Module 8 — Mechanism Design and Auctions

Core question

Can an institution choose a desirable outcome when the information needed to choose it is privately held by strategic participants?

Learning outcomes

You will be able to:

• define a mechanism by types, messages, outcomes, and transfers;
• distinguish dominant-strategy and Bayesian incentive compatibility;
• explain the revelation principle and VCG payments;
• derive symmetric first-price bidding in a benchmark auction;
• evaluate auction rules by efficiency, revenue, participation, robustness, and wider market effects.

1. Start from the design problem

Agent i has private type θ_i, sends message m_i, and receives outcome x(m) and transfer t_i(m). With quasi-linear utility:

A designer must specify:

Mechanism design reverses ordinary game theory: begin with a target, then choose the game whose equilibrium has the desired property.

2. Incentive and participation constraints

Dominant-strategy incentive compatibility

Truthful reporting is optimal for every report by others:

for all θ_i, θ̂_i, and θ_{−i}.

Bayesian incentive compatibility

Truth maximises expected utility given beliefs about others' types:

Individual rationality

Participation must beat the outside option:

Dominant-strategy truthfulness is more robust to beliefs but can restrict achievable outcomes more severely.

3. Revelation principle

If an outcome can be implemented by some mechanism and equilibrium, a direct mechanism exists in which agents report types and truthful reporting is an equilibrium producing the same outcome.

The principle lets theorists search over truthful direct mechanisms. It does not say:

• every desirable allocation is implementable;
• truth is the unique equilibrium;
• participants understand the mechanism;
• communication, computation, collusion, or privacy is costless.

4. Vickrey auction: pay the opportunity cost

One indivisible object is valued at 10 by A, 7 by B, and 4 by C. In a sealed-bid second-price auction:

• highest bid wins;
• winner pays the second-highest bid.

With truthful bids, A wins and pays 7, receiving utility 10−7=3.

Why truth is weakly dominant for A:

• if the highest rival bid is below 10, A wants to win and the payment is that rival bid, not A's own bid;
• if it is above 10, A wants to lose;
• misreporting only risks changing the win/loss decision in the wrong direction.

This logic assumes private values, enforceable payment, no spite or externality from who wins, and no binding budget below value.

5. VCG generalises the externality payment

Choose the outcome that maximises reported total value:

Agent i pays the harm their presence imposes on others:

The payment removes the gain from manipulating how one's report affects everyone else, producing dominant-strategy truthfulness under the benchmark.

VCG can fail practical desiderata: revenue may be low or negative, budget balance can fail, collusion and false identities can matter, and combinatorial valuation may be difficult to report or compute.

6. First-price auction: shade the bid

Suppose n risk-neutral bidders have independent private values uniformly distributed on [0,1]. In the symmetric equilibrium of a first-price sealed-bid auction:

With three bidders and value v=0.9, the equilibrium bid is 0.6. A higher bid increases the probability of winning but lowers surplus conditional on winning.

In a second-price auction, truthful bidding is weakly dominant; in a first-price auction, bidding depends on beliefs about rivals, risk attitude, number of bidders, and value distribution.

7. Revenue equivalence is conditional

Standard auctions have the same expected revenue when:

• bidders are risk-neutral;
• values are independent and identically distributed;
• the highest-value bidder wins;
• the lowest type receives zero expected surplus;
• participation and payment assumptions match.

Risk aversion, affiliated/common values, asymmetry, budgets, entry, reserve prices, and dynamic learning can break equivalence.

Optimal reserve in a benchmark

For distribution F with density f, Myerson's virtual value is:

For U[0,1], φ(v)=2v−1; the seller-optimal reserve is therefore v=1/2 under regularity. The reserve can leave the object unsold even when trade would create positive surplus: revenue and allocative efficiency differ.

8. A fundamental impossibility

In bilateral trade with private buyer and seller values whose distributions overlap, no mechanism can simultaneously guarantee efficiency, Bayesian incentive compatibility, individual rationality, and budget balance under the Myerson–Satterthwaite conditions.

The lesson is not “design is futile.” It is that a real institution must identify which desideratum to relax and who bears the resulting loss.

9. Current case: digital-ad auction rules affect product prices

Bergemann, Bonatti, and Wu model a platform with on- and off-platform consumer contact, data-enhanced matching, and auction-like ad allocation. Their platform-optimal managed campaign achieves efficient on-platform matching but conditions sponsored-product pricing on advertisers' off-platform prices; relative to data-augmented auctions, it raises off-platform prices and lowers consumer surplus (2025).

The example broadens auction evaluation: do not stop at auction revenue or click allocation. The mechanism can change advertisers' downstream product prices. The result is theoretical and rests on the model's data, contact, and commitment structure; it is not an estimate of one platform's realised price effect.

Frontier design result

Echenique and Núñez's “Price and Choose” mechanism sequentially lets agents define outcome prices before a final choice and implements efficient outcomes in a broad quasi-linear environment (2025). It illustrates that mechanism design remains an active search for simpler information and implementation structures—not only a catalogue of classic auctions.

10. Implementation checklist

Before recommending a mechanism, test:

1. incentive concept and equilibrium multiplicity;
2. participation and opt-out;
3. budget balance and payment risk;
4. collusion, shill bids, and false identities;
5. computational and communication burden;
6. privacy, auditability, and appeals;
7. dynamic entry and investment;
8. outcomes outside the auction itself.

Practice

1. Explain truthfulness in a second-price auction for every possible rival bid.
2. Derive b(v)=(n−1)v/n using a deviation by type v.
3. Calculate VCG allocation and payments for values (14,9,6,2).
4. Show how risk aversion changes first-price bid shading qualitatively.
5. Design an ad-auction evaluation with one auction outcome and two downstream-market outcomes.

Quick check

• A mechanism specifies messages, allocations, and transfers.
• IC aligns a participant's optimal message with the designer's interpretation.
• VCG charges opportunity cost but may violate other practical goals.
• Revenue equivalence depends on a narrow common benchmark.
• Auction rules can alter prices and competition outside the auction.

Next: test where standard choice models fail and how evidence scales.