$(0.028)$ $(0.016)$ $(0.020)$ $(0.012)$ a This table reports the RMSE between
the estimated values of the learned policy, calculated using both CATEs and DR
scores. The error is defined as the difference between the true value of the
learned policy (calculated using true CATEs) and the estimated value
(calculated using either estimated CATEs $\hat{\tau}$ or DR scores
$\hat{\gamma}$.

Table A10. Comparison: RMSE of Estimated Policy Advantages, Modified Tree
Policy, Confounding

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR $0.063$ $0.046$ $0.044$ $0.035$
$0.021$ $0.021$ NDR $0.025$ $0.017$ $0.019$ $0.014$ $0.009$ $0.007$ $(0.050)$
$(0.042)$ $(0.035)$ $(0.033)$ $(0.020)$ $(0.021)$ $(0.017)$ $(0.014)$
$(0.013)$ $(0.012)$ $(0.007)$ $(0.007)$ CF $0.073$ $0.060$ $0.050$ $0.042$
$0.024$ $0.020$ CF $0.025$ $0.022$ $0.019$ $0.017$ $0.009$ $0.008$ $(0.069)$
$(0.058)$ $(0.049)$ $(0.041)$ $(0.024)$ $(0.020)$ $(0.023)$ $(0.021)$
$(0.018)$ $(0.016)$ $(0.009)$ $(0.008)$ CFTT $0.074$ $0.052$ $0.050$ $0.036$
$0.024$ $0.020$ CFTT $0.026$ $0.020$ $0.019$ $0.014$ $0.009$ $0.007$ $(0.060)$
$(0.046)$ $(0.044)$ $(0.034)$ $(0.024)$ $(0.020)$ $(0.021)$ $(0.018)$
$(0.015)$ $(0.013)$ $(0.009)$ $(0.007)$ BART $0.067$ $0.050$ $0.051$ $0.037$
$0.039$ $0.021$ BART $0.022$ $0.017$ $0.019$ $0.014$ $0.011$ $0.007$ $(0.063)$
$(0.049)$ $(0.050)$ $(0.037)$ $(0.039)$ $(0.021)$ $(0.022)$ $(0.017)$
$(0.019)$ $(0.014)$ $(0.011)$ $(0.007)$ SETTING 2 $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR
$0.070$ $0.098$ $0.047$ $0.078$ $0.022$ $0.031$ NDR $0.036$ $0.023$ $0.027$
$0.014$ $0.013$ $0.007$ $(0.047)$ $(0.051)$ $(0.035)$ $(0.031)$ $(0.018)$
$(0.018)$ $(0.021)$ $(0.018)$ $(0.014)$ $(0.014)$ $(0.008)$ $(0.007)$ CF
$0.073$ $0.070$ $0.052$ $0.056$ $0.023$ $0.028$ CF $0.035$ $0.027$ $0.026$
$0.016$ $0.012$ $0.009$ $(0.052)$ $(0.054)$ $(0.043)$ $(0.039)$ $(0.021)$
$(0.019)$ $(0.028)$ $(0.024)$ $(0.020)$ $(0.016)$ $(0.009)$ $(0.008)$ CFTT
$0.071$ $0.072$ $0.052$ $0.057$ $0.024$ $0.025$ CFTT $0.039$ $0.024$ $0.028$
$0.014$ $0.012$ $0.008$ $(0.052)$ $(0.049)$ $(0.041)$ $(0.038)$ $(0.021)$
$(0.018)$ $(0.023)$ $(0.021)$ $(0.017)$ $(0.014)$ $(0.009)$ $(0.007)$ BART
$0.062$ $0.122$ $0.044$ $0.088$ $0.032$ $0.022$ BART $0.029$ $0.021$ $0.020$
$0.015$ $0.014$ $0.010$ $(0.060)$ $(0.088)$ $(0.044)$ $(0.029)$ $(0.032)$
$(0.013)$ $(0.025)$ $(0.020)$ $(0.019)$ $(0.015)$ $(0.013)$ $(0.009)$ SETTING
3 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ NDR $0.081$ $0.046$ $0.055$ $0.042$ $0.023$
$0.034$ NDR $0.029$ $0.029$ $0.020$ $0.026$ $0.010$ $0.013$ $(0.045)$
$(0.046)$ $(0.030)$ $(0.032)$ $(0.017)$ $(0.014)$ $(0.022)$ $(0.021)$
$(0.015)$ $(0.012)$ $(0.007)$ $(0.006)$ CF $0.090$ $0.047$ $0.061$ $0.039$
$0.027$ $0.022$ CF $0.033$ $0.032$ $0.021$ $0.026$ $0.011$ $0.014$ $(0.047)$
$(0.047)$ $(0.036)$ $(0.032)$ $(0.020)$ $(0.016)$ $(0.023)$ $(0.029)$
$(0.017)$ $(0.018)$ $(0.010)$ $(0.010)$ CFTT $0.087$ $0.048$ $0.062$ $0.039$
$0.027$ $0.022$ CFTT $0.033$ $0.032$ $0.021$ $0.026$ $0.010$ $0.014$ $(0.046)$
$(0.048)$ $(0.035)$ $(0.031)$ $(0.020)$ $(0.016)$ $(0.022)$ $(0.027)$
$(0.017)$ $(0.016)$ $(0.010)$ $(0.009)$ BART $0.062$ $0.057$ $0.048$ $0.049$
$0.031$ $0.017$ BART $0.040$ $0.046$ $0.029$ $0.041$ $0.017$ $0.010$ $(0.048)$
$(0.057)$ $(0.043)$ $(0.039)$ $(0.027)$ $(0.010)$ $(0.027)$ $(0.041)$
$(0.025)$ $(0.038)$ $(0.016)$ $(0.010)$ a This table reports the RMSE between
the estimated values of the learned policy, calculated using both CATEs and DR
scores. The error is defined as the difference between the true value of the
learned policy (calculated using true CATEs) and the estimated value
(calculated using either estimated CATEs $\hat{\tau}$ or DR scores
$\hat{\gamma}$.

Table A11. Comparison: True $A_{i}$ vs Estimated $A_{i}$ (Tree-based Policy)

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.10 -0.10 -0.10 OR -0.04 -0.04 -0.04 NDR $-0.036$ $-0.029$
$-0.061$ $-0.046$ $-0.041$ $-0.068$ $-0.070$ $-0.070$ $-0.081$ NDR $-0.019$
$-0.017$ $-0.024$ $-0.024$ $-0.023$ $-0.029$ $-0.030$ $-0.030$ $-0.033$
$(0.004)$ $(0.055)$ $(0.005)$ $(0.003)$ $(0.041)$ $(0.003)$ $(0.001)$
$(0.022)$ $(0.001)$ $(0.001)$ $(0.019)$ $(0.002)$ $(0.001)$ $(0.014)$
$(0.001)$ $(0.000)$ $(0.007)$ $(0.000)$ CF $-0.040$ $-0.044$ $-0.056$ $-0.050$
$-0.052$ $-0.060$ $-0.073$ $-0.072$ $-0.074$ CF $-0.021$ $-0.024$ $-0.026$
$-0.024$ $-0.026$ $-0.028$ $-0.030$ $-0.031$ $-0.031$ $(0.004)$ $(0.056)$
$(0.004)$ $(0.003)$ $(0.042)$ $(0.003)$ $(0.001)$ $(0.022)$ $(0.001)$
$(0.001)$ $(0.019)$ $(0.001)$ $(0.001)$ $(0.014)$ $(0.001)$ $(0.000)$
$(0.007)$ $(0.000)$ CFTT $-0.038$ $-0.045$ $-0.064$ $-0.047$ $-0.056$ $-0.068$
$-0.071$ $-0.074$ $-0.076$ CFTT $-0.020$ $-0.024$ $-0.027$ $-0.023$ $-0.027$
$-0.030$ $-0.030$ $-0.031$ $-0.032$ $(0.004)$ $(0.055)$ $(0.004)$ $(0.003)$
$(0.042)$ $(0.003)$ $(0.001)$ $(0.022)$ $(0.001)$ $(0.001)$ $(0.019)$
$(0.002)$ $(0.001)$ $(0.014)$ $(0.001)$ $(0.000)$ $(0.007)$ $(0.000)$ BART
$-0.043$ $-0.047$ $-0.055$ $-0.050$ $-0.049$ $-0.058$ $-0.069$ $-0.064$
$-0.069$ BART $-0.018$ $-0.016$ $-0.017$ $-0.024$ $-0.023$ $-0.023$ $-0.029$
$-0.030$ $-0.029$ $(0.004)$ $(0.049)$ $(0.004)$ $(0.003)$ $(0.044)$ $(0.002)$
$(0.001)$ $(0.035)$ $(0.001)$ $(0.001)$ $(0.017)$ $(0.001)$ $(0.001)$
$(0.014)$ $(0.001)$ $(0.000)$ $(0.009)$ $(0.000)$ SETTING 2 N = 500 N = 1000 N
= 5000 SETTING 2 N = 500 N = 1000 N = 5000 $\tau$ $\hat{\tau}$ $\hat{\gamma}$
$\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ OR -0.23 -0.23 -0.23 OR -0.05 -0.05 -0.05 NDR
$-0.172$ $-0.175$ $-0.086$ $-0.195$ $-0.198$ $-0.128$ $-0.212$ $-0.210$
$-0.188$ NDR $-0.021$ $-0.020$ $-0.022$ $-0.025$ $-0.025$ $-0.025$ $-0.034$
$-0.033$ $-0.032$ $(0.009)$ $(0.052)$ $(0.005)$ $(0.006)$ $(0.038)$ $(0.004)$
$(0.002)$ $(0.020)$ $(0.002)$ $(0.003)$ $(0.023)$ $(0.002)$ $(0.002)$
$(0.017)$ $(0.001)$ $(0.001)$ $(0.009)$ $(0.000)$ CF $-0.172$ $-0.177$
$-0.131$ $-0.192$ $-0.197$ $-0.156$ $-0.211$ $-0.210$ $-0.190$ CF $-0.021$
$-0.022$ $-0.022$ $-0.025$ $-0.027$ $-0.023$ $-0.034$ $-0.035$ $-0.029$
$(0.009)$ $(0.052)$ $(0.005)$ $(0.006)$ $(0.039)$ $(0.004)$ $(0.002)$
$(0.020)$ $(0.002)$ $(0.003)$ $(0.023)$ $(0.001)$ $(0.002)$ $(0.017)$
$(0.001)$ $(0.001)$ $(0.009)$ $(0.000)$ CFTT $-0.173$ $-0.178$ $-0.127$
$-0.193$ $-0.198$ $-0.155$ $-0.211$ $-0.210$ $-0.195$ CFTT $-0.021$ $-0.024$
$-0.026$ $-0.025$ $-0.028$ $-0.027$ $-0.034$ $-0.035$ $-0.032$ $(0.009)$
$(0.051)$ $(0.005)$ $(0.006)$ $(0.038)$ $(0.004)$ $(0.002)$ $(0.020)$
$(0.002)$ $(0.003)$ $(0.023)$ $(0.002)$ $(0.002)$ $(0.017)$ $(0.001)$
$(0.001)$ $(0.009)$ $(0.000)$ BART $-0.171$ $-0.142$ $-0.054$ $-0.195$
$-0.181$ $-0.116$ $-0.209$ $-0.207$ $-0.189$ BART $-0.019$ $-0.017$ $-0.014$
$-0.025$ $-0.023$ $-0.016$ $-0.034$ $-0.033$ $-0.026$ $(0.009)$ $(0.047)$
$(0.003)$ $(0.006)$ $(0.040)$ $(0.003)$ $(0.002)$ $(0.030)$ $(0.002)$
$(0.003)$ $(0.019)$ $(0.001)$ $(0.002)$ $(0.016)$ $(0.001)$ $(0.001)$
$(0.011)$ $(0.000)$ SETTING 3 N = 500 N = 1000 N = 5000 SETTING 3 N = 500 N =
1000 N = 5000 OR $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.21 -0.21 -0.21 OR -0.11 -0.11 -0.11 NDR $-0.074$ $-0.074$
$-0.038$ $-0.093$ $-0.087$ $-0.045$ $-0.111$ $-0.111$ $-0.075$ NDR $-0.060$
$-0.051$ $-0.025$ $-0.067$ $-0.063$ $-0.039$ $-0.079$ $-0.075$ $-0.066$
$(0.011)$ $(0.050)$ $(0.004)$ $(0.007)$ $(0.036)$ $(0.003)$ $(0.003)$
$(0.018)$ $(0.001)$ $(0.008)$ $(0.026)$ $(0.003)$ $(0.005)$ $(0.019)$
$(0.002)$ $(0.002)$ $(0.009)$ $(0.001)$ CF $-0.079$ $-0.079$ $-0.055$ $-0.093$
$-0.095$ $-0.064$ $-0.111$ $-0.114$ $-0.092$ CF $-0.061$ $-0.053$ $-0.032$
$-0.068$ $-0.062$ $-0.041$ $-0.078$ $-0.073$ $-0.059$ $(0.011)$ $(0.047)$
$(0.004)$ $(0.007)$ $(0.035)$ $(0.003)$ $(0.003)$ $(0.018)$ $(0.002)$
$(0.008)$ $(0.024)$ $(0.002)$ $(0.005)$ $(0.018)$ $(0.002)$ $(0.002)$
$(0.009)$ $(0.001)$ CFTT $-0.079$ $-0.078$ $-0.053$ $-0.094$ $-0.093$ $-0.062$
$-0.111$ $-0.114$ $-0.092$ CFTT $-0.062$ $-0.056$ $-0.033$ $-0.068$ $-0.062$
$-0.042$ $-0.078$ $-0.073$ $-0.059$ $(0.011)$ $(0.047)$ $(0.004)$ $(0.007)$
$(0.035)$ $(0.003)$ $(0.003)$ $(0.018)$ $(0.002)$ $(0.008)$ $(0.024)$
$(0.002)$ $(0.005)$ $(0.018)$ $(0.002)$ $(0.002)$ $(0.009)$ $(0.001)$ BART
$-0.076$ $-0.059$ $-0.024$ $-0.092$ $-0.074$ $-0.032$ $-0.109$ $-0.108$
$-0.089$ BART $-0.059$ $-0.045$ $-0.013$ $-0.066$ $-0.056$ $-0.023$ $-0.077$
$-0.074$ $-0.064$ $(0.011)$ $(0.042)$ $(0.002)$ $(0.007)$ $(0.036)$ $(0.002)$
$(0.003)$ $(0.025)$ $(0.002)$ $(0.008)$ $(0.022)$ $(0.001)$ $(0.005)$
$(0.019)$ $(0.001)$ $(0.002)$ $(0.014)$ $(0.001)$ a This table reports the
true policy advantage calculated using the learned policies and the true CATEs
(column $\tau$) for the tree-based policy class, and compares them to the
estimated policy advantage calculated using both estimated CATEs
($\hat{\tau}$) and estimated DR scores ($\hat{\gamma}$). Panel A depicts
results for common outcome prevalence, and Panel B the rare outcome
prevalence.

