• Overview
• Details
• Exercises

Mechanism design studies how to engineer rules of a game so that selfish, rational agents reveal information and produce socially desirable outcomes. Core goals include incentive compatibility, efficiency, individual rationality, and budget balance. We sketch classic settings (single-parameter auctions) and pillars like VCG and Myerson’s optimal auction.

• Private values and types: Agents have types (valuations/costs). A mechanism maps reported types to outcomes and payments.
• Incentive compatibility (IC): Truth-telling is a dominant strategy (DSIC) or a Bayesian Nash equilibrium (BIC).
• Individual rationality (IR): Participation yields nonnegative utility ex ante/interim/ex post.
• Efficiency: Maximize social welfare (allocative efficiency), e.g., VCG mechanisms in quasilinear environments.
• Revenue maximization: Myerson’s virtual values and optimal auctions (reserve prices) under regularity assumptions.
• Impossibility trade-offs: Green–Laffont (can’t have efficiency + budget balance + DSIC in public goods, in general).
• Single-parameter domains: Monotone allocation + payment by integral gives DSIC (Archer–Tardos characterization).
• Auctions: First-price and all-pay (not DSIC), second-price (Vickrey, DSIC single-item), VCG for combinatorial auctions (intractability issues).
• Computational aspects: Complexity of welfare maximization, approximation mechanisms, and truthful-in-expectation relaxations.

1. Show that a monotone allocation rule in a single-parameter domain admits DSIC payments via the integral formula; derive the payment rule.
2. Compare first-price and second-price auctions for i.i.d. private values. Compute symmetric BNE for first-price in a simple two-bidder uniform case.
3. Explain how VCG achieves efficiency in a single-item setting and why it may fail budget balance in public projects.