Question

Consider the common-value auction described in this chapter, where \(Y=y_{1}+y_{2}\) and \(y_{1}\) and \(y_{2}\) are both uniformly distributed on \([0,10]\). Player \(i\) privately observes \(y_{i}\). Players simultaneously submit bids and the one who bids higher wins. (a) Suppose player 2 uses the bidding strategy \(b_{2}\left(y_{2}\right)=3+y_{2}\). What is player 1's best-response bidding strategy? (b) Suppose each player will not select a strategy that would surely give him/her a negative payoff conditional on winning the auction. Suppose also that this fact is common knowledge between the players. What can you conclude about how high the players are willing to bid? (c) Show that for every fixed number \(z>0\), there is no Bayesian Nash equilibrium in which the players both use the bidding strategy \(b_{i}=y_{i}+z\).

Short answer

(a) Player 1's best-response is \(b_1(y_1) = y_1 + 2\). (b) Bids cannot exceed \(y_i + 5\). (c) No equilibrium exists for strategies \(b_i = y_i + z\) with \(z > 0\).

Step-by-step solution

Calculating Expected Value

For player 1, who observes their own signal \(y_1\) and knows the auction format \(Y = y_1 + y_2\), we need to consider the expected value of \(Y\) given the observed \(y_1\). The expected value of \(y_2\) given a uniform distribution from 0 to 10 is 5. Therefore, the expected value of \(Y\) given \(y_1\) is \(E[Y|y_1] = y_1 + 5\).

Determining Payoff

Player 1's payoff when winning is \(E[Y|y_1] - b_1(y_1)\), where \(b_1(y_1)\) is player 1's bid. To ensure non-negative payoff from winning, player 1 requires \(b_1(y_1) \leq E[Y|y_1] = y_1 + 5\).

Finding Best-Response Strategy for Player 1

Given player 2's strategy is \(b_2(y_2) = 3 + y_2\), player 1 wins if \(b_1(y_1) > b_2(y_2) = 3 + y_2\). Substituting the expected value \(b_1(y_1) = y_1 + 2\) results in player 1 bidding as follows: \[b_1(y_1) = y_1 + (5 - 3) = y_1 + 2\].

Limitation of Player Bids Due to Common Knowledge

Since no player will bid higher than their expected value of \(Y\) given \(y_i\) (creating a non-negative payoff), and both players know this fact, it implies that they will not bid above \(y_i + 5\). Thus, each player will set an upper bid limit at \(y_i + 5\).

Checking for Bayesian Nash Equilibrium

Assume both players use the strategy \(b_i = y_i + z\). If \(z > 5\), the strategy results in a negative expected payoff for at least one player due to exceeding \(y_i + 5\), violating common knowledge of rational bidding. Hence, there cannot be a equilibrium where \(z>0\) because it encourages bids exceeding expected payout.

Key concepts

Common-Value Auction

In a common-value auction, the item's true value is the same for all bidders, but this value is not known with certainty until after the auction. Each player receives a private signal—a piece of information—that can help them estimate this true value. In our example, the auction value is stated as \(Y = y_1 + y_2\), where \(y_1\) and \(y_2\) are uniformly distributed between 0 and 10.
The challenge in a common-value auction is for each player to deduce the total value based on only partial information (their own signal) and to strategize their bid accordingly. It becomes a game of belief and estimation, where players must consider not only their own valuation but also what competitors might value and bid.

Bayesian Nash Equilibrium

A Bayesian Nash Equilibrium (BNE) extends the concept of Nash Equilibrium to scenarios where players have incomplete information about others, and make decisions based on their beliefs (often characterized by probability distributions). Each player chooses their best strategy given their information and beliefs about other players' strategies.
In our context, no BNE exists where players use strategies \(b_i = y_i + z\) if \(z > 0\). The reasoning is that any positive \(z\) forces a player to exceed their expected non-negative payoff threshold \(y_i + 5\), contradicting the rational behavior assumption in auction theory. The equilibrium requires each player's strategy to be a best response to the strategies of others, and with \(z > 0\), it generates a negative expected payoff for at least one of the players, breaking equilibrium conditions.

Expected Value

Expected value in this context is a statistical measure of the mean value of the auction's item as estimated by a player. For each player receiving a signal like \(y_1\), the expected value of \(Y\) given this information is calculated as \(E[Y|y_1] = y_1 + 5\), because \(y_2\) is uniformly distributed from 0 to 10 and thus has an average value of 5.
Using expected value helps each player set a rational bid that takes into account both their own valuation and a reasonable estimate of the opponent's signal. This is crucial, as the player aims to make a bid that is high enough to win, but not so high that it risks a negative payoff when the true value is revealed post-auction.

Bidding Strategies

When strategizing bids in a common-value auction, players need to balance between bidding high enough to win and ensuring a positive payoff. Given the opponent's strategy \(b_2(y_2) = 3 + y_2\), player 1 might choose a best-response strategy of \(b_1(y_1) = y_1 + 2\). This exploits the expected value and competitive bid effectively.
Strategically, players also must ensure that their bids never exceed \(y_i + 5\) to maintain non-negative payoffs. A higher bid than this expected value, suggested by the auction setup, leads to an unsustainable strategy, risking losses if the bid exceeds the actual common value after the fact. Crafting an optimal bidding strategy thus involves predicting opponent responses while ensuring maximum expected gain without overstepping fair value estimations.