Auctions: Auction types

We should consider the following auction types:

Two main classes:

Main Auction Formats

Mechanism: All bidders submit bids at the same time. The highest bidder wins and pays their bid.
Strategy: Bidders tend to underbid, generally bidding half of their valuation. This strategy is a Nash equilibrium.
Expected Revenue: Selena's revenue is (\frac{1}{2} \times \max(v_A, v_B)). The expected value of (\max(v_A, v_B)) is (\frac{2}{3}), so the expected revenue is (\frac{1}{3}), which is the same as the second-price auction.

Mechanism: All bidders submit bids simultaneously. The highest bidder wins but pays the second-highest bid.
Strategy: The optimal strategy is to bid truthfully (bid your actual valuation). This is known as a dominant strategy.
Expected Revenue: The expected revenue is the expected value of the second-highest valuation. For uniformly distributed valuations between 0 and 1, this is (\frac{1}{3}).

Mechanism: The auctioneer starts at a low price and continually raises it. Bidders can drop out at any point, and the last remaining bidder wins and pays the price at which the second-to-last bidder dropped out.
Strategy: Similar to the second-price auction, the dominant strategy is to stay in the auction until the price reaches your valuation.
Revenue Equivalence: The expected revenue is the same as in the second-price auction, making these two formats revenue equivalent.

Mechanism: The auctioneer starts at a high price and gradually lowers it until a bidder accepts the current price. The first bidder to accept wins and pays the accepted price.
Strategy: Bidders do not bid truthfully; instead, they wait until the price drops to around half their valuation before accepting.
Revenue Equivalence: This auction is strategically and revenue-equivalent to the first-price sealed-bid auction.

Historical Context: Candle auctions were used in England from the 16th to 18th centuries. The auction was conducted while a candle burned, and bids were accepted until the flame went out.
Mechanism: Bidders submit bids while a candle burns. The last bid made before the candle extinguishes is the winning bid.
Strategy: This type of auction introduces randomness, which can reduce strategic bidding and potentially encourage bidders to bid closer to their true valuations.
Revenue Impact: The unpredictability of when the auction ends can create urgency, potentially leading to higher final bids compared to other formats.

Mechanism:
- Items are posted for sale at a low price, sometimes starting for nothing.
- Bids are open.
- A timer is reset and counts down any time a new bid is placed.
- This continues until the timer reaches zero and a bid wins the prize.
Strategy: In penny auctions, bidders must be wary about when to place their bids, as each bid takes money. The key is to place bids in a way that maximizes the chance of winning while minimizing the cost from placing/spamming multiple bids.
Revenue Equivalence: Penny auctions are generally not revenue equivalent to traditional auction formats. The element of charging for each bid often leads to higher revenues for the auctioneer compared to other auction types, as bidders may place many low-cost bids in hopes of winning, thus generating more revenue.

Description: Just like a standard english auction, but also includes a buyout option where a bidder can purchase the item right away for a fixed price, thus ending the auction.