Research map / Optimization / Nonconvex and smooth optimization
Beyond Monotonicity: On the Convergence of Learning Algorithms in Standard Auction Games
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Analyzes convergence of gradient-based learning in symmetric first- and second-price auctions by casting the equilibrium problem as infinite-dimensional variational inequalities, showing non-monotonicity but a unique VI solution (the BNE) among uniformly increasing bid functions.
Problem
Why equilibrium learning algorithms consistently converge in Bayesian auction games with continuous types and actions, where no monotonicity or Minty guarantees are known and equilibrium problems are systems of differential equations.
Approach
Derives the Gateaux derivative of the ex-ante utility in function space for symmetric IPV first- and second-price auctions, casts equilibrium as a Stampacchia-type and Minty-type infinite-dimensional VI, and checks monotonicity, pseudo/quasi-monotonicity and Minty conditions; shows the BNE is the unique VI solution in the class of uniformly increasing bid functions, giving ex-post convergence guar
Key results
- Proposition 1 and 2: the Gateaux derivative operator for second-price and first-price auctions is neither monotone nor pseudo- or quasi-monotone (counterexamples with n=2, uniform priors).
- Second-price auction's BNE (identity bid) is a Minty-type solution, while for first-price auction (bid x/2 with n=2, uniform prior) no Minty solution exists (Lemma 5).
- The BNE is the unique solution to the symmetric VI within uniformly increasing bid functions, so any converging gradient-based learner reaches the BNE; numerical gradient flows in piecewise-linear bid spaces converge to the BNE for first-price auctions.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a second-price sealed-bid auction instead of a first-price auction | Symmetric independent-private-values auction with two bidders and uniform priors · Second-price sealed-bid auction | Minty condition | Helped |
| Use a first-price sealed-bid auction instead of a second-price auction | Symmetric independent-private-values auction with two bidders and uniform priors · First-price sealed-bid auction | Minty condition | Hurt |
| Apply the variational-inequality analysis to first- or second-price auctions | Symmetric independent-private-values auctions · First-price and second-price sealed-bid auctions | Monotonicity | No clear effect |
| Start gradient flow from different piecewise-linear bid functions | First-price auction with two bidders and uniform priors · First-price sealed-bid auction | Convergence to BNE | Helped |
| Start gradient flow from different piecewise-linear bid functions, including where Minty is violated | First-price auction with two bidders and uniform priors · First-price sealed-bid auction | Convergence to BNE | Helped |
| Use a gradient-based learning algorithm that converges within uniformly increasing bid functions | First- or second-price auction · Symmetric independent-private-values auctions | Equilibrium identification | Helped |
Surprising findings
- The first-price auction has no Minty-type solution even with a uniform prior, although its BNE is the unique VI solution.
- Monotonicity, pseudo-monotonicity and quasi-monotonicity all fail in standard auction games, despite consistent convergence of learning algorithms in practice.
Limitations
- Analysis restricted to symmetric independent private values with two bidders and uniform priors for counterexamples.
- Convergence results rely on gradient-based learning converging to a pure strategy profile within the uniformly increasing class; limit cycles or divergence not ruled out in other auction games.
- Numerical evidence limited to low-parameter piecewise-linear bid functions.
Experimental setup
Metrics: monotonicity pseudo-monotonicity quasi-monotonicity Minty condition violation (vector field)
Builds on: Minty condition extragradient convergence Stampacchia-type variational inequality Kolumbus and Nisan complete-information learning convergence
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