Consider the game among a couple and their male marriage counselor. The problem for the marriage counselor is to prove that he is unbiased. It is common-knowledge at the outset that the wife worries that a male marriage counselor is biased and will always blame the wife.
Indeed if 10 weeks in a row they come in for counseling and talk about the week’s petty argument (how to stack dishes in the dishwasher, whether it matters that the towels are not folded corner-to-corner, etc.) he everytime sides with the husband, eventually the wife will want to find a new counselor.
So what happens after 9 weeks of deciding for the husband? Now all parties know that the counselor is on his last leg. He must start siding with the wife in order to keep his job, even if the husband is actually in the right (i.e. even if throwing out the 3-day old soggy quesadilla in the refrigerator was the right thing to do.) But that means that he’s now biased in favor of the wife and so the husband will fire him.
We have just concluded that if he decides for the husband 9 times in a row he will be fired. So what happens on week 9 in the rare event that he has decided for the husband 8 times in a row. Same thing, he is strategically biased in favor of the wife and he will be fired.
By induction he is biased even on week 1.
(NB: my marriage is beautiful (no counseling) and there is nobody who can fold a towel faster than me.)