...I didn't rule it out in the absolute mathematical sense...
Ok, I just wanted to make sure. You're right that I favor mathematical proofs over reasoned speculation. But of course, well written proofs concerning game theory - especially those that have passed the scrutiny of peer-review are quite infrequent.
In any case, I don't trust a lot of the "common sense" statements scattered in this thread. In math there's been many "common sense" statements that were later proven to be wrong. In 1968 Kershner announced the complete list of "all eight" pentagons that tile a plane saying "no more shapes are possible". But just three years ago the math world was stunned when a new shape was found. Kershner's "common sense" was refuted by careful analytical study. I believe this is one of the shapes he overlooked (it has degrees of freedom):


Pride is pride, same bias. Frustration causing vulgar language is part of the same bias. What one likes and dislikes is also part of the bias. Is there any action of one’s part which is not biased?
In terms of objective truth, the answer to this question depends on the definition of "biased". If it is defined in a way which encompasses all human thoughts, then no, there is no thought or action which is not biased (under said definition).
In terms of perceived truth, it is an even simpler question. If one decides that nonbiased thoughts or actions do not exist, then they do not exist (from said perspective).
This is why I prefer to discuss objective truth, rather than perceived truth; a given person's perceived truth is unique, and can be wholly independent from objective truth, thereby rendering any attempt at substantive discussion futile.