@tomcollins: It's not his money either.  The person who originally had it (let's call him the Faucet, for agility, and similarly call the other guy the Splitter and you the Boolean, since you only get to say yes or no) asked you to "split" it somehow.

Somehow you seem to imply that by putting the Splitter in the favorable side of a Nash Equilibrium, the Faucet has effectively given him the whole $5,000, so all that is left for you is to try and extort all you can from him, from a position of no legitimacy and hardly any force.  That is the game theoretical, abstractly "rational" framing, the one that ignores all human emotion and common sense conventions about fairness.  Very few people in the world would have seen it this way in pre-Nash times.

As we know, there are emotions and instincts at play.  Issues of self-image.  Empathy.  Perhaps genetically or culturally inherited notions and heuristics regarding property and fairness.  I can fight those notions, try to abstract away my own human nature, in pursuit of a rational ideal.  That, even to the extent we assume it's possible, doesn't come without a cost.  Those factors modify the real payoff matrix.  It's not purely about money.  Not even for those who act like it was (more on this later).

And what is my motivation to fight my own instincts?  The payoff I get for that effort is warping the game so I'm at the extreme sucker end.  Is that rational?

Repressing or overcoming emotion is often a requisite of rationality.  But that doesn't mean it is, in itself, automatically a rational thing to do.  For that, a benefit needs to be derived.

I argue that people who opt for --let's call it-- the Nash strategy do that self-restraint work in order to defend a self-image as "rational" beings, for a culturally established notion of rationality.  Nothing wrong with that, but it's a tradeoff, not inherently *the* rational thing to do.

Because it gets recursive and messy.  Once you convince yourself that going against primal emotions is the "good" thing to do, and if you manage to hide the costs under the carpet, then you won't feel as much of a sucker, so it will really "cost" less to you.  For example, by convincing yourself that the Splitter is not being a selfish greedy pig, you may actually dampen the primal outrage of being abused, and by doing that you may be justifying a greedy Splitter position.

This recursiveness means that beliefs on the game may be self-realising.

tl;dr:  Emotions alter not only what seems rational, but what is rational.  Beliefs about rationality, in turn, alter emotions, and thus, recursively, what is rational.  Any attempt at rationalizing the game transforms it, in some sort of Heisenberg uncertainty principle.  So a static game theoretical analysis that conflates money with utility doesn't settle the debate of what is rational.  

tl;dr's tl;dr: For all practical purposes, there is no rational answer.  What's right for you depends on your personality and cultural baggage.