Very short signatures? (eg: 48bits?)
Suppose you have an asymmetric algorithm that creates a 48-bit signature. Then an attacker can simply test possible 48-bit signatures and find one that verifies in $2^{47}$ tries on average. Even with a relatively slow asymmetric algorithm that would be feasible, especially if the attacker can test the same signature for multiple tokens. And that is only a brute force attack, asymmetric systems usually allow much faster attacks.

If you can establish a symmetric secret, even a short MAC is much better, because while the attacker still has a $2^{-48}$ probability of success with a random guess, they can no longer verify those guesses offline.