Two people must split something. Each names a share. If the two demands fit, both get what they asked for; if they add to more than there is, both get nothing. Every split that exactly exhausts the cake is an equilibrium. Half and half is one. So is nine tenths against one tenth. Nothing in the game marks the equal split as special.
Ask what a rational bargainer should demand and the question comes back at you. The right demand depends on what the other person will demand, and theirs depends on yours. Any pair of demands that adds to the whole is stable, and there are as many of those as you care to name.
Braithwaite (1955) put this to the Cambridge philosophy faculty as an inaugural lecture, and it is roughly where game theory entered moral philosophy. If a theory of rational bargaining could deliver the equal split on its own, it would have derived a piece of justice from nothing but self-interest. It cannot. Every proposal that gets there does it by putting fairness in somewhere first.
Skyrms (1996) changes the question. Not which division is rational, but which one a population of people who are not reasoning about any of this actually ends up at.
A divided population is not a failure of the model. It is a stable, self-sustaining arrangement in which some people systematically get less, everyone is behaving optimally given everyone else, and no individual can do anything about it.
Nash's bargaining solution picks the equal split, but it does so by imposing axioms, one of which is symmetry: if the situation looks the same from both sides, the answer should treat both sides alike. That is not a discovery about rationality. It is fairness, assumed in advance and then recovered. Braithwaite saw this clearly, and it is why his lecture reads less like a solution than a warning.
Skyrms drops rationality and asks about a population that simply does more of whatever has been working. About six times in ten it reaches the equal split. That is a genuine result: fair division turns out to have the largest basin of attraction, which is a fact about the dynamics rather than an axiom anyone put in.
Fairness is not what stability requires. It is what stability usually happens to find.
The other four times matter just as much. Those populations end in permanent inequality that is perfectly stable, and the people asking for less are not making a mistake. Given what everyone around them is doing, asking for more would leave them with nothing.
Add a small chance of meeting someone who asks what you ask and fair division becomes almost universal. Skyrms takes this seriously: if people interact more with those like themselves, through neighbourhood, kinship or anything else, the equal split becomes overwhelmingly likely without anyone having reasoned towards it.
It is worth being clear about what that argument delivers. It does not show fair division is rational. It shows that a population with a particular interaction structure will probably arrive there. Whether that counts as an explanation of justice or a change of subject is exactly the live question.
Everyone here is identical, the cake is the same size every time, nobody remembers anyone, and there is no bargaining in the ordinary sense: no offers, no refusals, no talking. Real bargaining involves asymmetric need, outside options and the ability to threaten, all of which change the answer and none of which appear here.