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Wednesday 27 October 2010

Cardano’s Gambit

Gamblers ‘Nonymous

Investing is, up to a point, gambling. Most of us don’t think of it in that way but if we conceive of the universe of stocks as a gas of randomly moving particles buffeted this way and that by forces largely beyond their – and certainly beyond our – control then there’s no other conclusion that can be drawn.

However, we don’t really believe this. What we generally believe is that although randomness is pervasive in stocks there’s a pattern that lies beneath the surface which we, in spite all evidence to the contrary, can pick out. For the idea that there are repeatable patterns hidden within apparently random games of chance we can thank one of our more unlikely heroes. Meet Girolamo Cardano, medieval physician, professional gambler and mathematician extraordinaire.

God’s Will

For a very long time in human history there was no appreciation or investigation of probability, the mathematics that lies behind assessments of risk. For the most part people didn’t believe in chance: stuff happened and that was God’s will. The idea that there was some order in the chaos either seems not to have occurred or to have been literally unthinkable.

Gamblers, however, did have some vague understanding that there were patterns in the randomness and quite a lot of self-interest in figuring these out. It’s no surprise that gambling figures quite large in early accounts of advances in probability theory. In Cardano, who seems to have been addicted to gambling, the will to understand and the ability to do so came together.

Elementary Probability

In many ways what Cardano figured out is today regarded as almost trivial, but at the time it was revolutionary and it allowed him an insight into why and when he should take a risk and when he shouldn’t. Perhaps the simplest example is to do with dice. At the time it was regarded as a bit of a mystery why, when three dice were rolled, the sum of face-up numbers came to ten more often than nine, despite the fact that there were six ways of summing possible numbers to both.

The answer to this conundrum is almost childishly simple to our eyes. There are twenty seven ways of combining the possible sums to ten while there are only twenty five ways of doing so for nine: the trick is to realise that 3+3+4 is not the same as 3+4+3. Cardano was the first, by nearly a century, to figure out the mathematics behind this. Of course, he didn’t tell anyone – partly because mathematical tricks like this were regarded as trade secrets and partly, presumably, because he was using his knowledge to make money. This paper on the Complex and Unpredictable Cardano gives a fascinating review of his career and the issues discussed above.

Now, of course, the specific result of throwing any three dice is random but the statistical probability of throwing any particular triple is not. This insight, that surface randomness often hides deeper order is one of the most profound in the history of humanity and is really the whole basis of risk management.

Cardano’s Gambit

Of course, with his mathematical prowess Cardano was in a position to honestly outwit his fellow gamblers – and also to detect when they were cheating. Yet he also realised something else of immense importance – that although he could calculate probabilities correctly he couldn’t adjust for the imperfections of the real world – so that, for instance, an imperfect die would yield imperfect results. Still, Cardano’s gambit – his willingness to take risks on his understanding of the perfect probabilities – seems to have paid off for him and he ascended to wealth and power.

Translated forward four hundred years many of the same insights apply to investors in stocks, although these are masked and obfuscated by the thicket of experts who spend their lives working on such things. The statistics of stock price movements are more random even than the throw of an imperfect set of dice and far more influenced by the malign manoeuvrings of card sharps with their marked cards.

Most investors today find themselves in the same position as Cardano’s gambling opponents did in the fifteenth century, groping around for some idea about how to manage and measure the randomness inherent in their chosen card game, yet with the overwhelming suspicion that there’s more going on than meets the eye. Playing Cardano’s gambit in this world is far, far more difficult than figuring out the probabilities behind a simple game of chance.

Chasing Probability

As all good gamblers know, there are occasions when the luck deserts you and there are times when you need to know when to fold and quit. Loss aversion, where people chase their losses because they can’t cope with their pain, is not something that affects such people. Playing the game means accepting losses on the grounds that the underlying probabilities will eventually fall in your favour.

Unfortunately probability seems to be something that’s inherently difficult for the human brain to get comprehend. This isn’t a minor issue confined to investors either. Consider the proposition that most CEO's are men. Well, even without hard evidence we’d likely accept that. Would that mean that we also agree that most men are CEO's? Hardly.

