TOC
1. Introduction
Auction is one of the most popular examples of static games of incomplete information. An auction is a mechanism of allocating goods. It includes analyze bidders’ behaviors and analyze auctioneer’s “optimal” choice.
Advantages of auctions:
- a simple way of determining the market-based prices
- more flexible than setting a fixed price
- can usually achieve efficiencey by allocating the resources to those who value them most highly
Examples of auctions:
- Consumption goods: antiques, art, wine
- Government: treasury bills, mineral rights, assets, 5G spectrum
- Stocks: IPOs
- Internet: eBay, Amazon
Types of auctions:
- Number of objects: A single object v.s. many
- Open v.s. Sealed-bid
- One-sided v.s. Two-sided
- Private value v.s. Common value (whether the bidders have the same valuation for the object)
Four classical auctions
- English auction: ascending, open
- Dutch auction: descending, open
- First-price auction: simultaneous, sealed-bid
- Second-price auction: simultaneous, sealed-bid
2. Second-price Auction
Definition
A second-price auction: suppose there are potential buyers (or bidders), with valuations for an object. Let each be drawn from some distribution . Each bidder knows his own valuation but does not know other bidders’ valuations. The bidders simultaneouly submit bids . The highest bidder wins the object and pays the second highest bid, while other bidders obtain nothing. If there are more than one winners, the object is allocated randomly among them.
Payoff
Let be the highest bid of all players other than player : .
The bidder ’s payoff function is
where is the number of bids that equal .
Truth telling
Proposition: For each player , the strategy of bidding his valuation (i.e. weakly dominates all other strategies. Compare bidder ’s payoffs for different bids .
First, compare with
Then weakly dominates .
Second, compare with ,
Then weakly dominates .
By the following proposition is a Bayesian NE, where for all .
Proposition: Consider a strategy profile in a Bayesian game. Suppose for any player , any and
i.e., weakly dominates every . Then is a Bayesian NE.
Proof
Because , the weak dominance implies
for any and .
Then for each and for each , solves
Therefore, is a Bayesian NE.
3. First-price Auction
3.1 Representation & Strategy
Definition
Suppose there are two bidders (). The bidders’ valuation for an object are and , which are independently and uniformly distributed on . The valuation is bidder ’s private information, which is unkown to the other bidder. Bidders submit their bids and simultaneously. The higher bidder wins the object and pays the highest bid, while the other obtains nothing. If there is a tie, the winner is determined by a flip of a coin.
Normal-form representation of this static Bayesian game :
- , and each bid is
- , and each valuation is
- Player belives that is uniformly distributed on
- The payoff is
Strategy
Bidder ’s strategy is a functions from to . is a Bayesian NE iff for each player and each type , solves
3.2 Linear Bayesian NE
There may be many Bayesian NE in this game, we just focus on equilibria in the form of linear functions:
where and for
To solve for the Bayesian NE, we just need to find out and accordingly.
Assumptions:
- : a bidder with higher valuation is willing to bid higher
- : bids cannot be negative
- : for , bidder can never end up with a positive payoff given
Best Response
We need to determine each player’s best response given the others strategy. Suppose player adopts a linear strategy in equilibrium. We have
For any , player ’s best response maximizes
Since , we can restrict our attention to (i.e., is pointless, while is not rationals). Under these restriction, we know
Player ’s best response solves
FOC implies the potentially optimal choice is . It is the optimal choice iff iff .
The best response function of bidder is
We want the equilibrium bid to be a linear function on [0, 1]. There are three cases:
Selection
- Case 1 violates the assumption .
- Case 3 violates the assumption and , which imply .
Therefore, we have , and the best response is .
In a Bayesian NE,
for all
Then we have
for and .
Therefore,
The unique linear Bayesian NE is
Alternatively, if we can somehow guess that is a Bayesian NE, we can prove it directly:
- Suppose player has adopted the strategy
- Player ’s best response solves
- For any , the unique maximizer is . Thus, is a Bayesian NE.
4. Double Auction
4.1 Representation & Bayes NE
Consider a trading game called a double auction: there are a buyer and a seller who have their own private valuation for the good. Assume the buyer’s valuation of the good is , and the seller’s valuation is . Both and are private information, and independently drawn from uniform distribution on . The seller names an asking price , and the buyer simultaneously names an offer price . If , then trade occurs at a price ; otherwise no trade occurs.
Normal-form representation
- , and for
- , and for
- The buyer believes that is uniformly distributed on , and likewise for the seller.
- The buyer’s payoff is
and the seller’s payoff is
Bayesian Nash equilibrium
A strategy for the buyer is a function specifying the price the buyer will offer for each of the buyer’s possible valuations (likewise for the seller).
A pair of strategies is a Bayesian Nash equilibrium if the following two conditions hold:
- For buyer, for each , solves
- For seller, for each , solves
4.2 One-price equilibrium
There are many Bayesian Nash equilibria. Consider the following one-price equilibrium in which trade occurs at a single price.
The buyer’s strategy is
and the seller’s strategy is
Given the buyer’s strategy , consider whether the seller with type would want to deivate:
- For , the seller would prefer “no tradeing” to “trading at ”.
- For , the seller would prefer “trading at ” to “no trading”.
- For , the seller is indifferent between “trading” at and “no trading”, and will not deviate either.
An analogous argument applies for the buyer.
In equilibrium, trade occurs when and as shown above. However, trade would be efficient for all pairs of such that , but does not occur in the two shaded regions.
4.2 Linear Bayesian NE
Suppose the seller’s strategy is , and then is uniformly distributed on . Buyer’s expected payoff, given his type is
Maximizing yields buyer’s best reponse
which implies and
Thus, if the seller plays a linear strategy, the buyer’s best response is also linear.
Analogously, suppose the buyer’s strategy is , and then is uniformly distributed on .
Seller’s expected payoff, given his type , is
The best response function of the seller is
In equilibrium, the buyer’s strategy must be a best response to the seller’s strategy, i.e.
Analogously, the seller’s strategy must be a best response to the buyer’s strategy, i.e.
The linear equilibrium strategies are
and
Trade occurs iff
In both equilibrium, efficient trade does not always occur, but the most valuable trade (i.e. and ) does occur. The one-price equlibrium misses some valuable trades (such as and where is small), and achieves some trades that are worth next to nothing (such as and ). The linear equilibrium misses all trades that are worth next to nothing, but achieves all trades worth at least . The linear equilibrium dominates the one-price equilibrium in terms of the expected gains that the players receive.
Inefficiency
Myerson and Satterwaite show that for the uniform distributions, the linear equilibrium yields the highest expected gains for the players among all possible Bayesian NE in the double auction. This implies that there is no Bayesian NE of the double auction in which trade occurs iff it is efficient.
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