On fallacies and normative reasoning: when people's judgements follow probability theory

On fallacies and normative reasoning: when people’s judgements follow probability theory. Fintan J. Costello (fintan.costello@ucd.ie) Travis Mathison (tmathiso@gmail.com) School of Computer Science and Informatics University College Dublin Dublin 4, IRELAND Abstract In the first section of the paper we give some background by briefly describing four biases in people’s probabilistic rea- soning. In the second section we describe our model and show how it explains these biases. In the third section we show that this model predicts that bias will be cancelled when probability estimates are combined according to probability theory’s addition rule, which requires that The systematic conjunction and disjunction fallacies seen in people’s probability judgments appear to show that people do not reason according to the rules of probability theory. In an experiment examining people’s judgments of the probabil- ity of different medical conditions, we find evidence against this view. In this experiment people’s probability judgments closely followed the fundamental ‘addition rule’ of probability theory. This close match to probability theory comes alongside frequent occurrence of the conjunction and disjunction falla- cies in those same probability judgments. These results sup- port a model where people reason about probability via prob- ability theory but are subject to random variation or noise in the recall of items from memory. In this model the effect of random variation is cancelled out by the mathematical form of the addition rule, producing agreement with probability the- ory; however, noise is not cancelled out for conjunctive or dis- junctive comparisons, producing conjunction and disjunction fallacy responses. Keywords: Probabilistic reasoning; conjunction fallacy; ratio- nality P(A 1 ) + P(A 2 ) − P(A 1 and A 2 ) − P(A 1 or A 2 ) = 0 for all events A 1 and A 2 . In the fourth section we describe an experiment testing this prediction and showing that when people’s probability estimates are combined via the addition rule, the result is 0 just as required by probability theory. The final section draws some conclusions from this result. Background:Biases in Probabilistic Reasoning We consider four biases in probabilistic reasoning: conser- vatism, subadditivity, the disjunction fallacy, and the conjunc- tion fallacy. Perhaps the best known of these is the conjunc- tion fallacy. Probability theory’s ‘conjunction rule’ requires that, for any pair of events A 1 and A 2 where P(A 1 ) ≤ P(A 2 ) the relationship Introduction Probability theory provides a calculus of chance describing how to make optimal predictions under uncertainty. Up to the 1960s the standard view in psychology was that people’s probabilistic reasoning essentially followed probability the- ory. However, various systematic biases in people’s proba- bility judgements (many identified in the 1970s and 1980s by Tversky, Kahneman and colleagues) led researchers to con- clude that, in fact, people do not follow probability theory but instead estimate probabilities using various heuristics. While these heuristics often yield reasonable judgments, they can also produce strong biases in people’s probabilistic reasoning in certain cases (Tversky and Kahneman, 1973). In this paper we return to the view that people follow prob- ability theory when reasoning about uncertainty. We present a simple model where people estimate probabilities accord- ing to probablity theory, but are subject to random error or noise in recall from memory: we show that this model can explain various systematic biases seen in people’s probabilis- tic reasoning. Importantly, this model predicts that bias will be ‘cancelled’ for certain combinations of people’s probabil- ity estimates, and so those combinations should agree closely with probability theory. In an experiment testing this predic- tion by gathering people’s probability estimates for various medical conditions, we find extremely close agreement with probability theory, demonstrating the predicted cancellation. P(A 1 and A 2 ) ≤ P(A 1 ) must hold. This follows from the fact that A 1 and A 2 can only occur if A 1 itself occurs. People reliably violate this re- quirement for some events, giving probability estimates for conjunctions that are greater than the estimates they gave for one or other constituent of that conjunction. Tversky & Kah- neman’s original demonstration of this fallacy (Tversky and Kahneman, 1983) concerned Linda: “Linda is 31 years old, single, outspoken, and bright. She majored in philosophy. As a student she was deeply concerned with issues of discrimination and social jus- tice, and participated in anti-nuclear demonstrations” Participants in Tversky & Kahneman’s study read this de- scription and were asked to rank various statements by their probability. Two of these statements were • (A 1 ) Linda is a bank teller. • (A 1 and A 2 ) Linda is a bank teller and an active feminist.