First-price sealed-bid auction
First-price sealed-bid auction
A first-price sealed-bid auction (FPSBA) is a common type of auction. It is also known as blind auction. In this type of auction, all bidders simultaneously submit sealed bids so that no bidder knows the bid of any other participant. The highest bidder pays the price that was submitted.
Strategic analysis In a FPSBA, each bidder is characterized by their monetary valuation of the item for sale.
Suppose Alice is a bidder and her valuation is <math>a</math>. Then, if Alice is rational: She will never bid more than <math>a</math>, because bidding more than <math>a</math> can only make her lose net value. If she bids exactly <math>a</math>, then she will not lose but also not gain any positive value. If she bids less than <math>a</math>, then she may* have some positive gain, but the exact gain depends on the bids of the others.
Alice would like to bid the smallest amount that can make her win the item, as long as this amount is less than <math>a</math>. For example, if there is another bidder Bob and he bids <math>y</math> and <math>y<a</math>, then Alice would like to bid <math>y+\varepsilon</math> (where <math>\varepsilon</math> is the smallest amount that can be added, e.g. one cent).
Unfortunately, Alice does not know what the other bidders are going to bid. Moreover, she does not even know the valuations of the other bidders. Hence, strategically, we have a bayesian-game - a game in which agents do not know the payoffs of the other agents.
The interesting challenge in such a game is to find a Bayesian Nash equilibrium. However, this is not easy even when there are only two bidders. The situation is simpler when the valuations of the bidders are independent and identically distributed random variables, so that the valuations are all drawn from a known prior distribution.
Example Suppose there are two bidders, Alice and Bob, whose valuations <math>a</math> and <math>b</math> are drawn from a continuous uniform distribution over the interval [0,1]. Then, it is a Bayesian-Nash equilibrium when each bidder bids exactly half his/her value: Alice bids <math>a/2</math> and Bob bids <math>b/2</math>.
PROOF: The proof takes the point-of-view of Alice. We assume that she knows that Bob bids <math>f(b) = b/2</math>, but she does not know <math>b</math>. We find the best response of Alice to Bob's strategy. Suppose Alice bids <math>x</math>. There are two cases: <math>x\geq f(b)</math>. Then Alice wins and enjoys a net gain of <math>a-x</math>. This happens with probability <math>f^{-1}(x)=2x</math>. <math>x<f(b)</math>. Then Alice loses and her net gain is 0. This happens with probability <math>1-f^{-1}(x)</math>. All in all, Alice's expected gain is: <math>G(x) = f^{-1}(x)\cdot(a-x)</math>. The maximum gain is attained when <math>G'(x)=0</math>. The derivative is (see Inverse functions and differentiation): :<math>G'(x) = - f^{-1}(x) + (a-x)\cdot {1 \over f'(f^{-1}(x))}</math> and it is zero when Alice's bid <math>x</math> satisfies: :<math>f^{-1}(x) = (a-x)\cdot {1 \over f'(f^{-1}(x))}</math> Now, since we are looking for a symmetric equilibrium, we also want Alice's bid <math>x</math> to equal <math>f(a)</math>. So we have: :<math>f^{-1}(f(a)) = (a-f(a))\cdot {1 \over f'(f^{-1}(f(a)))}</math> :<math>a = (a-f(a))\cdot {1 \over f'(a)}</math> :<math>a f'(a) = (a-f(a))</math> The solution of this differential equation is: <math>f(a) = a/2</math>.
Generalization Denote by: * <math>v_i</math> – the valuation of bidder <math>i</math>; * <math>y_i</math> – the maximum valuation of all bidders except <math>i</math>, i.e., <math>y_i = \max_{j\neq i}{v_j}</math>. Then, a FPSBA has a unique symmetric BNE in which the bid of player <math>i</math> is given by: ::<math>E[y_i \mid y_i < v_i]</math>
Incentive-compatible variant The FPSBA is not incentive-compatible even in the weak sense of Bayesian-Nash-Incentive-Compatibility (BNIC), since there is no Bayesian-Nash equilibrium in which bidders report their true value.
However, it is easy to create a variant of FPSBA which is BNIC, if the priors on the valuations are common knowledge. For example, for the case of Alice and Bob described above, the rules of the BNIC variant are: The highest bidder wins; The highest bidder pays 1/2 of his/her bid. In effect, this variant simulates the Bayesian-Nash equilibrium strategies of the players, so in the Bayesian-Nash equilibrium, both bidders bid their true value.
This example is a special case of a much more general principle: the revelation-principle.
Comparison to second-price auction <div id='comparison-2nd-price'></div> The following table compares FPSBA to sealed-bid second-price auction (SPSBA):
{| class="wikitable" |- ! Auction: !! First-price !! Second-price |- | Winner: || Agent with highest bid || Agent with highest bid |- | Winner pays: || Winner's bid || Second-highest bid |- | Loser pays: || 0 || 0 |- | Dominant strategy: || No dominant strategy || Bidding truthfully is dominant strategy |- | Bayesian Nash equilibrium|| Bidder <math>i</math> bids <math>\frac{n-1}{n}v_i</math> || Bidder <math>i</math> truthfully bids <math>v_i</math> |- | Auctioneer's revenue In contrast, if the seller does not announce the reserve price before the sale but only after the sale, it is a secret reserve price auction.
Comparison to other auctions A FPSBA is distinct from the english-auction in that bidders can only submit one bid each. Furthermore, as bidders cannot see the bids of other participants, they cannot adjust their own bids accordingly.
A generalized-first-price-auction is a non-truthful auction mechanism for sponsored search (aka position auction).
A generalization of both 1st-price and 2nd-price auctions is an auction in which the price is some convex combination of the 1st and 2nd price.