First-Price Sealed-Bid Auction with Independent Private Values

• Content map: SMU H3 Game Theory Map

Setup

Definition:

First-Price Sealed-Bid Auction with Independent Private Values

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

Rules

• Start with an object and privately known values $v_i$; Players simultaneously submit sealed bids $b_i\geq 0$.
• The player who submits the highest bid wins the object.
• Values are independently drawn from $U[0,1]$; The highest bidder wins the object.
• The winner pays own bid.
• The winner’s payoff is $v_i-b_i$; Losing bidders receive payoff $0$.

Game Tree

• Win means $b\geq \frac{n-1}{n}v_j$ for every opponent $j\neq i$ when opponents use $b(v)=\frac{n-1}{n}v$.

Derivation (Verification by Expected Payoff FOC)

Candidate strategy

• Claim that the symmetric Bayesian Nash equilibrium bidding function is

• Let

• Suppose all opponents use $b(v)=av$.

Deviation payoff

• Bidder $i$ has value $v_i$ and chooses a deviation bid $b$.
• For $0\leq b\leq a$, bidder $i$ wins when every opponent value satisfies

• Since opponent values are independent and uniform,

• Expected payoff from bid $b$ is

First-order condition

• Since $a^{n-1}$ is constant, maximize

• Differentiate:

• The interior maximizer solves

• For $b<av_i$, $g'(b)>0$; For $b>av_i$, $g'(b)<0$. Hence $b=av_i$ is the best response on $[0,a]$.

Verification

• If every opponent uses $b(v)=\frac{n-1}{n}v$, bidder $i$‘s best response is

• The same argument applies to every bidder by symmetry.
• Therefore the candidate bidding function is a symmetric Bayesian Nash equilibrium.

Order-statistic intuition

• Let $Y=\max\{v_j:j\neq i\}$.
• Since there are $n-1$ opponents,

• Conditional on $Y\leq v_i$,

Nash Equilibrium

Result:

The symmetric Bayesian Nash equilibrium bidding function is

Social Optimum

• The object should go to the bidder with the highest value.
• Since $b(v)=\frac{n-1}{n}v$ is strictly increasing, the highest-value bidder also submits the highest bid.
• The equilibrium allocation is efficient.

Diagram (Bidding Function)

Insights

Insight:

• A bidder shades the bid below own value.
• The bid equals the expected strongest rival value conditional on winning.
• In a first-price auction, the trade-off is higher winning probability versus lower surplus if winning.