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Online Learning in Betting Markets: Profit versus Prediction
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Models binary betting markets with Kelly bettors and a profit-maximising bookmaker, showing profit and information elicitation conflict, and gives two online price-setting algorithms (stochastic approximation for unfair odds, follow-the-leader for fair odds) with regret bounds.
Problem
Bookmakers in binary betting markets must set prices without knowing bettor belief distributions; prediction markets aim to elicit beliefs while betting markets aim for profit, and the two goals are usually studied separately.
Approach
Unifies betting and prediction markets in one Kelly-bettor model, shows profit hinges on deviation between bettor mean belief and bookmaker belief (heavier tails give more profit), and designs online price updates per bet: stochastic approximation under unfair odds and follow-the-leader with clipped cumulative average under fair odds.
Key results
- Stochastic approximation pricing keeps regret at or below 10^2 across four initialisations while risk-balancing regret grows by over an order of magnitude.
- Follow-the-leader under fair odds keeps regret at or below 10^1 over 10^5 bettors, while LMSR regret grows faster and converges to the crowd average belief.
- Theory gives O(sqrt(T)) regret against the worst local maximiser (unfair odds) and O(sqrt(T log T)) high-probability regret against the global maximiser (fair odds).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Online SA instead of risk balancing for unfair-odds pricing | Bookmaker pricing · Unfair-odds simulation, 100,000 bettors | regret: Online SA: \(\leq 10^2\); risk balancing: larger by more than an order of magnitude | Helped |
| Use FTL instead of LMSR for fair-odds profit maximisation | Fair-odds bookmaker pricing · Fair-odds simulation, 100,000 bettors | regret: FTL: \(\leq 10^1\); LMSR grows faster than FTL | Helped |
| Set fair price to match the bookmaker's belief to the mean bettor belief | Fair-odds betting market · Binary betting market | expected profit: zero | Hurt |
| Increase bettor-belief dispersion so the distribution is SOSD over the less dispersed distribution | Bookmaker profit · Binary betting market | expected profit | Helped |
| Use different initial prices for Online SA | Online SA · Unfair-odds simulation with a unique maximiser | regret: four different initialisations; regret \(\leq 10^2\) | No clear effect |
| Use different initial prices for Online SA when the profit maximiser is not unique | Online SA · Multi-modal belief distribution | regret: global-maximiser process: \(\leq 10^1\) | Mixed |
Surprising findings
- Under fair odds, the profit-maximising price is not the crowd's average belief, so optimal bookmaker prices can deviate from the information-eliciting prediction market price.
- Optimal prices for maximising profit can lie far from the bookmaker's belief (e.g. up to 41.4% above the common belief in an example), so prices give no useful indication of where the common belief lies.
Limitations
- Assumes Kelly bettors and i.i.d. belief distributions.
- Restricted to binary outcomes; non-binary markets left to future work.
- Under unfair odds, SA converges only to local maxima and needs proper initialisation when maximiser is non-unique.
Experimental setup
Metrics: stochastic regret adversarial regret profit
Baselines: risk-balancing heuristic logarithmic market scoring rule (LMSR)
Builds on: Kelly criterion logarithmic market scoring rule stochastic approximation follow the leader
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