The psychology of dynamic probability judgment: order effect, normative theories, and experimental methodology

Abstract

The Bayesian model is used in psychology as the reference for the study of dynamic probability judgment. The main limit induced by this model is that it confines the study of revision of degrees of belief to the sole situations of revision in which the universe is static (revising situations). However, it may happen that individuals have to revise their degrees of belief when the message they learn specifies a change of direction in the universe, which is considered as changing with time (updating situations). We analyze the main results of the experimental literature with regard to elementary qualitative properties of these two situations of revision. First, the order effect phenomenon is confronted with the commutative property. Second, an apparent new phenomenon is presented: the redundancy effect that is confronted with the idempotence property. Finally, results obtained in this kind of experimental situations are reinterpreted in the light of pragmatic analysis.

Appendices

Appendix 1. The “Bayesian solution” is the solution computed using Bayes’ rule

Let the prior probabilities be P(_freed A), P(_freed B) and P(_freed C); they are equal to 1/3 (“all three equally likely”). The likelihoods are: P(“_executed B”|_freed A) = 1/2 because if A is freed, the warder can indicate that either B or C will be executed and we are supposing that the warder has no preference between B and C; P(“_executed B”|_freed B) = 0 because if B is freed, the warder cannot indicate B _executed) and: P(“_executed B”|_freed C) = 1 because if C is freed, the warder can only indicate B _executed).

Bayes’ rule gives:

$$\begin{aligned}{} & P{\left( {{}_{{{\text{freed}}}}A|``{}_{{{\text{executed}}}}B\hbox{''}} \right)} \\ & \quad = \frac{{P({}_{{{\text{freed}}}}A) \times P(``{}_{{{\text{executed}}}}B\hbox{''}|{}_{{{\text{freed}}}}A)}}{{P({}_{{{\text{freed}}}}A) \times P(``{}_{{{\text{executed}}}}B\hbox{''}|{}_{{{\text{freed}}}}A) + P(_{{{\text{freed}}}} B) \times P(``{}_{{{\text{sex}}\,{\text{executed}}}}B\hbox{''}|{}_{{{\text{freed}}}}B) + P({}_{{{\text{freed}}}}C) \times P(``{}_{{{\text{executed}}}}B\hbox{''}|{}_{{{\text{freed}}}}C)}},\quad \quad P({}_{{{\text{freed}}}}A|``{}_{{{\text{executed}}}}B\hbox{''}) \\ & \quad = \frac{{1/3 \times 1/2}}{{1/3 \times 1/2 + 1/3 \times 0 + 1/3 \times 1}} \\ & \quad = 1/3\quad {\text{and}}\;{\text{similarly}}\;P({}_{{{\text{freed}}}}C|``{}_{{{\text{executed}}}}B\hbox{''}) = 2/3. \\ \end{aligned}$$

This solution is called the Bayesian solution in the literature (for example: Falk 1992). In fact, from the Bayesian point of view there is no such a thing as a unique solution to a problem. The important notion is that judgement of probability must respect the coherence axioms. Thus, participants who respond, for example, 1/2 to the three prisoners problem because they think that the warder is not honest and that P(“_executed B”|_freed A) = 1 (if A is freed, the warder says that B will be executed because he prefers B to C), or any other solution because they have some other prior degree of belief than 1/3, can genuinely be Bayesian. Thus, we use “Bayesian solution” in quotation marks.

Appendix 2

The standard version of “three prisoners” can be found in Shimojo and Ichikawa (1989, p. 2).

Three men, A, B and C, were in jail. A knew that one of them was to be set free and the other two were to be executed. But he didn’t know who was the one to be spared. To the jailer who did know, A said, “since two out of the three will be executed, it is certain that either B or C will be, at least. You will give me no information about my own chances if you give me the name of one man, B or C, who is going to be executed”. Accepting this argument after some thinking, the jailer said “B will be executed”. Thereupon A felt happier because now either he or C would go free, so his chance had increased from 1/3 to 1/2. This prisoner’s happiness may or may not be reasonable. What do you think?

The version we used in the experiment was similar to Shimojo and Ichikawas (1989)’ version. The main difference is that we asked participants to give two judgements of probability. A first judgment took place before the warder released his message. This step was meant to know which prior degrees of belief participants used, and to see if they were equal to 1/3 as assumed by the experimenter. A second judgement was asked after the release of the warder’s message in order to analyse participants’ dynamic process of revision.