Table A12. Comparison: True adv vs Estimated Adv (Plug-in Policy)

Common Outcomes Rare OUtcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.10 -0.10 -0.10 OR -0.04 -0.04 -0.04 NDR $-0.073$ $-0.073$
$-0.111$ $-0.077$ $-0.077$ $-0.108$ $-0.084$ $-0.083$ $-0.103$ NDR $-0.026$
$-0.026$ $-0.043$ $-0.028$ $-0.029$ $-0.042$ $-0.031$ $-0.031$ $-0.039$
$(0.003)$ $(0.055)$ $(0.003)$ $(0.002)$ $(0.041)$ $(0.002)$ $(0.001)$
$(0.022)$ $(0.001)$ $(0.001)$ $(0.019)$ $(0.001)$ $(0.001)$ $(0.014)$
$(0.001)$ $(0.000)$ $(0.007)$ $(0.000)$ CF $-0.088$ $-0.095$ $-0.105$ $-0.094$
$-0.101$ $-0.103$ $-0.098$ $-0.100$ $-0.099$ CF $-0.032$ $-0.036$ $-0.039$
$-0.034$ $-0.038$ $-0.039$ $-0.035$ $-0.037$ $-0.037$ $(0.002)$ $(0.056)$
$(0.001)$ $(0.002)$ $(0.042)$ $(0.000)$ $(0.001)$ $(0.022)$ $(0.000)$
$(0.000)$ $(0.019)$ $(0.000)$ $(0.000)$ $(0.014)$ $(0.000)$ $(0.000)$
$(0.007)$ $(0.000)$ CFTT $-0.079$ $-0.082$ $-0.107$ $-0.086$ $-0.091$ $-0.104$
$-0.095$ $-0.097$ $-0.099$ CFTT $-0.029$ $-0.032$ $-0.040$ $-0.031$ $-0.035$
$-0.039$ $-0.034$ $-0.036$ $-0.037$ $(0.003)$ $(0.055)$ $(0.002)$ $(0.002)$
$(0.042)$ $(0.001)$ $(0.001)$ $(0.022)$ $(0.001)$ $(0.001)$ $(0.019)$
$(0.001)$ $(0.000)$ $(0.014)$ $(0.001)$ $(0.000)$ $(0.007)$ $(0.000)$ BART
$-0.083$ $-0.110$ $-0.095$ $-0.091$ $-0.106$ $-0.096$ $-0.098$ $-0.098$
$-0.097$ BART $-0.031$ $-0.035$ $-0.030$ $-0.034$ $-0.036$ $-0.033$ $-0.035$
$-0.036$ $-0.035$ $(0.003)$ $(0.049)$ $(0.001)$ $(0.002)$ $(0.044)$ $(0.001)$
$(0.001)$ $(0.035)$ $(0.000)$ $(0.001)$ $(0.017)$ $(0.001)$ $(0.000)$
$(0.014)$ $(0.001)$ $(0.000)$ $(0.009)$ $(0.000)$ N = 500 N = 1000 N = 5000 N
= 500 N = 1000 N = 5000 SETTING 2 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ OR -0.23 -0.23 -0.23 OR -0.05 -0.05 -0.05 NDR
$-0.174$ $-0.180$ $-0.115$ $-0.204$ $-0.208$ $-0.147$ $-0.217$ $-0.216$
$-0.199$ NDR $-0.020$ $-0.021$ $-0.042$ $-0.025$ $-0.024$ $-0.039$ $-0.034$
$-0.034$ $-0.040$ $(0.009)$ $(0.052)$ $(0.003)$ $(0.005)$ $(0.038)$ $(0.003)$
$(0.002)$ $(0.020)$ $(0.002)$ $(0.003)$ $(0.023)$ $(0.001)$ $(0.002)$
$(0.017)$ $(0.001)$ $(0.001)$ $(0.009)$ $(0.000)$ CF $-0.196$ $-0.199$
$-0.156$ $-0.214$ $-0.216$ $-0.175$ $-0.220$ $-0.219$ $-0.200$ CF $-0.022$
$-0.030$ $-0.035$ $-0.027$ $-0.033$ $-0.033$ $-0.036$ $-0.038$ $-0.034$
$(0.008)$ $(0.051)$ $(0.004)$ $(0.005)$ $(0.039)$ $(0.003)$ $(0.002)$
$(0.020)$ $(0.001)$ $(0.003)$ $(0.023)$ $(0.000)$ $(0.002)$ $(0.017)$
$(0.000)$ $(0.001)$ $(0.009)$ $(0.000)$ CFTT $-0.196$ $-0.200$ $-0.152$
$-0.212$ $-0.215$ $-0.175$ $-0.220$ $-0.219$ $-0.205$ CFTT $-0.024$ $-0.029$
$-0.039$ $-0.029$ $-0.032$ $-0.036$ $-0.037$ $-0.038$ $-0.037$ $(0.008)$
$(0.051)$ $(0.004)$ $(0.005)$ $(0.038)$ $(0.003)$ $(0.002)$ $(0.020)$
$(0.002)$ $(0.003)$ $(0.022)$ $(0.001)$ $(0.002)$ $(0.017)$ $(0.001)$
$(0.001)$ $(0.009)$ $(0.000)$ BART $-0.152$ $-0.185$ $-0.074$ $-0.214$
$-0.220$ $-0.133$ $-0.220$ $-0.227$ $-0.203$ BART $-0.018$ $-0.034$ $-0.027$
$-0.023$ $-0.033$ $-0.026$ $-0.035$ $-0.041$ $-0.031$ $(0.009)$ $(0.046)$
$(0.002)$ $(0.005)$ $(0.040)$ $(0.002)$ $(0.002)$ $(0.030)$ $(0.002)$
$(0.003)$ $(0.019)$ $(0.001)$ $(0.002)$ $(0.016)$ $(0.000)$ $(0.001)$
$(0.011)$ $(0.000)$ SETTING 3 N = 500 N = 1000 N = 5000 SETTING 3 N = 500 N =
1000 N = 5000 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.21 -0.21 -0.21 OR -0.11 -0.11 -0.11 NDR $-0.074$ $-0.068$
$-0.076$ $-0.109$ $-0.109$ $-0.078$ $-0.159$ $-0.159$ $-0.107$ NDR $-0.061$
$-0.056$ $-0.051$ $-0.073$ $-0.067$ $-0.058$ $-0.082$ $-0.078$ $-0.074$
$(0.011)$ $(0.050)$ $(0.002)$ $(0.007)$ $(0.036)$ $(0.002)$ $(0.003)$
$(0.018)$ $(0.001)$ NDR $(0.008)$ $(0.026)$ $(0.002)$ $(0.005)$ $(0.019)$
$(0.002)$ $(0.002)$ $(0.009)$ $(0.001)$ CF $-0.088$ $-0.095$ $-0.082$ $-0.127$
$-0.124$ $-0.090$ $-0.167$ $-0.167$ $-0.128$ CF $-0.060$ $-0.064$ $-0.047$
$-0.072$ $-0.073$ $-0.054$ $-0.079$ $-0.078$ $-0.068$ $(0.011)$ $(0.047)$
$(0.002)$ $(0.007)$ $(0.035)$ $(0.002)$ $(0.003)$ $(0.018)$ $(0.001)$ CF
$(0.008)$ $(0.024)$ $(0.002)$ $(0.005)$ $(0.018)$ $(0.001)$ $(0.002)$
$(0.009)$ $(0.001)$ CFTT $-0.088$ $-0.089$ $-0.080$ $-0.125$ $-0.122$ $-0.088$
$-0.167$ $-0.167$ $-0.129$ CFTT $-0.062$ $-0.065$ $-0.049$ $-0.073$ $-0.074$
$-0.055$ $-0.079$ $-0.078$ $-0.068$ $(0.011)$ $(0.047)$ $(0.002)$ $(0.007)$
$(0.035)$ $(0.002)$ $(0.003)$ $(0.018)$ $(0.001)$ CFTT $(0.008)$ $(0.024)$
$(0.002)$ $(0.005)$ $(0.018)$ $(0.001)$ $(0.002)$ $(0.009)$ $(0.001)$ BART
$-0.053$ $-0.103$ $-0.053$ $-0.101$ $-0.132$ $-0.058$ $-0.174$ $-0.187$
$-0.139$ BART $-0.008$ $-0.043$ $-0.030$ $-0.046$ $-0.064$ $-0.034$ $-0.078$
$-0.084$ $-0.077$ $(0.011)$ $(0.041)$ $(0.001)$ $(0.007)$ $(0.036)$ $(0.001)$
$(0.003)$ $(0.025)$ $(0.001)$ BART $(0.008)$ $(0.022)$ $(0.001)$ $(0.006)$
$(0.019)$ $(0.001)$ $(0.002)$ $(0.014)$ $(0.001)$ a This table reports the
true policy advantage calculated using the learned policies and the true CATEs
(column $\tau$) for the plug-in policy class, and compares them to the
estimated policy advantage calculated using both estimated CATEs
($\hat{\tau}$) and estimated DR scores ($\hat{\gamma}$). The left and right
panels depict common and rare outcome prevalence, respectively.

Table A13. Comparison: RMSE of Estimated Policy Advantages, Plug-in Policy, No
Confounding

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR $0.063$ $0.048$ $0.044$ $0.034$
$0.019$ $0.016$ NDR $0.020$ $0.017$ $0.014$ $0.011$ $0.006$ $0.005$ $(0.063)$
$(0.031)$ $(0.045)$ $(0.024)$ $(0.019)$ $(0.014)$ $(0.020)$ $(0.011)$
$(0.014)$ $(0.007)$ $(0.006)$ $(0.004)$ CF $0.085$ $0.054$ $0.053$ $0.040$
$0.019$ $0.019$ CF $0.021$ $0.018$ $0.015$ $0.013$ $0.006$ $0.006$ $(0.085)$
$(0.053)$ $(0.053)$ $(0.040)$ $(0.019)$ $(0.019)$ $(0.021)$ $(0.018)$
$(0.015)$ $(0.013)$ $(0.006)$ $(0.006)$ CFTT $0.066$ $0.047$ $0.044$ $0.034$
$0.019$ $0.018$ CFTT $0.020$ $0.015$ $0.014$ $0.011$ $0.006$ $0.006$ $(0.066)$
$(0.044)$ $(0.044)$ $(0.033)$ $(0.019)$ $(0.018)$ $(0.020)$ $(0.014)$
$(0.014)$ $(0.010)$ $(0.006)$ $(0.006)$ BART $0.061$ $0.046$ $0.038$ $0.033$
$0.018$ $0.018$ BART $0.019$ $0.015$ $0.013$ $0.011$ $0.006$ $0.006$ $(0.054)$
$(0.046)$ $(0.035)$ $(0.033)$ $(0.018)$ $(0.018)$ $(0.017)$ $(0.015)$
$(0.013)$ $(0.011)$ $(0.006)$ $(0.006)$ SETTING 2 $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR
$0.060$ $0.073$ $0.038$ $0.069$ $0.016$ $0.023$ NDR $0.032$ $0.022$ $0.020$
$0.021$ $0.008$ $0.014$ $(0.060)$ $(0.042)$ $(0.038)$ $(0.024)$ $(0.016)$
$(0.014)$ $(0.032)$ $(0.019)$ $(0.020)$ $(0.010)$ $(0.008)$ $(0.006)$ CF
$0.063$ $0.076$ $0.040$ $0.057$ $0.017$ $0.023$ CF $0.034$ $0.030$ $0.021$
$0.025$ $0.009$ $0.014$ $(0.062)$ $(0.051)$ $(0.040)$ $(0.032)$ $(0.017)$
$(0.015)$ $(0.034)$ $(0.026)$ $(0.021)$ $(0.013)$ $(0.009)$ $(0.007)$ CFTT
$0.064$ $0.079$ $0.039$ $0.061$ $0.016$ $0.021$ CFTT $0.034$ $0.027$ $0.021$
$0.023$ $0.009$ $0.013$ $(0.064)$ $(0.051)$ $(0.039)$ $(0.031)$ $(0.016)$
$(0.015)$ $(0.034)$ $(0.024)$ $(0.021)$ $(0.013)$ $(0.009)$ $(0.007)$ BART
$0.083$ $0.088$ $0.038$ $0.110$ $0.019$ $0.024$ BART $0.033$ $0.030$ $0.025$
$0.036$ $0.009$ $0.026$ $(0.038)$ $(0.081)$ $(0.031)$ $(0.034)$ $(0.015)$
$(0.014)$ $(0.020)$ $(0.030)$ $(0.015)$ $(0.028)$ $(0.008)$ $(0.007)$ SETTING
3 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ NDR $0.060$ $0.037$ $0.039$ $0.041$ $0.016$
$0.058$ NDR $0.031$ $0.022$ $0.021$ $0.021$ $0.009$ $0.015$ $(0.060)$
$(0.037)$ $(0.039)$ $(0.027)$ $(0.016)$ $(0.012)$ $(0.031)$ $(0.020)$
$(0.021)$ $(0.011)$ $(0.009)$ $(0.006)$ CF $0.062$ $0.047$ $0.041$ $0.051$
$0.016$ $0.048$ CF $0.034$ $0.029$ $0.021$ $0.025$ $0.009$ $0.014$ $(0.062)$
$(0.045)$ $(0.041)$ $(0.033)$ $(0.016)$ $(0.015)$ $(0.034)$ $(0.025)$
$(0.021)$ $(0.015)$ $(0.009)$ $(0.007)$ CFTT $0.065$ $0.047$ $0.040$ $0.052$
$0.016$ $0.048$ CFTT $0.033$ $0.027$ $0.021$ $0.024$ $0.009$ $0.013$ $(0.064)$
$(0.046)$ $(0.040)$ $(0.034)$ $(0.016)$ $(0.014)$ $(0.033)$ $(0.023)$
$(0.021)$ $(0.014)$ $(0.009)$ $(0.007)$ BART $0.063$ $0.044$ $0.045$ $0.043$
$0.023$ $0.046$ BART $0.034$ $0.031$ $0.024$ $0.036$ $0.009$ $0.026$ $(0.036)$
$(0.044)$ $(0.025)$ $(0.037)$ $(0.014)$ $(0.012)$ $(0.020)$ $(0.031)$
$(0.015)$ $(0.028)$ $(0.008)$ $(0.007)$ a This table reports the RMSE between
the estimated values of the learned policy, calculated using both CATEs and DR
scores. The error is defined as the difference between the true value of the
learned policy (calculated using true CATEs) and the estimated value
(calculated using either estimated CATEs $\hat{\tau}$ or DR scores
$\hat{\gamma}$.

