Antipodal Signalling has a bit error probability of $P_b = \frac{1}{2} \mathrm{erfc}\big(\sqrt{\frac{E_s}{N_0}}\big)$ and on-off-signalling has a bit error probability of $\frac{1}{2} \mathrm{erfc}\big(\sqrt{\frac{E_s}{2 N_0}}\big)$. Why does this correspond to a higher SNR of 3 dB which we need for on-off-signalling to achieve the same bit error probability as with antipodal signalling?
I know that the factor 2 corresponds to 3 dB. But the complementary error function does not decrease by a factor of 2 but by a factor of $\sqrt{2}$