Luis Schmitz

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Auction Simulator

Compare first-price vs. second-price auctions and explore revenue equivalence.

Setup Parameters

Number of Bidders5
Number of Auctions100

Metrics

Avg Revenue
0.00
Avg Winning Bid
0.00
Efficiency
0.0%
Auctions Run
0

Revenue Distribution

Run simulation to see revenue distribution

What to Notice

  • Configure your auction parameters and run simulations to compare first-price vs. second-price auction performance and explore revenue equivalence.
  • Try different numbers of bidders to see how competition affects bidding strategies and revenues.
  • First-price auctions encourage bid shading - bidders bid below their true valuation to maximize profit.
  • Second-price auctions incentivize truthful bidding since the winner pays the second-highest bid.

Math & Strategy

1. Basic Auction Model

Consider n risk-neutral bidders with private values v_i drawn independently from distribution F on [0, \bar{v}].

Each bidder i submits sealed bid b_i. Highest bidder wins.

2. First-Price Sealed-Bid Auction (FPSB)

Winner pays their own bid. Bidder i with value v_i chooses bid b_i to maximize expected payoff:

\pi_i(b_i) = (v_i - b_i) \cdot \Pr(\text{win with bid } b_i)

Probability of winning with bid b_i is probability that all other bids are below b_i:

\Pr(\text{win}) = [G(b_i)]^{n-1}

where G(b) is the CDF of the bid distribution.

2.1 Symmetric Equilibrium

In symmetric equilibrium, all bidders use the same bidding function β(v). For uniform distribution v_i \sim U[0,1]:

\beta(v) = \frac{n-1}{n} \cdot v

This creates bid shading: bidders bid below their true value to maximize profit.

3. Second-Price Sealed-Bid Auction (SPSB)

Winner pays the second-highest bid. Bidder i's expected payoff:

\pi_i(b_i) = \Pr(\text{win with bid } b_i) \cdot E[v_i - \max_{j \neq i} b_j | \text{win}]
3.1 Dominant Strategy

Truthful bidding b_i = v_i is a weakly dominant strategy:

  • If v_i > \max_{j \neq i} b_j: bidder wins regardless of bid above second-highest
  • If v_i < \max_{j \neq i} b_j: bidder loses regardless of bid below highest
  • Payment is independent of own bid (depends only on others' bids)

4. Revenue Equivalence Theorem

Under standard assumptions (risk neutrality, independent private values, symmetric bidders), expected revenue is the same across auction formats:

E[R_{FPSB}] = E[R_{SPSB}] = E[v_{(2)}]

where v_{(2)} is the second-highest value.

4.1 Intuition

Revenue equivalence arises because:

  • FPSB: Lower bids but winner pays own bid
  • SPSB: Higher bids (truthful) but winner pays second-highest
  • These effects exactly offset in expectation

5. Efficiency and Welfare

Both auction formats are efficient in equilibrium: the bidder with highest value wins.

Total expected welfare:

E[W] = E[v_{(1)}]

where v_{(1)} is the highest value.

6. Risk Aversion Effects

With risk-averse bidders, revenue equivalence breaks down:

  • FPSB: Risk aversion increases bids (insurance against losing)
  • SPSB: Truthful bidding remains optimal regardless of risk preferences
  • Result: E[R_{FPSB}] > E[R_{SPSB}] with risk aversion

7. Number of Bidders Effects

As n \to \infty:

  • Bid shading in FPSB decreases: \beta(v) = \frac{n-1}{n} v \to v
  • Competition intensifies, driving bids toward values
  • Revenue approaches the maximum possible: E[v_{(1)}]

References:Wikipedia - Auction Theory,Revenue Equivalence Theorem