Mathsport 2019: Establishing a performance edge in P2P betting

University of St Andrews & DMP Statistical Solutions UK Ltd.

• Betting basics
• The P2P world of gambling
• Genuine edges can be hard to establish
• Determining your power
• Real-world example
• Recap

We're basically considering odds over time.

• There are different types and somewhat related to probability of an outcome
• Fractional: e.g. 2/1 bet £1 to perhaps profit £2
• Decimal: the multiples of your stake returned (including stake)

NB - these aren't standard statistical odds \( p/(1-p) \)

Odds may be directly related to probabilities:

• Consider a fair coin
• Rational/informed/numerate players will only agree on decimal odds of 2, factional 1/1
• For games with calculable probs, rational odds are clear

These situations are in the minority - usually \( p \) is estimated

Gambling has gotten very modern:

• We can play on both sides now - you can “sell” or “buy” odds
• Acting as bookie, we lay - we sell some odds \( X \), so make money if A doesn't happen (we're gambling against)
• Acting as punter, we back - we offer to buy odds \( X \), so make money if A does happen (we're gambling for)
• There is almost direct analogy with going long/short on stocks

• There are 1000s of methods you might devise to make money - but which ones are actually effective?
• To assess this, you need a good grasp on variance and statistical fundamentals
• Power and Gambler's Ruin are classics that need transferring to the P2P area

• This basically quantifies our ability to detect signal of specific sizes for a system (that comprises signal and noise)
• Closely related to type-1 error (false positives) - which drives our thresholds for \( p \)-values in statistical testing
• In short: “Given an effect of size \( X \), how likely are we to find it”
• Relatedly, “How much data do we need to be confident in finding the signal?”

The basic problem is commonplace in statistics - the building blocks are Bernoulli/Binomial:

• Tests and CIs on \( p \)
• Some complications:
  • We have “portfolios” of bets which are a mix of RVs
  • Independence issues
  • Market forces through time and stochasticity in matching

• Assume we have a system that operates at odds \( \omega \) - without edge, we have implied \( p = 1/\omega \)
• Over the course of 1000 plays, the sampling distribution for \( p \) is roughly Normal (if \( p \) isn't very small or large)
• The central 95% of this is

\[ p \pm z_{\alpha = 0.025} \sqrt(p(1-p)/1000) \]

• Power calculations require some speculative size of signal
• Here I'll assume a 10% edge is some minimum requirement to be profitable (commisions eat into things)
• 10% edge here means probability of winning is 10% less than odds imply (I'm laying)
• The same bet-portfolio applies i.e. I need to know the distribution of betting odds achieved

Take \( Y \) to be the outcome (1 a win, 0 a loss), \( P(Y=1) = p \) and \( P(Y=0) = 1-p \)

For laying, the return \( R \) is given by

\[ R = \begin{cases} -S(\omega - 1) & \mbox{if } Y = 1\\ S & \mbox{if } Y = 0\\ \end{cases} \]

With a few sums

• \( E[R] = 0 \) if \( p \) = \( \omega^{-1} \) (not accounting for commission)
• \( V[R] = \omega-1 \)

The second is of particular interest. If we assume independence of events then we can get the variance of many such events by simple summing.

More generally, we'd have a mix of positions

• \( V[R_i] = \omega_i-1 \)
• so variance of the return for many such bets under independence: \( \sum_n V[R_i] \)

So we can go directly to the return distribution - which will be also predicable in shape for large numbers of bets (Normal - yay for the statistical free lunch that is CLT).

• We're assuming independence, but generally multiple positions might be sought within a single event
• Normality is asymptotic - so long odds, with small N won't be so predicable
• However you need the distribution of \( p \) or \( \omega \) for your system
• This could easily be estimated by running the system on historical data

• Data are collected from global horse racing events with reasonable liquidity
• This is 200+ events per day
• Odds data are collected on 1/3 second basis for a period prior to, and during, event
• Currently approximately 2TB of uncompressed data per annum
• Automated system(s) trained on historic data will take positions pre-play

• Analysis over historic data provides a distribution of positions taken
• Entry rate (i.e. positions filled) is approximately 40%
• Projected return is approx 16% per position, post-commission
• How much data is required to establish this is genuine?

• From which we can derive a variance of returns after 1000 bets of about 8200.
• Suggesting an approximate Normal(0, 90) return distribution
• So really, unless our system is thought to achieve more than 180 units on 1000 bets, it will still look like noise at this point.

The simulated returns look like the following, which is about 2500 trades:

Is it any good?

• Apparent returns are about 16% per position taken, post-commission
• No edge at all seems likely to return in the order of 280 units - we've hit 400
• \( p \) is approximately 0.002 under \( N(0, 140) \). An edge seems likely (even not accounting for commission), but magnitude is uncertain (and this is not real trading at this point).

Is it any good?

• How much actual trading do we need to do to be confident there really is a double digit return?
• We'd calculate low power for say 4000 future bets.

• Bookies effectively profit by offering poor odds - refer bookies over-round
• P2P levies a commision on winnings - say 5% for Betfair (variable - up to 60%), 2% for Betdaq (some caveats) and others
• This can be substantial! Consider a system offering 10 units won to 9 units lost - commission is actually now 50% of profits.
• Hence this really needs consideration in all calculations

• The general availability of lay betting is dangerous
  • Liabilities are easily large (\( S(\omega-1) \))
  • Medium to long odds laying gives the appearance of success - it is very easy to be fooled by randomness
• You should know the properties of the Null position before you start betting your house
  • Simple simulations and sums are informative

• Power is quite low for any form of longer odds gambling
• Even large data isn't sufficient to offer confidence for long-odds approaches
• Consider Premier league - some 380 games per season - years may be required
• These are also for filled bets - making a lot of bets doesn't equal lots of data