个体物品拍卖理论：机制设计与激励兼容

机械设计

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机械设计理论是经济学与规划领域中的一个重要分支，主要探讨如何通过设计机制来引导和激励参与者的决策行为，确保资源配置的有效性和公平性。在Debasis Mishra的《机制设计理论》一书中，作者从理论基础到实际应用，深入探讨了这个领域的关键概念。 1. **私人信息与效用转移**：在理论框架中，卖家通常拥有关于物品价值的私有信息，即每个买家对物品的估值。机制设计者的目标是设计出一个拍卖或其他交易方式，使得卖家能够通过信息揭示来最大化收益，同时确保买家的诚实披露其真实偏好。 2. **偏好模型**：机制设计的核心是理解买方的偏好结构，如单峰、凸偏好或无差异曲线等。这有助于构建满足公平性和效率要求的决策规则，如社会选择函数，它定义了从所有人的偏好中选择最佳结果的方式。 3. **主导策略激励兼容性**：一个重要的设计原则是保证机制下的最优行为是参与者的主导策略（dominant strategy），即无论其他参与者做什么，他们都应该选择报告真实价值，这样可以避免策略性欺骗，实现市场效率。 4. **贝叶斯激励兼容性**：当信息不完全时，考虑贝叶斯推理的机制设计更为复杂。即使存在不确定性，理想机制应该使参与者在任何可能的信号下都有报告真实信息的动机。 **转移与近似线性关系**：机制设计中引入了支付转移的概念，用来平衡各方的利益。例如，支付函数不仅要保证卖方向买家收取合理费用，还要确保买家愿意参与。在近似线性关系的假设下，机制设计者寻求一种方式，使分配规则既有效率又能激励兼容。 - **Vickrey-Clarke-Groves (VCG)机制**：这是一种著名的拍卖机制，其中支付根据每个参与者对整体福利的影响来确定，鼓励参与者如实报价，因为他们的私人信息不会影响自身的支付。VCG机制适用于单一物品拍卖、组合拍卖以及赞助搜索广告等领域。 - **Aﬃne最大化器分配规则**：这些规则强调在满足效率的前提下，通过简单的加权平均或其他线性方法分配资源，确保机制的实施性。总结来说，Debasis Mishra的《机制设计理论》提供了一套完整的框架，用于理解和设计在经济交易中如何利用信息和支付手段来优化资源配置，确保市场效率和参与者的激励相一致。该书涵盖了理论基础、具体模型、实例分析以及关键机制的详细讨论，是机械设计理论研究的重要参考资料。

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the contingencies. This in turn induces a Bayesian game with messages playing the role of

actions. Given a message proﬁle, the mechanism chooses an outcome.

Definition 2 A mechanism is a collection of message spaces and a decision rule: M ≡

, . . . , M

, φ), where

• for every i ∈ N, M

is the message space of agent i and

• φ : M

× . . . × M

→ X

is the decision rule.

A mechanism is a direct mechanism if M

= D

for every i ∈ N.

So, in a direct mechanism, every agent communicates a type from his type space to the

mechanism designer. Hence, if F is an scf, then ((D

, . . . , D

), F ) is a direct mechanism -

for simplicity, we will just refer to F as a (direct) mechanism.

We denote the outcome of agent i in a mechanism (M

, . . . , M

, φ) by φ

, . . . , m

) for

each (m

, . . . , m

) ∈ M

× . . . × M

The message space of a mechanism can be quite complicated. Consider the sale of a single

object by a “price-based” procedure. The mechanism designer announces a price and asks

every buyer to communicate if he wants to buy the object at the announced price. The price

is raised if more than one buyer expresses interest in buying the object, and the procedure

is repeated till exactly one buyer shows interest. The message space in such a mechanism is

quite complicated. Here, a message must specify the communication of the buyer (given his

type) for every contingent price.

1.3 Dominant Strategy Incentive Compatibility

We now introduce the notion of incentive compatibility. The idea of a mechanism and

incentive compatibility is often attributed to the works of Hurwicz - see (Hurwicz, 1960).

The goal of mechanism design is to design the message space and outcome function in a way

such that when agents participate in the mechanism they have (best) actions (messages) that

they can choose as a function of their private types such that the desired outcome is achieved.

The most fundamental, though somewhat demanding, notion of incentive compatibility in

mechanism design is the notion of dominant strategies.

