First-Price Common Value Auction with Independent Signals

• Content map: SMU H3 Game Theory Map

Setup

Definition:

First-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 own bid; losing bidders receive payoff $0$.

Derivation (Verification by Expected Payoff FOC)

Candidate strategy

• Candidate symmetric strategy:

• Suppose bidder $2$ uses $b(v_2)=v_2$.
• Bidder $1$ with signal $v_1$ considers bid $b$.
• Bidder $1$ wins when $v_2<b$.
• Since $v_2\sim U[0,100]$,

• Expected payoff is

• Therefore

• Compute:

• Maximise

• The first-order condition is

• Hence

• For $b<v_1$, the derivative is positive; For $b>v_1$, 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 main efficiency issue is not allocation across bidders, but surplus loss through payments and informational rents.

Diagram (Best Response Functions)

Insights

Insight:

• Bid your signal.
• A bid reveals information about the bidder’s signal.
• The winning event is informative about the opponent’s signal.
• In this common-value first-price example, the symmetric best response is $b(v)=v$.