主题：两个女生和一台二手电视机引来的game theory 和 auction 问题 -- Ready-Go

What is fair and what is unfair is a

philosophical question, which depends

on people's ideology and cannot be

settled by game theory.

What they are disputing her is that the

joint utility derived from the TV by A

and B decreases because B is leaving.

Because the second-hand market is

illiquid and transaction cost, they

cannot return to their original state.

Someone must bear this loss of value,

the central question is how to divide

this loss fairly.

I can think of at least 2 ways to

define fairness

1. The loss of value is caused by B, so

B should bear the whole cost.

2. Everyone should bear equal share of

the loss.

Your second friend's argument that the

Dutch Auction format is unfair is based

on the second definition. Because B has

to pay the transaction cost to sell the

TV, and A has to pay transaction cost

to buy the TV, the TV worths more to B

to A. Therefore whatever auction format

you choose, B will be better off than

A, so it is unfair.

Now on auction formats.

1. Dutch auction is equevalant to first-

price sealed bid.

2. This auction game is highly

irregular, if you want to model it

realistically.

(1). The bidder not only need to pay

their bid, but also (160-x)/2, which

depends on their bids. The Revenue

Equvalence theorem breaks down. Assume

it is a symmetric private value

auction, (more on that later), and

uniform distribution of types, denote V

as valuation, you can prove the optimal

strategy in Dutch auction is 3V/4

(higher than the standard case which is

V/2), while the optimal strategy in

English Auction (and Vickey Auction) is

2V-160 (lower than the standard case

which is V). Dutch auction generates

more revenue than English auction. B is

better off, and A is worse off in Dutch

Auction. (Because A is more likely to

win the auction.)

Furthermore, if V is less than 80, they

want to bid a negative number, the

Vickey auction is not viable.

(2). However, because of the second

hand market, there is a large part of

common value in this auction. Milgrom

and Weber (1982) shows that in this

case, English auction generates more

revenue than VIckey Auction, and Vickey

Auction generates more revenue than

first-price auction. So B is worse off

and A is better off in Dutch auction.

(3). Because we know B probably values

the TV less than B, the valuation is

asymmetric. A will bid more agressively

in first-price auctin than in English

auction. However, I don't know how this

will affect A and B's utility.

In conclusion, as to the choice of

auction format, the answer is I don't

know. It needs further assumptions and

calibration about how A and B's

valuation is distributed and correlated.

I can do the research for them, if they

give me a fair payment.