Question

Consider the following simple model of a common values auction. Two buyers cach obtain a private signal about the value of an object. The signal can be cither high \((H)\) or low \((L)\) with equal probability. If both obtain signal \(H,\) the object is worth 1 otherwise, it is worth 0 a. What is the expected value of the object to a buyer who sees signal \(L\) ? To a buyer who sees signal \(H ?\) b. Suppose buyers bid their expected value computed in part (a). Show that they earn negative profit conditional on observing signal \(H-\) an example of the winner's curse.

Short answer

Answer: The expected value of the object for a buyer who sees signal L is 0, and for a buyer who sees signal H, it is 1/4. If buyers bid their expected values, there could be a winner's curse, as shown by the negative profit earned by a buyer with signal H in some scenarios.

Step-by-step solution

Determine the probability for each possible outcome

We first determine the probability of each possible outcome of the buyers receiving the signals. Since there are two buyers and each can get either High (H) or Low (L) with equal probability, there are four possible outcomes: 1. Both buyer A and buyer B get signal H (HH): The probability is \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\). 2. Buyer A gets signal H and buyer B gets signal L (HL): The probability is \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\). 3. Buyer A gets signal L and buyer B gets signal H (LH): The probability is \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\). 4. Both buyer A and buyer B get signal L (LL): The probability is \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\).

Calculate the expected value for a buyer who sees signal L

For a buyer who sees signal L, the possible outcomes are either LH or LL. In case of LH, the value of the object is 0, and in case of LL, the value of the object is also 0. Therefore, the expected value of the object for a buyer who sees signal L is \(0 \times \frac{1}{4} + 0 \times \frac{1}{4} = 0\).

Calculate the expected value for a buyer who sees signal H

For a buyer who sees signal H, the possible outcomes are either HH or HL. In case of HH, the value of the object is 1, and in case of HL, the value of the object is 0. Therefore, the expected value of the object for a buyer who sees signal H is \(1 \times \frac{1}{4} + 0 \times \frac{1}{4} = \frac{1}{4}\).

Analyze profits for buyers if they bid their expected values

If both buyers bid their expected values, a buyer with signal L will bid 0, and a buyer with signal H will bid \(\frac{1}{4}\). Suppose buyer A gets signal H and buyer B gets signal L. If they bid their expected values, buyer A will win the auction by bidding \(\frac{1}{4}\), but the actual value of the object is 0 (since buyer B got signal L). Therefore, buyer A has a negative profit of \(- \frac{1}{4}\). Under this scenario, a winner's curse occurs.

Conclusion

For a buyer who sees signal L, the expected value of the object is 0, and for a buyer who sees signal H, the expected value of the object is \(\frac{1}{4}\). If buyers bid their expected values, there can be a situation where they earn negative profit conditional on observing signal H, which is an example of the winner's curse.

Key concepts

Expected Value

In the context of common values auctions, understanding the expected value is crucial for making informed bidding decisions. The expected value is a calculation that helps bidders determine what the average outcome is likely to be, given the imperfect information they have. In our exercise scenario, each buyer receives one of two signals about the value of the object: high (H) or low (L). To find the expected value, we consider all possible outcomes and their probabilities. For a buyer observing signal L, the expected value is derived from two possibilities: if the other buyer also sees L (LL), the object is worth 0, or if the other sees H (LH), it's again worth 0. Thus, with probabilities of \( rac{1}{4} \) for each outcome, the expected value is 0. Meanwhile, a buyer seeing signal H has outcomes of HH, where the object is worth 1, or HL, where it's worth 0. With probabilities \( rac{1}{4} \) each, the expected value is \( rac{1}{4} \). This highlights how crucial understanding expected values is in adjusting bidding strategies.

Private Signal

In auctions, a private signal is an individual insight or piece of information that a bidder has, which helps them estimate the value of the auctioned item. This information isn't shared with other bidders, and it provides a personal estimate of potential value. In our exercise, each buyer might receive either a high (H) or low (L) signal about the item's value. This signal is private; meaning one bidder doesn’t know what the other is seeing. The strategy then, is to use this private signal to predict the value while considering the potential signals of other bidders based on probabilities. The individuality of private signals in common values auctions prompts different expectations and can lead to varying outcomes in terms of bidding. It's significant because one might interpret a high signal as an indication to bid more aggressively, without knowing the real signal of others, possibly inflating their outcome forecast.

Winner's Curse

The winner's curse is a phenomenon that occurs in common value auctions when a bidder wins by placing the highest bid, often because they've overestimated the item's value based on their private signal. This leads to the unfortunate situation where the winner realizes they paid more than the item's actual worth, hence the 'curse.' In our scenario, if a buyer sees a high signal (H) and accordingly bids based on their expectation of value \( rac{1}{4} \) but the object is actually worth 0, owing to the signal the other bidder received (low, L), the result is a negative profit. Thus, the expectation was misleading, and the winner ends up at a loss. The winner's curse highlights the importance of cautious bidding strategies. It underscores the risk of overvaluing items based on skewed expectations and affirms that in auctions, being wary of how private signals might mislead is key to avoiding unnecessary losses.