Table A14. Comparison: RMSE of Estimated Policy Advantages, Tree Policy, No
Confounding

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR $0.091$ $0.036$ $0.061$ $0.030$
$0.026$ $0.016$ NDR $0.024$ $0.011$ $0.018$ $0.009$ $0.008$ $0.005$ $(0.091)$
$(0.026)$ $(0.061)$ $(0.023)$ $(0.026)$ $(0.015)$ $(0.024)$ $(0.010)$
$(0.018)$ $(0.008)$ $(0.008)$ $(0.005)$ CF $0.092$ $0.031$ $0.063$ $0.026$
$0.025$ $0.016$ CF $0.022$ $0.011$ $0.017$ $0.009$ $0.007$ $0.005$ $(0.092)$
$(0.028)$ $(0.063)$ $(0.025)$ $(0.025)$ $(0.016)$ $(0.022)$ $(0.011)$
$(0.017)$ $(0.009)$ $(0.007)$ $(0.005)$ CFTT $0.092$ $0.032$ $0.065$ $0.028$
$0.026$ $0.016$ CFTT $0.024$ $0.011$ $0.017$ $0.009$ $0.008$ $0.005$ $(0.092)$
$(0.026)$ $(0.065)$ $(0.023)$ $(0.026)$ $(0.015)$ $(0.024)$ $(0.010)$
$(0.017)$ $(0.008)$ $(0.008)$ $(0.005)$ BART $0.077$ $0.027$ $0.056$ $0.023$
$0.025$ $0.016$ BART $0.021$ $0.009$ $0.017$ $0.008$ $0.007$ $0.005$ $(0.076)$
$(0.025)$ $(0.056)$ $(0.022)$ $(0.025)$ $(0.016)$ $(0.021)$ $(0.009)$
$(0.017)$ $(0.008)$ $(0.007)$ $(0.005)$ SETTING 2 $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR
$0.083$ $0.095$ $0.051$ $0.076$ $0.020$ $0.025$ NDR $0.039$ $0.032$ $0.027$
$0.029$ $0.011$ $0.017$ $(0.083)$ $(0.035)$ $(0.051)$ $(0.026)$ $(0.020)$
$(0.015)$ $(0.039)$ $(0.018)$ $(0.026)$ $(0.013)$ $(0.011)$ $(0.006)$ CF
$0.078$ $0.073$ $0.052$ $0.055$ $0.020$ $0.024$ CF $0.038$ $0.030$ $0.027$
$0.026$ $0.011$ $0.013$ $(0.078)$ $(0.041)$ $(0.052)$ $(0.031)$ $(0.020)$
$(0.015)$ $(0.038)$ $(0.020)$ $(0.026)$ $(0.014)$ $(0.011)$ $(0.007)$ CFTT
$0.082$ $0.078$ $0.054$ $0.059$ $0.020$ $0.022$ CFTT $0.039$ $0.027$ $0.026$
$0.024$ $0.010$ $0.013$ $(0.081)$ $(0.041)$ $(0.053)$ $(0.030)$ $(0.020)$
$(0.015)$ $(0.039)$ $(0.019)$ $(0.026)$ $(0.014)$ $(0.010)$ $(0.007)$ BART
$0.074$ $0.142$ $0.053$ $0.112$ $0.020$ $0.026$ BART $0.035$ $0.053$ $0.027$
$0.058$ $0.011$ $0.026$ $(0.069)$ $(0.036)$ $(0.050)$ $(0.026)$ $(0.020)$
$(0.014)$ $(0.034)$ $(0.023)$ $(0.025)$ $(0.016)$ $(0.010)$ $(0.007)$ SETTING
3 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ NDR $0.085$ $0.050$ $0.054$ $0.052$ $0.023$
$0.041$ NDR $0.038$ $0.032$ $0.027$ $0.029$ $0.010$ $0.017$ $(0.085)$
$(0.036)$ $(0.053)$ $(0.026)$ $(0.023)$ $(0.015)$ $(0.037)$ $(0.018)$
$(0.026)$ $(0.013)$ $(0.010)$ $(0.006)$ CF $0.080$ $0.048$ $0.053$ $0.043$
$0.022$ $0.027$ CF $0.039$ $0.029$ $0.027$ $0.025$ $0.010$ $0.014$ $(0.079)$
$(0.033)$ $(0.053)$ $(0.025)$ $(0.022)$ $(0.014)$ $(0.039)$ $(0.020)$
$(0.026)$ $(0.015)$ $(0.010)$ $(0.007)$ CFTT $0.077$ $0.052$ $0.052$ $0.048$
$0.022$ $0.027$ CFTT $0.040$ $0.028$ $0.029$ $0.025$ $0.010$ $0.013$ $(0.077)$
$(0.031)$ $(0.052)$ $(0.024)$ $(0.022)$ $(0.014)$ $(0.040)$ $(0.019)$
$(0.028)$ $(0.015)$ $(0.010)$ $(0.007)$ BART $0.067$ $0.071$ $0.049$ $0.075$
$0.022$ $0.029$ BART $0.037$ $0.053$ $0.030$ $0.058$ $0.010$ $0.026$ $(0.064)$
$(0.034)$ $(0.047)$ $(0.026)$ $(0.022)$ $(0.012)$ $(0.036)$ $(0.023)$
$(0.027)$ $(0.017)$ $(0.010)$ $(0.007)$ a This table reports the RMSE between
the estimated values of the learned policy, calculated using both CATEs and DR
scores. The error is defined as the difference between the true value of the
learned policy (calculated using true CATEs) and the estimated value
(calculated using either estimated CATEs $\hat{\tau}$ or DR scores
$\hat{\gamma}$.

Table A15. Comparison: RMSE of Estimated Policy Advantages, Modified Trees, No
Confounding

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR $0.067$ $0.045$ $0.042$ $0.034$
$0.018$ $0.018$ NDR $0.025$ $0.016$ $0.016$ $0.011$ $0.006$ $0.006$ $(0.052)$
$(0.042)$ $(0.037)$ $(0.033)$ $(0.018)$ $(0.018)$ $(0.019)$ $(0.014)$
$(0.013)$ $(0.010)$ $(0.006)$ $(0.006)$ CF $0.067$ $0.055$ $0.042$ $0.040$
$0.019$ $0.019$ CF $0.021$ $0.018$ $0.015$ $0.013$ $0.006$ $0.006$ $(0.065)$
$(0.054)$ $(0.042)$ $(0.040)$ $(0.019)$ $(0.019)$ $(0.020)$ $(0.018)$
$(0.014)$ $(0.013)$ $(0.006)$ $(0.006)$ CFTT $0.066$ $0.048$ $0.040$ $0.035$
$0.019$ $0.019$ CFTT $0.022$ $0.016$ $0.015$ $0.012$ $0.006$ $0.006$ $(0.059)$
$(0.047)$ $(0.038)$ $(0.035)$ $(0.019)$ $(0.019)$ $(0.020)$ $(0.015)$
$(0.014)$ $(0.011)$ $(0.006)$ $(0.006)$ BART $0.059$ $0.049$ $0.040$ $0.036$
$0.019$ $0.019$ BART $0.019$ $0.017$ $0.014$ $0.012$ $0.006$ $0.006$ $(0.057)$
$(0.049)$ $(0.039)$ $(0.036)$ $(0.019)$ $(0.019)$ $(0.018)$ $(0.017)$
$(0.014)$ $(0.012)$ $(0.006)$ $(0.006)$ SETTING 2 $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR
$0.071$ $0.102$ $0.040$ $0.082$ $0.018$ $0.027$ NDR $0.036$ $0.034$ $0.023$
$0.030$ $0.010$ $0.016$ $(0.046)$ $(0.051)$ $(0.031)$ $(0.028)$ $(0.015)$
$(0.015)$ $(0.022)$ $(0.026)$ $(0.017)$ $(0.014)$ $(0.008)$ $(0.006)$ CF
$0.073$ $0.080$ $0.042$ $0.059$ $0.018$ $0.024$ CF $0.036$ $0.032$ $0.023$
$0.025$ $0.010$ $0.013$ $(0.044)$ $(0.054)$ $(0.031)$ $(0.033)$ $(0.015)$
$(0.016)$ $(0.023)$ $(0.027)$ $(0.016)$ $(0.014)$ $(0.008)$ $(0.007)$ CFTT
$0.072$ $0.086$ $0.042$ $0.064$ $0.019$ $0.022$ CFTT $0.036$ $0.030$ $0.023$
$0.025$ $0.010$ $0.012$ $(0.045)$ $(0.055)$ $(0.031)$ $(0.032)$ $(0.015)$
$(0.015)$ $(0.023)$ $(0.026)$ $(0.017)$ $(0.014)$ $(0.008)$ $(0.007)$ BART
$0.068$ $0.102$ $0.034$ $0.115$ $0.017$ $0.025$ BART $0.031$ $0.031$ $0.022$
$0.038$ $0.009$ $0.024$ $(0.043)$ $(0.094)$ $(0.032)$ $(0.036)$ $(0.015)$
$(0.014)$ $(0.020)$ $(0.030)$ $(0.016)$ $(0.032)$ $(0.008)$ $(0.008)$ SETTING
3 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ NDR $0.080$ $0.046$ $0.053$ $0.041$ $0.018$
$0.033$ NDR $0.038$ $0.033$ $0.022$ $0.030$ $0.009$ $0.016$ $(0.044)$
$(0.045)$ $(0.030)$ $(0.033)$ $(0.014)$ $(0.012)$ $(0.024)$ $(0.027)$
$(0.018)$ $(0.015)$ $(0.008)$ $(0.006)$ CF $0.092$ $0.044$ $0.061$ $0.039$
$0.022$ $0.023$ CF $0.038$ $0.031$ $0.023$ $0.026$ $0.010$ $0.013$ $(0.045)$
$(0.044)$ $(0.029)$ $(0.031)$ $(0.014)$ $(0.014)$ $(0.024)$ $(0.027)$
$(0.017)$ $(0.016)$ $(0.008)$ $(0.007)$ CFTT $0.086$ $0.048$ $0.059$ $0.042$
$0.022$ $0.024$ CFTT $0.037$ $0.030$ $0.022$ $0.025$ $0.010$ $0.013$ $(0.045)$
$(0.047)$ $(0.028)$ $(0.034)$ $(0.014)$ $(0.013)$ $(0.023)$ $(0.025)$
$(0.017)$ $(0.015)$ $(0.008)$ $(0.007)$ BART $0.053$ $0.048$ $0.036$ $0.042$
$0.019$ $0.021$ BART $0.032$ $0.033$ $0.022$ $0.038$ $0.008$ $0.024$ $(0.038)$
$(0.047)$ $(0.026)$ $(0.040)$ $(0.014)$ $(0.012)$ $(0.020)$ $(0.032)$
$(0.016)$ $(0.032)$ $(0.008)$ $(0.007)$ a This table reports the RMSE between
the estimated values of the learned policy, calculated using both CATEs and DR
scores. The error is defined as the difference between the true value of the
learned policy (calculated using true CATEs) and the estimated value
(calculated using either estimated CATEs $\hat{\tau}$ or DR scores
$\hat{\gamma}$.

