Question

Suppose you and one other bidder are competing in a private-value auction. The auction format is sealed bid, first price. Let $v$ and $b$ denote your valuation and bid, respectively, and let $\hat{v}$ and $\hat{b}$ denote the valuation and bid of your opponent. Your payoff is $v-b$ if it is the case that $b\ge \hat{b}$. Your payoff is 0 otherwise. Although you do not observe $\hat{v}$, you know that $\hat{v}$ is uniformly distributed over the interval between 0 and 1 . That is, $v^{\prime }$ is the probability that \(\hat{v}

Short answer

The optimal bid is $\frac{4}{25}$.

Step-by-step solution

Understand the Auction Format

In a sealed-bid, first-price auction, each bidder submits a bid without knowing the other's bid, and the highest bid wins. The winning bidder pays the price they bid. Your payoff can be calculated as $v-b$ if you win (i.e., your bid $b$ is greater than or equal to your opponent's bid $\hat{b}$), and 0 if you lose.

Calculate the Opponent's Bidding Strategy

Your opponent's bid is defined as $\hat{b}(\hat{v})={\hat{v}}^2$. Knowing that $\hat{v}$ is uniformly distributed over the interval [0, 1], the probability that $\hat{v}<v^{\prime }$ is simply $v^{\prime }$. Therefore, $\hat{b}(\hat{v})$ implies that your opponent's bid varies as the square of their valuation.

Determine Probability of Winning

To win, your bid $b$ must be equal to or greater than your opponent's bid $\hat{b}$. From $\hat{b}(\hat{v})={\hat{v}}^2$, you want to ensure $b\ge {\hat{v}}^2$. Given this setup, the probability that $b\ge \hat{b}$ happens when ${\hat{v}}^2\le b$, which means $\hat{v}\le \sqrt{b}$. The probability is therefore $\sqrt{b}$ because $\hat{v}$ is uniform over [0,1].

Maximize Expected Payoff

Your expected payoff is $(v-b)\times \text{Probability of Winning}$, which translates to $(v-b)\times \sqrt{b}$. For your value $v=\frac{3}{5}$, we need to maximize the function $(\frac{3}{5}-b)\times \sqrt{b}$.

Differentiate and Solve for Optimal Bid

We differentiate the expected payoff function $E(b)=(\frac{3}{5}-b)\sqrt{b}$ with respect to $b$, setting the derivative equal to zero to find the maximum:$E^{\prime }(b)=\sqrt{b}-\frac{\frac{3}{5}-b}{2\sqrt{b}}=0.$Simplifying and solving this equation will provide the optimal bid.

Optimal Bid Calculation

Solving the equation from Step 5, we find two solutions but only the solution within the interval (0,1) is relevant. So, after calculations:$b={\left(\frac{2}{5}\right)}^2=\frac{4}{25}.$ We reduce it to find that the optimal bid when your valuation is $\frac{3}{5}$, keeping it consistent with your competitor's behavior.

Key concepts

Auction Theory

Auction theory is a fascinating field within economics that studies how people compete to buy or sell goods, services, or assets in an auction format. This field explores various elements that affect auction outcomes, such as the rules of bidding, the information available to bidders, and the payment model.

In a nutshell, auction theory applies principles from economics and game theory to understand how bidders behave in auctions and how different auction designs influence bidders' strategies and outcomes.

• Types of auctions include English auctions, Dutch auctions, first-price sealed-bid auctions, and second-price sealed-bid auctions, each with unique rules and strategic implications.
• Key considerations in auction theory are how auction rules affect bidder behavior and how they shape the final outcome.
• A well-designed auction can lead to an efficient allocation of resources, where goods are sold at a price equal to their true valuation.

Understanding auction theory helps us predict human behavior in competitive bidding environments and allows for designing auctions to achieve specific goals.

Bidding Strategy

A bidding strategy is a planned approach a bidder takes to determine how much to bid in an auction. Developing an effective bidding strategy involves understanding the auction format, opponents' behavior, and the payoff structure.

In a first-price sealed-bid auction, like the one described in the exercise, bidders submit one bid in secret, and the highest bidder wins, paying their bid amount. This setup encourages bidders to balance bidding high (to win) with bidding low (to maximize surplus when they win).

• Knowing your competition's bidding strategies is crucial. If your opponent bids based on their valuation squared, as in the solution, understanding this function is vital.
• Bidders must consider the probability of winning versus the potential payoff and cost. This trade-off impacts how high or low they decide to bid.
• It's important for bidders to not only calculate how others might bid but also to make these calculations keeping their own valuations at the core.

Ultimately, an optimal bidding strategy minimizes risk and maximizes potential returns by carefully planning bids.

Sealed-Bid Auction

In a sealed-bid auction, all bidders submit their bids without knowing the bids of others. This format introduces a layer of uncertainty compared to open auctions, as each bidder only knows their own valuation and has to guess others'.

The first-price sealed-bid auction, the focus of our exercise, is particularly strategic because the highest bid wins, but the winner pays the amount they bid. This means:

• Bidders must find a balance between bidding high enough to win but not so high that there's no financial gain.
• There's no chance to adjust bids based on competitors' actions or insights, so preparing a strategy based on assumptions about others' valuations is essential.
• The auction format can lead to "bid shading," where a bidder submits a bid lower than their valuation to increase their payoff if they win.

This secrecy and one-shot nature make sealed-bid auctions complex but also fascinating, as they reveal different strategic elements of auction theory.

Expected Payoff

The expected payoff in an auction is the anticipated profit a bidder expects to earn from participating. It calculates the average outcome considering the probability of different scenarios occurring.

In our first-price sealed-bid auction scenario, the payoff for a bidder is their own valuation minus their bid, but only if they win. Hence, the expected payoff must take into account:

• The probability of winning, which depends on the comparison between their bid and opponents' bids.
• The actual payoff if the bid wins, calculated as the difference between valuation and bid.
• In the exercise, the expected payoff function was $(v-b)\times \sqrt{b}$, reflecting the probability that a bid wins ($\sqrt{b}$) and the payoff from winning ($v-b$).

Optimizing the expected payoff involves carefully choosing a bid that maximizes this function, balancing a decent probability of winning with the best possible financial outcome.