Table 3.

WAIC model fits and standard errors for all models, based on hierarchical Bayesian fitting. Bold numbers highlight the winning model of each class. For the parameter-free BI model, the Akaike Information Criterion (AIC) was calculated precisely. WAIC differences are relative to next-best model of the same class, and include estimated standard errors of the difference as an indicator of meaningful difference. In the RL model, “$\alpha$” refers to the classic RL formulation in which ${\alpha }_{+}={\alpha }_{-}$. “${\alpha }_c$” refers to the model in which factual and counterfactual learning rates were separate, but positive and negative outcomes were not differentiated (${\alpha }_{+c}={\alpha }_{-c}$; Section 4.5.1).

	Free parameters (count)		(W)AIC	WAIC difference
BI	–	(0)	31,959	2668 $\pm 0$
	$\beta$	(1)	$29,291\pm 206$	$868\pm 78$
	$\beta$, $p$	(2)	$28,423\pm 201$	$4769\pm 132$
	$\beta$, $p$, $p_{reward}$	(3)	$23,654\pm 203$	$51\pm 10$
	$\beta$, $p$, $p_{reward}$, $p_{switch}$	(4)	23,603$\pm$200	0
RL	$\alpha$, $\beta$	(2)	$26,678\pm 200$	$438\pm 44$
	$\alpha$, $\beta$, ${\alpha }_c$	(3)	$26,240\pm 201$	$1429\pm 78$
	$\alpha$, $\beta$, ${\alpha }_c$, $p$	(4)	$24,811\pm 190$	$42\pm 13$
	${\alpha }_{+}$, $\beta$, ${\alpha }_{+c}$, $p$, ${\alpha }_{-}$	(5)	$24,769\pm 213$	$1260\pm 73$
	${\alpha }_{+}$, $\beta$, ${\alpha }_{+c}$, $p$, ${\alpha }_{-}$, ${\alpha }_{-c}$	(6)	$23,509\pm 211$	$17\pm 10$
	${\alpha }_{+}={\alpha }_{+c}$, ${\alpha }_{-}={\alpha }_{-c}$, $\beta$, $p$	(4)	23,492$\pm$201	0