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Computing Conjunctive Probabilities by Combining Constituent Probabilities: An Evaluation of the Weighted Average Model

English title Computing Conjunctive Probabilities by Combining Constituent Probabilities: An Evaluation of the Weighted Average Model
Applicant Rieskamp Jörg
Number 126721
Funding scheme Project funding (Div. I-III)
Research institution Fakultät für Psychologie Universität Basel
Institution of higher education University of Basel - BS
Main discipline Psychology
Start/End 01.01.2010 - 31.12.2011
Approved amount 133'086.00
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Keywords (8)

probability estimates; conjunctive probability; judgment processes; cognitive modeling; learning and experience; decision making; probability judgments; conjunctive probabilties

Lay Summary (English)

Lay summary
In everyday life the outcomes of people's decisions are often uncertain. Different events will determine whether a decision will lead to beneficial or detrimental outcomes. For instance, whether a skiing trip is an enjoyable success depends on the weather conditions, in particular on snowfall and on having sunny days. Often the best outcome of a decision can only be reached when a series of events occur. The present project examines how people assess the probability that combinations of events occur. The first part of the current research project examines how people combine single event probabilities (i.e., constituent probabilities) to assess the probability of joined events (i.e., conjunctive probabilities). For instance, the conjunctive probability that it will snow tonight and that the sun will shine tomorrow, is a function of the probability of both single events alone. Probability theory says that the conjunctive probability should be determined by multiplying the constituent probabilities. However, past research has shown that people do not access probabilities according to the normative rule. Therefore we explore alternative models for probability judgments. More specifically we examine the hypothesis that people access conjunctive probabilities by determining an average of the constituent probabilities.In a second part of the project we examine whether the averaging model also leads to accurate probability judgments, when the constituent probabilities can only be accessed with some error. Computer simulations have shown that when the objective constituent probabilities are only approximately known, then it can happen that the averaging rule produces conjunctive probabilities that are higher correlated with the objective conjunctive probabilities than the conjunctive probabilities estimated by the normative multiplicative rule. We will explore whether this result is also observed in behavioral data. Accordingly people should be better in assessing objective conjunctive probabilities by using an averaging rule than by following the normative rule. If this could be demonstrated it will have important implications for teaching people how to assess the probabilities of joint events. In summary, the project should lead to a better understanding of how people make decisions when the outcome of their decision depends on the occurrence of multiple events.
Direct link to Lay Summary Last update: 21.02.2013

Responsible applicant and co-applicants



Hierarchical Bayesian parameter estimation for cumulative prospect theory
Nilsson H, Rieskamp J, Wagenmakers EJ (2011), Hierarchical Bayesian parameter estimation for cumulative prospect theory, in JOURNAL OF MATHEMATICAL PSYCHOLOGY, 55(1), 84-93.

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Jenny, M. A., Rieskamp, J., & Nilsson, H. Choices in a conjunctive probability task: Evidence for the configural weighted average model. Poster presented at the 32nd Annual Meeting of the Society for Judgment and Decision Making 24.11.2011 Seattle, Washington, US
Jenny, M. A., Rieskamp, J., & Nilsson, H. (2011). Describing conjunctive probability assessment from experience with weighted averaging. Poster presented at the Bayesian Modeling for Cognitive Science Workshop 22.08.2011 Amsterdam, Netherlands
Jenny, M. A., Rieskamp, R., & Nilsson, H. (2011). The queen of hearts and the ace of spades: Describing conjunctive probability assessment from experience with weighted averaging. Paper presentation at the 10th Annual Summer Interdisciplinary Conference, 07.07.2011 Caldes de Boi, Spain
Jenny, M. A., Rieskamp, J., & Nilsson, H. The queen of hearts and the ace of spades: Describing conjunctive probability assessment from experience with weighted averaging. Paper presented at the 53. Meeting of Experimental Psychologists 13.03.2011 Halle, Germany
Jenny, M. A., Nilsson, H., & Rieskamp, J. (2010). Probability theory reconsidered: Describing probability assessment with weighted averaging. Poster 06.11.2010 St. Louis, Missouri, USA
Rieskamp, J., Nilsson, H., & Wagenmakers, E.J. (2010). The methodological advantages of hierarchical Bayesian estimation process: Illustrated with prospect theory. 47th congress of the German Psychology Association. 23.08.2010 Bremen, Germany
Jenny, M. A., Nilsson, H., & Rieskamp, J. Probability theory versus configural weighting and averaging: How do we assess conjoint probabilities? Poster präsentiert an der 52. Tagung experimentell arbeitender Psychologen 15.03.2010 Saarbrücken, Deutschland


Title Year
Cognitive Science Society Award for the Best Student Paper at ASIC 2011 2011

Associated projects

Number Title Start Funding scheme
138174 Generalizing the weighted average hypothesis: The cognitive processes underlying probability updating 01.01.2012 Project funding (Div. I-III)


This project deals with how people assess subjective probabilities. Because probabilities play a crucial role in much of human decision making, the question of how they are subjectively assessed has attracted extensive attention. The common finding is that though subjective probabilities tend to correlate with true objective probabilities, there are systematic deviations. More specifically, this project deals with how people combine subjective constituent probabilities (e.g., the probability that it will snow tomorrow) to form subjective conjunctive probabilities (e.g., the probability that it will snow tomorrow AND that my car will break down on the way to work). We explore a hypothesis suggesting that people estimate subjective con-junctive probabilities by taking a weighted average of the subjective probabilities of the con-junction’s two constituents. This rule is in opposition to normative probability theory, accord-ing to which the constituent probabilities have to be multiplied. If people follow this weighted average rule they will systematically overestimate conjunctive probabilities. This is exactly what has been found in past empirical research. However, this does not imply that the weighted av-erage rule is a poor rule to use for combining constituent probabilities. Instead, Juslin, Nils-son, and Winman (2009) showed by computer-based simulations that the weighted average rule has rational properties. If true constituent probabilities are not perfectly known but are assessed with some error, the true conjunctive probabilities are often better approximated by taking a weighted average of the two constituent probabilities than by multiplying them.Our proposed project will explore the conditions under which people follow the weighted average rule. For most decision problems people are equipped with a repertoire of decision strategies that they use depending on the characteristics of the decision situation (e.g., Rieskamp & Otto, 2006). We look at how the probability that the weighted average rule is selected is influenced by experience (i.e., exposure to the task environment) and the structure of the task. Furthermore, the current project will study how decisions are affected by subjective conjunctive probabilities and how subjective conjunctive probabilities are integrated when evaluating options. This integration will have an impact on people’s decisions. We will examine how different rules for combining constituent probabilities affect the quality of decisions and how this process depends on the level of individuals’ experiences. Obviously, this project has the potential to increase our understanding of both how people derive subjective conjunctive probabilities from subjective constituent probabilities and how people integrate subjective probabilities for making decisions. A less obvious, but maybe more intricate contribution is the following. It is often assumed that if decision makers follow the normative solution to a problem they will make better decisions. Therefore, great effort has been put into developing tools to aid this process. We will explore how the two rules for com-bining constituent probabilities into conjunctive probabilities (the weighted average rule and the normative product rule) affect the quality of decisions at different levels of experience. Our hypothesis, based on the findings by Juslin et al. (2009), is that it will be more beneficial to use the weighted average rule than the normative product rule, at least when experience is limited. This would imply that following the normative rule will only lead to good decisions under cer-tain circumstances (for similar ideas see, e.g., Gigerenzer, Todd, & the ABC Research Group, 1999). In turn, this would indicate that tools developed to help decision makers follow norma-tive solutions might sometimes hinder them in their efforts to make optimal decisions.