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Lecture 1 on Auctions Optimal Bidding.

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### Lecture 1 on AuctionsOptimal Bidding

Auctions serve the dual purpose of eliciting preferences and allocating resources between competing uses. A less fundamental but more practical reason for studying auctions is that the value of goods exchanged each year by auction is huge. We describe the main kinds of auctions, define strategic and compare the outcomes from using different types.

Read Chapters 19 and 20 of Strategic Play.

Auctions

Auctions are widely used by companies, private individuals and government agencies to buy and sell commodities.

They are also used in competitive contracting between an (auctioneer) firm and other (bidder) firms up or down the supply chain to reach trading agreements.

Merger and acquisitions often have an auction flavor about them.

To begin our discussion we compare two sealed bid auctions, where there are two bidders with known valuations of 2 and 4 respectively.

First price sealed bid auction

- In a first price sealed auction, players simultaneously submit their bids, the highest bidder wins the auction, and pays what she bid for the item.
- Highway contracts typically follow this form.
- For example, if the valuation 4 player bids 5 and the valuation 2 player bids 1, then the former wins the auction, pays 5 and makes a net loss of 1.

Dominance for the row player

The column player does not have a dominant strategy.

There is a dominant strategy for the row player to bid 1.

By iterative dominance the column player bids 2.

Iterative dominance in a first price auction

The column player does not have a dominant strategy.

There is a dominant strategy for the row player to bid 1.

By iterative dominance the column player bids 2.

Second price sealed bid auction

- In a second priced sealed bid auction, players simultaneously submit their bids, the highest bidder wins the auction, and pays the second highest bid.
- This is similar to Ebay, although Ebay is not sealed bid.
- For example, if the valuation 4 player bids 5 and the valuation 2 player bids 1, then the former wins the auction, pays 1 and makes a net profit of 3.

Dominance in a second price auction

The row player has a dominant strategy of bidding 2.

The column player has a dominant strategy of bidding 4.

Thus both players have a dominant strategy of bidding their (known) valuation.

Comparing auction mechanisms

Comparing the two auctions, the bidder place different bids but the outcome is the same, the column player paying 2 for the item.

How robust is this result, that the form of the auction does not really matter?

Do people know their own valuations?

If not, are the valuations related?

What do bidders know about the valuations of their rivals?

Is everyone risk neutral?

How many items are up for sale, and how many units is each player bidding on?

Is collusion an issue?

Bidding strategies

- Recall that a strategy is a complete description of instructions to be played throughout the game, and that the strategic form of a game is the set of alternative strategies to each player and their corresponding expected payoffs from following them.
- Two games are strategically equivalent if they share the same strategic form.
- In strategically equivalent auctions, the set of bidding strategies that each potential bidders receive, and the mapping to the bidder’s payoffs, are the same.

Descending auctions are strategically equivalent to first-price auctions

- During the course of a descending auction no information is received by bidders.
- Each bidder sets his reservation price before the auction, and submits a market order to buy if and when the limit auctioneer\'s limit order to sell falls to that point.
- Dutch auctions and first price sealed bid auctions share strategic form, and hence yield the same realized payoffs if the initial valuation draws are the same.
- Rule 1: Pick the same reservation price in Dutch auction that you would submit in a first price auction

English auction

- In an English auction bidders compete against each other by successively raising the price at which they are willing to pay for the auctioned object.
- The bidding stops when nobody is willing to raise the price any further, and the item is sold to the person who has bid the highest price, at that price.

Second-price versus ascending auctions

- When there are only 2 bidders, an ascending auction mechanism is strategically equivalent to the second price sealed bid auction (because no information is received during the auction).
- More generally, both auctions are (almost) strategically equivalent if all bidders have independently distributed valuations (because the information conveyed by the other bidders has no effect on a bidder’s valuation).
- In common value auctions the 3 mechanisms are not strategically equivalent if there are more than 2 players.
- Rule 2: If there are only two bidders, or if valuations are independently distributed, choose the same reservation price in English, Japanese and second price auctions.

Bidding in a second-price auction

- If you know your own valuation, there is a general result about how to bid in a second price sealed bid auction, or where to stop bidding in an ascending auction.
- Bidding should not depend on what you know about the valuations of the other players, nor on what they know about their own valuations.
- It is a dominant strategy to bid your own valuation.
- A corollary of this result is that if every bidder knows his own valuation, then the object will be sold for the second highest valuation.
- Rule 3 : In a second price sealed bid auction, bid your valuation if you know it.

Proving the third rule

- Suppose you bid above your valuation, win the auction, and the second highest bid also exceeds your valuation. In this case you make a loss. If you had bid your valuation then you would not have won the auction in this case. In every other case your winnings would have been identical. Therefore bidding your valuation dominates bidding above it.
- Suppose you bid below your valuation, and the winning bidder places a bid between your bid and your valuation. If you had bid your valuation, you would have won the auction and profited. In every other case your winnings would have been identical. Therefore bidding your valuation dominates bidding below it.
- The proof is completed by combining the two parts.

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