Breast Cancer

OK, now how about “most women with breast cancer have a positive mammogram”? Again, we’d likely agree. So what about “most women with a positive mammogram have breast cancer”? Not so, most women with a positive mammogram do not have breast cancer. Most diagnosticians don’t get this right when presented with the raw statistics – 1% probability that woman of 40 has breast cancer, 80% chance if she’s got it that the mammogram detects it and a 9.6% chance she’ll test positive if she hasn’t got it. The experts reckon that, based on these numbers, a woman with a positive mammogram has a 75% chance of having breast cancer.

They’re wrong, and not a bit wrong. They're massively wrong.

No Vexation Without Representation

People just don’t think like this. Gerd Gigerenzer suggests that it’s a representational problem – we simply don’t think about these issues probabilistically but if the problems are presented in terms he refers to as “natural frequencies” the difficulty goes away. Consider:
“10 out of 1000 women at age forty who participate in routine screening have breast cancer. 8 out of every 10 women with breast cancer will get a positive mammogram. 95 out of 990 women without breast cancer will also get a positive mammogram.”
How many women with a postive mammogram will actually have breast cancer? You can find a discussion of this with the answers in the appendix to this paper, but the accuracy of diagnoticians improves dramatically.

Base Rate Neglect

This representational problem, technically known as base rate neglect, seems to be behind our fundamental difficulty in dealing with probabilities and is one of the factors that leads investors to make all sorts of strange gambles. In truth Cardano’s gambit is a simple one: understand the odds, minimise the risks and maximise the chances. In investing terms: analyse the fundamentals, manage you emotional biases and build the best margin of safety you can.

Cardano’s skills led him from poverty to riches and the support of some of the most powerful men in Renaissance Italy. Unfortunately he wasn’t so lucky with his children, who led him a merry dance before one of them betrayed him to the Inquisition. He ended his days back in poverty and certifiably crazy to boot. There is more to life than gambling, no matter how good at it you are.


The correct answer is about 8%: so at least 9 out of 10 positive mammograms are false positives.

The Book of My Life (New York Review Books Classics)Cardano's Cosmos: The Worlds and Works of a Renaissance AstrologerCalculated Risks: How to Know When Numbers Deceive You

Related articles: Pascal's Wager - For Richer, For Poorer, Laplace's Hammer: the End of Economics, Utility, The Deus Ex Machina of Economics


  1. You wrote: 10 out of 100 women at age forty who participate in routine screening have breast cancer.
    Surely that should be 10 out of 1000? Otherwise, there is a 10% probability that a woman of 40 has breast cancer, not 1%.

    Tricky probabilities.....

  2. Hi No'am

    It's not the stats that are the trouble, it's the proofreading ... Broken link to the reference paper also fixed.


  3. A more nuanced view of share-price behavior, and of the matter of underlying order and surface randomness, comes from Benjamin Graham, Warren Buffett's mentor: "In the short term, the market is a voting machine. In the long term, it is a weighing machine."

  4. >He ended his days back in poverty

    According to wiki he secured lifetime annuity from Pope Gregory XIII. I don't think annuity form Pope count as poverty.

  5. According to wiki he secured lifetime annuity from Pope Gregory XIII. I don't think annuity form Pope count as poverty.

    Good place to start research but rarely an end point. The UK government provides an annuity to its pensioners but that doesn't mean they're not poor ...

    Cardano was tortured by the Inquisition in 1570, moved to Rome in 1572 and died in 1576. Gregory XIII received the pontificate in 1573 so the annuity was awarded in that period somewhere. That Cardano needed this at all given his former wealth and status is significant in itself.

    Remarkably Cardano's autobiography is still in print, The Book of My Life. Extraordinary man, but I'm not sure he'd be someone you'd want to have as a friend.

  6. Great post Psy-Fi,
    Since I embarked on the journey of developing the Texas Holdem Investing theory I'm a big proponent of the similarities between investing and gambling (specifically poker in my case).
    Great so see a luminary like yourself thinking along the same lines.
    The Masked Financier