Luis Schmitz

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Auction Simulator

See how First-Price and Second-Price auctions behave on the same scenarios (1,000 runs).

Setup

Number of bidders5
Runs: 1000 (fixed)

Metrics

Average revenue (First-Price)
0.00
Average revenue (Second-Price)
0.00
Efficiency (First-Price)
—
Efficiency (Second-Price)
—
Auctions run
0

What prices do we see most often?

Run 1,000 auctions to compare First-Price and Second-Price side-by-side.

What this shows (plain English)

  • Second-Price: best move is to bid your value; you pay the 2nd-highest bid.
  • First-Price: you pay your own bid, so people usually bid a bit below their value.
  • In these simple settings, both designs earn about the same revenue on average.

Math & Strategy

Model & Assumptions

Single good; n\ge 2 bidders with independent private values (IPV). Each value v_i is i.i.d. from continuous CDF F with density f, support[\underline v,\overline v] (here: [0,100] or truncated Normal). Bidders are risk-neutral unless noted.

Order-stat convention used here: v_{(1)} = highest, v_{(2)} = second-highest.

Second-Price (Vickrey) Auction

  • Dominant strategy: b_i=v_i (truthful bidding).
  • Payment: winner pays the highest rival bid = v_{(2)}.

First-Price Auction: Symmetric BNE

In a strictly increasing symmetric equilibrium \beta(v), type v wins with prob.\,F(v)^{n-1}. Expected payoff:

\max_{b}\;(v-b)\,F\!\big(\beta^{-1}(b)\big)^{\,n-1} \quad\Rightarrow\quad \text{at } b=\beta(v):\;\; U(v)=(v-\beta(v))\,F(v)^{\,n-1}.

Envelope/FOC yield the standard ODE (risk-neutral case) and closed form:

\beta'(v)=(n-1)\,\frac{f(v)}{F(v)}\,[\,v-\beta(v)\,], \qquad \beta(v)=v-\frac{\int_{\underline v}^{\,v} F(t)^{\,n-1}\,dt}{F(v)^{\,n-1}}.

For Uniform on [0,1]: \beta(v)=\tfrac{n-1}{n}v (we scale to [0,100] in the UI).

Revenue Equivalence (RET)

Under IPV, risk-neutrality, symmetry, continuous types, plus (i) efficient allocation (highest value wins w.p. 1) and (ii) zero information rent for the lowest type, any two “standard” auctions have the same expected revenue:

E[R_{\text{First-Price}}]=E[R_{\text{Second-Price}}]=E[v_{(2)}].

For Uniform on [0,1], with our convention\;E[v_{(1)}]=\tfrac{n}{n+1},\;E[v_{(2)}]=\tfrac{n-1}{n+1}. Scaling to [0,100] gives E[R]\approx 100\cdot\tfrac{n-1}{n+1}.

Efficiency & Welfare

Under the RET conditions, allocation is efficient: the highest value wins (efficiency ≈ 100%). Expected total surplus is E[v_{(1)}].

Beyond the Baseline

  • Risk aversion (private values): In First-Price, risk-averse bidders bid more aggressively (shade less) so revenue rises; Second-Price stays truthful regardless of risk attitude:\;E[R_{\text{FP}}]>E[R_{\text{SP}}] (for sufficient risk aversion).
  • Optimal reserves (Myerson): Virtual value \phi(v)=v-\tfrac{1-F(v)}{f(v)}. Optimal (truthful) mechanism allocates to the highest \phi(v) above 0 and sets reserver^* solving \phi(r^*)=0. For Uniform [0,100],\phi(v)=2v-100\;\Rightarrow\;r^*=50.
  • Interdependent/common values: With affiliated signals, more information release (e.g., ascending English) can increase revenue (linkage principle). Our simulator sticks to IPV.
  • General F: The First-Price equilibrium is given by the integral formula above; the closed form \tfrac{n-1}{n}v is specific to Uniform.

Connecting to the Plots

  • Explainer curve: purple line b=v (Second-Price), green dashed \beta(v) (First-Price). As n\to\infty, \beta(v)\to v (shading disappears).
  • Revenue histograms: Under RET conditions, both centers are near E[v_{(2)}]. Reserves shift mass right; risk-averse FPSB shifts its mean above SPSB.