**Abstract**:

**Abstract**
**Eye movements function to bring detailed information onto the high-resolution region of the retina. Previous research has shown that human observers select fixation points that maximize information acquisition and minimize target location uncertainty. In this study, we ask whether human observers choose the saccade endpoint that maximizes gain when there are explicit rewards associated with correctly detecting the target. Observers performed an 8-alternative forced-choice detection task for a contrast-defined target in noise. After a single saccade, observers indicated the target location. Each potential target location had an associated reward that was known to the observer. In some conditions, the reward at one location was higher than at the other locations. We compared human saccade endpoints to those of an ideal observer that maximizes expected gain given the respective human observer's visibility map, i.e., d′ for target detection as a function of retinal location. Varying the location of the highest reward had a significant effect on human observers' distribution of saccade endpoints. Both human and ideal observers show a high density of saccades made toward the highest rewarded and actual target locations. But humans' overall spatial distributions of saccade endpoints differed significantly from the ideal observer as they made a greater number of saccade to locations far from the highest rewarded and actual target locations. Suboptimal choice of saccade endpoint, possibly in combination with suboptimal integration of information across saccades, had a significant effect on human observers' ability to correctly detect the target and maximize gain.**

*f*noise. They found that the number of saccades and distribution of saccade endpoints were consistent with the predictions of an optimal observer that chooses each endpoint so as to reduce target-location uncertainty.

^{2}. Eye position was monitored using an SR Research Eyelink1000 tower-mount eye tracker with a sampling rate of 1000 Hz, controlled using the Eyelink Toolbox Matlab interface (Cornelissen, Peters, & Palmer, 2002).

*SD*= 0.66°), masked by additive Gaussian white noise covering the same 2° diameter patch. Mean luminance was 40 cd/m

^{2}. Signal contrast was at most 50%. Noise

*SD*was 16.7% contrast, and noise values were clipped at three

*SD*s above and below the mean, so that luminance values always lay within the display gamut. The observer sat at a distance of 42 cm from the monitor. Head position was constrained by a chin rest firmly attached to the eye tracker. Each block of trials began with a nine-point spatial calibration of the eye tracker.

*c*at retinal location

*r*,

*θ*(in polar coordinates relative to the fovea). We assume a Weibull psychometric function where

*p*is the probability of a correct response. Several studies have shown that the steepness parameter,

*β*, changes as a function of eccentricity (Geisler, Perry, & Najemnik, 2006; Michel & Geisler, 2011; Najemnik & Geisler, 2005). To determine the effect of both eccentricity (

*r*) and direction (

*θ*) on

*β*, we fit Weibull psychometric functions independently to the data for each of the 25 stimulus locations (each with a single threshold parameter

*α*and steepness parameter

*β*). We regressed, for each subject, the fit

*β*s onto

*r*and

*θ*. Regression slopes were not significantly different from zero. The fit

*β*s are shown in Figure 2 with the results of the regression of

*β*onto eccentricity. The steepness parameters did not vary systematically with eccentricity or direction. Therefore, we treat

*β*in Equation 1 as constant across the visual field.

*a*and

_{i}*β*are parameters fit to the data. The sin

*θ*term allows

*g*to account for the vertical meridian asymmetry in which stimuli are more detectable at the bottom than at the top of the visual field (Liu, Heeger, & Carrasco, 2006; Previc, 1990). The sin2

*θ*term allows for the vertical/horizontal asymmetry in which stimuli are more detectable along the horizontal than the vertical meridian (Carrasco, Talgar, & Cameron, 2001). The exponential form allowed

*α*to model the increase of threshold in the periphery.

*p*values from Equation 1 were converted to

*d*′ (Wickens, 2002): where Φ

^{−1}is the inverse of the cumulative standard normal distribution. Visibility maps for each subject are shown in Figure 3A for detectability at the stimulus contrast used in the main experiment: Michelson contrast = 16%.

