### Extended Data Figure 7-1

Parameter recovery analysis to evaluate the weight-estimation method. In experiment 3, weights for direct and intermediate reaches were highly uncorrelated. To examine whether this finding arose because the weights were actually uncorrelated or because the method used to estimate the weights was unreliable, we performed a parameter recovery analysis. We observed good recovery of our simulation weights (compare the median recovered weight to the simulated weight, black open circles), for both (*A*) direct reaches, and (*B*) intermediate reaches. The dashed gray line reflects the line y = x; all median recovered weight values fall close to this line. *C*, We also compared the recovered weights from direct and intermediate reaches to each other on each simulation repetition, where in each simulation the same weight value was used to generate both direct and intermediate reaches. Note that these points cluster also along the line y = x, suggesting that if we see a lack of relationship between the weights estimated on direct and intermediate reaches across participants, it is likely because these weights were indeed uncorrelated rather than because of an artifact introduced by the weight-estimation procedure.

For the parameter recovery analysis, we simulated the ability to accurately recover the underlying weights associated with frequency and reward biases. Biases were simulated in reach preference (direct reaches, *D*_{choice}) and reach direction (intermediate reaches, *I*_{bias}). These biases were determined by choosing a relative weighting of likelihood and reward. On each simulated trial, we calculated the preference for choosing the more frequent target based on the current trial’s frequency ratio and the reward ratio separately (e.g., if the frequency ratio is 2:1, the preference for choosing the more frequent target is 0.67). We calculated the distance this preference was from equal preference (i.e., 0.5), and scaled that distance by the predetermined frequency or reward weighting; this yielded a frequency-biased and reward-biased preference for each target:
Target1_preferencefrequency=(frequencyTarget1frequencyTarget1+frequencyTarget2−0.5)*frequency_weight+0.5
Target1_preferencereward=(rewardTarget1rewardTarget1+rewardTarget2−0.5)*reward_weight+0.5.

Then, the objective utility (probability times reward) was computed for each of the two targets based on the scaled preference values:
EV[Target1]=Target1_preferencefrequency*Target1_preferencereward
EV[Target2]=Target2_preferencefrequency*Target2_preferencereward.

These expected values were normalized by the sum of the expected values for the two targets to yield the probability of choosing each target. On direct reaches, a random number was drawn; if the random number was lower than the probability of choosing Target 1, we recorded a direct-reach choice of Target 1; otherwise, we recorded a choice of Target 2. On intermediate reaches, we converted the probability of choosing Target 1 into a reach direction by multiplying by 30°, then subtracting 15° to center reaches about the midline. In this way, we could build a binary distribution of direct reaches and a continuous distribution of intermediate reaches, for every possible combination of frequency and reward ratios presented during experiment 3 (200 trials were simulated for each condition). Finally, we applied our weight estimation approach as described in Materials and Methods (computing regressions to estimate the relationship between frequency ratio and bias and reward ratio and bias separately, then finding the best weighting of these regressions to explain the bias observed when both frequency and reward ratios were not equal to 1) to recover the degree to which frequency was weighted across simulated trials (*D*_{choice} _{and} *I*_{bias}). We repeated this simulation for 100 iterations at each of nine frequency-reward weightings (0.1–0.9). We compared the recovered weighting to the actual weighting for both direct and intermediate reaches separately, as well as comparing the recovered weightings for direct reaches versus intermediate reaches (since on each repetition both direct and intermediate reaches were simulated from the same underlying weights). Download Figure 7-1, EPS file.