Correction to: Nature Human Behaviour https://doi.org/10.1038/s41562-018-0526-x, published online 11 February 2019.

In the version of this article initially published online, axis labels contained errors in Figs. 3 and 4, ‘Prop model’ should have been capitalized in the sentence beginning ‘Next, we examined the possibility that the Prop-V model…’ and errors occurred in the first two equations in Methods. Additionally, extraneous parentheses were present in text around the sentence beginning ‘In this paper, we adopt …’, which should have appeared as a separate paragraph before the sentence beginning ‘Methodologically, the sources of suboptimality…’. The errors have been corrected in the PDF and HTML versions of this article.

Fig. 3
figure 1

Original and corrected.

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Fig. 4
figure 2

Original and corrected.

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Equation 1 original:

$$\begin{array}{*{20}{l}}{p}\left( {{\bf{a}}_1, \ldots ,{\bf{a}}_N|k,b,\sigma ,\beta } \right) &= & \mathop {\prod}\limits_{i = 1}^N {{p}\left( {{\bf{a}}_i|k,b,\sigma ,\beta } \right)} \\ &= & \mathop {\prod}\limits_{i = 1}^N {{\int} {{p}\left( {{\bf{a}}_i|k,b,\eta _i,\beta } \right){p}\left( {\eta _i|\sigma } \right)\eta _i} } \\ &= & \mathop {\prod}\limits_{i = 1}^N {{\int} {\left[ {\mathop {\prod}\limits_{t = 1}^{T_i} {{p}\left( {{\it{a}}_{it}|k,b,\eta _i,\beta } \right)} } \right]{p}\left( {\eta _i|\sigma } \right)\eta _i} } \end{array}$$

Corrected:

$$\begin{array}{lll}p({\mathbf{a}}_{1},...,{\mathbf{a}}_{N}|k,b,\sigma,\beta) &= & \mathop{\prod} \limits_{i=1}^{N}p({\mathbf{a}}_{i}|k,b,\sigma,\beta) \\ &= & \mathop{\prod} \limits_{i=1}^{N} \int p({\mathbf{a}}_{i}|k,b,\eta_i,\beta)p(\eta_i|\sigma) {\mathrm{d}} \eta_i\\ &= & \mathop{\prod} \limits_{i=1}^{N} \int \left[\mathop{\prod} \limits_{t=1}^{T_i}p(a_{it}|k,b,\eta_i,\beta)\right]p(\eta_i|\sigma) {\mathrm{d}} \eta_i\end{array}$$

Equation 2 original:

$$\begin{array}{*{20}{l}}p\left( {a_{it} = 1|k,b,\eta _i,\beta } \right)& = & \frac{1}{{1 + e^{ - \beta \left( {\theta _{it} + \eta _i - r_{it}^ \ast } \right)}}} \\ \theta _{it} &= & k\left( {T_i - t + 1} \right) + b\left( {{\mathrm{Num}} - {\mathrm{V}}} \right)\;{\mathrm{or}}\\ \theta _{it}& = & k\frac{{T_i - t + 1}}{{T_i}} + b\left( {{\mathrm{Prop}} - {\mathrm{V}}} \right)p\left( {\eta _i|\sigma } \right) \sim {\cal N}\left( {0,\sigma ^2} \right)\end{array}$$

Corrected:

\(\begin{array}{l}p\left( {a_{it} = 1|k,b,\eta _i,\beta } \right) = \frac{1}{{1 + e^{ - \beta \left( {\theta _{it} + \eta _i - r_{it}^ \ast } \right)}}}\\ \theta _{it} = k\left( {T_i - t + 1} \right) + b\left( {{\mathrm{Num-V}}} \right)\,{\mathrm{or}}\,\theta _{it} = k\frac{{T_i - t + 1}}{{T_i}} + b\left( {{\mathrm{Prop-V}}} \right)\\ p\left( {\eta _i|\sigma } \right) \sim {\cal N}\left( {0,\sigma ^2} \right)\end{array}\)