Why is the fair coin toss considered fair?

If there is only blind chance - no actor doing that distribution - it seems that random results are considered acceptable. Often people are being told that "Life is not fair." This is an indication that we have a desire for fairness, but also accept it if circumstances on which nobody (no actor) has influence are random instead of fair. In social situations the concept of fairness seems to be different and strongly so, with many humans valuing fairness quite highly.

The ultimatum game in economics is an example where one sees massive discrepancies between the "rationally self-interested" optimal strategy and what is considered fair - And how much that is worth to people on average (hint:
a lot).

The ultimatum game is a two player game. The players get some amount of money which the first player splits among them. The second player can only accept or refuse the offer. In case of refusal both get nothing.

Rational self-interested actors are predicted to always accept any offer, because they can only gain by accepting.

On of the foundational experiments of Behavioral Economics is exactly this game. And the statement is more or less: "rationally self-interested agents" are not a good model for real humans.
Even when the possibility of refusal was removed, humans tend to distribute "fair". Remove the choice of the second player turns the game into the dictator game and removes any incentive from the "dictator" to be fair.

Real humans seem to have a concept of "fairness" - for which model agents would need an external valuation. In practice this valuation of "fairness" is high enough to lead to a refusal of most offers that are not about equal.

In a social environment one of the concepts underlying fairness seem to be: Equal pay for equal work. This is seen in nearly all social animals and could be the basis for the labor theory of value (LTV) - Which states that all labor, time at work, is equal and the labor infused into a product or services provides the limit around which value and prizes vary. This may be one basis of the fairness valuation in the ultimatum game, where there is no productive labor performed.

The LTV has its own problems, on which I will come to talk later. But I think the equality of work is a more central concept that may shed some light on higher concepts like "fairness" or "justice" and maybe even how those evolved.