The Principle of Indifference by Sorin Bangu

Probability Assignments and the Principle of Indifference.
A Reassessment of the Eliminative Strategies


A central and controversial component of the ‘classical’ conception of probability, the Principle of Indifference (hereafter PI) claims that equipossibility entails equiprobability. A more complete version can be formulated as follows:

Given a null state of background information, equal regions of the space of possible outcomes should be assigned equal probabilities (Howson and Urbach 2006, p. 266).

As is well known, the principle plays an important role not only in physics (in the foundations of statistical mechanics) but also in our everyday probabilistic inferences (e.g., in predicting the outcomes of coin-tossing). Yet many philosophers and scientists also hold that PI is subject to serious objections. These objections are of two kinds. The first kind has to do with the (logical) inconsistencies introduced by the use of the principle. It has long been claimed that the principle must be rejected because it leads to paradoxes – especially when employed in infinitary (continuous) contexts.[1] The second kind of criticism, advanced by Reichenbach (1949, §§ 68-71), maintains that the principle is dispensable in probabilistic inferences. Consequently, he proposed to pursue what I call here an eliminative strategy. In a nutshell, the guiding idea of this type of strategy is to show that reliance on the a priori principle is not necessary in order to infer the correct observed frequencies of outcomes.

My goal in this paper is to analyze in detail two attempts to implement the eliminative strategy.[2] While I agree that eliminativism is a very promising kind of strategy available when dealing with PI[3], I’ll maintain that both attempts proposed so far are fraught with problems. The paper is divided in two sections, each aiming to highlight the difficulties faced by those attempts. In the first section I show that Reichenbach’s implementation of the eliminative strategy fails; more precisely, I’ll show that one of the premises of his eliminative argument just assumes PI. In order to prove this, I’ll examine the same example taken by Reichenbach (following Poincaré) as paradigmatic in showing the effectiveness of PI, namely its employment in predicting the correct probabilities in the game of roulette. Since there is presumably nothing special about roulette (in the sense that the eliminative strategy can be adapted to other cases in which PI seems effective), it is natural to expect that the failure in this case will have similar consequences for other potential eliminative attempts.

In the second section I turn to another attempt to eliminate PI, Gillies’ recent heuristic approach (Gillies (2000)). Despite the fact that Gillies allows PI a certain role in our probabilistic inferences – namely, in conjuring probabilistic hypotheses – I’ll construe his view as another attempt to dispense with the role of PI in probabilistic reasoning. My reason for doing this is Gillies’ emphasis on the incapacity of the principle to justify those hypotheses. Although I agree that this line of thought is quite promising, I’ll close by raising doubts with regard to its cogency. More precisely, I show that the alternative method of justification / rejection of probabilistic hypotheses endorsed by Gillies (in essence, the method of statistical relevance tests) is subject to the same kind of difficulties as the method of a priori justification involving the application of the principle.
[1] van Fraassen, for instance, notes that “the great failure of symmetry thinking” is revealed in those situations “where indifference disintegrated into paradox.” (1989, p. 293).
[2] With one exception, none of the analyses of PI listed below pays attention to these strategies; instead, they focus exclusively on the relation between PI and the (Bertrand type) paradoxes. See Mikkelson (2004), Bartha and Johns (2000), Gillies (2000), Castell (1998), Marinoff (1994), Schlessinger (1993). The notable exception is Strevens (1998) and I’ll take issue with some of his points in the first section.
[3] Another line of objections to PI is of course based on its role in deriving the Bertrand type paradoxes. While I’ll be saying something about this role in section 2, the focus of this paper is not on the paradoxes.

The Money Pump Argument by Stuart Yasgur.

The Money Pump and the Justification of the Transitivity Condition
Stuart Yasgur

Rationality is said to require that agents have transitive preferences. The justification of the transitivity condition, as it will be referred to here, is widely thought to be provided by the money pump argument, though there is less consensus about the exact form of this justification.

An excerpt of a larger paper that examines the main justifications of the transitivity condition, this paper focuses on the consequentialist justification. It is important to note that since this paper focuses on the justification of the transitivity condition, it will differ significantly from papers which attempt to characterize the decisions within the money pump itself.