*β*across locations). We determined the magnitude of the error by calculating the median absolute deviation of

*d*′, for a signal contrast of 16%, as a function of eccentricity across the 1,000 fits for each subject. The result is shown in Figure 3B. We also calculated the signed deviations of the bootstrapped fits from the empirically measured visibility maps (Figure 3A) by taking the median of the difference between the bootstrapped

*d*′ values and the

*d*′ value of the visibility map at each horizontal and vertical coordinate. The signed error of the fits does not show a consistent spatial pattern across subjects and the median absolute deviations of the fit (Figure 3B) are relatively small, suggesting that our model of the detectability map will suffice for guiding our ideal observer.

^{2}. A saccade was deemed complete when either velocity or acceleration fell below threshold. The ideal-observer model we use requires pre- and post-saccadic noise to be uncorrelated. Thus, once saccade initiation was detected, new random noise fields were displayed in each ring. The target was displayed for 200 ms following termination of the saccade, so that subjects were prevented from gaining target information by making additional eye movements. When the target disappeared, leaving the eight noise fields, a question mark appeared at the center of the screen. The observer indicated the target position by fixating the corresponding ring.

*i*, the ideal observer computes the posterior probability of the target occurring at each location by Bayes' rule (Coombs, Dawes, & Tversky, 1970). In our case (see Supplement for derivation), this simplifies to where

*n*is the number of potential target locations,

*p*(

*i*) is the prior probability of the target occurring at position

*i*, and

**w**

_{1}= (

*w*

_{1,1}, ···,

*w*

_{n}_{,1}) is the vector of noisy template responses from each potential target location collected during the the first fixation (

*F*

_{1}, the central fixation), each with corresponding detectability value

*n*= 8, and

*p*(

*i*) = 1/8, and a reward

*V*is awarded for correctly detecting the target at each position

_{i}*i*. The ideal observer selects the target position,

*I*, with maximal expected gain:

*p*(

*c*|

*i*,

*F*

_{2},

**w**) is the probability of being correct having chosen position

_{1}*i*, following a saccade to location

*F*

_{2}, and given the template responses

**w**from the first fixation. The derivation of

_{1}**w**). It computes updated posterior probabilities where the values of

_{2}*I*, for which expected gain is maximized:

*SD*= 2.6°) to regions near potential target locations. The highest density of saccades is found near the potential target positions with the highest rewards and the actual target location (suggesting that both rewards and target information are being factored into the choice of saccade endpoint, Eckstein, Beutter, & Stone, 2001; Findlay, 1997).

*r*between the 289 bin counts for reward condition

_{ij}*i*and the corresponding bin counts for condition

*j*. We computed the average

*r̄*of the correlation coefficients corresponding to the 10 distinct pairs of reward conditions (

*ij*).

*r̄*across distribution pairs as above. We repeated this procedure 1,000 times to generate a distribution of

_{perm}*r̄*values for simulated datasets that shared the pooled spatial distribution of saccades of our dataset, but no effect of reward conditions. The mean correlation coefficients from the actual data (

_{perm}*r̄*) were always lower than the lowest mean correlation coefficient from the permuted datasets. To derive a

*p*value, we calculated the mean and

*SD*of the

*r̄*values, and calculated a

_{perm}*p*value assuming the

*r̄*values were normally distributed. All

_{perm}*r̄*values are significant (

*p*< 0.001), i.e., reward conditions had a significant effect on the choice of saccade endpoint.

*SD*= 2.58°) than the human observers. (Note that each ideal observer is based on a visibility map, and thus there is one such ideal observer per human observer in our study.) Most of the ideal observers' saccade endpoints were near the target position with the highest reward or the actual target location (Figure 6). The ideal observers' saccade endpoints were not as diffusely distributed as the human observers'. Fewer saccades were made to target positions far from the positions with the highest reward and that containing the target, as compared to the human observers.

*t*tests indicate that human and ideal-observer saccade magnitudes differ significantly (all eight

*p*s < 0.001, two tailed) for every subject and both unequal- and equal-rewards conditions. In seven out of eight comparisons, the ideal observer makes shorter saccades.

*r*, of the saccade. Thus we define saccade endpoint variance,

*σ*

^{2}, as

*θ*. In the direction of the saccade, variance is lowest along the horizontal meridian and lower for upward than downward saccades. Perpendicular to the saccade, variance is lowest along the horizontal and vertical meridians and highest at the oblique angles. We model these effects, following van Beers, by setting where

*σ*and

_{r}*σ*are the radial and transverse standard deviations of saccade endpoint, respectively. The parameters

_{t}*a*were adjusted to fit the across-subjects average data reported by van Beers (2007) leading to the variability maps and example error ellipses shown in Figure 8.

