# Mapping Expected Margins to Probabililties

I'm a sucker for a colourful chart, and today's is based on simulations using an earlier model of Home and Away team scoring, constrained by bookmaker-based empirical realities.

Specifically, I've simulated results using the scoring model from that earlier blog, assuming that:

• Neither the Home or Away team is expected to create fewer than 14 nor more than 40 Scoring Shots
• Combined, the Home and Away teams are expected to produce at least 40 and no more than 55 Scoring Shots

Those constraints are based on an analysis of real-world bookmaker-inferred scoring assumptions, derived from their over/under and line markets. Simply put, we can use these two markets for a bookmaker to infer that a bookmaker's:

• expected Home team score is equal to one half of his or her over/under aggregate team score minus his/her line market handicap for the Home team
• expected Away team score is equal to one half of his or her over/under aggregate team score plus his/her line market handicap for the Home tea,

This analysis, coupled with the earlier, empirically-based observation that both Home and Away teams convert Scoring Shots to goals at about a 53% rate, allows us to translate Home and Away team scoring assumptions into Scoring Shot assumptions, which is what we need for the scoring model alluded to earlier.

The simulations produce the following chart, which relates the Home team victory probability to the Home team's expected victory margin and is colour-coded based on the expected aggregate score in the game. It suggests that:

• Home teams are less likely to win when they're X point underdogs in a game that's expected to be low-scoring than when they're X point underdogs in a game that's expected to be high-scoring
• Conversely, Home teams are more likely to win when they're X point favourites in a game that's expected to be low-scoring than when they're X point favourites in a game that's expected to be high-scoring. (For the simulated scoring model used here, the standard deviation of the final game margin is positively correlated with total game score, so greater variability is beneficial for the underdog.)

This chart also suggests that we should see greater variability in a bookmaker's probability assessments (and hence head-to-head prices) for Home teams in games where he/she assesses them as about 9- to 50-point underdogs or 9- to 50-point favourites since this is where the chart above most spreads for a given expected Home team margin. This variability will result from his/her different assumptions about the expected aggregate score in a game.

There is, indeed, some empirical evidence for the suggested phenomenon. My own Line market and Head-to-Head data for the TAB Bookmaker for the period 2012 to 2015 reveals that:

• In games where his Line market handicap is -50.5 or less, the average implicit Home team probability range (applying the Overround Equalising assumption to Head-to-Head prices) for a given handicap is 0.8%
• In games where his Line market handicap is between -49.5 and -8.5, the average implicit Home team probability range for a given handicap is 3.5%
• In games where his Line market handicap is between -7.5 and +7.5, the average implicit Home team probability range for a given handicap is 1.8%
• In games where his Line market handicap is between 8.5 and 49.5, the average implicit Home team probability range for a given handicap is 4.1%
• In games where his Line market handicap is +50.5 or more, the average implicit Home team probability range (using the Overround Equalising assumption) for a given handicap is 1.8%

Broadly, the variability in the range of implicit probabilities associated with a particular line market handicap is consistent with the chart above.

### CONCLUSION

Based on the Home and Away team scoring model derived in an earlier blog, wider and narrower ranges of Home team probabilities can logically be associated with games where the Home team's expected victory margin, positive or negative, lie in a given range.

In particular, especially narrow or large expected victory or loss margins are associated with narrower victory probability spans, while margins in the ranges of approximately -49.5 to -8.5 and +8.5 to +49.5 are associated with broader probability spans. Empirical TAB data supports this proposition.

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