and he reckoned it as part of his debts just as though he had given it to me. I list about twenty-five aurei or even a few more which I had, and played on, giving my clothes and my rings as security.

I returned home in sadness (as was natural), especially since there was no hope of getting money from home because uprisings and plots were raging at Milan. And so (and now I tell the truth, there being no reason why I should lie), I contrived for myself a certain art; I do not now remember what it was, since thirty-eight years have passed, but I think it took its rise in geomancy, by which I kept in mind on up to twenty-four plays all the numbers whereby I should win and all those whereby I should lose; by chance the former were far more numerous than the latter, even in the proportion (if I am not mistaken) of seven to one; and I do not recall now in what order these were against me.

But when I saw that I could not safely hold more numbers in my memory, I admonished my young servant, whose name was Jacob, that when he saw I had won back my clothes and my money he was to call me. I threatened that if he did not do it I would beat him severely. He promised and we went. As the game went on I won and lost in all the plays just as I had foreseen and after the third play I realized that there was no trickery or deceit about it. They laid down money freely and I accepted the wagers, but he was delighted be the example of the previous day and also on account of the marked cards (as I have said).

Thus his thoughts were inflamed by his youthful ardor; but the result was otherwise, for, on those plays in which I saw (as it were, without any real foreknowledge) that I would win, I did not rehect any amount of money and made large bets of my own, and in the other cases, where I knew he would win, I refused if he was the first to wager, and wagered very meagerly myself: thus the result was that within twenty plays I regained my clothes, my rings, and money and also what he had added besides. As for the clothes, the rings, and a collar for the boy, I sent them home piecemeal. Out of the total number there remained four deals; I played and won, and also came out victor in a few deals which were not contained in the number.

He was already perturbed and full of admiration, since he saw that in all the plays in which we played for high stakes I came out the victor, and in those in which he won I myself wagered little and when he wished to wager a great deal I refused. So (he said) I believe some demon is advising you, or that you know by some enchantment what is going to happen. What happened after that I remember that I have narrated elsewhere. (pp. 32–34)

He also gives a bit of extra detail in Chapter 17, the chapter on fraud in card games:

There are also some who smear the cards with soap so that they may slide easily and slip past one another. This was the trick practiced upon me by the well-known Thomas Lezius of Venice, patrician, when in my youth I was addicted to gambling. (p. 27)

Extraordinary, isn't it?

Arnauld: Logic (1662), Part IV, chs. 13–16

19th-century portrait of Arnauld, the main author of the Logic.

The last four chapters of the Port-Royal Logic deal with reasoning under uncertainty. They touch briefly on issues of evidential support, balanced odds, and fair games.

In retrospect, these remarks are difficult not to read as a precursors of later probability theory. Many authors have pointed this out, including Ian Hacking and Lorraine Daston.

I'm reading the 1996 translation by Jill Buroker, published by Cambridge UP.

The content of the four chapters on probability, or plausibility, are:

• Chapter 13, "Some Rules for directing reason well in beliefs about events that depend on human faith," argues that the standard of geometric proof and mathematical certainty only applies to matters of "immutable essence," and that judgment about human affairs should be made by meditating on the available evidence and testimonies.
• Chapter 14, "Application of the preceding rule to the beliefs about miracles," claims that this means that there are several instances in which it is reasonable to believe miracles took place, even if this cannot be proven beyond all reasonable doubt.
• Chapter 15, "Another remark on the same subject of beliefs about events," adds that this explains why certain deeds are attested by two notaries, and that the technique of weighing the evidence also has applications to authorship attribution for ancient (religious) manuscripts.
• Chapter 16, "The Judgments we ought to make concerning future events," argues that the doctrine not only applies to reasoning about the past, but also about the prediction of the future; a number of fair and unfair games are described as examples, and a version of Pascal's wager is then put forward.

Quotes follow below.

Don't Hold Your Breath

In chapter 13, we are told that the methodology of geometric proof works for geometry,

But if we try to use the same rules for beliefs about human events, we will always judge them falsely, except by chance, and we will make a thousand fallacious inferences about them. (p. 263)

Instead, we have to look for circumstantial evidence:

In order to decide the truth about an event and to determine whether or not to believe in it, we must not consider it nakedly and in itself, as we would a proposition of geometry. But we must pay attention to all the accompanying circumstances, internal as well as external. I call those circumstances internal that belong to the fact itself, and those external that concern the persons whose testimony leads us to believe in it. Given this attention, if all the circumstances are such that it never or only rarely happens that similar circumstances are consistent with the falsity of the belief, the mind is naturally led to think that it is true. Moreover, it is right to do so, above all in the conduct of life, which does not require greater certainty than moral certainty, and which even ought to be satisfied in many cases with the greatest probability.