Table A16. Comparison: True $A_{i}$ vs Estimated $A_{i}$ (Tree-based Policy,
No Confounding)

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.10 -0.10 -0.10 OR -0.04 -0.04 -0.04 NDR $-0.032$ $-0.035$
$-0.058$ $-0.046$ $-0.042$ $-0.066$ $-0.077$ $-0.076$ $-0.082$ NDR $-0.022$
$-0.022$ $-0.026$ $-0.024$ $-0.024$ $-0.029$ $-0.030$ $-0.029$ $-0.032$
$(0.005)$ $(0.057)$ $(0.005)$ $(0.003)$ $(0.040)$ $(0.003)$ $(0.001)$
$(0.018)$ $(0.001)$ $(0.001)$ $(0.018)$ $(0.002)$ $(0.001)$ $(0.013)$
$(0.001)$ $(0.000)$ $(0.006)$ $(0.000)$ CF $-0.032$ $-0.037$ $-0.046$ $-0.047$
$-0.041$ $-0.054$ $-0.077$ $-0.074$ $-0.077$ CF $-0.022$ $-0.021$ $-0.024$
$-0.024$ $-0.024$ $-0.026$ $-0.030$ $-0.029$ $-0.030$ $(0.005)$ $(0.057)$
$(0.004)$ $(0.003)$ $(0.040)$ $(0.003)$ $(0.001)$ $(0.018)$ $(0.001)$
$(0.001)$ $(0.018)$ $(0.001)$ $(0.001)$ $(0.013)$ $(0.001)$ $(0.000)$
$(0.006)$ $(0.000)$ CFTT $-0.033$ $-0.040$ $-0.052$ $-0.047$ $-0.043$ $-0.061$
$-0.077$ $-0.076$ $-0.081$ CFTT $-0.022$ $-0.022$ $-0.025$ $-0.024$ $-0.025$
$-0.028$ $-0.030$ $-0.029$ $-0.031$ $(0.005)$ $(0.057)$ $(0.004)$ $(0.003)$
$(0.040)$ $(0.003)$ $(0.001)$ $(0.018)$ $(0.001)$ $(0.001)$ $(0.019)$
$(0.001)$ $(0.001)$ $(0.013)$ $(0.001)$ $(0.000)$ $(0.006)$ $(0.000)$ BART
$-0.035$ $-0.047$ $-0.046$ $-0.049$ $-0.046$ $-0.056$ $-0.078$ $-0.076$
$-0.078$ BART $-0.021$ $-0.018$ $-0.018$ $-0.025$ $-0.024$ $-0.024$ $-0.030$
$-0.029$ $-0.029$ $(0.004)$ $(0.049)$ $(0.004)$ $(0.003)$ $(0.036)$ $(0.003)$
$(0.001)$ $(0.017)$ $(0.001)$ $(0.001)$ $(0.017)$ $(0.001)$ $(0.001)$
$(0.012)$ $(0.001)$ $(0.000)$ $(0.006)$ $(0.000)$ SETTING 2 N = 500 N = 1000 N
= 5000 SETTING 2 N = 500 N = 1000 N = 5000 $\tau$ $\hat{\tau}$ $\hat{\gamma}$
$\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ OR -0.23 -0.23 -0.23 OR -0.11 -0.11 -0.11 NDR
$-0.171$ $-0.160$ $-0.083$ $-0.195$ $-0.191$ $-0.124$ $-0.216$ $-0.216$
$-0.195$ NDR $-0.055$ $-0.050$ $-0.029$ $-0.066$ $-0.062$ $-0.039$ $-0.079$
$-0.079$ $-0.064$ $(0.009)$ $(0.055)$ $(0.004)$ $(0.006)$ $(0.037)$ $(0.003)$
$(0.002)$ $(0.016)$ $(0.002)$ $(0.008)$ $(0.027)$ $(0.003)$ $(0.005)$
$(0.019)$ $(0.002)$ $(0.002)$ $(0.008)$ $(0.001)$ CF $-0.178$ $-0.173$
$-0.118$ $-0.196$ $-0.192$ $-0.151$ $-0.216$ $-0.217$ $-0.198$ CF $-0.054$
$-0.050$ $-0.033$ $-0.066$ $-0.062$ $-0.044$ $-0.079$ $-0.079$ $-0.068$
$(0.009)$ $(0.053)$ $(0.005)$ $(0.006)$ $(0.037)$ $(0.004)$ $(0.002)$
$(0.016)$ $(0.002)$ $(0.008)$ $(0.027)$ $(0.003)$ $(0.005)$ $(0.019)$
$(0.002)$ $(0.002)$ $(0.008)$ $(0.001)$ CFTT $-0.178$ $-0.169$ $-0.111$
$-0.196$ $-0.192$ $-0.146$ $-0.216$ $-0.217$ $-0.200$ CFTT $-0.053$ $-0.049$
$-0.034$ $-0.066$ $-0.063$ $-0.045$ $-0.079$ $-0.079$ $-0.069$ $(0.009)$
$(0.053)$ $(0.005)$ $(0.005)$ $(0.037)$ $(0.003)$ $(0.002)$ $(0.016)$
$(0.002)$ $(0.008)$ $(0.027)$ $(0.003)$ $(0.005)$ $(0.019)$ $(0.002)$
$(0.002)$ $(0.008)$ $(0.001)$ BART $-0.169$ $-0.141$ $-0.032$ $-0.196$
$-0.178$ $-0.087$ $-0.216$ $-0.216$ $-0.194$ BART $-0.055$ $-0.045$ $-0.008$
$-0.066$ $-0.056$ $-0.010$ $-0.080$ $-0.078$ $-0.054$ $(0.009)$ $(0.046)$
$(0.002)$ $(0.005)$ $(0.033)$ $(0.002)$ $(0.002)$ $(0.015)$ $(0.002)$
$(0.008)$ $(0.024)$ $(0.001)$ $(0.005)$ $(0.018)$ $(0.001)$ $(0.002)$
$(0.008)$ $(0.001)$ SETTING 3 N = 500 N = 1000 N = 5000 SETTING 3 N = 500 N =
1000 N = 5000 OR $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.21 -0.21 -0.21 OR -0.11 -0.11 -0.11 NDR $-0.076$ $-0.064$
$-0.041$ $-0.095$ $-0.090$ $-0.050$ $-0.115$ $-0.112$ $-0.077$ NDR $-0.056$
$-0.052$ $-0.030$ $-0.066$ $-0.060$ $-0.040$ $-0.079$ $-0.078$ $-0.064$
$(0.011)$ $(0.053)$ $(0.004)$ $(0.007)$ $(0.036)$ $(0.003)$ $(0.003)$
$(0.015)$ $(0.001)$ $(0.008)$ $(0.027)$ $(0.003)$ $(0.005)$ $(0.019)$
$(0.002)$ $(0.002)$ $(0.008)$ $(0.001)$ CF $-0.085$ $-0.074$ $-0.050$ $-0.098$
$-0.093$ $-0.063$ $-0.115$ $-0.114$ $-0.092$ CF $-0.055$ $-0.050$ $-0.034$
$-0.066$ $-0.059$ $-0.045$ $-0.079$ $-0.078$ $-0.068$ $(0.011)$ $(0.048)$
$(0.003)$ $(0.007)$ $(0.033)$ $(0.002)$ $(0.003)$ $(0.014)$ $(0.002)$
$(0.008)$ $(0.027)$ $(0.003)$ $(0.005)$ $(0.019)$ $(0.002)$ $(0.002)$
$(0.008)$ $(0.001)$ CFTT $-0.084$ $-0.076$ $-0.043$ $-0.099$ $-0.093$ $-0.057$
$-0.115$ $-0.114$ $-0.092$ CFTT $-0.056$ $-0.051$ $-0.035$ $-0.065$ $-0.059$
$-0.046$ $-0.079$ $-0.078$ $-0.068$ $(0.011)$ $(0.049)$ $(0.003)$ $(0.007)$
$(0.034)$ $(0.002)$ $(0.003)$ $(0.014)$ $(0.002)$ $(0.008)$ $(0.027)$
$(0.003)$ $(0.005)$ $(0.019)$ $(0.002)$ $(0.002)$ $(0.008)$ $(0.001)$ BART
$-0.080$ $-0.060$ $-0.018$ $-0.096$ $-0.080$ $-0.026$ $-0.115$ $-0.113$
$-0.088$ BART $-0.056$ $-0.045$ $-0.008$ $-0.066$ $-0.052$ $-0.011$ $-0.079$
$-0.077$ $-0.054$ $(0.011)$ $(0.042)$ $(0.002)$ $(0.007)$ $(0.030)$ $(0.002)$
$(0.003)$ $(0.014)$ $(0.002)$ $(0.008)$ $(0.024)$ $(0.001)$ $(0.005)$
$(0.018)$ $(0.001)$ $(0.002)$ $(0.008)$ $(0.001)$ a This table reports the
true policy advantage calculated using the learned policies and the true CATEs
(column $\tau$) for the tree-based policy class, and compares them to the
estimated policy advantage calculated using both estimated CATEs
($\hat{\tau}$) and estimated DR scores ($\hat{\gamma}$). Panel A depicts
results for common outcome prevalence, and Panel B the rare outcome
prevalence.

Table A17. Comparison: True adv vs Estimated Adv (Plug-in Policy, No
Confounding)

Common Outcomes Rare OUtcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.10 -0.10 -0.10 OR -0.04 -0.04 -0.04 NDR $-0.071$ $-0.067$
$-0.107$ $-0.081$ $-0.080$ $-0.105$ $-0.093$ $-0.093$ $-0.099$ NDR $-0.028$
$-0.028$ $-0.041$ $-0.029$ $-0.029$ $-0.038$ $-0.033$ $-0.033$ $-0.036$
$(0.003)$ $(0.057)$ $(0.003)$ $(0.002)$ $(0.040)$ $(0.002)$ $(0.001)$
$(0.018)$ $(0.001)$ $(0.001)$ $(0.018)$ $(0.001)$ $(0.001)$ $(0.013)$
$(0.001)$ $(0.000)$ $(0.006)$ $(0.000)$ CF $-0.088$ $-0.089$ $-0.099$ $-0.096$
$-0.095$ $-0.099$ $-0.098$ $-0.098$ $-0.098$ CF $-0.032$ $-0.034$ $-0.037$
$-0.034$ $-0.034$ $-0.035$ $-0.035$ $-0.035$ $-0.035$ $(0.002)$ $(0.057)$
$(0.001)$ $(0.002)$ $(0.040)$ $(0.000)$ $(0.001)$ $(0.018)$ $(0.000)$
$(0.000)$ $(0.018)$ $(0.000)$ $(0.000)$ $(0.013)$ $(0.000)$ $(0.000)$
$(0.006)$ $(0.000)$ CFTT $-0.084$ $-0.083$ $-0.100$ $-0.092$ $-0.093$ $-0.100$
$-0.098$ $-0.097$ $-0.098$ CFTT $-0.031$ $-0.032$ $-0.037$ $-0.033$ $-0.033$
$-0.036$ $-0.035$ $-0.035$ $-0.035$ $(0.003)$ $(0.057)$ $(0.001)$ $(0.002)$
$(0.040)$ $(0.001)$ $(0.001)$ $(0.018)$ $(0.000)$ $(0.001)$ $(0.019)$
$(0.000)$ $(0.000)$ $(0.013)$ $(0.000)$ $(0.000)$ $(0.006)$ $(0.000)$ BART
$-0.085$ $-0.112$ $-0.093$ $-0.092$ $-0.107$ $-0.097$ $-0.098$ $-0.099$
$-0.097$ BART $-0.031$ $-0.037$ $-0.030$ $-0.034$ $-0.036$ $-0.032$ $-0.035$
$-0.035$ $-0.035$ $(0.003)$ $(0.048)$ $(0.001)$ $(0.002)$ $(0.036)$ $(0.001)$
$(0.001)$ $(0.017)$ $(0.000)$ $(0.000)$ $(0.017)$ $(0.001)$ $(0.000)$
$(0.012)$ $(0.000)$ $(0.000)$ $(0.006)$ $(0.000)$ N = 500 N = 1000 N = 5000 N
= 500 N = 1000 N = 5000 SETTING 2 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ OR -0.23 -0.23 -0.23 OR -0.11 -0.11 -0.11 NDR
$-0.171$ $-0.174$ $-0.110$ $-0.207$ $-0.206$ $-0.142$ $-0.220$ $-0.221$
$-0.203$ NDR $-0.059$ $-0.057$ $-0.048$ $-0.072$ $-0.072$ $-0.054$ $-0.084$
$-0.084$ $-0.071$ $(0.009)$ $(0.055)$ $(0.003)$ $(0.005)$ $(0.037)$ $(0.003)$
$(0.002)$ $(0.016)$ $(0.001)$ $(0.008)$ $(0.027)$ $(0.002)$ $(0.005)$
$(0.019)$ $(0.002)$ $(0.002)$ $(0.008)$ $(0.001)$ CF $-0.197$ $-0.204$
$-0.141$ $-0.216$ $-0.217$ $-0.168$ $-0.221$ $-0.222$ $-0.204$ CF $-0.063$
$-0.063$ $-0.048$ $-0.077$ $-0.076$ $-0.056$ $-0.085$ $-0.086$ $-0.073$
$(0.007)$ $(0.053)$ $(0.003)$ $(0.005)$ $(0.036)$ $(0.003)$ $(0.002)$
$(0.016)$ $(0.001)$ $(0.008)$ $(0.027)$ $(0.002)$ $(0.005)$ $(0.019)$
$(0.002)$ $(0.002)$ $(0.008)$ $(0.001)$ CFTT $-0.195$ $-0.200$ $-0.135$
$-0.215$ $-0.215$ $-0.163$ $-0.221$ $-0.223$ $-0.206$ CFTT $-0.064$ $-0.062$
$-0.050$ $-0.077$ $-0.076$ $-0.057$ $-0.085$ $-0.085$ $-0.074$ $(0.008)$
$(0.053)$ $(0.003)$ $(0.005)$ $(0.037)$ $(0.003)$ $(0.002)$ $(0.016)$
$(0.002)$ $(0.008)$ $(0.027)$ $(0.002)$ $(0.005)$ $(0.019)$ $(0.002)$
$(0.002)$ $(0.008)$ $(0.001)$ BART $-0.092$ $-0.165$ $-0.056$ $-0.206$
$-0.227$ $-0.101$ $-0.220$ $-0.232$ $-0.201$ BART $-0.018$ $-0.045$ $-0.021$
$-0.042$ $-0.062$ $-0.019$ $-0.084$ $-0.090$ $-0.060$ $(0.010)$ $(0.046)$
$(0.001)$ $(0.005)$ $(0.033)$ $(0.002)$ $(0.002)$ $(0.015)$ $(0.001)$
$(0.008)$ $(0.024)$ $(0.001)$ $(0.006)$ $(0.018)$ $(0.001)$ $(0.002)$
$(0.008)$ $(0.001)$ SETTING 3 N = 500 N = 1000 N = 5000 SETTING 3 N = 500 N =
1000 N = 5000 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.21 -0.21 -0.21 OR -0.11 -0.11 -0.11 NDR $-0.080$ $-0.079$
$-0.078$ $-0.112$ $-0.111$ $-0.081$ $-0.164$ $-0.164$ $-0.107$ NDR $-0.059$
$-0.060$ $-0.050$ $-0.072$ $-0.071$ $-0.054$ $-0.084$ $-0.084$ $-0.071$
$(0.011)$ $(0.053)$ $(0.002)$ $(0.007)$ $(0.036)$ $(0.002)$ $(0.003)$
$(0.015)$ $(0.001)$ $(0.008)$ $(0.027)$ $(0.002)$ $(0.005)$ $(0.019)$
$(0.002)$ $(0.002)$ $(0.008)$ $(0.001)$ CF $-0.089$ $-0.091$ $-0.075$ $-0.125$
$-0.126$ $-0.086$ $-0.170$ $-0.170$ $-0.124$ CF $-0.064$ $-0.066$ $-0.049$
$-0.077$ $-0.075$ $-0.057$ $-0.085$ $-0.085$ $-0.073$ $(0.011)$ $(0.048)$
$(0.002)$ $(0.007)$ $(0.033)$ $(0.002)$ $(0.003)$ $(0.014)$ $(0.001)$
$(0.008)$ $(0.027)$ $(0.002)$ $(0.005)$ $(0.019)$ $(0.002)$ $(0.002)$
$(0.008)$ $(0.001)$ CFTT $-0.079$ $-0.073$ $-0.068$ $-0.120$ $-0.120$ $-0.080$
$-0.170$ $-0.170$ $-0.124$ CFTT $-0.064$ $-0.065$ $-0.051$ $-0.077$ $-0.075$
$-0.058$ $-0.085$ $-0.085$ $-0.074$ $(0.011)$ $(0.049)$ $(0.002)$ $(0.007)$
$(0.034)$ $(0.002)$ $(0.003)$ $(0.014)$ $(0.001)$ $(0.008)$ $(0.027)$
$(0.002)$ $(0.005)$ $(0.019)$ $(0.002)$ $(0.002)$ $(0.008)$ $(0.001)$ BART
$-0.048$ $-0.099$ $-0.050$ $-0.079$ $-0.117$ $-0.058$ $-0.169$ $-0.187$
$-0.124$ BART $-0.016$ $-0.044$ $-0.023$ $-0.044$ $-0.063$ $-0.020$ $-0.084$
$-0.090$ $-0.059$ $(0.011)$ $(0.042)$ $(0.001)$ $(0.008)$ $(0.030)$ $(0.001)$
$(0.003)$ $(0.014)$ $(0.001)$ $(0.008)$ $(0.024)$ $(0.001)$ $(0.006)$
$(0.018)$ $(0.001)$ $(0.002)$ $(0.008)$ $(0.001)$ a This table reports the
true policy advantage calculated using the learned policies and the true CATEs
(column $\tau$) for the plug-in policy class, and compares them to the
estimated policy advantage calculated using both estimated CATEs
($\hat{\tau}$) and estimated DR scores ($\hat{\gamma}$). The left and right
panels depict common and rare outcome prevalence, respectively, and treatment
is assigned randomly with $P(W)=0.2$.

