In the first part, we left Bruno de Finetti in 1949 as he established that for four distinct individual possibilities (like which team will win a championship), any usual ordering of “tickets” that was “quasi-additive” was also “probabilistic.” He conjectured that this would be true for any finite number of quasi-additively ordered propositions, and invited the community to help prove him right or wrong.

One who accepted de Finetti’s invitation was Leonard Savage, who later developed his own landmark axiomatization of subjective probability. Savage gave the obscure 1949 paper to Charles Kraft, John Pratt and Abraham Seidenberg. They showed in 1959 that de Finetti’s conjecture was wrong if there are five or more basic outcomes.

That’s the “broken” part of the story. De Finetti’s conjecture, scribbled in haste to rebut his friend George Polya, is false. Many people scrambled to fix it. The first were Kraft, Pratt and Seidenberg themselves in 1959. A famous second time was five years later, by Dana Scott. Oddly, neither solution was satisfactory to its authors. Odder still was that De Finetti himself may have come within a whisker of repairing it back at the beginning in 1949.

**Kraft, Pratt and Seidenberg break it and then fix it**

Suppose we have five NBA teams, the Bucks, Celtics, Knicks, Lakers and Mavericks. Adding the rest of the teams won’t help if de Finetti’s conjecture doesn’t work for these five. Let’s think about choosing “tickets” that pay off if one of these teams wins the championship. We’re not pricing them. We have our choice between two tickets at a time, and we’re saying which one we’d pick.

Four pairs of tickets will tell the tale.

Someone prefers “Bucks or Knicks or Lakers” to “Celtics or Mavericks,” strictly so. They aren’t indifferent between them; they think the first ticket is definitely more likely than the second to pay.

They also prefer “Celtics or Knicks” to “Bucks or Lakers,” “Bucks or Mavericks” to “Knicks or Lakers,” and finally, “Lakers” to “Bucks or Knicks.”

Kraft, Pratt and Seidenberg wrote out a complete quasi-additive ordering of all 31 distinct tickets involving these five teams to show that these choices were part of a quasi-additive ordering. No probability distribution agrees with these four choices. Suppose there were. Try to add up the probabilities in the four inequalities.

p(Bucks) + p(Knicks) +p(Lakers) > p(Celtics) + p(Mavericks)

p(Celtics) + p(Knicks) > p(Bucks) + p(Lakers)

p(Bucks) + p(Mavericks) > p(Knicks) + p(Lakers)

p(Lakers) > p(Bucks) + p(Knicks)

Both sides add up to:

2 p(Bucks) + p(Celtics) + 2 p(Knicks) + 2 p(Lakers) + p(Mavericks) The sum is greater than itself! Oops. There are no such numbers. There is no such probability.

Kraft, Pratt and Seidenberg then looked for some additional property which guaranteed that an ordering of possible tickets would have a probability distribution that agreed with it. They went back to De Finetti’s work in the 1930’s, described in part 1, where he “padded out” the teams with coin tosses.

If you flip a fair coin, not only is it equally likely to come up heads or tails, but the outcome of the coin toss has nothing to do with the outcome of the playoffs. If you like the Celtics better than the Bucks, then you presumably also like Celtics and heads better than Bucks and heads and better than Bucks and tails.

Consider, then, tickets which pay on combinations of teams and the outcome of a single coin toss (one particular toss referred to by all tickets):

(Bucks, H) or (Knicks, H) or (Lakers, H) > (Celtics, H) or (Mavericks, H)

(Celtics, H) or (Knicks, T) > (Bucks, H) or (Lakers, H)

(Bucks, T) or (Mavericks, H) > (Knicks, H) or p(Lakers, T)

(Lakers, T) > (Bucks, T) or (Knicks, T)

On each side, the four tickets can be combined into a single ticket. Each side’s ticket pays off on just the same events as the other side’s,

Bucks (no matter how the coin lands), or Celtics and heads, or Knicks, or Lakers, or Mavericks and heads

Nevertheless, one ticket is supposedly strictly preferred to its duplicate. That is incoherent as gambling advice, and a violation of quasi-additivity.

Kraft, Pratt and Seidenberg proved an important fact: To ensure a quasi-additivity violation can be derived from probability disagreement with a finite number of possibilities, the required number of coin tosses is always finite. It is possible, then, to extend any probability disagreeing quasi-additive domain to build another real-life finite domain where quasi-additivity is violated.

**Why wasn’t that a solution to the problem?**

The quest to repair de Finetti’s 1949 conjecture is not merely a mathematical task to specify when orderings are “probabilistic,” but also to make suitable rebuttal for de Finetti to have used against Polya in 1949. That’s a rhetorical objective. We are looking for something normative, something disturbing about orderings that aren’t probabilistic, or something attractive about those that are.

So, the Kraft, Pratt and Seidenberg repair is “inaesthetic” rather than wrong. Their auxiliary experiments with the coins seem “*ad hoc.*” The very fact that the coin tosses have nothing to do with the NBA, which is why the tosses accomplish anything, makes people question why they’re tossing coins when the original question was about basketball. The repair ideally should be something that’s wrong with the choices I actually make, not something that would be wrong if somebody shows up with a coin.

