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Smooth Quadratic Prediction Markets
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Proposes the Smooth Quadratic Prediction Market, replacing the DCFMM's Bregman-divergence fee with a quadratic L/2||r||^2 fee, showing it has better worst-case loss while agents' trades implement steepest gradient descent (incremental incentive compatibility).
Problem
Prediction markets based on the DCFMM implicitly use FTRL to price securities; the question is whether other learning algorithms can inspire market designs with equal or better guarantees, e.g. higher market-maker profit and worst-case loss.
Approach
Decompose the DCFMM payment into a linear price term plus a Bregman-divergence fee, and replace the Bregman fee with a quadratic fee (L/2)||r||^2 derived from L-smoothness of the convex conjugate cost function; agents' expected-return maximization then yields general steepest descent updates on the market state.
Key results
- Theoretically shows the Smooth Quadratic Prediction Market has better worst-case loss than the DCFMM for any fixed trade history, since its payments upper-bound the DCFMM's Bregman-based payments.
- Shows that with l2 norm and a CIIP cost function, agents' trades converge to their belief at rate 1/t under incremental incentive compatibility.
- Simulations with softmax (L=1), q0=(10,20,10), and belief (1/6,1/6,2/3) show convergence under l1, l2, and l-infinity norms, including under budget constraints (B=0.01) and buy-only constraints.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the DCFMM Bregman-divergence fee with a smoothness-based quadratic fee | Smooth Quadratic Prediction Market vs. DCFMM · AD securities, fixed trade history | worst-case monetary loss | Helped |
| Use the ℓ2-based Smooth Quadratic Prediction Market with repeated traders sharing belief μ | Smooth Quadratic Prediction Market · Repeated trades with a shared belief | convergence of market prices to belief: at a rate of 1/t | Helped |
| Use an ℓp-based quadratic fee and general steepest-descent trading updates | Smooth Quadratic Prediction Market · Repeated trades with a shared belief | convergence of market prices to belief: at a rate of 1/t | Helped |
| Impose a natural budget constraint on traders | Smooth Quadratic Prediction Market with softmax prices · Budget-bounded trading | convergence of market prices to belief: The rate of convergence was proportional to the size of the budget. | Helped |
| Restrict trades to nonnegative bundles | ℓ2-based Smooth Quadratic Prediction Market · Buy-only market | convergence of market prices to belief | Helped |
| Vary the norm used in the quadratic fee (ablation) | Smooth Quadratic Prediction Market · Market-state updates toward μ | convergence path | Mixed |
| Decrease smoothness as trading volume increases and add a liquidity fee | Adaptive-liquidity Smooth Quadratic Prediction Market · Adaptive liquidity | no-arbitrage guarantee | Helped |
Surprising findings
- The Smooth Quadratic Prediction Market does not satisfy standard incentive compatibility, only an incremental version.
Limitations
- Softmax is not l1-smooth, so the l1-norm experiments are only used for comparison.
- Adaptive liquidity extension lacks proofs of worst-case loss, information incorporation, and incentive compatibility.
Experimental setup
Metrics: worst-case monetary loss convergence of market state to trader belief
Baselines: DCFMM
Builds on: Duality-based Cost Function Market Maker Follow-The-Regularized-Leader sparsemax Volume-parameterized market making framework
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