**A. Equivalence.** For each observer, we estimated 12 eccentricity-contrast pairs

that the observer judged to be equally discriminable. We compare observers' judgments to actual discrimination performance for that observer measured in the calibration task. Suppose, for example, that an observer judges

, that is, a low contrast target at

eccentricity is as discriminable as a medium contrast target at

. Based on calibration performance, we estimate that probability correct for

was 0.93 while that for

was 0.61. We plot these probabilities on the vertical colored axes for L (red) and M (blue) and connect them by a straight line with an arrow at the end corresponding to the eccentricity-contrast pair whose eccentricity varied in the staircase procedure. If the line segment is horizontal then the observer correctly judged the pairs to be equally discriminable. If the line segment is significantly slanted, the observer is in error. We plotted each of the 12 pairs judged equally discriminable (four for each possible pair of contrasts) in this way. The labels L, M, and H, or the colors red, blue, gray, respectively denote low, medium, and high contrasts. Black denotes an insignificant probability difference while magenta denotes a significant probability difference. The overall significance level is .05, Bonferroni corrected for 12 conditions (that is, each test had a size of .0042 = .05/12). The observers' judgments exhibit large, patterned failures.

**B. Transitivity.** Each panel corresponds to one observer. The three axes of the inverse Y configuration are for the three contrasts. L, M, H denotes low, medium, and high contrasts, respectively. On each axis, the distance of a point to the center represents the eccentricity of a target, ranging from

to

on a log scale (see inset). Lines connect eccentricity-contrast pairs of subjective indifference. For each observer, we started from

, used the low-to-medium equivalence transformation to locate the eccentricity

for

, and then used the medium-to-high mapping to move to the next and so on. If the low-to-medium, medium-to-high, and high-to-low mappings satisfy transitivity, the fourth point should fall on the starting point. A gap between them implies intransitivity. Observers that significantly failed the transitivity test are plotted in blue with six mapping lines. The observer that passed the transitivity test is plotted in black ending at the third mapping line. Notice that all the intransitive mapping lines spiral outward from the center. See text.

**C. Dominance.** Percentage of errors for each observer and their median in the equi-contrast trials (top) and equi-eccentricity trials (bottom). Error bar denotes the 95% confidence interval. Dashed lines mark the chance levels.

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