For statisticians, though, worrying about somebody “showing up with a coin” has the situation backwards. The coin tosses in the counterexample have performed the role of evidence. The coin tosses changed my opinion about the four comparisons. I strictly preferred them before, but now that I can toss a coin, I am indifferent. But it is absurd that a coin toss would be evidence for anything else, an occasion for belief change.

I wanted to isolate the NBA teams from the rest of the universe, not from just coin tosses, but from everything in the world that is happening which I judge to be independent of the NBA playoffs. My only warrant for ignoring those indisputably real events is my sense that it would be absurd if they made any difference to my opinion. Yet, they demonstrably can make a difference to people with quasi-additive but unprobabilistic beliefs.

Kraft, Pratt and Seidenberg’s demonstration does “repair” de Finetti’s conjecture fully, both in stating conditions for probability agreement and by providing rhetorical or normative force to de Finetti’s position in 1949. In the event, however, Kraft, Pratt and Seidenberg didn’t make much rhetorical argument.

**Scott repairs the conjecture a second time, and shows how close de Finetti came to getting it right**

About five years later, Dana Scott called attention to another aspect of the Kraft, Pratt and Seidenberg counterexample, that there was some finite combination of preference comparisons where each team appears the same number of times on the left as on the right. Scott’s theorem says that any system of ordered “tickets” is probabilistic **unless** such a combination of preference comparisons exists.

Fantastic, no coins. But what does that mean, and, wait a minute, any ordering? What happened to quasi-additivity? Well, any quasi-additivity violation leads immediately to a pair of comparisons forbidden by Scott:

a > b

b or c > a or c

One *a*, one *b* and one *c* appear on each side.

As to what his theorem meant, Scott acknowledged that his result in a sense restated Kraft, Pratt and Seidenberg’s. Like them, Scott didn’t claim that there was any normative case to be made for his result, either.

There is nevertheless a case to be made. Chief among Bruno de Finetti’s distinctive contributions to probability theory is his gambling semantics, that compound “tickets” should have properties consistent with collections of the individual equivalently paying “tickets.”

A fundamental operation in the de Finetti scheme is the “ticket exchange.” I will exchange any compound ticket (Bucks or Knicks) for the collection tickets that pay off the same amount under the same circumstances (a Bucks ticket and a Knicks ticket), in either direction of exchange.

Scott’s theorem invites us to build two portfolios of tickets. At each step, we can choose either one of two tickets, like “Bucks or Knicks or Lakers” *versus* “Celtics or Mavericks.” We like the first one better? It goes into our portfolio. But let’s put the rejected ticket in a second portfolio.

We make the four Kraft, Pratt and Seidenberg choices, and of the two portfolios we thus create, we strictly prefer one to the other. As with any portfolio, we accept the de Finetti exchanges, and now have two piles of tickets, each ticket paying on an individual team. We must strictly prefer one pile to the other.

But they are the same pile, two Bucks tickets, a Celtics, two Knicks, two Lakers and a Mavericks. This contradicts that one portfolio is better than the other, and conflicts with our expressed preferences in building the two portfolios.

So, although Scott did not pursue the normative or rhetorical aspect of his result, there clearly is an argument he could have made, solidly grounded in de Finetti’s work, that demands probability agreeing orderings. Scott’s Theorem, suitably explained, is a natural and intuitive second repair of de Finetti’s conjecture.

As pointed out earlier, Scott’s Theorem doesn’t single out quasi-additivity, of which de Finetti made so much in 1949. Oddly, de Finetti made a mistake in his 1949 paper, which is available here in translation. Sharp-eyed readers will see that in the third paragraph of section 2, de Finetti’s so-called “’additive’” property is **not** quasi-additivity, but a weaker property called “monotonicity.” There are well-known ways to order propositions that are monotonic but not quasi-additive (for example, by using a popular uncertainty tool called “possibility,” which produces orderings that are generally probability-**dis**agreeing).

What makes that slip interesting is that Scott’s Theorem can easily be rewritten as a monotonicity requirement for portfolios. De Finetti wrote about comparing individual tickets in 1949, but some part of his mind was thinking about the right property for proving his point – if he had required the property to hold for the right “ticket structure,” the portfolio, rather than only for pairs of tickets. A correct conjecture, and a successful rebuttal to Polya, eluded de Finetti by only a few hasty pen strokes.

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This report reflects work presented formally by the blogger in 2005 at the International Symposium on Imprecise Probabilities and Their Applications (ISIPTA proceedings, pages 332 ff.) and in 2006 at the Information Processing and Managment of Uncertainty conference (IPMU proceedings, pages 1996ff.). These papers are available online (links checked October 2015) at

http://leo.ugr.es/sipta/isipta05/proceedings/papers/s033.pdf

http://www.math.s.chiba-u.ac.jp/~yasuda/open2all/Paris06/IPMU2006/HTML/FINALPAPERS/P216.PDF