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The second-price auction had a magic property: bid your true value, always. VCG extends that magic to selling many things at once. The trick is to charge each winner not what they bid, but the cost their winning imposes on everyone else.

VCG mechanisms

Vickrey-Clarke-Groves takes the second-price idea to any allocation problem. Choose the allocation that maximizes total value, then charge each participant the externality it creates: the welfare others lose because it took part. Under that rule, honest bidding is a weakly dominant strategy, whatever anyone else does.

From earlier lessons

Two quick ones before we start.

In a second-price auction you should bid...

The revelation principle says the designer can focus on mechanisms where...

Predict, then play

Two identical units are for sale. Bidder A wants the pair; B and C each want a single unit. Set the values and watch VCG pick the efficient allocation and charge each winner its externality.

Why did that happen?

Take a guess, then reveal the reason

Paying your own bid would make you shade it, as in a first-price auction. VCG instead charges you the externality: how much worse off everyone else is because you took a unit. Your report can change which allocation is chosen and therefore the payment. But, once others’ reports are fixed, the welfare they could achieve without you is a constant. VCG makes your payoff equal to the reported total welfare of the chosen allocation, with your true value substituted for your report, minus that constant. So reporting your true value makes the mechanism choose the allocation that is best for you. The second-price auction is just this rule for a single item, where the externality you impose is the value of the runner-up you displaced.

The math

Bidder $A$ values the pair at $a$ (a single unit is worthless to $A$ ); $B$ and $C$ value one unit at $b$ and $c$ . The efficient allocation maximizes total value: give both to $A$ (welfare $a$ ) or one each to $B$ and $C$ (welfare $b+c$ ). With $a=10, b=8, c=6$ , splitting wins: $b+c = 14 > 10$ .

Definition, VCG (Vickrey-Clarke-Groves)

Choose the allocation maximizing total reported value. Each participant pays the externality it imposes: the maximum welfare the others could achieve without it, minus the welfare the others actually get in the chosen allocation. For quasilinear utility, fix the other reports and write their best welfare without $i$ as $H_{-i}$ . If report $b_i$ induces allocation $x(b_i,b_{-i})$ , then

u_i=v_i(x(b_i,b_{-i})) + \sum_{j \ne i} b_j(x(b_i,b_{-i})) - H_{-i}.

The final term is fixed. Reporting $b_i=v_i$ therefore makes VCG maximize the rest of the expression, so truth-telling is weakly dominant. A single-item VCG is exactly the second-price auction.

$B$ ’s payment: without $B$ , the best for others is $A$ takes the pair, welfare $10$ ; with $B$ present, others get $C$ ‘s $6$ . So $B$ pays $10 - 6 = 4$ . Symmetrically $C$ pays $10 - 8 = 2$ . In this example each payment is below the winner’s value, so each keeps surplus. Note VCG can run a deficit and is vulnerable to collusion and shill bids; weakly dominant truthfulness is its strength, not budget balance.

Work it out

Compute the pivot payments

Total value is maximized by giving one unit each to $B$ and $C$ : welfare $8 + 6 = 14$ , beating $A$ ‘s pair value of $10$ . So $B$ and $C$ win.

$B$ ’s VCG payment is the externality it imposes. Remove $B$ : the others ( $A$ and $C$ ) do best if $A$ takes the pair, worth $10$ . Keep $B$ in the efficient allocation: the others get only $C$ ‘s $6$ (since $A$ is shut out). The difference, $10 - 6 = 4$ , is $B$ ‘s payment. For $C$ : without $C$ , others get $A$ ‘s $10$ ; with $C$ , others get $B$ ‘s $8$ ; payment $10 - 8 = 2$ . A report from $B$ can change the allocation and its payment, but its utility is its true value plus the others’ reported welfare in that allocation, less the fixed $10$ absence welfare. Reporting $8$ makes VCG maximize exactly that quantity, so honesty is dominant.

Key idea

VCG allocates efficiently and charges each winner the externality it imposes on others, which makes truthful bidding weakly dominant. It generalizes the second-price auction to many goods, at the cost of possible deficits and vulnerability to collusion.

Check yourself

Check yourselfapply

In VCG, what does a winner actually pay?

Check yourselfanalyse

Why is truthful bidding a weakly dominant strategy under VCG?

Check yourselfbrings back: auctions

How does the single-item second-price auction fit into VCG?

Sources

Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16(1), 8-37.
Clarke, E. H. (1971). Multipart pricing of public goods. Public Choice, 11, 17-33.
Groves, T. (1973). Incentives in teams. Econometrica, 41(4), 617-631.