Second-Price Common Value Auction with Independent Signals

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Second-Price Common Value Auction with Independent Signals

• Players: Two bidders: bidder $1$ and bidder $2$.
• Strategies: Each bidder observes own signal $v_i\in[0,100]$; Each bidder chooses a bid $b_i\geq 0$; A symmetric strategy is a function $b(v)$.

Rules

• Two bidders observe independent signals and simultaneously submit sealed bids.
• Signals are independently drawn from $U[0,100]$.
• The common value is:

• The highest bidder wins; the winner pays the second-highest bid; losing bidders receive payoff $0$.

Derivation (Verification by Expected Payoff FOC)

Candidate strategy

• Candidate symmetric strategy:

• Suppose bidder $2$ uses $b(v_2)=2v_2$.
• Bidder $1$ with signal $v_1$ considers bid $b$.
• Bidder $1$ wins when

• If bidder $1$ wins, the price is $2v_2$ and payoff is

• Since $v_2\sim U[0,100]$,

• Expected payoff is

• Therefore

• Compute:

• Maximise

• The first-order condition is

• Hence

• For $b<2v_1$, the derivative is positive; For $b>2v_1$, the derivative is negative.
• The candidate strategy is a best response to itself.

Nash Equilibrium

Result:

The symmetric equilibrium bidding function is

Social Optimum

• Since $V=v_1+v_2$ is common after both signals are known, either bidder has the same ex-post value for the object.
• The auction mainly determines payments and information aggregation, not ex-post allocation efficiency.

Diagram (Best Response Functions)

Insights

Insight:

• Truthful private-value bidding does not apply in common-value auctions.
• The winner pays the expected opponent signal inferred from the second-highest bid.
• The bidder accounts for the information contained in winning.
• In this example, the second-price common-value bid is twice the private signal.