In his Philosophical Theories of Probability, Donald Gillies proposes what he calls an intersubjective theory of probability. A better name for it would be group-strategy model of probability.

Subjectivists such as Bruno di Finetti ask the reader to consider the following sort of game:

• Some potential event is identified.
• Our hero must choose a real number (negative or positive) q, a betting quotient.
• The nemesis, who is rational, must choose a stake S, which is a positive or negative sum of money or zero.
• Our hero must, under any circumstance, pay the nemesis q·S. (If the product q·S is negative, then this amounts to the nemesis paying money to our hero.)
• If the identified event occurs, then the nemesis must pay our hero S (which, if S is negative, then amounts to taking money from our hero). If it does not occur, then our hero gets nothing.

Di Finetti argues that a rational betting quotient will capture a rational degree of personal belief, and that a probability is exactly and only a degree of personal belief.

Gillies asks us to consider games of the very same sort, except that the betting quotients must be chosen jointly amongst a team of players. Such betting quotients would be at least examples of what Gillies calls intersubjective probabilities. Gillies tells us that these are the probabilities of rational consensus. For example, these are ostensibly the probabilities of scientific consensus.

Opponents of subjectivists such as di Finetti have long argued that the sort of game that he proposes fails in one way or another to be formally identical to the general problem for the application of personal degrees of belief. Gillies doesn’t even try to show how the game, if played by a team, is formally identical to the general problem of group commitment to propositions. He instead belabors a different point, which should already be obvious to all of his readers, that teamwork is sometimes in the interest of the individual.

Amongst other things, scientific method is about best approximation of the truth. There are some genuine, difficult questions about just what makes one approximation better than another, but an approximation isn’t relevantly better for promoting such things as the social standing as such or material wealth as such of a particular clique. It isn’t at all clear who or what, in the formation of genuinely scientific consensus, would play a rôle that corresponds to that of the nemesis in the betting game.

Karl Popper, who proposed to explain probabilities in terms of objective propensities (rather than in terms of judgmental orderings or in terms of frequencies), asserted that

Causation is just a special case of propensity: the case of propensity equal to 1, a determining demand, or force, for realization.

Gillies joins others in taking him to task for the simple reason that probabilities can be inverted — one can talk both about the probability of A given B and that of B given A, whereäs presumably if A caused B then B cannot have caused A.

Later, for his own propensity theory, Gillies proposes to define probability to apply only to events that display a sort of independence. Thus, flips of coins might be described by probabilities, but the value of a random-walk process (where changes are independent but present value is a sum of past changes) would not itself have a probability. None-the-less, while the value of a random walk and similar processes would not themselves have probabilities, they’d still be subject to compositions of probabilities which we would previously have called probabilities.

In other words, Gillies has basically taken the liberty of employing a foundational notion of probability, and permitting its extension; he chooses not to call the extension probability, but that’s just notation. Well, Popper had a foundational notion of propensity, which is a generalization of causality. He identified this notion with probability, and implicitly extended the notion to include inversions.

Later, Gillies offers dreadful criticism of Keynes. Keynes’s judgmental theory of probability implies that every rational person with sufficient intellect and the same information set would ascribe exactly the same probability to a proposition. Gillies asserts

[…] different individuals may come to quite different conclusions even though they have the same background knowledge and expertise in the relevant area, and even though they are all quite rational. A single rational degree of belief on which all rational being should agree seems to be a myth.

So much for the logical interpretation of probability, […].

No two human beings have or could have the same information set. (I am reminded of infuriating claims that monozygotic children raised by the same parents have both the same heredity and the same environment.) Gillies writes of the relevant area, but in the formation of judgments about uncertain matters, we may and as I believe should be informed by a very extensive body of knowledge. Awareness that others might dismiss as irrelevant can provide support for general relationships. And I don’t recall Keynes ever suggesting that there would be real-world cases of two people having the same information set and hence not disagreeing unless one of them were of inferior intellect.

