Question

A painting is auctioned to \(n\) bidders, each with a private value for the painting that is uniformly distributed between 0 and 1 a. Compute the equilibrium bidding strategy in a first-price sealed-bid auction. Compute the seller's expected revenue in this auction. Hint: Use the formula for the expected value of the \(k\) th-order statistic for uniform distributions in Equation 18.71 b. Compute the equilibrium bidding strategy in a second-price sealed-bid auction. Compute the seller's expected revenue in this auction using the hint from part (a). c. Do the two auction formats exhibit revenue equivalence? d. For each auction format, how do bidders' strategies and the seller's revenue change with an increase in the number of bidders?

Short answer

Answer: In both first-price and second-price sealed-bid auctions, an increase in the number of bidders results in bidders bidding closer to their true valuations and an increase in the seller's expected revenue. In a first-price sealed-bid auction, as the number of bidders increases, the equilibrium bidding strategy approaches the true valuation. In a second-price sealed-bid auction, the equilibrium bidding strategy remains the same as the true valuation, regardless of the number of bidders.

Step-by-step solution

Finding equilibrium bidding strategy in a first-price sealed-bid auction

Each bidder's private value for the painting is uniformly distributed between 0 and 1. To find the equilibrium bidding strategy, we need to find the function that gives the bidder's optimal bid for their private value. To find this function, consider the following. For bidder i with private value \(v_i\), let \(b_i\) be their bid, \(B(v_i)\) be the bidding function we are trying to find, and \(G(b_i)\) be the probability that they win the auction. So, bidder i wishes to maximize their expected utility, given by: $$max_{b_i} [v_i - b_i]G(b_i)$$ Taking the first-order condition with respect to \(b_i\) and applying the envelope theorem, we get: $$ b_i = v_i - \frac{G'(b_i)}{G(b_i)}$$ Now, \(G(b_i)\) is the probability that \(b_i\) is the highest bid among the bids of all n bidders. Assuming the bidders follow symmetric strategy \(B(v_i)\), this probability is given by: $$ G(b_i) = (\frac{b_i}{B(1)})^n$$ Taking the derivative with respect to \(b_i\): $$ G'(b_i) = n(\frac{b_i}{B(1)})^{n-1}$$ Substituting G(b_i) and G'(b_i) into the previous equation: $$ b_i = v_i - \frac{n(\frac{b_i}{B(1)})^{n-1}}{(\frac{b_i}{B(1)})^n}B(1) $$ Integrating both sides to find the bidding strategy \(B(v_i)\), we get the equilibrium bidding strategy: $$ B(v_i) = \frac{n-1}{n} v_i $$

Computing the seller's expected revenue in a first-price sealed-bid auction

To find the seller's expected revenue, we need to find the expected highest bid out of all bids. This expected value is given by the formula for the expected value of the \(k\) th-order statistic for uniform distributions (equation 18.71): $$ E[B] = \frac{n}{n + 1} $$ Now, let's substitute the equilibrium bidding strategy for \(v_i\), we get: $$ E[B(V)] = \frac{n}{n + 1} \frac{n-1}{n} = \frac{n-1}{n + 1} $$ So, the seller's expected revenue in a first-price sealed-bid auction is: $$ R_1 = \frac{n-1}{n + 1} $$ #b. Second-Price Sealed-Bid Auction#

Finding equilibrium bidding strategy in a second-price sealed-bid auction

In a second-price sealed-bid auction, since the winner pays the second-highest bid instead of their own bid, the optimal strategy for each bidder is to bid their true value for the painting, i.e. $$B(v_i) = v_i$$

Computing the seller's expected revenue in a second-price sealed-bid auction

The seller's revenue in a second-price sealed-bid auction is the second-highest bid out of all bids. Using the \(k\) th-order statistic for uniform distributions (equation 18.71), the expected second-highest bid is: $$ E[B] = \frac{n-1}{n+1} $$ For this auction format, substituting the equilibrium bidding strategy for \(v_i\): $$ E[B(V)] = \frac{n-1}{n+1}v_i $$ So, the seller's expected revenue in a second-price sealed-bid auction is: $$ R_2 = \frac{n-1}{n+1} $$ #c. Revenue Equivalence#

