A buyer and a seller each have private information about their own valuations of asingle object that the seller may sell to the buyer. The buyer's valuation, denotedV, and the seller's valuation, denoted v,, are independently drawn from a uniformdistribution on the interval [0, 1]. The buyer names an offer price, Pb (a nonnegativenumber), and the seller simultaneously names an asking price, p, (a nonnegativenumber). If p, < Ps, there is no trade; the payoffs to both the buyer and the seller are0. If p. > Ps, trade occurs at the price p = tp); the payoff to the buyer is v, – pand the payoff to the seller is p- vs.2 (a) Specify the strategic situation as a Bayesian game.(b) Find all the values of a such that the game has a Nash equilibrium in which thebuyer offers æ if v, > r and 0 otherwise, and the seller asks r if v, < x and 1otherwise.(c) (optional) Show that the game has a Nash equilibrium in which the buyer usesthe linear offer price function p(vb)price function p,(v.) = v, +.vr +5 and the seller uses the linear ask%3D

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ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

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Question

A buyer and a seller each have private information about their own valuations of a
single object that the seller may sell to the buyer. The buyer's valuation, denoted
V, and the seller's valuation, denoted v,, are independently drawn from a uniform
distribution on the interval [0, 1]. The buyer names an offer price, Pb (a nonnegative
number), and the seller simultaneously names an asking price, p, (a nonnegative
number). If p, < Ps, there is no trade; the payoffs to both the buyer and the seller are
0. If p. > Ps, trade occurs at the price p = tp); the payoff to the buyer is v, – p
and the payoff to the seller is p- vs.
2

(a) Specify the strategic situation as a Bayesian game.
(b) Find all the values of a such that the game has a Nash equilibrium in which the
buyer offers æ if v, > r and 0 otherwise, and the seller asks r if v, < x and 1
otherwise.
(c) (optional) Show that the game has a Nash equilibrium in which the buyer uses
the linear offer price function p(vb)
price function p,(v.) = v, +.
vr +5 and the seller uses the linear ask
%3D

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Step 1: Define Bayesian Game

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Step 2: Explain the strategic situation by Bayesian game.

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Step 3: Derive the Nash Equilibrium Strategy:

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Step 4: Explain the game of Nash equilibrium

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