Second-Price Sealed-Bid Auction with Independent Private Values

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Second-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 submits a bid $b_i\geq 0$.

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 and pays the second-highest bid.
• Values are independently drawn from $U[0,1]$; The highest bidder wins the object.
• The winner pays the second-highest bid.
• Losing bidders pay nothing.

Derivation (Verification by Iterated Deletion)

Candidate strategy

• Claim that the weakly dominant bidding function is

• Let $m$ be the highest bid among bidder $i$‘s opponents.
• If bidder $i$ wins, the price paid is $m$, not $b_i$.

Delete overbidding

• Compare $b_i>v_i$ with $b_i=v_i$.
• If $m<v_i$, both bids win and give payoff $v_i-m>0$.
• If $m>b_i$, both bids lose and give payoff $0$.
• If $v_i<m<b_i$, overbidding wins and gives payoff $v_i-m<0$, while truthful bidding loses and gives payoff $0$.
• Thus every bid $b_i>v_i$ is weakly dominated by $b_i=v_i$.

Delete underbidding

• Compare $b_i<v_i$ with $b_i=v_i$.
• If $m<b_i$, both bids win and give payoff $v_i-m>0$.
• If $m>v_i$, both bids lose and give payoff $0$.
• If $b_i<m<v_i$, underbidding loses and gives payoff $0$, while truthful bidding wins and gives payoff $v_i-m>0$.
• Thus every bid $b_i<v_i$ is weakly dominated by $b_i=v_i$.

Verification

• If $m<v_i$, winning gives positive surplus $v_i-m>0$.
• If $m>v_i$, winning gives negative surplus $v_i-m<0$.
• Bidding $b_i=v_i$ wins exactly when winning is profitable.
• After iterated deletion of weakly dominated bids, only $b_i=v_i$ remains for each type.

Derivation (Verification by Expected Payoff FOC)

Candidate strategy

• Suppose all opponents use truthful bidding:

• Bidder $i$ has value $v_i$ and chooses a deviation bid $b$.
• Let

Expected payoff

• Since there are $n-1$ opponents and values are independent uniform draws,

• For $0\leq b\leq 1$, bidder $i$ wins when $M\leq b$ and pays $M$.
• Expected payoff is

• Compute:

First-order condition

• Differentiate:

• The interior maximizer solves

• For $b<v_i$, the derivative is positive; For $b>v_i$, the derivative is negative.
• For $b>1$, the bidder wins for sure, giving the same payoff as $b=1$, so such bids cannot improve on $b=v_i$.
• Hence truthful bidding is bidder $i$‘s best response when all opponents bid truthfully.

Verification

• The same FOC argument applies to every bidder by symmetry.
• Therefore $b_i(v_i)=v_i$ is a symmetric Bayesian Nash equilibrium.

Nash Equilibrium

Result:

The weakly dominant strategy equilibrium is truthful bidding:

Social Optimum

• The object should go to the bidder with the highest value.
• Since bids equal values, the highest-value bidder wins.
• The equilibrium allocation is efficient.

Insights

Insight:

• In a second-price auction, the bid determines whether the bidder wins, not the price conditional on winning.
• This separates the winning decision from the payment decision.