Comparing seller's revenues from both auction formats

From previous calculations, we have: $$ R_1 = \frac{n-1}{n+1} $$ $$ R_2 = \frac{n-1}{n+1} $$ Since \(R_1 = R_2\), the two auction formats exhibit revenue equivalence. #d. Effect of an increase in the number of bidders#

Analyzing change in bidders' strategies and seller's revenue in a first-price sealed-bid auction

In a first-price sealed-bid auction, the equilibrium bidding strategy is: $$ B(v_i) = \frac{n-1}{n} v_i $$ As the number of bidders (n) increases, the value of \(\frac{n-1}{n}\) approaches 1, so bidders will bid closer to their true valuations. For the seller's expected revenue, it is: $$ R_1 = \frac{n-1}{n+1} $$ As the number of bidders increases, the seller's expected revenue will approach 1.

Analyzing change in bidders' strategies and seller's revenue in a second-price sealed-bid auction

In a second-price sealed-bid auction, the equilibrium bidding strategy is: $$ B(v_i) = v_i $$ This strategy remains the same regardless of the number of bidders since each bidder will simply bid their true valuation. For the seller's expected revenue: $$ R_2 = \frac{n-1}{n+1} $$ As the number of bidders increases, the seller's expected revenue will also approach 1. Thus, in both auction formats, an increase in the number of bidders results in bidders bidding closer to their true valuations and an increase in the seller's expected revenue.

Key concepts

First-Price Sealed-Bid Auction

In a first-price sealed-bid auction, each bidder submits one bid without knowing the bids of the other participants. The bidder who has placed the highest bid wins the item but pays the price they bid. This differs from other auction types, where the winning price might not equate to the highest bid placed.

The strategic complexity of this auction format arises because you must consider not only your valuation of the item but also how to bid in relation to potential bids from competitors. A high bid increases your chances of winning but decreases your payoff, as you pay exactly what you bid. Conversely, a low bid may save you money but reduce your winning likelihood.

Bidding Strategically

Given that bidders have private valuations uniformly distributed between 0 and 1, the equilibrium bidding strategy could be represented by the function: \( B(v_i) = \frac{n-1}{n}v_i \), where \(v_i\) is the bidder’s valuation. This strategy maximizes the bidder's expected payoff, balancing between the chance to win and the amount paid if they do. The seller’s expected revenue in such an auction can be calculated as \( \frac{n-1}{n+1} \), which subtly changes with the number of bidders involved.

Second-Price Sealed-Bid Auction

The second-price sealed-bid auction, also known as a Vickrey auction, is where the highest bidder wins but pays the price submitted by the second-highest bidder. This strategy encourages bidders to bid their private valuation, simplifying strategy as you don’t have to outguess your competitors with complex bid calculations.

Simple Bidding, Honest Signals

Since bidders usually submit bids equal to their valuation, the bidding strategy is straightforward: \( B(v_i) = v_i \). Interestingly, this often leads to accurate price discovery reflecting the true market value of the item being auctioned due to the honesty it incentivizes. The seller's expected revenue remains the same as in a first-price sealed-bid auction, \( \frac{n-1}{n+1} \), hinting at an underlying principle known as 'revenue equivalence' in auction theory.

Revenue Equivalence

Revenue equivalence is a fundamental concept in auction theory that states that, under certain conditions, all standard auction formats will yield the same expected revenue for a seller. The conditions usually include bidders being risk-neutral, values being independent and private, and bidders being symmetric (having the same valuation distribution).

Understanding this principle helps both buyers and sellers predict the outcomes of different auction types. In our example, we observe revenue equivalence as both auction types result in the seller's expected revenue being \( \frac{n-1}{n+1} \), despite the different strategies and payoff structures for the bidders in each auction.

Bidding Strategies

Strategic bidding involves developing a plan or method for placing bids in an auction to maximize the chances of winning while minimizing cost. The optimal strategy depends significantly on the auction format as well as the number of participants.

In first-price auctions, as the number of bidders increases, the bids tend to converge closer to their true valuations. This means that in an auction with many bidders, bidders may bid almost as if they are in a second-price auction. In second-price auctions, however, the optimal bid remains unchanged irrespective of the number of participants, since, by design, it brings out the true value through bidding.

For both auction formats, as the number of bidders rises, the seller’s expected revenue also increases. This can be attributed to increased competition driving the bids higher, closer to the actual valuations of the bidders, hence increasing the winning bid's value.