After objecting that the traditional subjective theory doesn’t satisfactorily cover all manner of judgmental probability, and claiming that his intersubjective notion can describe probabilities imputed by groups, Gillies takes another shot at Keynes:

When Keynes propounded his logical theory of probability, he was a member of an elite group of logically minded Cambridge intellectuals (the Apostles). In these circumstances, what he regarded as a single rational degree of belief valid for the whole of humanity may have been no more than the consensus belief of the Apostles. However admirable the Apostles, their consensus beliefs were very far from being shared by the rest of humanity. This became obvious in the 1930s when the Apostles developed a consensus belief in Soviet communism, a belief which was certainly not shared by everyone else.

Note the insinuation that Keynes thought that there were a single rational degree of belief valid for the whole of humanity, whereäs there is no indication that Keynes felt that everyone did, should, or could have the same information set. Rather than becoming obvious to him in the 1930s, it would have been evident to Keynes much earlier that many of his own beliefs and those of the other Apostles were at odds with those of most of mankind. Gillies' reference to embrace of Marxism in the ’30s by most of the Apostles simply looks like irrelevant, Red-baiting ad hominem to me. One doesn’t have to like Keynes (as I don’t), Marxism (as I don’t) or the Apostles (as I don’t) to be appalled by this passage (as I am).

While I was doing some research to-day, I ran across yet another article that classified John Maynard Keynes as a subjectivist when it came to probability theory. I feel moved to explain why this is incorrect.

First, let me explain something about the general issue. There is an outstanding question about just what a probability is. One could take many courses about probability without ever being alerted to the question. The textbook and lecturer might not ever touch on that basic question, or might present a definition of probability as if it is universally accepted by all Wise People. But Wise People are not in agreement. When it comes to answers to the basic question, the two dominant answers are very different one from another.

One answer is provided by the frequentists, who say that a probability is some sort of frequency of occurrence. They don’t agree amongst themselves as to the precise answer, but the gist of their answers is that if a process is repeated m times, where m is satisfactorily large, and results in some particular outcome n of those times, then the probability of that outcome is n/m.

One problem with this notion of probability is that it is only useful in cases where we are concerned with a sufficiently large sample. If one is concerned only with a single instance, then there is actually no logic to get us from a mere probability to a course of action. A single patient won’t have average mortality; she will either live or die.

Another answer is provided by the subjectivists, who assert that a probability is a degree of belief, formed subject to certain rationality constraints. These constraints can be largely motivated in terms of avoiding probability assignments under which believers would accept gambles that they are sure to lose. The rationality constraints themselves are ostensibly objective — rules that should hold for everyone; amongst other things, these rules are to constrain the evolution of one’s degrees of belief, as new information is introduced. The subjectivism is present in that one supposedly gets to start with any degrees of belief that don’t violate these rules.

One immediate consequence of this notion of probability is that probabilities become largely unarguable. There is no real contradiction in Tim claiming that there is an 80% chance of rain and Bob claiming that there is a 20% chance; each is describing his respective belief per se. (The rationality constraints force a convergence of belief at the limit, but that could take forever.)

The subjectivist notion is often defined in terms such as degree of rational belief or rational degree of belief; it’s best to be wary of such terms. The rationality constraints themselves only preclude certain sorts of irrationality; aspects of the degrees of belief permitted are at best not irrational. And if we are not somehow required to assign some quantity to that belief, then the assignment violates Ockham’s Razor.

Now, Keynes’s position is that we can make meaningful statements about the plausibility of uncertain outcomes for which frequencies are unknown or otherwise inapplicable. And he certainly wants to impose rationality constraints much like those of the subjectivists. But he sees no requirement that one always assign a quantity to belief. Indeed, he sees no reason to treat the set of possible outcomes as even necessarily totally ordered; that is to say that he holds that, when asked to compare the likelihood of two events, sometimes one can only shrug, rather than making claims that one event is more likely or that the two are equally likely.

Under Keynes’s theory, a rational person says no more about the probability of an event than the application of objective rules to the information set yields, and any other rational person with the same information set would reach exactly the same conclusions about probabilities (except, perhaps, where one person halted consideration where the other continued). Keynes rejects the very thing that is subjective in the subjectivist framework.