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Edited by: Adam Kepecs, Cold Spring Harbor Laboratory, USA

Reviewed by: O'Dhaniel A. Mullette-Gillman, National University of Singapore, Singapore; Kenway Louie, New York University, USA

*Correspondence: Benjamin Y. Hayden,

This article was submitted to Decision Neuroscience, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In his accompanying commentary on our recent paper, Dr. Padoa-Schioppa identifies two putative errors in our manuscript (Piantadosi and Hayden,

First, he argues that some commodities are inherently incommensurate, such as different juice flavors (“there is no parametric dimension along which two flavors can be assigned a scalar value”). This argument seems appealing because it is difficult to think of a single algorithmic function that would describe a juice flavor as a single number. So in a colloquial sense, juices could indeed by called incommensurate.

However, most models of choice assume, either tacitly or explicitly, that there is an intermediate stage during which each dimension is represented in a scalar manner and may there be deformed. A famous example is prospect theory (Kahneman and Tversky,

These idea of an intermediate stage is a critical part of several famous economic models (Bernoulli,

Second, Dr. Padoa-Schioppa is concerned that there are well-defined value comparisons to which our methods do not apply. We clearly acknowledged this point in our original manuscript. Our formalization provides a way to recognize utility functions it does apply to; this is a strength of our approach.

However, while the example he gives was not discussed in our manuscript, it is straightforwardly covered using the approach we advocate. Indeed, the set of contexts to which our arguments apply is somewhat larger than we stated in the original article. Specifically, it applies to any context in which the choice can be modeled by a utility equation and then that equation can be rearranged to avoid the utility stage.

Dr. Padoa-Schioppa discusses a choice between gambles with probability (P_{1}) of reward (R_{1}) and fixed costs (C_{1}), where choice is determined by:

Following a method much like the one we presented (that is, first computing relative differences and then rearranging the terms), this choice is equivalent to:

where the uppercase letters are the average value in each dimension and the lowercase are half the difference.

To translate into prose, the decision-maker does the essentially same thing as in the examples used in our paper: she computes a normalization term that depends on average values, and then computes the ratio of the dimension-free relative differences for gains, and asks whether that ratio is greater or less than the relative difference for costs.

Thus, this example thus does not challenge our arguments, but instead enhances them. Moreover, it endorses our bigger (and ultimately very simple) point: given the power and flexibility of algebra, it is often possible to create process models that have no utility stage but make identical predictions to ones that do through a simple rearrangement of terms.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by a R01 (NIDA 037229) to BH.