2. Consider the following sequential game. The first move is one of chance. Play

Question

2. Consider the following sequential game. The first move is one of chance. Player 1 observes the move by Nature, but Player 2 does not. Nature (11) ul 1,2 2,0 0,1 3,0 0,0 3,1 2,2 (a) Does this game have any separating perfect Bayesian equilibria? Show your analysis, and if there is such an equilibrium, report it. In searching for perfect Bayesian equilibria, check directly by (1) checking each possible separating strategy by player 1, (2) update player 2's beliefs L and R by Bayes Rule when possible (and formulate beliefs when it is not), (3) determine player 2's best response to player 1, given the updated beliefs, and (4) check if player 1 would stay with the specified strategy. (b) Does this game have any pooling perfect Bayesian equilibria? Show your anal- ysis, and if there is such an equilibrium, report it. In searching for the perfect Bayesian equilibria, follow the same procedure for (a), except of course check each possible pooling strategy by player 1. (c) Create the normal form for this game, and find the pure strategy Nash equil bria. Confirm that any pure strategy PBE found in problems (a) and (b) are Nash equilibria

Explanation / Answer

• So far we have studied two types of games: 1)

sequential move (extensive form) games where

players

take turns

choosing actions and 2)

strategic form (normal form) games where players

simultaneously

choose their actions.

• Of course it is possible to combine both game

forms as, for example, happens in the game of

football.

• The transformation between game forms may

change the set of equilibria, as we shall see.

• We will also learn the concept of

subgame

perfectionConsider the strategic move by which Plaintiff pre-pays his lawyer the

costs of a trial, c, up-front (Plaintiff puts money in a non-refundable

retainer account guaranteeing the attorney’s future availability for

trial).

• In this case the payoff to Give Up changes from –k to –k-

c.

• Plaintiff now goes to trial so long as pw>0, i.e. if there is any positive

probability p of winning the amount w.

• This further implies that the Defendant prefers to go to trial only if

s>pw+d.

• Defendant is now willing to settle for any amount s<pw+d.

• So the settlement range is [pw, pw+d]. If the Plaintiff can make a

take- it- or-leave-it offer, what is the settlement amount s?

• Outcome is Plaintiff sues, offers to settle, and a settlement is reachedonsider a game between a Plaintiff and a Defendant.

• Plaintiff moves first, deciding whether to file a lawsuit against

Defendant at cost to the Plaintiff of k>0. If a lawsuit is filed,

the Plaintiff makes a take-

it-or-leave it settlement offer of s>0.

• Defendant accepts or rejects. If Defendant rejects, Plaintiff has

to decide whether or not to go to trial at cost c>0 to the Plaintiff

and at cost d>0 to the Defendant.

• If the case goes to trial, Plaintiff wins the amount w with

probability p and loses (payoff=0) with probability 1-

p.

• Assume that pw < c, and this fact is common knowledge-

a

critical assumption.

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