Accumulation is late and brief in preferential choice.

Preferential choices are often explained using models within the evidence accumulation framework: value drives the drift rate at which evidence is accumulated until a threshold is reached and an option is chosen. Although rarely stated explicitly, almost all such models assume that decision makers have knowledge at the onset of the choice of all available attributes and options. In reality however, choice information is viewed 8 piece-by-piece, and is often not completely acquired until late in the choice, if at all. Across four eye-tracking experiments, we show that whether the information was acquired early or late is irrelevant in predicting choice: 10 all that matters is whether or not it was acquired at all. Models with potential alternative assumptions were 11 posited and tested, such as 1) accumulation of instantaneously available information or 2) running estimates 12 as information is acquired. These provided poor ﬁts to the data. We are forced to conclude that participants either are clairvoyant, accumulating using information before they have looked at it, or delay accumulating 14 evidence until very late in the choice, so late that the majority of choice time is not time in which evidence is ac- 15 cumulated. Thus, although the evidence accumulation framework may still be useful in measurement models, 16 it cannot account for the details of the processes involved in decision making. 17 In explain which characterises 20 the decision tracking accuracy (their gaze location could be measured for less 305 than 68% of the time averaged across all trials). The stimuli were two gambles with three equally likely outcomes, 306 presented at the top and bottom of the screen. The three payouts of each option were presented side-by-side as 307 white text within a solid grey circle. There were two within-subject conditions, counterbalanced for order. In the 308 commensurate condition, there were 42 trials, in a third of trials all attributes were displayed in pounds, yen and “Q” 309 respectively. In the incommensurate condition, on every trial, the three attributes of each option were presented in the 310 three difference currencies. Here, we only analyse the commensurate condition. Participants were told the exchange 311 rates for pounds, yen and Q to dollars. Here we analysed the equivalent dollar values. The eye-tracker was calibrated 312 at the beginning and every 21 trials. Participants pressed the up key if they preferred the top lottery, and the down key 313 if they preferred the bottom lottery. Areas of interested were deﬁned horizontal distance of 320 pixels, and a vertical 314 distance of 340 pixels apart on the screen. 315

the currently attended information. However, this dependency is incorporated as a bias towards the currently attended information, and that the drift rate is still inherently relative (based on a summary difference between all values). That is, it is assumed at all times, that the value of all other information is known (KAO) so that relative drift rates can still 91 be calculated from onset.

92
The question this paper asks is, how well do evidence accumulation models perform if we constrain their assump-93 tions and do not allow information to be used before there is any way a decision maker could know it? That is, 94 once we have ruled out clairvoyance, what can we conclude about evidence accumulation? We think that people are The simplest and most common existing assumption is that the drift rate is defined by the difference in value (or 110 subjective ratings) between the two options 1 . This is the assumption of accumulation of complete information from 111 the beginning of the trial and of course implies KAO. We use this as our baseline. In multi-attribute choices, we 112 assume that an object's value is represented as the mean value across its attributes.

113
To provide a concrete example across each of the following assumptions, we use the hypothetical trial described 114 in Figure 2 and accompanying diagram. On this trial, participants had to choose between two options each with 3 115 attributes, which were 3 equally likely monetary outcomes (labelled a, b and c for Option 1 and x, y and z for Option 116 2). Formally, the value difference on this trial is given by where V i indicates the value of attribute i. Therefore, on this trial ∆ VALUE equals (21+78+84)/3−(85+76+32)/3 = Here, we implement this by multiplying each value by the number of times it was fixated in a trial and divide by the 125 total number of fixations to each item. We then take the difference between them.

126
For the example trial in Figure 2, this is would be formally represented as where f i is the frequency of fixations to attribute i across the trial and N = i∈a,b,c f i + j∈x,y,z f j is the to-128 tal number of fixations. Therefore, on the example trial ∆ WEIGHTED VALUE = 1/6 ((1 × 21 + 2 × 78 + 1 × 84) −

129
(1 × 85 + 1 × 76 + 0 × 32)) = 16.67 (because 78 is fixated twice and 32 is never fixated). Here, we implement this using the prior v which is set to the mean of all values across the experiment (using 140 the mean is sufficient instead of a uniform distribution because of the linear combination of values). In our Figure 2  in Appendix A we considered a model where v was learned over previous trials-this made no difference.) As each 144 attribute is fixated, its value is updated from the prior 50, to the true value and the drift rate is recalculated. Thus for 145 each fixation we calculate the estimated value difference and then average this over fixations: where [V i ] n is the participant's understanding of value for attribute i on fixation n and N is the total number of fixations. Therefore, on this example trial, to have a value v. Note that this model excludes information about when in the trial information was attended.
where N is the final fixation on that trial. Therefore, for the example trial ∆ FINAL VALUE would be equal to Our last model is a more general test of whether it matters when an attribute value was acquired. Therefore, we 161 also include time known terms for each attribute (a, b, c, x, y, z). Formally, this is defined as where p i equals the proportion of fixations for which the participant knew the attribute value. of option values 10,29 -that more attention to an alternative increases the probability of choosing that alternative.

