Our Gemara on the top of Amud Aleph makes the following observation about human nature: it is less embarrassing to voluntarily shame oneself than when another shames you. In our Gemara’s scenario, because the goal is to induce public reflection and repentance, the strategy is to have others place ashes on the Nasi’s head to maximize his shame, as representative of the community.
This principle is used elsewhere in Shas, sometimes to mitigate shame. For example, in certain situations where a Sanhedrin is in judicial gridlock, the elder of the court declares that they are at an impasse and acquits the defendant. The Gemara (Sanhedrin 41a) asks, why does the elder embarrass himself, should it not be more appropriate for one of the junior judges to announce that they could not adjudicate the trial? The Gemara answers because in any case people will hear about it and it will reflect poorly on the Chief Justice, it is less embarrassing for him to disclose their shortcoming than another.
Another interesting application of this principle comes from Tzror Hamor (Bereishis 38:1). The verse states “And Yehuda went down”. The Midrash tells us that Yehuda was lowered in his supremacy as leader of the brothers as a result of the Yosef debacle and seeing his father’s unrelenting grief. However, Tzror Hamor notes the grammar on the verse is not reflexive nor passive, but active. If so, this would imply that Yehuda voluntarily demoted himself, based on the principle that it is less humiliating to bring shame on oneself than have it imposed.
There is research that shows interpersonal trauma is more damaging than non-personal kinds. For example, the humiliation and pain of being mugged, beaten or molested is more painful than suffering a similar loss due to so-called natural causes. Thus losing money or limb due to a hurricane or earthquake, while plenty disturbing, feels different than a person malignantly inflicting it upon you.
I wonder if it is related to this same principle, at least on one side of the coin. That is, when it is inflicted by another person it feels much worse. In lomdishe terms we can say the main part of the principle is not that self inflicted shame feels less painful, rather the key point is that shame inflicted by others feels worse.