Why's the unique sub-game perfect equilibrium, that the first player should offer around $1.25 to player 2?
Please see the bolden phrase below. Let's abbreviate player $j$ to $Pj$. Even if this is the unique sub-game perfect equilibrium, it feels unnecessarily risky and irrational to me. Why wouldn't you simply offer $2.5 (50% of 5) to P2?
You don't know if P2 is rational, reasonable, or sane. I can't remember if this book cites Keith Chen's study, but How We Learn Fairness | The New Yorker
A monkey in isolation is happy to eat either a grape or a slice of cucumber. But a monkey who sees that she’s received a cucumber while her partner has gotten a grape reacts with anger: she might hurl her cucumber from her cage. Some primates, Brosnan and de Waal concluded, “dislike inequity.”
Similarly, even if offering $1.25 to P2 is rational and P2 should rationally accept it, P2 may reject it out of a sense of inequity and unfairness.
AN APPLICATION: EYE-TRACKING AND BACKWARD INDUCTION IN BARGAINING GAMES
The example I would like to discuss is based on studies of alternating-offer bargaining games ( Johnson et al., 2002). It is not an example of neuroeconomics per se, because the only variable measured is indirect—namely, which information people look for on a computer screen. However, the hope is that in future studies such measures of cognitive process will be linked directly to brain activity and other psycho-physiological correlates to form a tentative picture of the mechanisms and behavioral outputs involved.
Alternating-offer bargaining games are fashionable to study as models of how bargaining could be studied noncooperatively with mathematically precise results (and later linked to cooperative approaches). In the games we study, bargaining occurs over three stages, between two players. At each of the three stages, the first player offers a division of a known sum of money to a second player. If the offer is accepted by this second player, the game ends. If the offer is rejected, the proposed amount is divided by two and the second player who rejected the offer can make the next offer (offering rights alternate).
Thus, in the first stage, player 1 offers a division of $5 to player 2. If rejected, player 2 offers a division of $2.50 to player 1. If rejected, player 1 offers a division of $1.25 to player 1. If that final offer is rejected they earn zero. (Note that the last round is an ultimatum.)
If both players have mutual knowledge that the players are self-interested and rational, the unique sub-game perfect equilibrium is that the first player should offer around $1.25 to player 2, who should accept it because if s/he does not, s/he will have only $1.25 at the next round to divide, thus the maximum alternative amount.
Paul Slovic, The Irrational Economist (2010), p 82.