ECO290E: Game Theory
Lecture 12: Applications of incomplete information games
Review of Lecture 11
(Static) games of incomplete information is analyzed by Bayesian games.
A strategy for a player is a complete action plan, which specifies her action for all possible types.
Nash equilibrium of Bayesian games is named as a Bayesian Nash equilibrium (BNE).
Instead of considering strategies, it is often easier to derive BNE by solving optimal action for each type.
Lemon with Certification
What happens if the seller can issue certification of quality (miles, accident, etc) of the car?
Suppose there are 10 car owners, k = 1,2, …, 10, and their valuations of own car is vk = 10k. You can interpret that seller 1 has the worst car while 10 owns the best car.
Assume there are many buyers, and the market price for each car becomes 20% higher than the expected
valuations for buyers.
Analysis without Certification
What if the seller cannot issue any verifiable information about the quality of their own cars?
Suppose every cars are sold in the market.
Then the expected value for buyers becomes 55×1.2 = 66, which is smaller than vk for k = 7, 8, 9, 10.
So, the seller 7 through 10 exit from the market.
The expected value of remaining cars becomes 35×1.2 = 42, smaller than vk for k = 5, 6.
This argument continues until all the seller except for 1, the owner of the worst car, exit from the market…
Full Information Disclosure
Suppose each seller can issue certification for free. Then, what is the Bayesian Nash equilibrium?
The seller 10 should issue certification since she can sell the car by p10 = 120.
Given the seller 10 issues certification, the seller 9 should also have an incentive to issue certification, since there is no gain not to reveal the quality of the car.
This argument continues until k = 2 (k = 1 is indifferent).
Thus, all the private information is voluntary disclosed!
⇒ What if issuing certification incur cost, say C > 0?
Double Auction
A buyer names a asking price (pb) and a seller names an offer price (ps).
If pb ≥ ps, then trade occurs at price p = (ps + pb) / 2.
If pb < ps, then no trade occurs.
The buyer’s valuation (vb) and the seller’s valuation (vs) are private information, and these are independently and
uniformly distributed on [0, 1].
Trade is efficient if and only if vb > vs.
Impossibility of Efficient Trade
A buyer (/seller) has an incentive to understate (/
overstate) the value in hopes of increasing her surplus.
In a (linear) Bayesian Nash equilibrium,
Buyer’s strategy: 2/3∙vb + 1/12
Seller’s strategy: 2/3∙vs + 1/4
Profitable trade may not happen (with positive probability).
There are other BNE with non-linear strategies.
Analysis: see the handout. (Gibbons 3.2.C)
Variety of Auctions
Selling single objects or multiple objects.
Open auctions: all bids are publicly observable.
English (ascending-price) auction
Dutch (descending price) auction
Sealed-bid auctions: only seller can observe bids.
First price auction
Second price (Vickrey) auction
Assumptions
Private value ⇔ Common (interdependent) value
Each bidder knows her valuation, i.e., type = value of the good.
Independent value ⇔ Affiliated value
Type (valuation) for each player is independently drawn.
Seller’s valuation is normalized to be 0.
No secondary market (no other resale possibility)
Second Price Auction
The winner is the bidder with the highest bid while she will pay the second highest bid.
Equivalent to English auction under the assumptions of private and independent value.
Is there (weakly) dominant strategy?
Yes, it is optimal for each bidder to bid her own valuation.
In English auction, it is optimal for each bidder to stay until the price reaches her own valuation.
First Price Auction
The winner is the bidder with the highest bid while she will pay her own bid.
Truthful-bidding CANNOT be optimal ⇒ no gain from winning.
Strategically equivalent to Dutch auction.
Suppose there are n bidders; each bidder’s value (type) is uniformly and independently distributed between 0 and 1.
What is the Bayesian Nash equilibrium?
Equilibrium of First Price Auction (1)
Bidders simultaneously and independently submit bids b1 and b2.
I The painting is awarded to the highest bidder i∗ with max bi ,
I who must pay her own bid, bi∗.
To derive a Bayesian Nash equilibrium, we assume the bidding
strategy in equilibrium is i) symmetric, and ii) linear function of xi. That is, in equilibrium, player i chooses
β(xi) = c + θxi. (1)
Now suppose that player 2 follows the above equilibrium strategy, and we shall check whether player 1 has an incentive to choose the same linear strategy (1). Player 1’s optimization problem, given she received a valuation x1, is
Equilibrium of First Price Auction (2)
Since x2 is uniformly distributed on [0, 1] by assumption, we obtain Pr{b1 > β(x2)} = Pr{b1 > c + θx2}
= Pr⇢ b1 − c
θ > x2
"
= b1 − c θ .
The first equality comes from the linear bidding strategy (1), the third equality is from the uniform distribution. Substituting it into (2), the expected payoff becomes a quadratic function of b1.
maxb1 (x1 − b1)
b1 − c θ
Taking the first order condition, we obtain du1
db1 = 1
θ[−2b1 + x1 + c] = 0 ⇒ b1 = c 2 +
x1
2 . (3)
Revenue Equivalence Theorem
We already verify that (for each possible combination of types), second price = English, first price = Dutch.
Comparing expected revenues, second price = first price.
In the first price auction, it can be shown that the expected payment is equal to the second highest value.
⇒ Four auction formats yield the same expected revenue!
(General) Revenue Equivalence Theorem
Any auction formats that results in the same (1) allocation and (2) the expected payoff of the lowest value type,
Revenue Equivalence: Illustration
Open
Dutch
(Descending)
English
(Ascending)
Sealed-Bid
First-Price
Second-Price
Strong
Weak
Revenue Equivalence Theorem
Further Exercises
Consider why the second price and English auction will no longer be equivalent when valuations are inter-
dependent; also consider why the first price and Dutch auction will still be equivalent.
In the second price auction, find a BNE other than the one in which every bidder takes weakly dominant
strategy.
Study revelation principle (see the handout of Gibbons).