But if, on the contrary, these circumstances are such that they are often consistent with the falsity of the belief, reason would require either that we remain in suspense, or that we view as false whatever we are told when its truth does not look likely, even if it does not look completely impossible. (p. 264)

Cloister of the Hôpital Cochin, inhabiting the former site of the Port-Royal abbey.

 

A Perversion of Reason

This idea is echoed in chapter 15:

Since we should be satisfied with moral certainty in matters not susceptible of metaphysical certainty, so too when we cannot have complete moral certainty, the best we can do when we are committed to taking sides is to embrace the most probable, since it would be a perversion of reason to embrace the less probable. (p. 270)

We also hear that negative evidence "weaken or destroy in the mind the grounds for belief" (p. 270).

A couple of things worth noting about these quotes:

• The process described here is a mental therapy, not a decision calculus; you pay close attention, and your "mind is naturally led" to a certain belief.
• The focus is on human affairs, that is, on practical matters.
• As in frequentist statistics, the contrast is not between true and false, but between confirmed and unconfirmed; we thus "remain in suspense" if the evidence is insufficient.

Set the Record Straight

Chapter 16 pours scorn on "many people" for entertaining a certain "illusion":

This is that they consider only the greatness and importance of the benefit they desire or he disadvantage they fear, without considering in any way the likelihood or probability that this benefit or disadvantage will or will not come about. (p. 273)

That is, thinking only of utility while ignoring probability. Moreover,

This is what attracts so many people to lotteries: Is it not highly advantageous, they say, to win twenty thousand crowns for one crown? Each person thinks he will be the happy person who will win the jackpot. No one reflects that if it is, for example, twenty thousand crows, it may be thirty thousand times more probable for each individual to lose rather than to win it.

The flaw in this reasoning is that in order to decide what we ought to do to obtain some good or avoid some harm, it is necessary to consider not only the good or harm itself, but also the probability that it will or will not occur, and to view geometrically the proportion all these things have when taken together. This can be clarified by the following example.

There are game in which, if ten persons each put in a crown, only one wins the whole pot and all the others lose. This each person risks losing only a crown and may win nine. If we consider only the gain and loss in themselves, it would appear that each person has the advantage. But we must consider in addition that if each could win nine crowns and risks losing only one, it is also nine times more probability for each person to lose one crown and not to win nine. Hence each has nine crowns to hope for himself, one crown to lose, nine degrees of probability of losing a crown, and only one of winning the nine crowns. This puts the matter at perfect equality. (pp. 273–74)

This argument is then pushed a little further to deal with some more extreme bets; to the fear of lightning, which is allegedly irrational and has to be "set straight" (p. 275); and finally to a version of Pascal's wager.

Daston: Classical Probability in the Englightenment (1988)

I've been wanting to read Lorraine Daston's book on the early history of probability for about eight years now, but there was always something else that seemed more important. Recently, however, I finally got around to checking it out of the library along with Abraham De Moivre's The Doctrine of Chances (1718).

There are two things that I have found interesting about the narrative of the book so far: First, Daston's claim that, despite appearances, gambling in fact was not the real engine behind the emergence of probability theory; and second, her discussion of the controversy surrounding the proper definition of expectation.

Gambling or Law?

With respect to the claim about gambling, she cites contract law (and to some extent, criminal law) as a plausible interpretation of what the people of the enlightenment were really talking about when they talked about "gambling." Specifically, she cites the problem of determining a fair price for an aleatory contract (such as an insurance policy) as an important new question that gave meaning to the young subject.

As the name suggests, contracts of this kind were from the onset conceptualized as a kind of gamble. All the dice-throwing in the classical text about probability may thus have been a kind of crypto-legal pedagogy (just like a math problem about dividing a cake might really aim at teaching you something about accounting).

Reasonable Expectation

With respect to the second point, she discusses at length Daniel Bernoulli's solution to the St. Petersburg problem: Assume that the utility of money decreases linearly as a function of your capital.

This leads to a logarithmic utility function and thus aligns Bernoulli's solution with Kelly gambling. It corresponds to the intuition that bankruptcy is qualitatively different from other levels of bankroll, and that one should not adopt a gambling strategy that assigns a positive (or even high) probability to going bankrupt.

A similar and much simpler example of the same ambiguity comes up in the comparison of extreme gains that have very low probability relative to modarately large gains that have substantial probability. Gambling a lot of money on lotteries of the first kind tends to lead to bankruptcy with quite high probability.