170
Therefore we calculate the proportion of time where attention is directed to each option.

171
Here, attentional bias is defined as where T i is the amount of time in milliseconds the participant fixated within the area of interest around attribute i on 173 each trial.

174
We estimate an additional version of each of the above models that includes the attentional bias A as a predictor. The most striking result is that, for every experiment, the final value model explains more variance than the updat-187 ing value model. That is, model fits worsen when they assume that the drift rate value is updated as new information is 188 learned. This conclusion is supported by the fact that the final value plus history model accounts for no more variance 189 than the final value (alone) model. In other words, adding in information about when an attribute value was acquired 190 does not improve model performance. Furthermore, using this information to constrain accumulation models actually 191 results in a substantially worse fit to the data.

192
In three of the four experiments, the final value model explains more variance than the fixation weighted model.

193
That is, assuming that evidence is only accumulated based upon the currently attended information makes the model Here, we identify that a key assumption of evidence accumulation models is inherently impossible and wrong: that 211 decision makers have perfect knowledge at the onset of a choice-KAO. We examine a number of alternative evidence 212 accumulation models which remove this assumption. These alternatives retain the attractive properties of evidence 213 accumulation models, including neural plausibility and the ability to predict choices and reaction times simultaneously.

214
However, the results across four experiments show that the best performing models are ones which ignore the time 215 when information is acquired.

216
This has serious implications for accumulation-based models of value-based choice 7,9,[12][13][14][15]31 . We suggest that 217 accumulation models, although they fit choice and reaction time data well, are fundamentally missing something. 218 We are forced to a strong conclusion: If adding knowledge about when information becomes available to evidence 219 accumulation models makes their fit worse, we must conclude that, if there is an accumulation process, it does not 220 begin until about the time the final fixation is made. Effectively, our result confines the accumulation process to a 221 small fraction of time at the end of a choice, because allowing it to start any earlier results in significantly poorer 222 choice predictions. (Appendix B considers further how early, exactly, the accumulation process could start.) While the 223 starting point for many avenues of research is an accumulation model, we should, perhaps, be looking more stringently 224 at testing those underlying assumptions.

225
There are additional implications for research using process tracing methods 32 . A great deal of work has focused 226 upon fitting models of choice to reaction time data, information search, and eye-tracking. However, these results 227 suggest that this process tracing work may have been based upon a fundamental misunderstanding of the underlying 228 process itself. For example, how are we to interpret findings testing the effect of choice difficulty upon drift rates 229 and reaction times in light of the suggestion that evidence is not being accumulated over the majority of the decision 230 time? This is not to say that process tracing is not valuable, but that research on properties of that process is very 231 often structured around a particular model (or class of model). Such work is therefore inherently reliant upon the 232 assumptions underlying that model, and when those assumptions prove to be faulty, the conclusions of process tracing 233 efforts need to be revisited. A preferential approach might be to increase focus on model free tests, and independent 234 characterisations of process data.

235
The conclusion from this paper is that it is perhaps futile to fit accumulator models like multialternative decision  These were comprised of five types of snack: crisps, fruit candy, sweet carbonated drinks, health and sports drinks, 262 and chocolate. This experiment was split into two parts. In the first, the participant rated the desirability of 50 snack 263 items on a 1-9 scale. In the second, the 50 stimuli were paired to create 100 binary choice trials. Choice pairs were 264 created such that the rating difference between the two items was 3 or less, and so that an individual snack item was 265 not present in more than 5 trials. At the end of the experiment one trial was randomly selected and the subject was 266 given the item they chose.

Posters 44
268 Useable data was collected from 53 participants. The experiment was displayed on a widescreen monitor (1920 269 x 1080 resolution, 60Hz refresh). Additional data was collected from 13 participants but 12 were excluded due to a 270 programming error and one because their gaze location could be measured for less than 70% of the time across all 271 trials. The stimuli were chosen from the International Affective Picture System 30 . The pictures were all positive in 272 affect (average ratings between 5=neutral and 7=mildly positive for both males and females) and had differences in 273 value ratings of no more than 1.5 between male and female raters. After visual inspection, a further 7 images were 274 removed for containing sexual images and 32 images were removed because they had a portrait aspect ratio. The 200 275 stimuli for each participant were randomly sampled without replacement from the 253 pictures that met these criteria.

