One person offers a share of a known quantity, say, $100, to a second.  The second person knows that they can accept or reject the offer.  If player 2 accepts, both receive the reward in the agreed proportion.  If the second player rejects the offer, both receive nothing.

Just because I know you love graphs...

From Gamelab, Harvard:

Number of players: 30 (15 games)
Mean proposal: 39 ± 10.7
Distribution of proposals:

Notes
The vast majority (73%) offers 40 or 50 points.
Surprisingly one offer of 40 is rejected
Comments
The rational strategy is to offer 1 point, and to accept everything. In reality, offers below 30% get mostly rejected. In a vast majority of studies conducted with different incentives in different countries, some 60-80% of proposers offer between 40% and 50% of the total sum, and only 3% of proposers offer less than 20%. Conversely, some 50% of responders reject offers below 30% of the total.

The 'rational',  material, utility-maximising response for the proposee is to accept anything offered, even one unit out of 100.

People don't.  There appears to be a payoff in depriving the maker of an unfair offer of his/her share, even at cost to oneself.