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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

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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 $x=3155$.

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Thank you. The plaintiff makes an offer, the defendant accepts or rejects. I'm presuming that would alter the game so that the defendant would be best to accept anything less than ~10,244 –  Hal Nov 26 '13 at 2:39
    
In the subgame perfect equilibrium, yes. Indeed, the timing may have an influence on the outcome, but this is an often observed fact in game theory. –  Nameless Nov 26 '13 at 10:21
    
The set of Nash equilbria (anything between the two expected values) is the same independent of timing though. –  Nameless Nov 26 '13 at 11:03
    
really? If the plaintiff makes an offer 1 below the defendant's expected payout, and the defendant can now only accept or reject it - and if he rejects it he goes to trial and loses 1 more. Wouldn't he have to accept it? –  Hal Nov 28 '13 at 3:29
1  
It's a sequential game, so it goes the other way around (backward induction). If the last player (defendant) threatens to accept only a proposal $y\le t< 10244$, say $t=5000$, then the plaintiff (believing the threat) will not demand more than that (because then he expects only $3155$ from the trial. Again, this hinges on a non-credible threat, as you correctly point out. This is why those Nash equilibria are not subgame perfect Nash equilibria. –  Nameless Nov 28 '13 at 15:02

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