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There are two kinds of auctions
- Open Auction - the bidders observe some dynamic process that evolves until a winner emerges.
- English Auction- the players bid increasing prices until no player is willing to bid. At which point the highest bidder pays.
- Dutch Auction - the price starts really high and goes progressively down. Once a bidder declares they will buy, the auction ends.
- Sealed Bid - bidders write down their bids and submit them without knowing the bids of their opponents. Highest bidder wins.
- First-Price - bidders write down the bid in an envelope. Bidding prices are revealed simultaneously and the highest bidder wins, playing the price equal to their bid.
- Second Price - similar to first-price but the highest bidder does not pay their bidding price, rather they pay the second highest bidding price
- Open Auction - the bidders observe some dynamic process that evolves until a winner emerges.
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To frame an auction as a Bayesian game, we will set the type of the player to be the value of player
obtaining the good. - Each player knows the type distribution of other players., encapsulated in the cumulative distribution function
, where . - Each player’s strategy is some bidding price
that is dependent on his valuation . - The payoff is
if the player loses and if they win and pay price , the payoff is . - We assume Independent Private Values, that is, types are sampled independently from the type distribution
- Each player knows the type distribution of other players., encapsulated in the cumulative distribution function
Second-Price Sealed Bid
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(Tadelis 13.1) In the second-price sealed bid auction, each player has a weakly dominant strategy which is to bid his true valuation. That is
is a Bayesian Nash Equilibrium in weakly dominant strategies. - Intuition: It is never worse to bid your true valuation and it’s also strictly better.
- This is because even if you win, you don’t have to pay your bidding price
- And if you bid something lower than your bidding price, either you could’ve win by bidding the maximum amount (i.e.,
) or you would still have lost (in which case there is no difference).
- Additionally, it is worse to bid higher than your true valuation (since in the case you do win, you will have to pay more.)
- Intuition: It is never worse to bid your true valuation and it’s also strictly better.
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This result implies three things
- Bidders in this scenario do not care about the probability distribution over their opponent types. Hence, this auction works even when players have no idea about their opponent’s preferences
- Independence of type valuation is not necessary. Players can still bid truthfully even when their beliefs about the type distribution are incorrect.
- The auction is Pareto optimal. The person who values the item most will get the item.
English Auction
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For the English Auction, we reframe the problem as follows: The price raises continuously over time. If a player taps out (is unwilling to pay the price), they are permanently out of the game. The last player wins the item. This is called the Button-Auction Model (tapping out in the game is reminiscent to releasing a button).
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(Tadelis 13.2) In the button-auction model, it is a weakly dominant strategy for each player to keep playing as long as
, and to stop playing once . This results in a Bayesian Nash equilibrium in weakly dominant strategies that is outcome-equivalent to the second price, sealed bid auction. - If the player taps out while
they forego any opportunity to win the item at a price lower than their valuation. - If the player keeps playing while
, then they may win the item but have to pay more than they are willing to for the item. - The player ends up playing the second highest price (when the last player tapped out).
- If the player taps out while
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It remains appealing in the same way as Second-Price Sealed Bid auctions are.
First-Price Sealed-Bit Auctions and Dutch Auctions
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(Tadelis Claim 13.1) In a first-price sealed bid auction, it is a dominated strategy for a player to bid their valuation.
- Bidding the valuation means the players’ payoff is zero — losing implies a payoff of
and winning implies paying the same amount the item is worth - Players will lower their bid to increase payoff at the cost of the probability of winning.
- Bidding the valuation means the players’ payoff is zero — losing implies a payoff of
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Assume that a higher player valuation implies a higher bid. In other words
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The assumption allows us to calculate the payoff as
And assuming a symmetric strategy for all players, we have that 1, assuming
Where the second term in the RHS of the equation above denotes how much below the valuation player
bid. As expected the optimal bid is to shade the bid depending on the knowledge of other players’ types
Revenue Equivalence
- The Revenue Equivalence Theorem states that any auction game that satisfies four conditions will yield the same expected revenue, and will yield each type of bidder the same expected payoff.
- Each bidder’s type is drawn from a well-behaved distribution
- Bidders are risk neutral.
- The bidder with the highest type wins.
- The bidder with the lowest possible type has an expected payoff of
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Common Value
- Common Value means that the value of obtaining the resource is not solely determined by the player’s type — but it can also be determined by other player’s types.
- The winner’s curse - players will tend to win when they receive signals (information) that are optimistic, and which means that they have over-estimated the value of the good and are thus overpaying
- Thus players must bid with the winner’s curse in mind to avoid overpaying for the object.