To begin, I will present a basic version of the transitivity condition and a statement of the money pump argument. There are a number of variations of the transitivity requirement, but here we can avoid complications and deal with a weak version, though the arguments in the paper apply equally to stronger versions:

Take the expression x>y to mean that the agent prefers x to y and the expression x > y to mean that the agent prefers x to y or is indifferent between the two. Given this, we can define transitivity as follows: an agent’s preferences are taken to be transitive if, for all triples of alternatives (x, y, z), x > y and y > z imply x > z. Correspondingly an agent’s preferences are taken to be intransitive if for a triple of alternatives (x, y, z), x > y, y > z, and z > x.

Generic version of the money pump argument:
An agent prefers x to y, y to z, and z to x. The agent also prefers more money to less. The agent is offered the opportunity to switch from z to y for a small amount of money, and he accepts. He is then offered the opportunity to switch from y to x for a small amount of money, and he accepts. And, he is offered the opportunity to switch from x to z for a small amount of money, which he accepts.

There is also an extended version of the money pump that is quite common, in which the cycle is repeated until the agent looses all of his money. Though it is easy to slip between the two, the potential justificatory force of each differs, so here they will be dealt with separately. Unless otherwise mentioned, I will be focusing on the basic version.

There are a few things to notice about the money pump argument. First, the agent has intransitive preferences. Second, the agent always moves from a less preferred option to a more preferred option. Third, in terms of his own preferences, the agent is unambiguously worse off at the end of the cycle than he was at the beginning of the cycle. I.e., in terms of his preferences over x, y, and z, the agent is no better off and no worse off, but in terms of his preference for more money rather than less he is worse off.

It should also be clear that the money pump is not an argument. It is an example, but examples on their own are not arguments. To establish that the money pump example justifies the transitivity condition, we must understand the force of the example, and this is where views begin to diverge.

Arguably, the consequentialist justification of the transitivity condition is the most plausible. As was already mentioned, the agent in the money pump is made unambiguously worse off, and it is thought to be these consequences themselves that justify the claim that transitivity is a requirement of rationality.

Consequentialist justifications of requirements of rationality take the following form:
· P1: If X leads an agent to suffer negative consequences, then X is irrational.
· P2: X leads an agent to suffer negative consequences, in suitable circumstances.
· C: Therefore X is irrational.

Much can be said to make arguments of this form more specific, but the general version should suffice to make the point at hand. The first thing to note is that the money pump offers a case in which P2 holds, and if P1 holds, then the conclusion follows. Next, notice that if we take X to be ‘false beliefs’, then P2 would also hold. If P1 holds, then it would follow that having false beliefs is irrational. Having false beliefs is not irrational, therefore P1 does not hold; and therefore the money pump does not offer a consequentialist justification for considering intransitive preferences irrational.

Further, because of the gap between preferences and the consequences of choices based on them, there does not seem to be a way to refine P1 so that it would apply to intransitive preferences but not false beliefs.[1]

Conclusion:
In the longer version of this paper I argue that despite its currency in the literature the money pump does not justify the transitivity condition. However, my own view is that the transitivity condition, suitably qualified, is a genuine requirement of rationality that can be justified based on a broader understanding of the relationship between rationality and value. Rather by discussing the money pump’s limited justificatory force I hope to bring into focus the need to reexamine the justification of one of the basic conditions of preference theory; and I am particularly interested in people’s thoughts about possible consequentialist justifications.
[1] Consider the following example:
· P1`: If X leads an agent to suffer negative consequences even when he is ideal in every other way, then X is irrational.
· P2`: X leads an agent to suffer negative consequences in suitable circumstances, even when he is ideal in every other way.
· C: Therefore X is irrational.

Since P2` still holds for false beliefs, P1` should as well. But it does not.


[1] Consider the following example:
· P1`: If X leads an agent to suffer negative consequences even when he is ideal in every other way, then X is irrational.
· P2`: X leads an agent to suffer negative consequences in suitable circumstances, even when he is ideal in every other way.
· C: Therefore X is irrational.
Since P2` still holds for false beliefs, P1` should as well. But it does not.