Figure A0. True vs Estimated Ai: Trees Learned from DR scores, No Confounding

PANEL A: Common Outcome Prevalence

Setting 1

ct-.2inNDR

(a)

ct-.2inCF

(b)

ct-.2inCFTT

(c)

ct-.2inBART

(d)

Setting 2

(e)

(f)

(g)

(h)

Setting 3

(i)

(j)

(k)

(l)

PANEL B: Rare Outcome Prevalence

Setting 1

ct-.2inNDR

(m)

ct-.2inCF

(n)

ct-.2inCFTT

(o)

ct-.2inBART

(p)

Setting 2

(q)

(r)

(s)

(t)

Setting 3

(u)

(v)

(w)

(x)

This figure depicts the values of the policy advantage of trees learned from
estimated CATEs, calculated using true CATEs $\tau$ (red line), estimated
CATEs $\hat{\tau}$ (blue dashed line), and estimated DR scores $\hat{\gamma}$
(green dotted line). The grey horizontal line is the mean value of the true
optimal (oracle) policy.  Figure A0. True vs Estimated Ai: Modified Trees
(Learned from CATEs), No Confounding

PANEL A: Common Outcome Prevalence

Setting 1

ct-.2inNDR

(a)

ct-.2inCF

(b)

ct-.2inCFTT

(c)

ct-.2inBART

(d)

Setting 2

(e)

(f)

(g)

(h)

Setting 3

(i)

(j)

(k)

(l)

PANEL B: Rare Outcome Prevalence

Setting 1

ct-.2inNDR

(m)

ct-.2inCF

(n)

ct-.2inCFTT

(o)

ct-.2inBART

(p)

Setting 2

(q)

(r)

(s)

(t)

Setting 3

(u)

(v)

(w)

(x)

This figure depicts the values of the policy advantage of trees learned from
estimated DR-scores calculated using true CATEs $\tau$ (red line), estimated
CATEs $\hat{\tau}$ (blue dashed line), and estimated DR-scores $\hat{\gamma}$
(green dotted line). The grey horizontal line is the mean value of the true
optimal (oracle) policy.  Figure A0. Comparison of CATE Histograms This figure
depicts the histograms of true and estimated CATEs for a sample dataset. The
data is generated according to Setting 3, with rare outcomes, mild
confounding, and N = 5000. Figure A1. Comparison of DR Score Histograms This
figure depicts the histograms of true and estimated double-robust scores for a
sample dataset. The data is generated according to Setting 3, with rare
outcomes, mild confounding, and N = 5000. Table A18. Ratio of True Advantages
of Learned Policies, trees/plugins

Panel A: Common Outcome Prevalence Random Treatment SETTING 1 SETTING 2
SETTING 3 N=500 N=1000 N=5000 N=500 N=1000 N=5000 N=500 N=1000 N=5000 NDR 0.46
0.56 0.83 1.00 0.94 0.98 0.95 0.85 0.70 CF 0.37 0.49 0.78 0.90 0.91 0.97 0.95
0.79 0.68 CFTT 0.39 0.50 0.78 0.91 0.91 0.98 1.06 0.82 0.68 BART 0.41 0.53
0.80 1.84 0.95 0.98 1.68 1.22 0.68 Moderate Overlap SETTING 1 SETTING 2
SETTING 3 N=500 N=1000 N=5000 N=500 N=1000 N=5000 N=500 N=1000 N=5000 NDR 0.50
0.60 0.84 0.99 0.96 0.98 1.00 0.85 0.70 CF 0.46 0.53 0.74 0.88 0.90 0.96 0.89
0.73 0.66 CFTT 0.48 0.55 0.74 0.88 0.91 0.96 0.90 0.75 0.66 BART 0.52 0.55
0.70 1.12 0.91 0.95 1.43 0.91 0.63 Panel B: Rare Outcome Prevalence Random
Treatment SETTING 1 SETTING 2 SETTING 3 N=500 N=1000 N=5000 N=500 N=1000
N=5000 N=500 N=1000 N=5000 NDR 0.79 0.83 0.90 0.94 0.91 0.94 0.96 0.92 0.94 CF
0.69 0.71 0.84 0.86 0.86 0.93 0.87 0.85 0.93 CFTT 0.71 0.74 0.84 0.84 0.85
0.93 0.87 0.85 0.93 BART 0.68 0.73 0.84 3.01 1.58 0.94 3.41 1.51 0.94 Moderate
Overlap SETTING 1 SETTING 2 SETTING 3 N=500 N=1000 N=5000 N=500 N=1000 N=5000
N=500 N=1000 N=5000 NDR 0.72 0.83 0.96 1.07 0.98 1.01 0.97 0.92 0.96 CF 0.67
0.71 0.85 0.97 0.90 0.94 1.02 0.95 1.00 CFTT 0.70 0.75 0.87 0.87 0.85 0.92
1.00 0.94 0.99 BART 0.59 0.70 0.83 1.07 1.08 0.97 7.14 1.44 0.99 a This table
reports the ratios of the true values of learned policies, i.e. the value of
the tree-based policy divided by the value of the plug-in policy. Both values
are calculated using the true simulated CATEs.

Table A19. Percentage of Oracle Policy Achieved, Continuous Outcomes

Panel A: Common Outcomes SETTING 1 N = 500 N = 1000 N = 5000 Tree M.Tree
$\hat{\tau}<0$ Tree M.Tree $\hat{\tau}<0$ Tree M.Tree $\hat{\tau}<0$ NDR 0.02
0.01 0.00 NDR 0.02 0.01 0.00 0.01 0.01 0.00 CF 0.01 0.01 0.01 CF 0.01 0.01
0.01 0.01 0.02 0.01 CFTT 0.02 0.02 0.00 CFTT 0.01 0.02 0.00 0.01 0.02 0.01
BART 0.01 0.02 0.02 BART 0.01 0.02 0.02 0.01 0.01 0.02 SETTING 2 N = 500 N =
1000 N = 5000 Tree M.Tree $\hat{\tau}<0$ Tree M.Tree $\hat{\tau}<0$ Tree
M.Tree $\hat{\tau}<0$ NDR 0.89 0.88 0.88 NDR 0.89 0.88 0.88 NDR 0.89 0.89 0.88
CF 0.88 0.88 0.88 CF 0.88 0.88 0.88 CF 0.89 0.89 0.88 CFTT 0.88 0.88 0.88 CFTT
0.88 0.88 0.88 CFTT 0.89 0.89 0.88 BART 0.89 0.88 0.88 BART 0.88 0.89 0.88
BART 0.88 0.89 0.88 SETTING 3 N = 500 N = 1000 N = 5000 Tree M.Tree
$\hat{\tau}<0$ Tree M.Tree $\hat{\tau}<0$ Tree M.Tree $\hat{\tau}<0$ NDR 0.68
0.67 0.70 NDR 0.77 0.79 0.83 NDR 0.85 0.87 0.94 CF 0.63 0.62 0.65 CF 0.72 0.74
0.78 CF 0.84 0.84 0.90 CFTT 0.63 0.60 0.62 CFTT 0.73 0.73 0.77 CFTT 0.84 0.84
0.90 BART 0.77 0.41 0.49 BART 0.81 0.64 0.74 BART 0.83 0.86 0.94 Panel A: Rare
Outcomes SETTING 1 N = 500 N = 1000 N = 5000 Tree M.Tree $\hat{\tau}<0$ Tree
M.Tree $\hat{\tau}<0$ Tree M.Tree $\hat{\tau}<0$ NDR 0.21 0.41 0.35 NDR 0.26
0.45 0.39 NDR 0.37 0.47 0.43 CF 0.23 0.41 0.40 CF 0.28 0.47 0.47 CF 0.38 0.48
0.48 CFTT 0.23 0.39 0.37 CFTT 0.27 0.44 0.42 CFTT 0.37 0.48 0.46 BART -0.05
0.37 0.32 BART -0.01 0.43 0.34 BART 0.15 0.48 0.37 SETTING 2 N = 500 N = 1000
N = 5000 Tree M.Tree $\hat{\tau}<0$ Tree M.Tree $\hat{\tau}<0$ Tree M.Tree
$\hat{\tau}<0$ NDR 0.84 0.86 0.86 NDR 0.85 0.86 0.86 NDR 0.86 0.86 0.86 CF
0.84 0.86 0.86 CF 0.85 0.86 0.86 CF 0.86 0.86 0.86 CFTT 0.84 0.86 0.86 CFTT
0.85 0.86 0.86 CFTT 0.86 0.86 0.86 BART 0.82 0.86 0.82 BART 0.83 0.86 0.82
BART 0.85 0.86 0.82 SETTING 3 N = 500 N = 1000 N = 5000 Tree M.Tree
$\hat{\tau}<0$ Tree M.Tree $\hat{\tau}<0$ Tree M.Tree $\hat{\tau}<0$ NDR 0.48
0.32 0.39 NDR 0.61 0.47 0.55 NDR 0.75 0.79 0.83 CF 0.47 0.44 0.49 CF 0.59 0.64
0.69 CF 0.76 0.80 0.88 CFTT 0.47 0.45 0.50 CFTT 0.59 0.63 0.68 CFTT 0.76 0.81
0.88 BART 0.16 0.62 0.66 BART 0.24 0.76 0.83 BART 0.55 0.82 0.95 a This table
reports the true policy advantage calculated using the learned policies and
the true CATEs, as a proportion of the oracle optimal policy. Panel A depicts
results for common outcome prevalence, and Panel B the rare outcome
prevalence. The Tree column is the percentage of the oracle advantage achieved
by the tree-based policies, the M.tree columns corresponds to our modified
policy tree learned from estimated CATEs, and the $\hat{\tau}<0$ column is the
percentage of the advantage achieved by plug-in policies. Results are for the
simulations with continuous outcomes.

