There are two parties in talks to settle a law suit.
The expected value of going to court for the plaintiff is $\$3,155$
The expected value of going to court for the defendant is $-\$10,244$
The defendant could give up $3,155 + \$1$ and settle the matter - and at first blush it seems the plaintiff ought to accept.
The plaintiff could demand $10,244 - \$1$ and settle the matter - and at first blush it seems the defendant ought to accept.
Assuming every dollar is equally valued by each party - what's the equilibrium point here? How is it calculated? Is it just a matter of splitting the difference between $-3,155$ and $-10,244$?
Thank you
The game description is a bit incomplete: what happens if they cannot agree on a settlement? I assume that the defendant makes one settlement offer, and if this is rejected by the plaintiff, they go to court. I assume the expected values are common knowledge, and that both parties are risk neutral (expected utility maximizers).
There are many equilibria to this game. One is:
The plaintiff rejects any offer $x<10244$, the defendant offers $x=10244$. Outcome: plaintiff receives $10244 as a settlement.
Proof: This is a sequential game (defendant offers, then plaintiff accepts/rejects). We solve backwards. The plaintiff rejects any offer $x<10244$. Anticipating this, the defendant cannot offer $x<10244$, because that leads to trial, where he expects a greater loss. On the other hand, he will not offer $x>10244$, because he knows (in equilibrium) that the plaintiff accepts 10244.$\Box$
Indeed, there is a continuum of equilibria: a Nash equilibrium is characterized by an acceptance threshold $a$ of the plaintiff (he rejects any offer that is less than $a$), and an offer $x$ by the defendant.
Any $10244\ge a\ge 3155$ and $x=a$ is a Nash equilibrium.
Proof: Exact same reasoning as above.$\Box$
Because of this multiplicity of equilibria in sequential games, Reinhard Selten proposed the subgame perfect Nash equilibrium concept. It is a stronger concept, which basically rules out empty threats, thereby reducing the set of equilibria.
The only two subgame perfect equilibria are $a=3155$ or $a=3156$ and $x=a$.
Why? Because it is an empty threat of the plaintiff to reject an offer $x>3156$. After all, the plaintiff only expects $3155$ in trial, and the defendant knows this. Given that the offer, say, $x=4000$ is made, and given that both know after rejection there will be a trial where the plaintiff expects less, the defandant just doesn't believe empty threats like above. We have two equilibria, because the plaintiff is indifferent between accepting and rejecting the offer $x=3155$.