General Mechanism

Micro EC487_Advanced_Microeconomics

Background

This is a topic from Mechanism Design .

General Mechanism is different from Direct Mechanism

---

In General Mechanism, the agent is able to send any message $m$ from the message space $\mathcal{M}$. The mechanism then maps this message to an outcome through a function:

$g:\mathcal{M}\to X$

Key features:

• The message space $\mathcal{M}$ can be arbitrary—not necessarily related to the type space $\Theta$
• The agent must strategically choose which message $m$ to send based on their private type $\theta$
• The agent’s optimization problem: $m^{\ast }(\theta )=\arg {\max }_{m\in \mathcal{M}}u(\theta ,g(m))$

This framework is very flexible but can be complex to analyze, as we need to consider the agent’s full strategic response across the entire message space.

---

Direct Mechanism

In contrast, a Direct Mechanism simplifies the problem by setting the message space equal to the type space: $\mathcal{M}=\Theta$. The agent is simply asked to report their type directly.

$g:\Theta \to X$

Key features:

• The agent reports $\hat{\theta }\in \Theta$ (a reported type, which may or may not be truthful)
• Much simpler structure—no need to analyze complex message spaces
• If the mechanism satisfies Incentive Compatibility (IC), then truthful reporting $\hat{\theta }=\theta$ is optimal for the agent

---

Comparison Summary

Aspect	General Mechanism	Direct Mechanism
Message Space	Arbitrary $\mathcal{M}$	Type space $\Theta$
Agent’s Action	Strategically choose $m\in \mathcal{M}$	Report type $\hat{\theta }$
Complexity	Can be complex	Simple and tractable
Analysis	More difficult	Easier to characterize

---

The Revelation Principle

The crucial result that connects these two approaches is the Revelation Principle:

For any general mechanism and its equilibrium outcome, there exists an equivalent direct mechanism where truthful reporting is a Bayesian-Nash equilibrium.

Implications:

1. We can restrict attention to direct mechanisms without loss of generality when searching for optimal mechanisms
2. Any outcome achievable by some general mechanism can also be achieved by an incentive-compatible direct mechanism
3. This dramatically simplifies mechanism design problems

Intuition: If an agent of type $\theta$ optimally sends message $m^{\ast }(\theta )$ in a general mechanism, we can construct a direct mechanism that produces the same outcome by directly asking the agent to report $\theta$ and then computing $g(m^{\ast }(\theta ))$.

---

Connection to A Simple Screening Example

The screening problem in A Simple Screening Example implicitly applies the Revelation Principle:

• The seller designs a menu of contracts $\{(q(\theta ),p(\theta )){\}}_{\theta \in \Theta }$
• The buyer chooses which contract to accept, effectively revealing their type
• This is a direct mechanism where truthful type revelation is incentive compatible

The fact that we can focus on designing such menus (rather than more complex message spaces) is justified by the Revelation Principle.

---

Further Reading

• Mechanism Design