Table A20. Comparison: True $A_{i}$ vs Estimated $A_{i}$ (Tree-based Policy,
Continuous Outcomes)

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.13 -0.13 -0.13 OR -0.06 -0.06 -0.06 NDR $-0.003$ $-0.002$
$-0.027$ $-0.003$ $0.004$ $-0.026$ $-0.001$ $-0.001$ $-0.020$ NDR $-0.012$
$-0.008$ $-0.017$ $-0.015$ $-0.012$ $-0.019$ $-0.021$ $-0.020$ $-0.023$
$(0.007)$ $(0.051)$ $(0.003)$ $(0.005)$ $(0.039)$ $(0.002)$ $(0.002)$
$(0.021)$ $(0.001)$ $(0.003)$ $(0.016)$ $(0.001)$ $(0.002)$ $(0.012)$
$(0.001)$ $(0.001)$ $(0.006)$ $(0.000)$ CF $-0.002$ $-0.011$ $-0.018$ $-0.002$
$-0.001$ $-0.014$ $-0.001$ $-0.000$ $-0.006$ CF $-0.014$ $-0.013$ $-0.016$
$-0.016$ $-0.014$ $-0.017$ $-0.022$ $-0.021$ $-0.022$ $(0.007)$ $(0.051)$
$(0.002)$ $(0.005)$ $(0.040)$ $(0.001)$ $(0.002)$ $(0.021)$ $(0.000)$
$(0.003)$ $(0.016)$ $(0.001)$ $(0.002)$ $(0.012)$ $(0.001)$ $(0.001)$
$(0.006)$ $(0.000)$ CFTT $-0.002$ $-0.014$ $-0.029$ $-0.002$ $-0.006$ $-0.026$
$-0.001$ $-0.005$ $-0.015$ CFTT $-0.014$ $-0.014$ $-0.019$ $-0.016$ $-0.014$
$-0.019$ $-0.022$ $-0.021$ $-0.023$ $(0.007)$ $(0.051)$ $(0.003)$ $(0.005)$
$(0.039)$ $(0.002)$ $(0.002)$ $(0.021)$ $(0.001)$ $(0.003)$ $(0.016)$
$(0.001)$ $(0.002)$ $(0.012)$ $(0.001)$ $(0.001)$ $(0.006)$ $(0.000)$ BART
$-0.002$ $-0.022$ $-0.024$ $-0.002$ $-0.010$ $-0.022$ $-0.001$ $-0.003$
$-0.013$ BART $0.003$ $-0.054$ $0.001$ $0.001$ $-0.035$ $-0.000$ $-0.009$
$-0.009$ $-0.009$ $(0.007)$ $(0.042)$ $(0.003)$ $(0.005)$ $(0.040)$ $(0.002)$
$(0.002)$ $(0.032)$ $(0.001)$ $(0.003)$ $(0.040)$ $(0.002)$ $(0.002)$
$(0.036)$ $(0.001)$ $(0.001)$ $(0.027)$ $(0.000)$ SETTING 2 N = 500 N = 1000 N
= 5000 SETTING 2 N = 500 N = 1000 N = 5000 $\tau$ $\hat{\tau}$ $\hat{\gamma}$
$\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ OR -0.17 -0.17 -0.17 OR -0.05 -0.05 -0.05 NDR
$-0.147$ $-0.143$ $-0.111$ $-0.147$ $-0.144$ $-0.129$ $-0.147$ $-0.146$
$-0.142$ NDR $-0.042$ $-0.040$ $-0.040$ $-0.042$ $-0.041$ $-0.041$ $-0.043$
$-0.042$ $-0.042$ $(0.006)$ $(0.011)$ $(0.003)$ $(0.004)$ $(0.007)$ $(0.003)$
$(0.002)$ $(0.003)$ $(0.001)$ $(0.002)$ $(0.005)$ $(0.001)$ $(0.002)$
$(0.004)$ $(0.000)$ $(0.001)$ $(0.002)$ $(0.000)$ CF $-0.147$ $-0.143$
$-0.128$ $-0.146$ $-0.145$ $-0.136$ $-0.146$ $-0.146$ $-0.144$ CF $-0.042$
$-0.041$ $-0.041$ $-0.042$ $-0.041$ $-0.042$ $-0.042$ $-0.042$ $-0.042$
$(0.006)$ $(0.010)$ $(0.004)$ $(0.004)$ $(0.007)$ $(0.003)$ $(0.002)$
$(0.003)$ $(0.001)$ $(0.002)$ $(0.005)$ $(0.001)$ $(0.002)$ $(0.004)$
$(0.000)$ $(0.001)$ $(0.002)$ $(0.000)$ CFTT $-0.146$ $-0.144$ $-0.129$
$-0.146$ $-0.145$ $-0.135$ $-0.146$ $-0.146$ $-0.143$ CFTT $-0.042$ $-0.040$
$-0.040$ $-0.042$ $-0.041$ $-0.041$ $-0.042$ $-0.042$ $-0.042$ $(0.006)$
$(0.010)$ $(0.004)$ $(0.004)$ $(0.007)$ $(0.003)$ $(0.002)$ $(0.003)$
$(0.001)$ $(0.002)$ $(0.005)$ $(0.001)$ $(0.002)$ $(0.004)$ $(0.000)$
$(0.001)$ $(0.002)$ $(0.000)$ BART $-0.147$ $-0.127$ $-0.098$ $-0.146$
$-0.138$ $-0.121$ $-0.146$ $-0.145$ $-0.141$ BART $-0.040$ $-0.035$ $-0.039$
$-0.041$ $-0.037$ $-0.041$ $-0.042$ $-0.041$ $-0.042$ $(0.006)$ $(0.009)$
$(0.003)$ $(0.004)$ $(0.007)$ $(0.002)$ $(0.002)$ $(0.004)$ $(0.001)$
$(0.002)$ $(0.005)$ $(0.001)$ $(0.002)$ $(0.004)$ $(0.001)$ $(0.001)$
$(0.003)$ $(0.000)$ SETTING 3 N = 500 N = 1000 N = 5000 SETTING 3 N = 500 N =
1000 N = 5000 OR $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.18 -0.18 -0.18 OR -0.22 -0.22 -0.22 NDR $-0.121$ $-0.090$
$-0.053$ $-0.138$ $-0.110$ $-0.071$ $-0.153$ $-0.136$ $-0.115$ NDR $-0.105$
$-0.105$ $-0.039$ $-0.132$ $-0.136$ $-0.047$ $-0.163$ $-0.165$ $-0.093$
$(0.008)$ $(0.030)$ $(0.003)$ $(0.006)$ $(0.022)$ $(0.003)$ $(0.002)$
$(0.012)$ $(0.001)$ $(0.011)$ $(0.058)$ $(0.004)$ $(0.007)$ $(0.043)$
$(0.003)$ $(0.003)$ $(0.023)$ $(0.002)$ CF $-0.113$ $-0.084$ $-0.046$ $-0.130$
$-0.101$ $-0.058$ $-0.151$ $-0.130$ $-0.091$ CF $-0.102$ $-0.104$ $-0.063$
$-0.129$ $-0.137$ $-0.088$ $-0.166$ $-0.165$ $-0.139$ $(0.008)$ $(0.031)$
$(0.002)$ $(0.006)$ $(0.024)$ $(0.002)$ $(0.002)$ $(0.013)$ $(0.001)$
$(0.011)$ $(0.058)$ $(0.004)$ $(0.007)$ $(0.044)$ $(0.003)$ $(0.003)$
$(0.023)$ $(0.002)$ CFTT $-0.113$ $-0.085$ $-0.044$ $-0.131$ $-0.100$ $-0.056$
$-0.152$ $-0.131$ $-0.091$ CFTT $-0.102$ $-0.102$ $-0.063$ $-0.128$ $-0.134$
$-0.083$ $-0.165$ $-0.165$ $-0.134$ $(0.008)$ $(0.031)$ $(0.002)$ $(0.006)$
$(0.024)$ $(0.002)$ $(0.002)$ $(0.013)$ $(0.001)$ $(0.011)$ $(0.057)$
$(0.004)$ $(0.007)$ $(0.044)$ $(0.003)$ $(0.003)$ $(0.023)$ $(0.002)$ BART
$-0.138$ $-0.113$ $-0.022$ $-0.145$ $-0.128$ $-0.046$ $-0.150$ $-0.148$
$-0.105$ BART $-0.034$ $-0.153$ $-0.012$ $-0.053$ $-0.116$ $-0.028$ $-0.120$
$-0.108$ $-0.102$ $(0.008)$ $(0.026)$ $(0.002)$ $(0.005)$ $(0.023)$ $(0.002)$
$(0.002)$ $(0.017)$ $(0.002)$ $(0.012)$ $(0.096)$ $(0.005)$ $(0.008)$
$(0.084)$ $(0.005)$ $(0.003)$ $(0.064)$ $(0.003)$ a This table reports the
true policy advantage calculated using the learned policies and the true CATEs
(column $\tau$) for the tree-based policy class, and compares them to the
estimated policy advantage calculated using both estimated CATEs
($\hat{\tau}$) and estimated DR scores ($\hat{\gamma}$). Panel A depicts
results for common outcome prevalence, and Panel B the rare outcome
prevalence.

Table A21. Comparison: True adv vs Estimated Adv (Plug-in Policy, Continuous
Outcomes)

Common Outcomes Rare OUtcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.13 -0.13 -0.13 OR -0.06 -0.06 -0.06 NDR $-0.000$ $0.003$
$-0.065$ $-0.000$ $-0.000$ $-0.059$ $-0.001$ $-0.001$ $-0.048$ NDR $-0.021$
$-0.021$ $-0.031$ $-0.023$ $-0.022$ $-0.030$ $-0.025$ $-0.025$ $-0.029$
$(0.007)$ $(0.051)$ $(0.002)$ $(0.005)$ $(0.039)$ $(0.001)$ $(0.002)$
$(0.021)$ $(0.001)$ $(0.003)$ $(0.016)$ $(0.001)$ $(0.002)$ $(0.012)$
$(0.001)$ $(0.001)$ $(0.006)$ $(0.000)$ CF $-0.001$ $-0.007$ $-0.046$ $-0.001$
$-0.015$ $-0.035$ $-0.002$ $-0.006$ $-0.017$ CF $-0.024$ $-0.027$ $-0.031$
$-0.027$ $-0.029$ $-0.029$ $-0.028$ $-0.029$ $-0.028$ $(0.007)$ $(0.051)$
$(0.000)$ $(0.005)$ $(0.040)$ $(0.000)$ $(0.002)$ $(0.021)$ $(0.000)$
$(0.003)$ $(0.016)$ $(0.000)$ $(0.002)$ $(0.012)$ $(0.000)$ $(0.001)$
$(0.006)$ $(0.000)$ CFTT $-0.000$ $-0.012$ $-0.056$ $-0.001$ $-0.010$ $-0.047$
$-0.001$ $-0.006$ $-0.033$ CFTT $-0.021$ $-0.024$ $-0.032$ $-0.024$ $-0.025$
$-0.030$ $-0.027$ $-0.028$ $-0.028$ $(0.007)$ $(0.051)$ $(0.001)$ $(0.005)$
$(0.039)$ $(0.001)$ $(0.002)$ $(0.021)$ $(0.000)$ $(0.003)$ $(0.016)$
$(0.001)$ $(0.002)$ $(0.012)$ $(0.000)$ $(0.001)$ $(0.006)$ $(0.000)$ BART
$-0.002$ $-0.096$ $-0.056$ $-0.002$ $-0.090$ $-0.051$ $-0.002$ $-0.064$
$-0.039$ BART $-0.018$ $0.004$ $-0.034$ $-0.020$ $0.002$ $-0.032$ $-0.022$
$-0.014$ $-0.031$ $(0.007)$ $(0.042)$ $(0.001)$ $(0.005)$ $(0.040)$ $(0.001)$
$(0.002)$ $(0.032)$ $(0.000)$ $(0.003)$ $(0.040)$ $(0.001)$ $(0.002)$
$(0.036)$ $(0.001)$ $(0.001)$ $(0.027)$ $(0.000)$ N = 500 N = 1000 N = 5000 N
= 500 N = 1000 N = 5000 SETTING 2 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$
$\hat{\tau}$ $\hat{\gamma}$ OR -0.17 -0.17 -0.17 OR -0.05 -0.05 -0.05 NDR
$-0.146$ $-0.147$ $-0.114$ $-0.146$ $-0.146$ $-0.131$ $-0.146$ $-0.146$
$-0.142$ NDR $-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$
$-0.043$ $-0.043$ $(0.006)$ $(0.011)$ $(0.003)$ $(0.005)$ $(0.007)$ $(0.002)$
$(0.002)$ $(0.003)$ $(0.001)$ $(0.002)$ $(0.005)$ $(0.000)$ $(0.002)$
$(0.004)$ $(0.000)$ $(0.001)$ $(0.002)$ $(0.000)$ CF $-0.145$ $-0.146$
$-0.131$ $-0.146$ $-0.146$ $-0.137$ $-0.146$ $-0.147$ $-0.144$ CF $-0.043$
$-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$
$(0.006)$ $(0.010)$ $(0.003)$ $(0.005)$ $(0.007)$ $(0.003)$ $(0.002)$
$(0.003)$ $(0.001)$ $(0.002)$ $(0.005)$ $(0.000)$ $(0.002)$ $(0.004)$
$(0.000)$ $(0.001)$ $(0.002)$ $(0.000)$ CFTT $-0.145$ $-0.147$ $-0.132$
$-0.146$ $-0.146$ $-0.137$ $-0.146$ $-0.147$ $-0.144$ CFTT $-0.043$ $-0.043$
$-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$ $-0.043$ $(0.006)$
$(0.010)$ $(0.003)$ $(0.005)$ $(0.007)$ $(0.003)$ $(0.002)$ $(0.003)$
$(0.001)$ $(0.002)$ $(0.005)$ $(0.000)$ $(0.002)$ $(0.004)$ $(0.000)$
$(0.001)$ $(0.002)$ $(0.000)$ BART $-0.146$ $-0.131$ $-0.101$ $-0.146$
$-0.140$ $-0.124$ $-0.146$ $-0.146$ $-0.142$ BART $-0.041$ $-0.037$ $-0.042$
$-0.041$ $-0.038$ $-0.043$ $-0.041$ $-0.040$ $-0.043$ $(0.006)$ $(0.009)$
$(0.002)$ $(0.005)$ $(0.007)$ $(0.002)$ $(0.002)$ $(0.004)$ $(0.001)$
$(0.002)$ $(0.005)$ $(0.001)$ $(0.002)$ $(0.004)$ $(0.001)$ $(0.001)$
$(0.003)$ $(0.000)$ SETTING 3 N = 500 N = 1000 N = 5000 SETTING 3 N = 500 N =
1000 N = 5000 $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ $\tau$ $\hat{\tau}$ $\hat{\gamma}$ $\tau$ $\hat{\tau}$
$\hat{\gamma}$ OR -0.18 -0.18 -0.18 OR -0.22 -0.22 -0.22 NDR $-0.126$ $-0.103$
$-0.078$ $-0.149$ $-0.126$ $-0.090$ $-0.169$ $-0.155$ $-0.126$ NDR $-0.084$
$-0.082$ $-0.083$ $-0.119$ $-0.123$ $-0.087$ $-0.181$ $-0.182$ $-0.121$
$(0.008)$ $(0.030)$ $(0.002)$ $(0.005)$ $(0.022)$ $(0.002)$ $(0.002)$
$(0.012)$ $(0.001)$ $(0.011)$ $(0.058)$ $(0.003)$ $(0.008)$ $(0.044)$
$(0.002)$ $(0.003)$ $(0.023)$ $(0.001)$ CF $-0.117$ $-0.094$ $-0.061$ $-0.141$
$-0.115$ $-0.070$ $-0.163$ $-0.145$ $-0.100$ CF $-0.106$ $-0.114$ $-0.099$
$-0.150$ $-0.161$ $-0.117$ $-0.191$ $-0.191$ $-0.163$ $(0.008)$ $(0.031)$
$(0.002)$ $(0.005)$ $(0.024)$ $(0.001)$ $(0.002)$ $(0.013)$ $(0.001)$
$(0.011)$ $(0.058)$ $(0.003)$ $(0.007)$ $(0.044)$ $(0.002)$ $(0.003)$
$(0.023)$ $(0.001)$ CFTT $-0.112$ $-0.088$ $-0.059$ $-0.139$ $-0.113$ $-0.069$
$-0.162$ $-0.145$ $-0.100$ CFTT $-0.109$ $-0.110$ $-0.098$ $-0.147$ $-0.156$
$-0.114$ $-0.191$ $-0.192$ $-0.159$ $(0.008)$ $(0.031)$ $(0.002)$ $(0.005)$
$(0.024)$ $(0.001)$ $(0.002)$ $(0.012)$ $(0.001)$ $(0.011)$ $(0.057)$
$(0.003)$ $(0.007)$ $(0.044)$ $(0.002)$ $(0.003)$ $(0.023)$ $(0.001)$ BART
$-0.088$ $-0.116$ $-0.049$ $-0.133$ $-0.151$ $-0.069$ $-0.169$ $-0.183$
$-0.131$ BART $-0.144$ $-0.168$ $-0.100$ $-0.180$ $-0.202$ $-0.134$ $-0.206$
$-0.213$ $-0.189$ $(0.009)$ $(0.026)$ $(0.001)$ $(0.006)$ $(0.023)$ $(0.002)$
$(0.002)$ $(0.017)$ $(0.001)$ $(0.010)$ $(0.096)$ $(0.003)$ $(0.006)$
$(0.084)$ $(0.003)$ $(0.002)$ $(0.064)$ $(0.002)$ a This table reports the
true policy advantage calculated using the learned policies and the true CATEs
(column $\tau$) for the plug-in policy class, and compares them to the
estimated policy advantage calculated using both estimated CATEs
($\hat{\tau}$) and estimated DR scores ($\hat{\gamma}$). The left and right
panels depict common and rare outcome prevalence, respectively.

