Questions: Vickrey-Clarke-Groves (VCG) Mechanisms

Question 1

In a VCG mechanism, how is agent i's payment determined?

• A. Agent i pays their own reported valuation for the item they receive
• B. Agent i pays the second-highest reported valuation among all agents
• C. Agent i pays the difference between the total value others would have received without i and the total value others actually receive with i
• D. Agent i pays a fixed fee equal to the expected social welfare gain from their participation

Answer: (C)

VCG payments equal the externality each agent imposes: the total welfare of all *other* agents in the counterfactual where i doesn't exist, minus the total welfare of all other agents in the actual outcome with i present. This 'externality tax' makes truth-telling dominant because an agent's own report affects only which allocation is chosen, not the payment formula — which depends on others' reports. Option A describes a first-price auction (not incentive-compatible). Option B is correct only for the single-item Vickrey auction, a special case of VCG.

Question 2

A bidder in a VCG auction believes she can secure a better outcome by bidding above her true valuation. Is this a rational strategy?

• A. Yes — a higher bid wins more often and VCG payments are still bounded by the second-highest bid
• B. No — overbidding cannot increase her utility because the mechanism always chooses the efficient allocation and she pays her externality regardless
• C. Yes — overbidding shifts the allocation in her favor without changing her payment
• D. No — but only because overbidding violates the mechanism's rules and results in disqualification

Answer: (B)

In a VCG mechanism, truth-telling is a *dominant strategy*: misreporting never improves and can only harm an agent's utility. Her payoff is the value she receives minus her externality payment. Overbidding can distort the allocation chosen by the mechanism, potentially forcing an allocation where she wins an item whose true value to her is less than the externality she pays. Underbidding can cause the mechanism to assign her a worse allocation. The payment formula — based on others' reports — is unchanged by her own report, so her best response to any reports by others is to report truthfully.

Question 3

The Vickrey second-price single-item auction is a special case of the VCG mechanism.

• T. True
• F. False

Answer: True

In a single-item VCG auction, the winner is the highest bidder (efficient allocation). The externality the winner imposes equals: the value the second-highest bidder would have received if the winner didn't exist (they'd win and get their value), minus the value the second-highest bidder actually receives with the winner present (zero, since they lose). So the winner pays exactly the second-highest bid — the Vickrey payment rule. VCG generalizes this to multiple goods, combinatorial settings, and public goods by applying the same externality logic.

Question 4

In a VCG mechanism, agent i's payment depends in part on their own reported valuation.

• T. True
• F. False

Answer: False

Agent i's payment is computed from *other* agents' reported valuations — specifically, the difference in welfare of all other agents in two scenarios (with and without agent i). Agent i's own report affects only which allocation is chosen (the efficient one given all reports), not the payment formula. This separation is crucial: it means an agent cannot manipulate their payment by changing their report, only the allocation. The incentive to report truthfully follows from the fact that the allocation maximizing social surplus given truthful reports is also the one that maximizes i's own surplus.

Question 5

Why does making each agent pay exactly their externality on others cause truth-telling to be a dominant strategy in the VCG mechanism?

Model answer: Each agent's payoff is their true value for the allocation they receive minus their externality payment. Because the payment depends only on other agents' reports (not on i's own report), agent i's payment is fixed regardless of what i reports. Given a fixed payment, i's best response is to maximize the value of the allocation they receive — which means they want the mechanism to choose the allocation with the highest total reported value that includes i's true value. Reporting truthfully achieves this: when i reports their true value, the mechanism picks the allocation that genuinely maximizes total surplus including i's contribution. Any misreport can only distort the allocation away from i's true optimum, reducing i's payoff.

The mechanism decouples 'what allocation do I get?' from 'how much do I pay?' in a crucial way: i controls the allocation through their report, but i's payment is determined by others' reports alone. This makes truth-telling optimal regardless of what others report — the definition of dominant-strategy incentive compatibility, which is stronger than Bayesian incentive compatibility.