^{1}

^{1}

*Correspondence:

This article was submitted to Frontiers in Cognition, a specialty of Frontiers in Psychology.

This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and other Frontiers conditions are complied with.

How is information integrated across the attributes of an option when making risky choices? In most descriptive models of decision under risk, information about risk, and reward is combined multiplicatively (e.g., expected value; expected utility theory, Bernouli,

Here I argue that information integration in risky decision-making may be additive. Integration is additive in other domains and, if cognitive processes are shared, integration may be additive in risky choice too. Further, although valuations of risky prospects show multiplicative integration of risk and reward, integration is additive for judgments of attractiveness and, if risky decisions are based on attractiveness rather than valuation, integration in risky choice may be additive. Finally, I show that, for simple risky choices, an additive model can mimic a multiplicative model, and vice versa. Implications for the assessment of the stability of risky preference are profound – stable parameters in the multiplicative model will correspond with different stable parameters in the additive model and, further, the mode of integration itself may vary from time to time or context to context.

In a wide variety of decisions that do not involve risk, the additive model describes people's valuation of options better than the multiplicative model. For example, people average across descriptive adjectives when judging the likeability of a person (Anderson,

Buying prices, selling prices, bids, and certainty equivalents are often used to value risky options. For example, Tversky (

It is not obvious to me whether choices will be more closely linked to valuation or attractiveness judgments. To the best of my knowledge, no one has explicitly compared additive models with multiplicative models using choice data. It may be that an additive model proves successful.

The information integration process cannot be considered in isolation from other cognitive steps. For example, because the logarithmic transform turns summing into multiplying [log(

In the following modeling I show that a multiplicative and additive model can mimic one another. The valence _{i}_{i}_{i}_{i}_{i}

where _{1} + _{2} + _{3} = 1. The restricted model with _{1} + _{2} = 1/2 and _{3} = 0 is the additive model. The restricted model with _{1} = _{2} = 0 and _{3} = 1 is the multiplicative model.

To provide a complete model of choice, I use Luce's choice rule to give the probability of choosing gamble _{i}

ɸ is a free parameter which produces chance responding when ɸ = 0 and increasingly deterministic as ɸ increases. Utility is assumed to be a power function of money:

where γ is a free parameter greater than zero. When 0 < γ < 1, the utility function is concave. Subjective probability is assumed to follow the form suggested by Wu and Gonzalez (

where β is a free parameter greater than zero. When 0 < β < 1, the subjective probability function has an inverse-S-shape.

To illustrate how additive and multiplicative models can mimic one another, I generated data from a base model with γ = 1/2, β = 2/3, ɸ = 1, and _{1} = _{2} = _{3} = 1/3. The exact parameter values are not crucial to the argument. These values are loosely based on the well established findings of a concave utility function, an inverse-S-shaped probability weighting function, and probability matching. These particular

The choice set used is the set of all possible choices of the form “_{1} chance of _{1} otherwise nothing” or “_{2} chance of _{2} otherwise nothing” that can be constructed using probabilities .1, 0.3, 0.5, 0.7, and 0.9 and amounts 20, 40, 60, 80, and 100. (In modeling, amounts were scaled for convenience by dividing by 100 so that amounts lay on the same interval, roughly, as probabilities.) Raw data take the form of the probability of choosing Gamble 1 according to the base model.

Figure _{3} = 1 − _{1} − _{2} (without loss of generality), the points in the horizontal plane represent all possible mixtures of information integration. At the leftmost corner of the plot where _{1} = _{2} = 0 and _{3} = 1 (i.e., a purely multiplicative model) the fit is somewhat compromised. The other two corners of the surface represent a model where only probability is weighted or where only amount is weighted, and are also similarly badly fitting. But for a ridge in the middle of the surface (the area colored red), quite large variation in the information integration has a small effect. Figure _{3} [I constrained _{1} = _{2} = (1 − _{3})/2 here]. There is a ridge of roughly equal likelihood which passes through the base model at ɸ = 1, _{3} = 1/3 where the log likelihood of the data given the model is −200.972. At one end, where _{3} = 0 and the model is completely additive, the log likelihood is −201.060. At the other end, where _{3} = 1 and the model is completely multiplicative, the log likelihood is −201.308. This means that, whatever mode of integration one chooses for the model, it can be completely compensated for by varying the degree of determinism in the choice rule. The more additive the model, the higher the value of ɸ needed to compensate. This result holds for a purely additive base model and a purely multiplicative base model. In short, with choices between these simple gambles, one cannot discriminate between additive and multiplicative models of decision under risk.

This special issue is about the stability, or otherwise, of risky preferences across time or context. To assess the stability of preferences, one can identify and fit a model of risky choice to data from two or more times or contexts and then compare parameters across times or contexts (see Zeisberger et al.,

In closing, I note that the ability of additive and multiplicative models to mimic one another offers an explanation for the success of the decision by sampling model I have proposed elsewhere (Stewart et al.,