Table A22. RMSE of True Policy Advantages: Tree-based Policies (SD),
Continuous Outcomes

Panel A: Common Outcomes Plugin Policy SETTING 1 SETTING 2 SETTING 3 N = 500 N
= 1000 N = 5000 N = 500 N = 1000 N = 5000 N = 500 N = 1000 N = 5000 NDR
$0.131$ $0.131$ $0.130$ $0.020$ $0.020$ $0.019$ $0.058$ $0.033$ $0.011$
$(0.009)$ $(0.007)$ $(0.003)$ $(0.003)$ $(0.002)$ $(0.001)$ $(0.021)$
$(0.013)$ $(0.002)$ CF $0.131$ $0.130$ $0.129$ $0.021$ $0.020$ $0.019$ $0.073$
$0.044$ $0.018$ $(0.010)$ $(0.008)$ $(0.006)$ $(0.003)$ $(0.002)$ $(0.001)$
$(0.038)$ $(0.021)$ $(0.005)$ CFTT $0.131$ $0.130$ $0.130$ $0.021$ $0.020$
$0.019$ $0.076$ $0.046$ $0.018$ $(0.009)$ $(0.007)$ $(0.003)$ $(0.003)$
$(0.002)$ $(0.001)$ $(0.036)$ $(0.021)$ $(0.005)$ BART $0.129$ $0.129$ $0.128$
$0.020$ $0.020$ $0.020$ $0.100$ $0.051$ $0.011$ $(0.010)$ $(0.007)$ $(0.003)$
$(0.003)$ $(0.002)$ $(0.001)$ $(0.040)$ $(0.021)$ $(0.004)$ Tree Policy
SETTING 1 SETTING 2 SETTING 3 N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 N = 500 N = 1000 N = 5000 NDR $0.128$ $0.128$ $0.129$ $0.019$ $0.019$
$0.019$ $0.063$ $0.045$ $0.028$ $(0.008)$ $(0.006)$ $(0.003)$ $(0.003)$
$(0.002)$ $(0.001)$ $(0.024)$ $(0.016)$ $(0.005)$ CF $0.129$ $0.129$ $0.129$
$0.019$ $0.019$ $0.019$ $0.074$ $0.055$ $0.030$ $(0.008)$ $(0.006)$ $(0.003)$
$(0.003)$ $(0.002)$ $(0.001)$ $(0.032)$ $(0.024)$ $(0.007)$ CFTT $0.129$
$0.129$ $0.129$ $0.019$ $0.019$ $0.019$ $0.074$ $0.054$ $0.029$ $(0.008)$
$(0.006)$ $(0.003)$ $(0.003)$ $(0.002)$ $(0.001)$ $(0.031)$ $(0.023)$
$(0.006)$ BART $0.129$ $0.129$ $0.129$ $0.019$ $0.019$ $0.019$ $0.045$ $0.037$
$0.031$ $(0.008)$ $(0.006)$ $(0.003)$ $(0.003)$ $(0.002)$ $(0.001)$ $(0.017)$
$(0.012)$ $(0.008)$ Modified Tree Policy SETTING 1 SETTING 2 SETTING 3 N = 500
N = 1000 N = 5000 N = 500 N = 1000 N = 5000 N = 500 N = 1000 N = 5000 NDR
$0.129$ $0.129$ $0.129$ $0.020$ $0.019$ $0.019$ $0.065$ $0.041$ $0.023$
$(0.009)$ $(0.007)$ $(0.004)$ $(0.003)$ $(0.002)$ $(0.001)$ $(0.026)$
$(0.015)$ $(0.003)$ CF $0.130$ $0.129$ $0.128$ $0.020$ $0.020$ $0.019$ $0.081$
$0.053$ $0.030$ $(0.010)$ $(0.008)$ $(0.006)$ $(0.003)$ $(0.002)$ $(0.001)$
$(0.043)$ $(0.027)$ $(0.008)$ CFTT $0.129$ $0.129$ $0.128$ $0.020$ $0.020$
$0.019$ $0.084$ $0.055$ $0.030$ $(0.009)$ $(0.007)$ $(0.004)$ $(0.003)$
$(0.002)$ $(0.001)$ $(0.042)$ $(0.027)$ $(0.008)$ BART $0.129$ $0.129$ $0.129$
$0.019$ $0.019$ $0.019$ $0.117$ $0.072$ $0.026$ $(0.010)$ $(0.007)$ $(0.004)$
$(0.003)$ $(0.002)$ $(0.001)$ $(0.049)$ $(0.033)$ $(0.006)$ Panel B: Rare
Outcomes Plugin Policy SETTING 1 SETTING 2 SETTING 3 N = 500 N = 1000 N = 5000
N = 500 N = 1000 N = 5000 NDR $0.039$ $0.036$ $0.034$ $0.007$ $0.007$ $0.007$
$0.137$ $0.102$ $0.037$ $(0.010)$ $(0.007)$ $(0.003)$ $(0.001)$ $(0.001)$
$(0.000)$ $(0.034)$ $(0.026)$ $(0.010)$ CF $0.038$ $0.031$ $0.030$ $0.007$
$0.007$ $0.007$ $0.125$ $0.076$ $0.028$ $(0.014)$ $(0.006)$ $(0.001)$
$(0.001)$ $(0.001)$ $(0.000)$ $(0.056)$ $(0.036)$ $(0.008)$ CFTT $0.039$
$0.035$ $0.031$ $0.007$ $0.007$ $0.007$ $0.117$ $0.077$ $0.027$ $(0.012)$
$(0.007)$ $(0.002)$ $(0.001)$ $(0.001)$ $(0.000)$ $(0.046)$ $(0.032)$
$(0.007)$ BART $0.042$ $0.039$ $0.037$ $0.009$ $0.009$ $0.009$ $0.084$ $0.042$
$0.012$ $(0.012)$ $(0.007)$ $(0.003)$ $(0.002)$ $(0.002)$ $(0.001)$ $(0.042)$
$(0.018)$ $(0.003)$ Tree-Based Policy SETTING 1 SETTING 2 SETTING 3 N = 500 N
= 1000 N = 5000 N = 500 N = 1000 N = 5000 NDR $0.047$ $0.044$ $0.037$ $0.008$
$0.008$ $0.007$ $0.121$ $0.091$ $0.057$ $(0.008)$ $(0.007)$ $(0.004)$
$(0.002)$ $(0.001)$ $(0.001)$ $(0.046)$ $(0.032)$ $(0.017)$ CF $0.045$ $0.043$
$0.037$ $0.008$ $0.008$ $0.007$ $0.124$ $0.095$ $0.054$ $(0.009)$ $(0.007)$
$(0.004)$ $(0.002)$ $(0.001)$ $(0.001)$ $(0.047)$ $(0.036)$ $(0.016)$ CFTT
$0.046$ $0.043$ $0.037$ $0.008$ $0.008$ $0.007$ $0.124$ $0.096$ $0.055$
$(0.009)$ $(0.007)$ $(0.004)$ $(0.002)$ $(0.001)$ $(0.001)$ $(0.046)$
$(0.036)$ $(0.016)$ BART $0.062$ $0.059$ $0.050$ $0.010$ $0.009$ $0.008$
$0.186$ $0.169$ $0.104$ $(0.007)$ $(0.007)$ $(0.006)$ $(0.003)$ $(0.002)$
$(0.002)$ $(0.030)$ $(0.040)$ $(0.038)$ Modified Tree Policy SETTING 1 SETTING
2 SETTING 3 N = 500 N = 1000 N = 5000 N = 500 N = 1000 N = 5000 N = 500 N =
1000 N = 5000 NDR $0.036$ $0.033$ $0.031$ $0.007$ $0.007$ $0.007$ $0.154$
$0.124$ $0.049$ $(0.010)$ $(0.005)$ $(0.002)$ $(0.001)$ $(0.001)$ $(0.000)$
$(0.046)$ $(0.044)$ $(0.016)$ CF $0.037$ $0.031$ $0.030$ $0.007$ $0.007$
$0.007$ $0.136$ $0.089$ $0.043$ $(0.014)$ $(0.006)$ $(0.001)$ $(0.001)$
$(0.001)$ $(0.000)$ $(0.060)$ $(0.043)$ $(0.009)$ CFTT $0.038$ $0.033$ $0.030$
$0.007$ $0.007$ $0.007$ $0.132$ $0.091$ $0.043$ $(0.012)$ $(0.006)$ $(0.001)$
$(0.001)$ $(0.001)$ $(0.000)$ $(0.055)$ $(0.042)$ $(0.009)$ BART $0.039$
$0.034$ $0.031$ $0.007$ $0.007$ $0.007$ $0.098$ $0.056$ $0.040$ $(0.014)$
$(0.007)$ $(0.002)$ $(0.001)$ $(0.001)$ $(0.000)$ $(0.054)$ $(0.021)$
$(0.007)$ a This table reports RMSE of true values of the learned policies,
for continuous outcomes. The error is calculated as the difference between the
true value of the learned policy and the best possible (oracle) policy
advantage.

Table A23. Comparison: RMSE of Estimated Policy Advantages, Plug-in Policy,
Continuous Outcomes

