Subjects: What is an auction? Auction formats. True values & known values. Relationships between auction formats

Transcription

1 Auctions

2 Subjects: What is an auction? Auction formats True values & known values Relationships between auction formats Auctions as a game and strategies to win. All-pay auctions

3 What is an auction? An auction is a process of buying and selling goods or services by offering them up for bid, taking bids, and then selling the item to one of the buyers. When do we come across auctions? o History o Politics o Web o Auctions houses o Sports o And many more

4 Why auctions? Attraction to buyers. Seller s payoff can be larger than his fixed selling price. Buyers can get the item for a price cheaper than what the item s value is for them. When the seller cannot evaluate the value of the item.

5 Auction formats We can consider two major cases of auctions: A buyer is interested in buying an item and runs an auction between a set of sellers the sellers need to compete each other for who gets to sell the item. A seller auctioning one item to a set of buyers the buyers need to compete each other for who gets to purchase the item.

6 Ascending-bid (English) auctions Carried out interactively in real time open auction. Bidders present bids physically or electronically. The price is raised gradually. Bidders drop out until one bidder remains. The most common type of auction. Example: Sotheby s, the world's second oldest international auction house used the English auction format during most of its existence

7 Descending-bid (Dutch) auctions Carried out interactively in real time open auction. Bidders present bids physically or electronically. The price is lowered gradually from initial high value. Example: Flower auction located in Aalsmeer in Netherlands the largest flowers auction in the world.

8 First-price sealed-bid auctions Carried out simultaneously. Bidders present sealed bids. The bids are revealed altogether. The buyer with the highest bid wins. The buyer pays the value of his bid the first highest bid. Example: Foreign exchange markets

9 Second-price sealed-bid (Vickery) auctions Carried out simultaneously. Bidders present sealed bids. The bids are revealed altogether. Called in honor of William Vickery, who wrote the first gametheoretic analysis of auctions The buyer with the highest bid wins. The buyer pays the value of the second highest bid. Example: Stamp collecting business, ebay, Google ads

10 Example: Google ads auction A service provided by Google, in which they place ads according to their relevance to user s searches. Each user search opens an auction for the ads that will be presented to the user according to the search keywords. The advertisers compete in this auction by stating the price they are willing to pay per click on their ad. Wants more users to be exposed to their ads, and that their money will be spent only on relevant profitable users. Google Wants both the user and the advertiser to be satisfied from Google s service and continue to use it Wants to get relevant search results and ads. Advertiser User

11 Example: Google ads auction In order to satisfy the users and the advertisers, Google developed a more complicated auction system, based on the second price auction concept. Each ad gets a quality score computed by: Quality score= The number of clicks in the ad so far + The relevance of the ad to the user s search + The landing page quality (relevant, original content, minimum pop ups ) The final ad rank = the advertiser bid x the quality score. Finally, the winner (with the highest rank) pays the second highest price calculated as such: p i = the price that bidder i will pay per click q i = the quality score of bidder i s ad b i = the bid that bidder i stated p 1 q 1 = b 2 q 2 p 1 = b 2 q 2 q 1

12 True value We assume that each buyer has an intrinsic value for the item being auctioned. The intrinsic value is the highest value the buyer is willing to pay for the item. Symmetrically, the seller has an intrinsic value which is the lowest value that the seller is willing to sell the item. We will call this value the buyer s true value. If the true value is known should the auction still be executed?

13 Known true value Let s consider the case in which everybody s true value is known. Let s mark: x = the seller s true value. y = the highest true value among the buyers. We assume that y x so the sale will be possible. I ll buy the item for just above x I ll sell the item for just below y Being committed to a fixed price can be crucial Who believes whose commitment? This case is more relevant to Bargaining From now on we will assume that the true values are unknown and independent

14 Relationship between auction formats Type Open/ Sealed Progress Who wins? Winner s payment Ascending-bid Open Price raised gradually Highest price bidder Highest bid Descending Open Price lowered gradually Highest price bidder Highest bid First-price sealed bid Sealed One bid Highest price bidder Highest bid Second-price sealed bid Sealed One bid Highest price bidder Second highest bid If we find equivalence between formats, we can analyze one format and try to apply similar analysis to its equivalent format. Is there any equivalence?

