Betting on Transitivity in an Economic Setting (2014)

Betting on Transitivity in an Economic Setting Dennis Hebbelmann (Dennis.Hebbelmann@psychologie.uni-heidelberg.de) Momme von Sydow (Momme.von-Sydow@uni-heidelberg.de) Department of Psychology, University of Heidelberg, Hauptstr. 47 69117 Heidelberg, Germany Abstract Theories of causal reasoning and learning often implicitly assume that the structural implications of causal models and empirical evidence are consistent. However, for probabilistic causal relations this may not be the case. Mismatches between structural implications and empirical evidence may lead to distortions of empirical evidence. Previous work has shown that people may use the generative causal relations A → B and B → C to infer a positive relation between events A and C, despite data showing that these events are actually independent (von Sydow et al., 2009, 2010). Here we used an economic trial-by-trial learning scenario to investigate how transitive reasoning in intransitive situations with even negatively related distal events may relate to betting behavior. Experiment 1 shows that transitive reasoning does affect not only probability estimates but betting as well. Experiment 2 shows that the effect remains stable even after repeated betting and feedback. Keywords: causal induction, transitivity, Markov condition, betting, economic reasoning, coherence-based induction Transitive Reasoning in Probabilistic Causal Chains Causal model theory and causal Bayes nets (Pearl, 2000; Sloman, 2005; Spirtes, Glymour, & Scheines, 2001; Waldmann, 1996; Waldmann, Cheng, Hagmayer, & Blaisdell, 2008) build on the assumption of the Markov condition, stating that any node in a causal model is conditionally independent of all upstream nodes, given its parents (Hausman & Woodward, 1999; Spohn, 2001). This entails transitivity in causal chains: If A causes B, and B causes C, then A causes C (via B). If in a probabilistic causal chain the Markov condition holds, then the strength of the global relation A → C can be infered from the strength of the local relations A → B and B → C, which means that transitivity holds: When using the causal strength estimate ΔP (ΔP AB = P(B|A) – P(B|¬A); Jenkins & Ward, 1965), the global ΔP can be calculated by multiplying all local ΔPs that make up the causal chain (e.g., ΔP AC = ΔP AB * ΔP BC ). It is therefore not necessary to observe the global relation directly. Related research on transitive reasoning in the induction of causal chains has shown that people assume a transitive causal relation based on integrating single causal links (Ahn & Dennis, 2000; Baetu & Baker, 2009). This research corroborated the hypothesis that people reasoned transitively even if no information on the distal event was shown. Subsequent research started to investigate intransitive chains (von Sydow, Meder, & Hagmayer, 2009; von Sydow, Meder, Hagmayer, & Waldmann, 2010). This allows bringing bottom-up evidence (correspondence) and top- down inferences based on the structural assumptions about causal models (coherence) into conflict. They suggest a causal coherence hypothesis that coherence-based induction may distort bottom-up evidence about causal relations considerably if the bottom-up data do violate the structural assumptions of Bayes Nets. People are taken, at least by default, to assume a modular integration of single causal relations into larger causal networks, for instance implying transitivity in causal chains. This is predicted even when evidence to the contrary is available, but people may give up this default belief if the mismatch between coherence-based induction and correspondence- based induction gets very evident. Intransitive chains are at odds with structural implications of Bayes Nets and involve a violation of the Markov condition. In the philosophical debate it has been put into question whether all causal relations necessarily adhere to the Markov condition and, as a consequence, whether chains need to be transitive (Cartwright 2001, 2006; Sober & Steel, 2012). However, even strict advocates of the Markov condition have pointed out that on the level of our actually used categories causal chains may not adhere to the Markov condition (Hausman & Woodward, 1999; Spohn, 2001). For instance, this may be the case if a category is the product of mixing subclasses for which different causal relations hold. Von Sydow et al. (2009, 2010) showed in several formats (overview format, trial-by-trial format) that participants may infer the relation A → C from A → B and B → C, even if this is not warranted by the data presented to them: In the materials used, A → B and B → C were positive, while A and C were statistically independent from each other (ΔP AB = ΔP BC = .5, ΔP AC = 0). Participants in accordance with the causal coherence hypothesis still judged A → C in line with transitivity if they were presented with A → B and B → C first. This effect remained stable even when participants were able to directly assess the data about A → C. However, in many regards the boundary conditions of the causal coherence hypothesis need further exploration. For instance it is not clear whether people continue to infer a positive distal causal relation from positive local relations if it is clearly negative or if they completely switch to bottom-up induction due to the obvious mismatch. Causal Reasoning and Decision Making We here transfer the idea of the causal coherence hypothesis and intransitive chains to the field of decision making.