Question

Consider the following simple model of a common values auction. Two buyers each obtain a private signal about the value of an object. The signal can be either 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

In a common values auction with two buyers and equal probability of receiving high (H) or low (L) signals, the expected value of the object for a buyer who sees signal L is 0.5, and the same goes for a buyer who sees signal H (0.5). When buyers bid their expected value (0.5) and observe signal H, they experience the winner's curse. This is because their expected profit conditional on observing signal H is negative (-0.25).

Step-by-step solution

In this step, we determine the probabilities of the possible combinations of signals. Since each buyer receives a signal H or L with equal probability, we have a total of 4 possible combinations for the two buyers: - Buyer 1 gets H and Buyer 2 gets H: probability is \((0.5)(0.5)=0.25\). - Buyer 1 gets H and Buyer 2 gets L: probability is \((0.5)(0.5)=0.25\). - Buyer 1 gets L and Buyer 2 gets H: probability is \((0.5)(0.5)=0.25\). - Buyer 1 gets L and Buyer 2 gets L: probability is \((0.5)(0.5)=0.25\). #Step 2: Determine the expected value of the object for a buyer who sees signal L#

To calculate the expected value of the object for a buyer who sees signal L, we need to consider the probability distribution of the object's value given the L signal. The possible values are: - Value of the object is 0 (when the other buyer also receives signal L): probability is given by the L-L combination, which is \(0.5\). - Value of the object is 1 (when the other buyer receives signal H): probability is given by the L-H combination, which is \(0.5\). Now we can compute the expected value for a buyer with signal L: Expected Value = \((0)(0.5) + (1)(0.5) = 0.5\). #Step 3: Determine the expected value of the object for a buyer who sees signal H#

To calculate the expected value of the object for a buyer who sees signal H, the possible values are: - Value of the object is 0 (when the other buyer receives signal L): probability is given by the H-L combination, which is \(0.5\). - Value of the object is 1 (when the other buyer also receives signal H): probability is given by the H-H combination, which is \(0.5\). Now we can compute the expected value for a buyer with signal H: Expected Value = \((0)(0.5) + (1)(0.5) = 0.5\). #Step 4: Show that buyers earn negative profit conditional on observing signal H (the winner's curse)#

Buyers will bid their expected value, which is 0.5 for both signals H and L. When they observe signal H, the object is worth 1 if the other buyer also has signal H. In this case, their profit is \((1 - 0.5) = 0.5\). If the other buyer has signal L, the object is worth 0, and their profit is \((0 - 0.5) = -0.5\). The probability of the other buyer having signal H or L is equal (0.5). Therefore, the expected profit conditional on observing signal H is given by the average of the two cases: Expected Profit = \((0.5)(0.5) + (-0.5)(0.5) = 0 - 0.25 = -0.25\). So, buyers earn negative profit conditional on observing signal H, which is an example of the winner's curse.

Key concepts

Expected Value

Understanding the expected value is crucial when dealing with uncertainty. In the context of auctions, this concept helps bidders estimate the value of an item based on the probability of different outcomes and their corresponding values.

For example, in a common values auction with two possible signals, high (H) and low (L), each with equal probability, the expected value becomes an average of the possible values weighted by their probabilities. If a buyer sees signal L, they must consider that the item is worth either 0 or 1. Due to the equal chances of the other buyer's signal being H or L, the expected value is calculated as \( (0)(0.5) + (1)(0.5) = 0.5 \) for both signals.

Thus, the expected value serves as a guide for the buyer's bid. However, if every participant bids their expected value, the actual profit from winning the auction could differ from this estimate, leading to phenomena like the winner's curse.

Winner's Curse

The winner's curse is a paradoxical outcome in common value auctions where the winner tends to overpay. This occurs when the winning bid exceeds the intrinsic value of the auctioned item.

This phenomenon is particularly insidious because bidders base their decisions on signals that are suggestive but not definitive of the item's value. If a buyer who sees signal H bids their expected value of \( 0.5 \), they stand a chance of overpaying when their signal is not paired with signal H from the other buyer. As illustrated by the problem, the expected profit when observing signal H is negative (\( -0.25 \) on average), manifesting the winner's curse.

Therefore, bidders must adjust their strategies to account for the risk of paying more than the item's worth, especially in a high-uncertainty auction setting.

Private Signal

In auctions, a private signal is information that a bidder has regarding the value of the item up for bid, which others may not have. Each bidder receives a private signal that could be either high or low, representing their estimate of the item's value.

The exclusivity of private signals leads to asymmetry of information among the bidders. This can add complexity to one's bidding strategy, as it must be based not only on the private signal but also on the anticipation of the signals of others. As is confirmed by the exercise, even with a strong high signal, a buyer might not gain profit if the collective signals (the signals received by all bidders) don't align to indicate a high value.

Probability Distribution

Probability distribution outlines the likelihood of various outcomes in a random event. In the auction problem, where signals determine the object’s value, the probability distribution illustrates the value's possible states and their corresponding probabilities.

Here, with each signal (H or L) being equally likely, the distribution becomes a simple discrete one with only two values: 0 or 1. Each potential combination of received signals (HH, HL, LH, LL) has a probability of \( 0.25 \). The scenario becomes a clear application of how probability distribution underpins expected value calculations and the strategic thinking in auctions.