Mechanism Design Logic

Origin. Connects mechanism design (Hurwicz, 1960s) to logic. Pauly's coalition logic. Van der Hoek, Wooldridge: social choice in logic. Formal verification of mechanisms. Automated mechanism design foundations.

Models. Designing games for desired outcomes. Mechanism: game + rules. Designer: wants particular outcome (efficiency, fairness). Agents: strategic, self-interested. Logic: express and verify mechanism properties.

Formalism.

Social choice function: f: Preferences^n → Outcomes Aggregates preferences to outcome.

Mechanism: M = (S₁,...,Sₙ, g) where:

Sᵢ: strategy space for agent i
g: S₁ × ... × Sₙ → Outcomes (outcome function)

Implementation: f implemented by M if: equilibrium strategies yield f's output. Dominant strategy: works regardless of others. Nash: works given others' equilibrium play.

Properties (in logic):

Incentive compatibility: ∀i.∀θᵢ.∀θ'ᵢ. uᵢ(f(θ), θᵢ) ≥ uᵢ(f(θ'ᵢ, θ₋ᵢ), θᵢ) "Truth-telling is optimal"
Efficiency: ¬∃o'. ∀i. uᵢ(o') ≥ uᵢ(f(θ)) ∧ ∃i. uᵢ(o') > uᵢ(f(θ)) "No Pareto improvement"
Individual rationality: uᵢ(f(θ), θᵢ) ≥ uᵢ(default) "Participation is beneficial"

Verification: Check mechanism satisfies properties. Model check: finite types. Theorem prove: symbolic.

Auction logic: Vickrey auction: second-price sealed-bid. Verify: truthful bidding is dominant strategy.

Symbols.

Symbol	Unicode	Name	Meaning
f	—	Social choice	Preference aggregation
M	—	Mechanism	Game form
u	—	Utility	Payoff function
θ	U+03B8	Type	Private information
S	—	Strategy	Action space
g	—	Outcome	Result function
IC	—	Incentive compatible	Truthful

Metatheory. Revelation principle: can restrict to direct mechanisms. Gibbard-Satterthwaite: no perfect mechanism for ≥3 outcomes. Complexity: verification can be hard. Automated synthesis: search for mechanisms.

Applies to. Auction design. Voting systems. Resource allocation. Matching markets. Public goods. Regulatory design. Blockchain mechanisms.

Limitations. Rationality assumptions strong. Computational mechanism design challenges. Infinite types harder. Strategic behavior complex. Real mechanisms deviate from models. Verification vs synthesis gap.