_{i}*F*

_{1}(

*x*) and the empirical distribution

*F*

_{2}(

*x*):

*K*total bins. The test statistic is where (

*x*,

_{k}*y*) are the horizontal and vertical coordinates of the

_{k}*k*

^{th}bin, respectively, and the estimated bin probabilities are

*γ*(

_{i}*x*,

*y*). The test statistic Ψ is not invariant with respect to the point in the spatial array chosen as the origin. To remove this dependency, Ψ is computed starting from each of the four corners of the array, and the resulting four values of Ψ are averaged. We computed the

*γ*values using 2° × 2° bins.

_{i}*p*value for the human observer's Ψ is the probability of a sample from the bootstrapped distribution having that value of Ψ or greater. There is a significant difference between human and ideal-observer saccade endpoint distributions in all conditions (all

*p*s < 0.01).

*Q*quantile map is derived by fitting the eight-parameter psychometric function,

_{LOW}*p*(

*r*,

*θ*) (Equations 1−3), assuming

*β*to be constant across locations, to the 0.0275 quantile values. The

*Q*map is derived by fitting the 0.975 quantile values. The quantile maps for each subject are shown in Figure 9.

_{HIGH}*d*′ in the peripheral regions of the

*Q*maps result in longer saccades (mean = 10.36°,

_{LOW}*SD*= 1.83°) made in the direction of the highest rewarded position and actual target location. Higher peripheral values of

*d*′ in the

*Q*maps result in shorter saccades (mean across subjects = 4.79°,

_{HIGH}*SD*= 1.68°) made similarly into regions near the highest rewarded position and actual target location. The human observer's saccade magnitudes (mean = 9.94°,

*SD*= 2.6°) lie between the values for the low- and high-quantile-map simulations.

*Q*and

_{LOW}*Q*visibility maps. All subjects' average saccade magnitudes in the unequal-rewards conditions are significantly less than those of the low-quantile ideal observer. This suggests that their higher saccade magnitudes (as shown in Figure 7) can be attributed to error in the visibility map measurement. All subjects' saccade magnitudes in the equal-rewards condition are greater than those for either low or high quantile simulations suggesting that this result is not attributable to error in the visibility maps.

_{HIGH}*z*test. We did this by averaging the data across subjects and unequal-rewards conditions (by pooling target locations that were in identical angular positions relative to the position of the highest reward). All differences are significant (

*p*< 0.001).

*E*, was calculated for each subject as follows: i.e., the ratio of the average gain per trial for the human (

*H*) and ideal (

*I*) observers. Efficiency for each subject is shown in Figure 14A. The standard error of each subject's mean total efficiency was derived by bootstrapping. The

*z*tests show all total efficiencies to be significantly less than one (

*p*< 0.001). Note, however, that total efficiency is quite high, with three out of four subjects earning above 75% of the expected gain of the ideal observer.

*E*for the ideal integrator by substituting its average gain per trial for that of the human observer in the above equation. By comparing the ideal integrator's relative efficiency with that of the respective human observer, we can get some idea of the relative contributions of saccade endpoint and information integration on efficiency. The results are shown in Figure 14B.

_{rel}*E*less than one indicate suboptimal behavior.

_{rel}*E*is significantly less than one for all subjects (

_{rel}*p*< 0.001). Negative values of

*E*indicate that the human observer's choice of saccade endpoint resulted in gain lower than that of the random searcher. The

_{rel}*z*tests, with the

*SE*derived by bootstrapping, show

*E*for Subjects 1 and 2 to be significantly less than zero (

_{rel}*p*< 0.01). Subject 3 and 4's

*E*values are not significantly different from zero.

_{rel}*E*with that of the corresponding ideal integrator. The ideal integrator's

_{rel}*E*is higher for both subjects indicating that Subjects 1 and 2's negative

_{rel}*E*values are due, at least in part, to suboptimal integration. That the ideal integrator's

_{rel}*E*is lower than zero for all subjects suggests that the human observers' strategy for selecting an endpoint is worse than that of the random searcher's in terms of its effect on efficiency. We conclude that subjects' suboptimal choice of saccade endpoint, in combination with suboptimal integration across saccades, resulted in a significant loss of rewards.