276
All participants completed binary choice and strength-of-preference tasks in a counterbalanced order. Here, we 277 only analyse the binary choice data. Two landscape pictures (each 514 × 384px) were displayed side by side following 278 a fixation cross. The response scale was presented horizontally centered, below the stimuli. For the binary choice 279 task, two labels ("Option A" and "Option B") were shown underneath the appropriate stimuli. The current choice was 280 signified by a red, square marker (30 × 30px) above the label. The marker was initially centered equidistant between 281 the two images. To respond, the participants had to press the left mouse button. Reaction times were measured from 282 the start of the trial to the mouse click onset. A blank, black screen was displayed for 500ms between each trial.
Finally, participants had to rate their overall liking for each picture on a Likert scale, vertically displayed to the left of 284 each image. The eye-tracker was recalibrated at the beginning of each condition and then after every 25 trials.

285
Lottery 45 286 Useable data was collected from 54 participants. The data of an additional 5 subjects was excluded: 1 because they 287 failed to complete the task within a reasonable time frame and withdrew, and 4 because of poor quality eye tracking 288 data. The stimuli were gambles with three equally likely outcomes. Because the outcomes were equally likely, no 289 probabilities or likelihoods were displayed during the trials themselves. For each trial, two gambles were presented, 290 one on the top and one on the bottom of the screen. The three payouts of a single gamble were presented in horizontal 291 alignment and were displayed as white text within a solid grey circle. This was done to reduce the discriminability of 292 the numbers in peripheral vision, so that subjects had to directly fixate the number to determine its value. Each gamble 293 consisted of three possible outcomes which were always one low value (10-30), one medium value (40-60), and one 294 high value (70-90). The specific values were randomly drawn on each trial.

295
The main task consisted of 100 trials. Trials were presented in a random order, with trials from the different Usable data was collected from 46 participants from the California Institute of Technology participation pool. 303 Additional data had been collected from 14 participants but four were excluded due to incomplete data (they ran over 304 the experiment slot) and 10 because of poor eye tracking accuracy (their gaze location could be measured for less 305 than 68% of the time averaged across all trials). The stimuli were two gambles with three equally likely outcomes, 306 presented at the top and bottom of the screen. The three payouts of each option were presented side-by-side as 307 white text within a solid grey circle. There were two within-subject conditions, counterbalanced for order. In the 308 commensurate condition, there were 42 trials, in a third of trials all attributes were displayed in pounds, yen and "Q" 309 respectively. In the incommensurate condition, on every trial, the three attributes of each option were presented in the 310 three difference currencies. Here, we only analyse the commensurate condition. Participants were told the exchange 311 rates for pounds, yen and Q to dollars. Here we analysed the equivalent dollar values. The eye-tracker was calibrated 312 at the beginning and every 21 trials. Participants pressed the up key if they preferred the top lottery, and the down key 313 if they preferred the bottom lottery. Areas of interested were defined horizontal distance of 320 pixels, and a vertical 314 distance of 340 pixels apart on the screen.    In the updating value model, we assume that participants use the experiment-average attribute value as 412 a static prior for all trials. However, this assumption obviously assumes some clairvoyance at the beginning of the 413 experiment. Therefore, here we also include a model which assumes that participants use as a prior the updating mean 414 of all the values they have seen on earlier trials in the experiment. Figure A1 shows that adopting this more realistic 415 assumption makes almost no difference to model fit. The fixation weighted value difference and the updating value difference model both assume that people accu-418 mulate attribute value information as it is attended. The final value difference model assumes that accumulation is 419 deferred until all of the information that will be acquired has been acquired. Finding that this final value difference 420 model fits better allows us to reject the idea that accumulation begins as information is first acquired.

421
Here we explore the possibility that information acquisition begins later than the first fixations, but begins before 422 the fixation upon which all of the information that will be acquired is acquired. To explore the issue of when accumu-423 lation starts directly, we split each trial into early and late fixations and model separately the early and late fixations 424 with the updating value difference model. 425 Figure B1 splits each trial into early and late fixations. We took the late fixations and ran the updating value 426 difference model as if accumulation only occurred during the late fixations. The "after-split" lines in Figure B1 427 show how model fit changes accumulation is assumed to start earlier and earlier (from right to left). Assuming that 428 accumulation starts earlier than the last fixations makes the model fits slightly worse, or makes little difference. We    Table B1 shows that our estimates for the onset of accumulation are very different from those estimated by the 446 drift diffusion model-we estimate that people begin accumulating much later than is estimated by t 0 in the DDM.

447
Taking the food experiment as an example, we have a mean reaction time of 2210ms and an average fixation duration 448 of 242ms. Based on the drift diffusion models fits, we have an estimated t 0 of 532ms, which is the estimated time 449 at which accumulation begins. Subtracting this from the mean reaction time, we have an estimate of 1678ms as the