Proof : Suppose mechanism (M

, . . . , M

, φ) implements F in dominant strategies. Let

: D

→ M

be the dominant strategy of each agent i.

Fix an agent i ∈ N. Consider two types R

, R

∈ D

. Consider R

−i

to be the report of

other agents in the direct mechanism. Let s

) = m

and s

−i

) = m

−i

. Similarly, let

) = m

. Then, using the fact that F is implemented by our mechanism in dominant

strategies, we get

, m

−i

) R

, m

−i

)

But F

, R

−i

) = φ

, m

−i

) and F

, R

−i

) = φ

, m

−i

). This gives us the desired

relation: F

, R

−i

) R

, R

−i

), which establishes that F is strategy-proof. 

Thus, a social choice function F is implementable in dominant strategies if and only if the

direct mechanism F is strategy-proof. Revelation principle is a central result in mechanism

design. One of its implications is that if we wish to ﬁnd out what social choice functions can

be implemented in dominant strategies, we can restrict attention to direct mechanisms. This

is because, if some non-direct mechanism implements a social choice function in dominant

strategies, revelation principle says that the corresponding direct mechanism is also strategy-

proof.

Of course, a drawback is that a direct mechanism may leave out some equilibria of the

main mechanism. The original mechanism may have some equilibria that may get ruled out

because of restricting to the direct mechanism since it has a diﬀerent strategy space. In

general, this is a criticism of the mechanism design theory. Even in a direct mechanism,

incentive compatibility only insists that truth-telling is an equilibrium but there may be

other equilibria of the mechanism which may not implement the given social choice function.

These stronger requirement that every equilibria, truth-telling or non-truth-telling, must

correspond to the social choice function outcome is the cornerstone of the implementation

literature.

1.4 Bayesian Incentive Compatibility

Bayesian incentive compatibility can be attributed to Harsanyi (1968a,b,c), who introduced

the notion of a Bayesian game and a Bayesian equilibrium. It is a weaker requirement than

the dominant strategy incentive compatibility. While dominant strategy incentive compati-

bility required the equilibrium strategy to be the best strategy under all possible strategies

of opponents, Bayesian incentive compatibility requires this to hold in expectation. This

means that in Bayesian incentive compatibility, an equilibrium strategy must give the high-

est expected utility to the agent, where we take expectation over types of other agents. To

be able to take expectation, agents must have information about the probability distribu-

tions from which types of other agents are drawn. Hence, Bayesian incentive compatibility

is informationally demanding. In dominant strategy incentive compatibility the mechanism

designer needed information on the type space of agents, and every agent required no prior

information of other agents to compute his equilibrium. In Bayesian incentive compatibility,

every agent needs to know the distribution from which agents’ types are drawn.

Since we need to compute expectations, we will represent preferences over X by utility

functions over X. In particular, the type of agent i is a utility function u

: X → R. When

transfers are not allowed, such utility representation is possible since X ≡ A will be assumed

to ﬁnite - even if it is not ﬁnite, we will assume that such a representation is always possible.

When transfers are allowed, we will assume preferences will always admit a continuous (and

monotonic in transfers) utility representation. So, our domain D

is a domain of utility

functions now.

To understand Bayesian incentive compatibility, ﬁx a mechanism (M

, . . . , M

, φ). A

strategy of agent i ∈ N for such a mechanism is a mapping s

: D

→ M

. To deﬁne a

Bayesian equilibrium, let G

−i

(·|u

) denote the conditional distribution of types of agents

other than agent i when agent i has type u

. A strategy proﬁle (s

, . . . , s

) is a Bayesian

equilibrium if for all i ∈ N, for all u

∈ D

we have

−i

(φ

), s

−i

)))dG

−i

) ≥

−i

(φ

, s

−i

)))dG

−i

) ∀ m

∈ M

If all u

s are drawn independently, then we need not condition in the expectation.

A direct mechanism (social choice function) F is Bayesian incentive compatible if

) = u

for all i ∈ N and for all u

∈ D

is a Bayesian equilibrium, i.e., for all i ∈ N and

for all u

, u

∈ D

we have

−i

, u

−i

))dG

−i

) ≥ u

, u

−i

))dG

−i

)

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个体物品拍卖理论：机制设计与激励兼容

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