博弈论笔记 1

 2025-04-25   数学 博弈论 笔记

Review: Symmetric BNE in First-Price Auction (FPA)

Components

Equilibrium Strategy

S∗(θi)=E[Y∣Y≤θi]=∫0θiθfY(θ)FY(θi)dθS^*(\theta_i) = \mathbb{E}[Y \mid Y \leq \theta_i] = \int_0^{\theta_i} \frac{\theta f_Y(\theta)}{F_Y(\theta_i)} d\theta

Revenue Analysis in FPA

Expected Payment of Bidder 1
Conditional on valuation θ1\theta_1:

E[Payment∣θ1]=FY(θ1)⋅S∗(θ1)=∫0θ1θfY(θ)dθ\mathbb{E}[\text{Payment} \mid \theta_1] = F_Y(\theta_1) \cdot S^*(\theta_1) = \int_0^{\theta_1} \theta f_Y(\theta) d\theta

Total Expected Revenue (FPA)
By integrating over all θ1\theta_1:

E[Revenue]=∫0θmax ⁣FY(θ1)E[Y∣Y≤θ1]fθ(θ1)dθ1=∫0θmax ⁣y[1−Fθ(y)]fY(y)dy\begin{aligned} \mathbb{E}[\text{Revenue}] &= \int_0^{\theta_{\text{max}}} \! F_Y(\theta_1) \mathbb{E}[Y \mid Y \leq \theta_1] f_\theta(\theta_1) d\theta_1 \\ &= \int_0^{\theta_{\text{max}}} \! y[1 - F_\theta(y)] f_Y(y) dy \end{aligned}

For NN symmetric bidders:

E[Revenue]=∫0θmax ⁣y[N(1−Fθ(y))fY(y)]dy\mathbb{E}[\text{Revenue}] = \int_0^{\theta_{\text{max}}} \! y[N(1 - F_\theta(y))f_Y(y)] dy

Revenue Equivalence Between SPA and FPA

Claim
Under symmetric BNE in FPA and dominant strategies in SPA:

E[Revenue in SPA]=E[Revenue in FPA]\mathbb{E}[\text{Revenue in SPA}] = \mathbb{E}[\text{Revenue in FPA}]

Proof

  1. Define θ(2)\theta^{(2)}: Second-highest valuation among θ1,…,θN\theta_1, \dots, \theta_N

  2. CDF of θ(2)\theta^{(2)}:

    Pr⁡(θ(2)≤θ)=FθN(θ)+N(1−Fθ(θ))FθN−1(θ)\Pr(\theta^{(2)} \leq \theta) = F_\theta^N(\theta) + N(1 - F_\theta(\theta))F_\theta^{N-1}(\theta)

  3. Differentiate to find PDF:

    fθ(2)(θ)=N(1−Fθ(θ))⋅(N−1)FθN−2(θ)fθ(θ)=N(1−Fθ(θ))fY(θ)f_{\theta^{(2)}}(\theta) = N(1 - F_\theta(\theta)) \cdot (N-1)F_\theta^{N-2}(\theta)f_\theta(\theta) = N(1 - F_\theta(\theta))f_Y(\theta)

  4. Revenue Equivalence:

    E[Revenue in SPA]=∫0θmax ⁣θfθ(2)(θ)dθ=E[Revenue in FPA]\mathbb{E}[\text{Revenue in SPA}] = \int_0^{\theta_{\text{max}}} \! \theta f_{\theta^{(2)}}(\theta) d\theta = \mathbb{E}[\text{Revenue in FPA}]

Revenue Equivalence Theorem

Statement

Any auction mechanism satisfying:

  1. i.i.d. valuations
  2. Symmetric BNE with strictly increasing differentiable strategies
  3. Zero expected payment for θi=0\theta_i = 0
  4. Allocation to highest bidder

yields identical expected revenue.

Proof

Notation

Expected Payoff

E[Payoff∣θ1]=θ1FY(θ^1)−p(θ^1)\mathbb{E}[\text{Payoff} \mid \theta_1] = \theta_1 F_Y(\hat{\theta}_1) - p(\hat{\theta}_1)

Optimality Condition
At BNE θ^1=θ1\hat{\theta}_1 = \theta_1:

ddθ^1E[Payoff]∣θ^1=θ1=θ1fY(θ1)−p′(θ1)=0\frac{d}{d\hat{\theta}_1}\mathbb{E}[\text{Payoff}] \bigg|_{\hat{\theta}_1 = \theta_1} = \theta_1 f_Y(\theta_1) - p'(\theta_1) = 0

Integrate to Find Payment

p(θ1)=∫0θ1 ⁣θfY(θ)dθ=FY(θ1)E[Y∣Y≤θ1]p(\theta_1) = \int_0^{\theta_1} \! \theta f_Y(\theta) d\theta = F_Y(\theta_1) \mathbb{E}[Y \mid Y \leq \theta_1]

Conclusion
The expected payment depends only on allocation rules and valuation distributions, not on auction format.

Preview: Myerson's Optimal Auction

Setup

Objective
Design a mechanism maximizing seller's expected revenue.