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR $0.063$ $0.048$ $0.044$ $0.034$
$0.019$ $0.016$ NDR $0.020$ $0.017$ $0.014$ $0.011$ $0.006$ $0.005$ $(0.063)$
$(0.031)$ $(0.045)$ $(0.024)$ $(0.019)$ $(0.014)$ $(0.020)$ $(0.011)$
$(0.014)$ $(0.007)$ $(0.006)$ $(0.004)$ CF $0.085$ $0.054$ $0.053$ $0.040$
$0.019$ $0.019$ CF $0.021$ $0.018$ $0.015$ $0.013$ $0.006$ $0.006$ $(0.085)$
$(0.053)$ $(0.053)$ $(0.040)$ $(0.019)$ $(0.019)$ $(0.021)$ $(0.018)$
$(0.015)$ $(0.013)$ $(0.006)$ $(0.006)$ CFTT $0.066$ $0.047$ $0.044$ $0.034$
$0.019$ $0.018$ CFTT $0.020$ $0.015$ $0.014$ $0.011$ $0.006$ $0.006$ $(0.066)$
$(0.044)$ $(0.044)$ $(0.033)$ $(0.019)$ $(0.018)$ $(0.020)$ $(0.014)$
$(0.014)$ $(0.010)$ $(0.006)$ $(0.006)$ BART $0.061$ $0.046$ $0.038$ $0.033$
$0.018$ $0.018$ BART $0.019$ $0.015$ $0.013$ $0.011$ $0.006$ $0.006$ $(0.054)$
$(0.046)$ $(0.035)$ $(0.033)$ $(0.018)$ $(0.018)$ $(0.017)$ $(0.015)$
$(0.013)$ $(0.011)$ $(0.006)$ $(0.006)$ SETTING 2 $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR
$0.060$ $0.073$ $0.038$ $0.069$ $0.016$ $0.023$ NDR $0.032$ $0.022$ $0.020$
$0.021$ $0.008$ $0.014$ $(0.060)$ $(0.042)$ $(0.038)$ $(0.024)$ $(0.016)$
$(0.014)$ $(0.032)$ $(0.019)$ $(0.020)$ $(0.010)$ $(0.008)$ $(0.006)$ CF
$0.063$ $0.076$ $0.040$ $0.057$ $0.017$ $0.023$ CF $0.034$ $0.030$ $0.021$
$0.025$ $0.009$ $0.014$ $(0.062)$ $(0.051)$ $(0.040)$ $(0.032)$ $(0.017)$
$(0.015)$ $(0.034)$ $(0.026)$ $(0.021)$ $(0.013)$ $(0.009)$ $(0.007)$ CFTT
$0.064$ $0.079$ $0.039$ $0.061$ $0.016$ $0.021$ CFTT $0.034$ $0.027$ $0.021$
$0.023$ $0.009$ $0.013$ $(0.064)$ $(0.051)$ $(0.039)$ $(0.031)$ $(0.016)$
$(0.015)$ $(0.034)$ $(0.024)$ $(0.021)$ $(0.013)$ $(0.009)$ $(0.007)$ BART
$0.083$ $0.088$ $0.038$ $0.110$ $0.019$ $0.024$ BART $0.033$ $0.030$ $0.025$
$0.036$ $0.009$ $0.026$ $(0.038)$ $(0.081)$ $(0.031)$ $(0.034)$ $(0.015)$
$(0.014)$ $(0.020)$ $(0.030)$ $(0.015)$ $(0.028)$ $(0.008)$ $(0.007)$ SETTING
3 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ NDR $0.060$ $0.037$ $0.039$ $0.041$ $0.016$
$0.058$ NDR $0.031$ $0.022$ $0.021$ $0.021$ $0.009$ $0.015$ $(0.060)$
$(0.037)$ $(0.039)$ $(0.027)$ $(0.016)$ $(0.012)$ $(0.031)$ $(0.020)$
$(0.021)$ $(0.011)$ $(0.009)$ $(0.006)$ CF $0.062$ $0.047$ $0.041$ $0.051$
$0.016$ $0.048$ CF $0.034$ $0.029$ $0.021$ $0.025$ $0.009$ $0.014$ $(0.062)$
$(0.045)$ $(0.041)$ $(0.033)$ $(0.016)$ $(0.015)$ $(0.034)$ $(0.025)$
$(0.021)$ $(0.015)$ $(0.009)$ $(0.007)$ CFTT $0.065$ $0.047$ $0.040$ $0.052$
$0.016$ $0.048$ CFTT $0.033$ $0.027$ $0.021$ $0.024$ $0.009$ $0.013$ $(0.064)$
$(0.046)$ $(0.040)$ $(0.034)$ $(0.016)$ $(0.014)$ $(0.033)$ $(0.023)$
$(0.021)$ $(0.014)$ $(0.009)$ $(0.007)$ BART $0.063$ $0.044$ $0.045$ $0.043$
$0.023$ $0.046$ BART $0.034$ $0.031$ $0.024$ $0.036$ $0.009$ $0.026$ $(0.036)$
$(0.044)$ $(0.025)$ $(0.037)$ $(0.014)$ $(0.012)$ $(0.020)$ $(0.031)$
$(0.015)$ $(0.028)$ $(0.008)$ $(0.007)$ a This table reports the RMSE between
the estimated values of the learned policy, calculated using both cates and
scores. The error is defined as the difference between the true value of the
learned policy (calculated using true cates) and the estimated value
(calculated using either estimated cates $\hat{\tau}$ or scores
$\hat{\gamma}$.

Table A24. Comparison: RMSE of Estimated Policy Advantages, Tree Policy,
Continuous Outcomes

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR $0.077$ $0.030$ $0.059$ $0.028$
$0.033$ $0.020$ NDR $0.025$ $0.009$ $0.018$ $0.007$ $0.009$ $0.005$ $(0.077)$
$(0.018)$ $(0.058)$ $(0.015)$ $(0.033)$ $(0.008)$ $(0.025)$ $(0.008)$
$(0.018)$ $(0.007)$ $(0.008)$ $(0.005)$ CF $0.075$ $0.027$ $0.060$ $0.021$
$0.033$ $0.009$ CF $0.024$ $0.010$ $0.018$ $0.008$ $0.008$ $0.005$ $(0.075)$
$(0.022)$ $(0.060)$ $(0.017)$ $(0.033)$ $(0.008)$ $(0.024)$ $(0.010)$
$(0.018)$ $(0.008)$ $(0.008)$ $(0.005)$ CFTT $0.079$ $0.034$ $0.060$ $0.029$
$0.033$ $0.016$ CFTT $0.025$ $0.011$ $0.019$ $0.008$ $0.009$ $0.005$ $(0.078)$
$(0.020)$ $(0.059)$ $(0.015)$ $(0.033)$ $(0.007)$ $(0.025)$ $(0.009)$
$(0.019)$ $(0.007)$ $(0.009)$ $(0.005)$ BART $0.067$ $0.033$ $0.063$ $0.027$
$0.048$ $0.015$ BART $0.078$ $0.007$ $0.061$ $0.006$ $0.036$ $0.005$ $(0.064)$
$(0.024)$ $(0.063)$ $(0.018)$ $(0.048)$ $(0.009)$ $(0.053)$ $(0.007)$
$(0.050)$ $(0.006)$ $(0.036)$ $(0.005)$ SETTING 2 $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR
$0.010$ $0.037$ $0.007$ $0.019$ $0.003$ $0.006$ NDR $0.005$ $0.005$ $0.004$
$0.004$ $0.002$ $0.002$ $(0.009)$ $(0.007)$ $(0.006)$ $(0.005)$ $(0.003)$
$(0.002)$ $(0.005)$ $(0.004)$ $(0.003)$ $(0.004)$ $(0.002)$ $(0.002)$ CF
$0.010$ $0.021$ $0.008$ $0.013$ $0.003$ $0.004$ CF $0.005$ $0.005$ $0.004$
$0.003$ $0.002$ $0.001$ $(0.010)$ $(0.010)$ $(0.007)$ $(0.007)$ $(0.003)$
$(0.003)$ $(0.005)$ $(0.004)$ $(0.004)$ $(0.003)$ $(0.002)$ $(0.001)$ CFTT
$0.010$ $0.020$ $0.007$ $0.013$ $0.003$ $0.005$ CFTT $0.005$ $0.004$ $0.004$
$0.003$ $0.002$ $0.002$ $(0.009)$ $(0.010)$ $(0.007)$ $(0.007)$ $(0.003)$
$(0.003)$ $(0.005)$ $(0.004)$ $(0.004)$ $(0.003)$ $(0.002)$ $(0.001)$ BART
$0.022$ $0.050$ $0.011$ $0.026$ $0.004$ $0.006$ BART $0.008$ $0.005$ $0.007$
$0.004$ $0.004$ $0.002$ $(0.009)$ $(0.008)$ $(0.007)$ $(0.005)$ $(0.004)$
$(0.002)$ $(0.006)$ $(0.005)$ $(0.005)$ $(0.004)$ $(0.004)$ $(0.002)$ SETTING
3 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ NDR $0.055$ $0.073$ $0.042$ $0.069$ $0.022$
$0.039$ NDR $0.094$ $0.082$ $0.068$ $0.093$ $0.032$ $0.073$ $(0.045)$
$(0.024)$ $(0.030)$ $(0.017)$ $(0.015)$ $(0.012)$ $(0.094)$ $(0.049)$
$(0.068)$ $(0.037)$ $(0.032)$ $(0.021)$ CF $0.059$ $0.074$ $0.047$ $0.077$
$0.028$ $0.062$ CF $0.090$ $0.063$ $0.069$ $0.059$ $0.034$ $0.036$ $(0.052)$
$(0.033)$ $(0.036)$ $(0.025)$ $(0.018)$ $(0.013)$ $(0.090)$ $(0.050)$
$(0.068)$ $(0.043)$ $(0.034)$ $(0.025)$ CFTT $0.056$ $0.077$ $0.048$ $0.079$
$0.027$ $0.062$ CFTT $0.085$ $0.061$ $0.071$ $0.062$ $0.034$ $0.040$ $(0.049)$
$(0.033)$ $(0.036)$ $(0.024)$ $(0.018)$ $(0.013)$ $(0.086)$ $(0.047)$
$(0.071)$ $(0.042)$ $(0.034)$ $(0.025)$ BART $0.043$ $0.118$ $0.035$ $0.101$
$0.022$ $0.046$ BART $0.182$ $0.039$ $0.142$ $0.040$ $0.091$ $0.030$ $(0.035)$
$(0.021)$ $(0.031)$ $(0.017)$ $(0.022)$ $(0.013)$ $(0.138)$ $(0.032)$
$(0.127)$ $(0.031)$ $(0.090)$ $(0.024)$ a This table reports the RMSE between
the estimated values of the learned policy, calculated using both cates and
scores. The error is defined as the difference between the true value of the
learned policy (calculated using true cates) and the estimated value
(calculated using either estimated cates $\hat{\tau}$ or scores
$\hat{\gamma}$.

Table A25. Comparison: RMSE of Estimated Policy Advantages, Modified Trees,
Continuous Outcomes

Common Outcomes Rare Outcomes N = 500 N = 1000 N = 5000 N = 500 N = 1000 N =
5000 SETTING 1 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR $0.099$ $0.055$ $0.087$ $0.046$
$0.061$ $0.029$ NDR $0.018$ $0.013$ $0.012$ $0.009$ $0.005$ $0.006$ $(0.037)$
$(0.027)$ $(0.028)$ $(0.021)$ $(0.015)$ $(0.010)$ $(0.014)$ $(0.012)$
$(0.010)$ $(0.009)$ $(0.005)$ $(0.006)$ CF $0.080$ $0.054$ $0.064$ $0.041$
$0.030$ $0.018$ CF $0.020$ $0.019$ $0.014$ $0.013$ $0.006$ $0.006$ $(0.057)$
$(0.033)$ $(0.044)$ $(0.024)$ $(0.023)$ $(0.012)$ $(0.019)$ $(0.017)$
$(0.014)$ $(0.013)$ $(0.006)$ $(0.006)$ CFTT $0.107$ $0.056$ $0.092$ $0.044$
$0.056$ $0.023$ CFTT $0.023$ $0.016$ $0.015$ $0.011$ $0.006$ $0.006$ $(0.043)$
$(0.029)$ $(0.035)$ $(0.020)$ $(0.018)$ $(0.008)$ $(0.018)$ $(0.014)$
$(0.013)$ $(0.010)$ $(0.006)$ $(0.006)$ BART $0.084$ $0.057$ $0.080$ $0.047$
$0.055$ $0.031$ BART $0.059$ $0.017$ $0.052$ $0.011$ $0.026$ $0.006$ $(0.040)$
$(0.031)$ $(0.044)$ $(0.021)$ $(0.034)$ $(0.010)$ $(0.047)$ $(0.015)$
$(0.035)$ $(0.010)$ $(0.020)$ $(0.006)$ SETTING 2 $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ NDR
$0.008$ $0.033$ $0.005$ $0.017$ $0.002$ $0.005$ NDR $0.004$ $0.004$ $0.003$
$0.004$ $0.002$ $0.002$ $(0.008)$ $(0.007)$ $(0.005)$ $(0.005)$ $(0.002)$
$(0.002)$ $(0.004)$ $(0.004)$ $(0.003)$ $(0.004)$ $(0.002)$ $(0.002)$ CF
$0.009$ $0.018$ $0.007$ $0.011$ $0.003$ $0.004$ CF $0.005$ $0.004$ $0.004$
$0.003$ $0.002$ $0.001$ $(0.009)$ $(0.010)$ $(0.007)$ $(0.007)$ $(0.003)$
$(0.003)$ $(0.005)$ $(0.004)$ $(0.004)$ $(0.003)$ $(0.002)$ $(0.001)$ CFTT
$0.009$ $0.017$ $0.007$ $0.012$ $0.003$ $0.004$ CFTT $0.005$ $0.004$ $0.004$
$0.003$ $0.002$ $0.001$ $(0.009)$ $(0.010)$ $(0.006)$ $(0.007)$ $(0.003)$
$(0.003)$ $(0.005)$ $(0.004)$ $(0.004)$ $(0.003)$ $(0.002)$ $(0.001)$ BART
$0.018$ $0.047$ $0.009$ $0.024$ $0.004$ $0.006$ BART $0.008$ $0.005$ $0.007$
$0.004$ $0.004$ $0.002$ $(0.008)$ $(0.008)$ $(0.006)$ $(0.005)$ $(0.004)$
$(0.002)$ $(0.006)$ $(0.005)$ $(0.006)$ $(0.004)$ $(0.004)$ $(0.002)$ SETTING
3 $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$
$\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$ $\hat{\tau}$ $\hat{\gamma}$
$\hat{\tau}$ $\hat{\gamma}$ NDR $0.030$ $0.064$ $0.025$ $0.064$ $0.016$
$0.039$ NDR $0.103$ $0.063$ $0.076$ $0.063$ $0.030$ $0.071$ $(0.030)$
$(0.041)$ $(0.022)$ $(0.026)$ $(0.013)$ $(0.013)$ $(0.051)$ $(0.063)$
$(0.041)$ $(0.053)$ $(0.023)$ $(0.022)$ CF $0.043$ $0.071$ $0.028$ $0.074$
$0.018$ $0.058$ CF $0.098$ $0.068$ $0.074$ $0.061$ $0.028$ $0.033$ $(0.039)$
$(0.046)$ $(0.028)$ $(0.029)$ $(0.015)$ $(0.013)$ $(0.056)$ $(0.068)$
$(0.048)$ $(0.053)$ $(0.024)$ $(0.026)$ CFTT $0.042$ $0.070$ $0.027$ $0.074$
$0.017$ $0.058$ CFTT $0.100$ $0.067$ $0.075$ $0.061$ $0.028$ $0.037$ $(0.037)$
$(0.046)$ $(0.027)$ $(0.029)$ $(0.015)$ $(0.013)$ $(0.055)$ $(0.067)$
$(0.045)$ $(0.052)$ $(0.024)$ $(0.025)$ BART $0.037$ $0.064$ $0.027$ $0.067$
$0.020$ $0.044$ BART $0.107$ $0.079$ $0.072$ $0.053$ $0.061$ $0.021$ $(0.031)$
$(0.056)$ $(0.025)$ $(0.033)$ $(0.019)$ $(0.012)$ $(0.107)$ $(0.068)$
$(0.072)$ $(0.034)$ $(0.061)$ $(0.016)$ a This table reports the RMSE between
the estimated values of the learned policy, calculated using both cates and
scores. The error is defined as the difference between the true value of the
learned policy (calculated using true cates) and the estimated value
(calculated using either estimated cates $\hat{\tau}$ or scores
$\hat{\gamma}$.