_{rel}*d*′ in the periphery and, consequently, higher probability correct for target detection on the part of the ideal observer. The efficiency of the human observer, in this case, would be as low or lower than that shown in Figure 14. On the other hand, error in the measurement of the visibility map parameters tending toward the higher quantiles of the parameter distributions results in lower probability correct on the part of the ideal observer. Note that performance of the human observers, as shown in Figure 12, is relatively close to that of the ideal observers, particularly when the target appears at the highest rewarded, or adjacent, locations. A decrease in performance on the part of the ideal observer quickly leads to the situation in which the human observers' performance exceeds that of the respective ideal observer. This indeed results in higher efficiency for the human observer. However, performance of the human observer in excess of that of the ideal observer is indicative, not of high efficiency, but of either (a) error in the visibility map, such that it does not accurately represent the human observer's underlying response to the stimuli, or (b) error in the cost function that is optimized by the ideal observer. We address the possibility that the human observers may employ a cost function entirely different from that of the ideal observer in the Discussion. We conclude that the visibility maps, as measured, are more accurate than those that may be derived from a higher quantile of the parameter distribution and that efficiency, as measured, is not substantially affected by visibility map measurement error.

*d*′, of the target at each potential saccade endpoint. The observer may under- or overestimate

*d*′ by, for example, over- or underestimating the variance of the noise, respectively, and thus mistake variability in the noise images at a nontarget locations for the signal. An overestimation of

*d*′ across the visual field could result in higher saccade magnitudes, as found with the human observers, as, in the observer's mind, the eye may travel further from potential target locations with less risk of nondetection at distant locations. We simulated a subideal observer in our task that generates template responses to the stimulus at each potential target location using the actual

*d*′s from the respective observer's visibility map. It then overestimates

*d*′ at every location by a factor of two in the process of calculating posterior probabilities and the probability of a correct response at each potential saccade endpoint. The result, as predicted, is an increase in saccade magnitude similar to that of the human observers. However, overestimation of

*d*′ also resulted in an increase in the proportion of saccades made near to the actual target location regardless of the location of the highest reward. This is unlike the pattern we found with the human observers, who tended to make the highest density of saccades to regions near the highest rewarded position.

*I*, with the highest expected gain for choosing it as the location of the target (Equation 6): and then made a saccade to that location (

*I*).

*consideration set*(Roberts & Lattin, 1991) of possible saccade endpoints that includes only the potential target locations. A target location is then chosen given the first exposure to it at the initial point of fixation and the choice is confirmed by making a saccade to the chosen location. The strategy is risky in that if the initial choice turns out to be wrong, the saccade, and second exposure to the target, will have yielded little useful information about the target's actual location. The misplaced saccade will have placed most of the potential target locations far from fixation. The second exposure to the target will likely produce a low internal response. Consequently, the posterior probabilities of target occurrence at all locations will be nearly equal and the expected gain of any subsequent choice of target location will be relatively low. This is a problem that the present ideal observer inherently avoids by choosing saccades that maximize expected gain summed across all potential target locations. However, it may be that the timing constraints of our task make it impossible for a human observer to perform the calculation of the optimal saccade endpoint, thus making this risky, suboptimal strategy more feasible than the ideal observer's. Given limited time to gather information and make a decision, the best strategy may be to choose a saccade endpoint that maximizes expected gain at the position where it is already highest rather than maximizing gain across all locations. The current paradigm does not offer a fair comparison between the risky strategy described above and that of the current ideal observer as the former only saccades to potential target locations and the latter is free to saccade to any location. Thus, we leave it to future research to compare the two strategies in terms of their relative effectiveness in harvesting gains under various timing constraints.

*d*′ as a function of eccentricity in the main experiment (in which there were rewards and saccades were made) may have differed from that measured in the preliminary experiment. The paradigm could be improved by making the task used to measure the visibility maps more similar to the main experiment by employing rewards and allowing saccades.

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