15 Descending bid and First price auctions No communication: No bidder knows the other bidder s bid until the auction ends: On descending bid there is no communication between the bidders until a bidder accepts the price and the item is sold. On first price all the bids are sealed and revealed altogether at the end of the auction. The final price: b=the bidder with the highest true value v=b s true value On descending bid, the final price will be v since once the price reaches (lowered) to v, b will accept the price before all other bidders with lower true values. On first price, the final price will be v since b has only one chance to bid and so he will bid as much as he needs to get the item in order to win.

16 Ascending bid and Second price auctions Assume that the Ascending bid winner pays the second highest bid (since the auction will end when the second highest bidder drops out). When will a bidder drop out of the auction? Lose the opportunity to pay less for the item Lose = no gain Win = pay more than the true value Conclusions: Current price = true value The bidder should remain in the auction until the price reaches his true value. The bidder with the highest true value will remain in the auction the longest, and therefore will win the auction and will pay the second highest price. Equivalence to sealed bid when bidders bid their true values. The winner will be the bidder with the highest true value and will pay the second highest bid.

17 Second price auction as a Game Let s mark: v i = bidder i s true value b i = the amount bidder i bids We will consider the auction process as a game: The players = the bidders. The strategy = a function that maps the players true value (v i ) to a bid b i, s v i = b i. The payoff (for second price sealed bid): 0 if b i is not the winning bid v i b j if b i is the winning bid and b j is the second place bid. In case of a tie: we assume that there is a fixed ordering of the bidders which is agreed in advance. The winning bidder will be the one that comes first on the order.

18 Dominant strategy in Second price auction Claim: In a sealed bid second price auction, it is a dominant strategy for each bidder i to choose a bid b i = s v i = v i We will try to prove the claim by proving that no deviation b i from the bid b i = v i would improve i s payoff. First, let s notice that the value of bidder i s bid only affects whether i wins or loses. i s payoff will be determined either by loss (0) or by the second price bid. We will consider two cases: o Case A: deviation in which b i > v i = b i. o Case B: deviation in which b i < v i = b i.

19 Dominant strategy in Second price auction Case A : b i > v i : o This case will affect i s payoff only if i loses with v i and wins with b i. o According to this, the second highest bid is v i b j < b i o The payoff will be at most v i b j 0 Case B : b i < v i : o This case will affect i s payoff only if i wins with v i and loses with b i. o According to this, before deviating, the second highest bid was b i b k < v i o The payoff : Before deviating payoff= at most v i b k 0 After deviating payoff= 0 (i loses)

20 First price auctions and other formats On first price sealed bid as a game, the payoff is different than second price auction: If b i is not the winning bid then the payoff to i is 0. If b i is the winning bid then the payoff to i is v i b i On first price auction, the value b i affects both whether bidder i wins/loses and how much he will pay. Is bidding your true value a dominant strategy on this case? According to the new payoff definition the answer is NO. If i loses then the payoff is obviously 0. If i wins then he pays his bid so his payoff is v i b i = v i v i = 0.

21 First price auctions and other formats The optimal way to have positive payoff in a first price auction is to shade down the bid. Bid too far below the true value reduce the chance of winning Bid too close to the true value - small payoff Finding the optimal tradeoff is complex and depends on the knowledge of the other bidders and the distribution of possible true values among then. For example, when there is a large number of bidders, the competing bids are likely to be higher. In this case, the bidder should bid higher and closer to his true value in order to win. We will talk about first price auction strategy on the next presentation

22 All-pay auctions On all-pay auctions all bidders submit their bids, the highest bidder receives the item, and all the bidders pay their bids regardless of whether they win or lose. Examples of all-pay auction models in real life: Funding of election campaigns the funders donate money to a party that they identify with. The party s winning is their personal payoff. Eventually, all the funders pay regardless of the election outcome, and there is only one winner. Design competitions all the design firms invest lots of money on preliminary designs in order to impress the client and win the contract. In the end, only one firm wins the contract and the others are left without the spent money. Similarly to first price auctions, bidders need to shade down their bids and there is a tradeoff. The risk to pay without winning is large and affects the bids that are much lower.

23 Summary We saw examples for auctions in real life, and how big companies developed a selling format based on auctions guidelines (Google ads). We considered different auction formats and the equivalence between them. We defined auctions as a game according to game theory. We analyzed strategies for stating a bid on an auction.

24 Questions? To be continued