Evidence Accumulation Model

evidencedownloadMy first notice of this model was in the Sollner et al paper. My quick search finds that this 2004 paper, “Evidence accumulation in decision making:  Unifying the “take the best” and the “rational” models,” by Michael D. Lee and Tarrant Cummins  is the sole exposition of the model.

A simple but common type of decision requires choosing which of two alternatives has the greater (or the lesser) value on some variable of interest. Examples of these forced-choice decisions range from the everyday (e.g., deciding whether a red or a green curry will taste
better for lunch), to the moderately important (e.g., deciding whether Madrid or Rome will provide the more enjoyable holiday), to the very important (e.g., deciding whether cutting the red or the black wire is more likely to lead to the destruction of the world).

The rational approach is often viewed as a normative theory of decision. The motivation behind substantively rational decision models is to use all of the relevant available information. This can be done by evaluating the probability that Stimulus A is greater than Stimulus B, having considered whether or not the stimuli have each of the cues. If the probability is greater than .5, it is rational to choose Stimulus A; if the probability is less than .5, it is rational to choose Stimulus B; if the probability is exactly .5, it is rational to guess.

In developing their fast-and-frugal approach to modeling human decision making, Gigerenzer and Todd challenged the rational approach and emphasized procedural rationality. They argued that because human decision making processes evolved in competitive environments,
they needed to be fast, and because they evolved in changeable environments, they needed to have the robustness that comes from simplicity. In an environment in which one or a few pieces of information in a stimulus is highly predictive of the remaining pieces of information and the search for additional information is an effortful process, it is adaptive to consider only the first piece or first few pieces of information. Todd showed that many real-world stimulus domains have these sorts of information structures and developed a number of cognitive models—including the take the best (TTB) model of forced choice.

TTB by itself has shortcomings. The goal of the authors of this article is to develop and evaluate a unifying theoretical account of the TTB model and its rational alternative. The theoretical unification is achieved by viewing both as special cases of a sequential-sampling decision making process. Sequential-sampling processes have been extensively studied as models of human decision making, particularly in relation to elementary psychophysical tasks, such as judging which of two lines is longer. The basic idea is that the TTB model corresponds to the case in which the first piece of sampled evidence that favors one decision is sufficient, whereas the rational approach requires all of the available information to be sampled before a decision is made.

To overcome the frequentist problem that the validity of a cue is considered the same whether it has been accurate one hundred times or one time, the researchers proposed a Bayesian approach to estimating cue validities. The basic idea is to establish a prior distribution for
the validity of each cue and to modify these priors, using the evidence provided by a cue making correct or incorrect decisions. As a cue makes more decisions, the frequentist and Bayesian validity estimates converge toward the same value. When the available data are limited, however, the Bayesian approach is sensitive to the sample size and provides a better measure. Once validities for each cue have been estimated, the TTB decision model is straightforward. For a given pair of stimuli, the cue with the highest validity is examined. If this cue discriminates between the stimuli (i.e., one stimulus has the cue, and the other does not), the favored stimulus is chosen, and no further cues are examined. If, however, the cue does not discriminate, the cue with the next highest validity is examined. This process continues
until either a decision is made or all of the cues have been exhausted.
Figure 1 shows a particular sequential-sampling process, known as a random walk, accruing information in making a comparison between two stimuli. (Note that the log odds are just representations of probability and though they are monotonic, they help the statistical manipulations in the model work.) Each of the cues is examined, from the highest validity to the lowest, and the evidence provided by that cue is used to update the state of the random walk in favor of choosing Stimulus A or Stimulus B. If Stimulus A has the cue and Stimulus B does not, the random walk moves toward choosing A. If Stimulus B has the cue and Stimulus A does not, the random walk moves toward choosing B. If both stimuli either have or do not have the cue, the state of the random walk is unchanged. The important observation about Figure 1 is that the TTB and the RAT models correspond simply to different required levels of evidence being accrued before a decision is made. If a very small evidence threshold were set, the sequential-sampling process would choose Stimulus A, in agreement with the TTB choice. Alternatively, if a very large evidence threshold were set, the sequential sampling process would eventually choose Stimulus B (because the final log-odds are in its favor), in agreement with the RAT model. Thus, one unifying way of explaining experimental results in which people make decisions consistent with both fast-and-frugal and rational models, is that all of the people were using an evidence accrual decision process but that different people required different levels of evidence before making their decisions.

Lee and Cummins prepared experiments to evaluate the unified model. Many participants
made decisions consistent with both the TTB and the RAT models,  but to test the model formally, the researchers allowed the unified model to assume different evidence thresholds for different participants and examined the ability of these families of models to account for all the decisions made. The combined model was superior to the separate models.

Lee and Cummins suggest that it is useful to interpret the unified model as a natural extension of the fast-and-frugal approach. Effectively, what the evidence accrual model does is extend the TTB account of one-reason decision making to two-reason, three-reason, or many-reason decision making. In these extensions of the TTB model, it may be necessary for a stimulus to have two or more high-validity cues that its alternative does not, rather than just relying on the first discriminating cue. This seems reasonable: Whereas choosing a red curry may demand only one good reason, many people would prefer to buy a plane ticket to Madrid only after they had established several advantages over Rome. By assuming that people must reach some threshold of evidence before making a decision, it is possible to capture the different importance of different decisions in a natural way that begins at the fast-and-frugal and ends at
the rational. Furthermore, the evidence accrual formulation gives a sensible account of why decisions take longer in some environments than in others.

Lee, M. D.,& Cummins, T. D. R. (2004). “Evidence accumulation in decision making:  Unifying the “take the best” and the “rational” models.” Psychonomic Bulletin & Review, 11 (2), 343-352.

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