Game Theory 9 – chap15. The lemon market

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4. Bayesian Games
4.3 Applications: Lemons, Auctions, and Information Aggregation (Chapter 27)
Markets and Lemons

Application 1: Lemon Market (Akerlof, 1970)

“Information asymmetry may bring about a market collapse“

George Akerlof, Nobel Iaureate in 2001

Bilateral trade between a used-car seller and a buyer
Only the seller knows the exact quality of the car

QualityBuyerSeller
H10,0008,000
L5,0003,000

As \(v_{b} > v_{s}\), the used car should be traded if the buyer knows the exact quality

Suppose the buyer believes Pr(low quality) > 0.4 and makes a price offer to the seller

10,000∙Pr(H) + 5,000∙Pr(L) < 8,000

In this case, it is bad news to the buyer that his offer is accepted by the seller

Consider a bilateral trade between a used-car seller (Freddie) and a buyer (Jerry)
Only Freddie knows the exact quality of the car

QualityJerryFreddie
Peach3,0002,000
Lemon1,0000

Jerry knows only that the car is peach with probability q ∈ (0, 1)
Both traders decide whether to trade (T) or not (N) at a given market price p

J \ F\(T^{P}\)\(N^{P}\)J \ F\(T^{L}\)\(N^{L}\)
T3000 – p, p0, 2000T1000 – p, p0 ,0
N0, 20000, 2000N0, 00, 0
Peach (q)Lemon (1-q)

In Bayesian Normal Form, \(S_{J}\), \(S_{F}\) = {\(T^{P}T^{L}, N^{P}T^{L}, T^{P}N^{L}, N^{P}N^{L}\)}

To investigate which type of the car is traded in equilibrium, we first convert the given game into Bayesian normal form:

Jerry \ Freddie\(T^{P}T^{L}\)\(N^{P}T^{L}\)\(T^{P}N^{L}\)\(N^{P}N^{L}\)
T3000q + 1000(1-q) – p
p		(1000-p)(1-q)
2000q+p(1-q)		3000q-pq
pq		0
2000q
N0, 2000q0, 2000q0, 2000q0, 2000q

(1) When is only the lemon traded in eqbm?

(T, \(N^{P}T^{L}\))
1) (1000-p)(1-q) ≥ 0 ⇒ p ≤ 1000
2) 2000q+p(1-q) ≥ p (∵ p ≥ pq and 2000q ≥ 2000q + p(1-q)) ⇒ p ≤ 2000
By 1) and 2), p ≤ 1000.

(2) When are both cars traded in eqbm?

(T, \(T^{P}T^{L}\))
1) 3000q + 1000(1-q) – p ≥ 0
⇔ 3000q + 1000(1-q) ≥ p ≥ 2000 ⇒ q ≥ 1/2
2) p ≥ 2000q + p(1-q) (Always satisfied because p ≥ 2000)

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