(All mathematical process are extracted from lecture notes directly ! )
Please see at the end of the text part.
To explain why the four basic types of auction are equivalent in the eyes of a risk-neutral seller under an independent private value model, it is necessary to introduce the four types of auctions, the definition of risk neutral seller. Then, by describing and explaining the independent private value model carefully and specifically, which provides a clear picture on answering the question.
There are four common basic types of auctions, they are English auction; Dutch auction; First-Price Sealed- Bid auction; and Second-Price Sealed-Bid auction. English auction is based on open ascending price auction, which means that bids ends when no higher bid is forthcoming and all biding price information are available to bidders. This kind of auction is usually used in antiques and artwork. Dutch auction is based on open descending price action, which means the auctioneer set up an initial high price and then announces to decrease the price gradually. The auction ends once there is one bidder indicates his willingness to pay the auctioneer’s offer price. First-Price Sealed-Bid auction is an auction that has a deadline, each bidder submits a sealed bid without knowing the bids of others. Only bidders with the highest bidding price could obtain the auction item. The Second-Price Sealed-Bid auction is the same as First-Price Sealed- Bid auction except the rule is only bidders with the second highest bidding price could be rewarded that item.
Through definition, a risk-neutral individual is indifferent between any certainty and expected outcomes. The expected outcome is associated with risk and uncertainty, which comprises all kinds of scenarios. In this case, the risk neutral means individual values the same utility no matter under the certain outcome or the expected outcome.
In fact, the independent private values model consists one risk neutral seller and N risk neutral bidders. Further, any bidders should not have priority on access to relevant auction information of others’, they are all the same but might have different preferences on valuing the auction items. Of particular, it is necessary to recognize that even though bidders could value items as different price, but all of them in their minds are likely to accept items at no cost. It is because the auction items are normal goods, which satisfy the downward sloping demand curve. There is one extremely situation that all bidders value one item for zero price, which means the expected value of that item is zero. As discussions of risk- neutral individuals, who actually regard the expected outcome as the same as certain outcome, which means sell’s valuation is equal to zero. In this model, assumptions must be presented so that eliminating any logical errors. They are no transaction cost, no collusions, no any other cost relevant to auction. In this case, under the assumptions and discussions above, it is nature determines the auction outcome because the bidders are numbered so that v1 > v2 > _ _ _ > vN. The outcome of the auction is Pareto efficient since the goods are always allocated to people whom with the highest valuation of them.
The situation facing a bidder in an English auction is identical to that in a Second-Price Sealed-Bid auction.
Bidder i.s strategy bi (vi ): quitting price as a function of his valuation.
Bid = quitting price determined by valuation, which is independent.
) choose the same bid whether others.bids are observed.
Whoever has the highest bid (quitting price) wins, the price = second highest bid (quitting price) (The auction ends the moment the bidder with second-highest quitting price quits.)
(Lecture slides 18, Auction Theory)
Each bidder in an English auction has a dominant strategy
Bidder i.s dominant strategy b_i (vi ) = vi .
Suppose the last other bidder quits at ˆb (unknow to bidder i ).
If vi > ˆb,
bi (vi ) = vi ) get the item at price = ˆb, surplus = vi �ˆb;
bi (vi ) > vi ) same outcome as bi (vi ) = vi ;
bi (vi ) < vi ) risk losing the item & surplus.
If vi < ˆb,
bi (vi ) = vi ) not get the item;
bi (vi ) < vi ) same outcome as bi (vi ) = vi ;
bi (vi ) > vi ) risk getting the item at price ˆb > vi .
bi (vi ) = vi weakly dominates bi (vi ) > vi and bi (vi ) < vi
however others bid.
How bi (vi ) affects the prob of winning doesn’t matter .
(Lecture slides 19, Auction Theory)
All bidders quit at their valuations.
(v1 > v2 > _ _ _ > vN )
The auction ends when bidder 2 quits.
Bidder 1 receives the good at price = v2.
Outcome is Pareto e¢ cient.
The equilibirum price (for the English auction) PE = v2.
Seller.s expected revenue:
E[v2] = l + (
N + 1 � k
N + 1
)(h � l ) = 0 + (
N + 1 � 2
N + 1
)(1 � 0) =
N � 1
N + 1
(Lecture slides, 20,……)
Just extract Lecture slides 28 directly
Dutch & First-Price Sealed-Bid auctions are always equivalent.
English & Second-Price Sealed-Bid auctions are equivalent in an independent-private-values model.
The bidders.strategies & outcomes are identical.
The equilibrium prices
PE = PSSA = v2 not equal to PD = PFSA =
N � 1
But the expected price or revenue to the seller are the same.
N � 1
N � 1
N + 1
The outcomes are all Pareto efficient