1 Correction to: Review of Industrial Organization https://doi.org/10.1007/s11151-022-09875-w

In the original publication of the article, the appearance of \(\widetilde{\mu }\) and the inequalities

\(\mu \ge \widetilde{\mu }\) and \(\mu <\widetilde{\mu }\) throughout the paper are disorderly presented.

In Tables 1 and 2, the payoff combination “\(B, B\)” is placed without space.

The text between Lemma 2 and Lemma 3, Lemma 4 and Lemma 5, Lemma 6 and Lemma 7, Lemma 7 and Proposition 2 and incorrectly provided in italics.

The mathematical expressions in the paragraph below Lemma 8 “(\({\delta }_{pd}^{^{\prime}}\left({\sigma }_{1}\right)\) and \({V}_{pd}^{^{\prime}}\left({\sigma }_{1}\right)\))” are placed very closely that may cause confusion.

In the Appendix, the Proof of Lemma 8 and Proposition 3 (especially the first lines) on p. 18 is not properly presented. I quote the correct form of this proof right below:

Recall that the critical discount factors for the ringleader and the follower are

$$ \delta_{n}^{\prime } \left( \sigma \right) = \frac{{2\left( {1 - \sigma } \right) + a\sigma \mu }}{{2\left( {1 - a} \right)}}\quad {\text{and}}\quad \delta_{n}^{\prime \prime } \left( \sigma \right) = \frac{{2\sigma + a\mu \left( {1 - \sigma } \right)}}{{2\left( {1 - a} \right)}} , $$

respectively when both report under non-discrimination. Observe that \( \frac{{\partial \delta _{n}^{\prime } }}{{\partial \sigma }} < 0 \) and \( \frac{{\partial \delta _{n}^{{\prime \prime }} }}{{\partial \sigma }} > 0 \); the ringleader’s (follower’s) ICC loosens (tightens) with \(\sigma \).

For \(\sigma =\frac{1}{2}\), \( \delta _{n}^{\prime } \left( {\frac{1}{2}} \right) = \delta _{n}^{{\prime \prime }} \left( {\frac{{\text{1}}}{{\text{2}}}} \right) = \delta _{{{n}}} = \frac{{{\text{2 + a}}\mu }}{{{\text{4}}\left( {{\text{1}} - a} \right)}};\quad max \{ \delta _{n}^{\prime } ,\delta _{n}^{{\prime \prime }} \} \); is minimized; and the firms’ expected collusive payoffs are also equal: \( V_{n}^{\prime } \left( {\frac{1}{2}} \right) = V_{n}^{{\prime \prime }} \left( {\frac{{\text{1}}}{{\text{2}}}} \right){\text{ = }}\frac{{\pi \left( {{\text{2}} - a\mu } \right)}}{{{\text{2}}\left[ {{\text{1}} - \delta \left( {{\text{1}} - a} \right)} \right]}} \).

The critical discount factors for the ringleader and the follower are \( \delta _{{pd}}^{\prime } (\sigma ) = \frac{{1 - \sigma + a\sigma \mu }}{{1 - a}}\quad {\text{and}}\quad \delta _{{pd}}^{{\prime \prime }} (\sigma ) = \frac{\sigma }{{{\text{1}} - a^{\prime } }} \), respectively, given that the investigated follower reports under partial discrimination. Observe that \(\frac{\partial {\delta }_{pd}^{^{\prime}}}{\partial \sigma }<0\) and \(\frac{\partial {\delta }_{pd}^{{^{\prime}}{^{\prime}}}}{\partial \sigma }>0\); the ringleader’s (follower’s) ICC loosens (tightens) with \(\sigma \).

For \(\sigma ={\sigma }_{1}\equiv \frac{1}{2-a\mu }\), \({\delta }_{pd}^{^{\prime}}\left({\sigma }_{1}\right)={\delta }_{pd}^{{^{\prime}}{^{\prime}}}\left({\sigma }_{1}\right)=\frac{1}{\left(1-a\right)\left(2-a\mu \right)}\). For \(\sigma ={\sigma }_{1}\) the firms’ expected collusive payoffs are also equal: \({V}_{pd}^{^{\prime}}\left({\sigma }_{1}\right)={V}_{pd}^{{^{\prime}}{^{\prime}}}\left({\sigma }_{1}\right)\). It is easy to verify that

$$ \delta _{{pd}}^{\prime } \left( {\sigma _{1} } \right) = \frac{1}{{\left( {1 - a} \right)\left( {2 - a\mu } \right)}} > \delta _{n} = \frac{{2 + a\mu }}{{4\left( {1 - a} \right)}}. $$