RWC2019: The South Africa Problem

My initial attempt to base some RWC Outright probabilities on something tangible like previous results fell well short. Especially when it came to South Africa. It failed to identify the improvement in South Africa since their latest Head Coach appointment. The improvement is self evident. Previously they have received routine thumpings from New Zealand for example 57-0 in 2017. Since the Head Coach replacement they've played NZ 3 times. The aggregate score is 82-82, after 1 win 1 draw and 1 defeat. It would seem that the results post March 2018 are much more relevant when assessing South Africa. This drastically reduces my sample size for them (just 12 matches vs Big 6). So although using less matches I'll use extra information from those matches. In other words instead of a binary win/defeat I will use Points Differences. This will help capture the relative strengths better and close matches will get the reward they deserve. 

So here it is: 

	P	W	D	L	Avg PD	P[W]	P[WWW]	Dec Odds	Matchbook Odds
NZ	28	22	1	5	14.85	83%	57%	1.75	2.52
ENG	25	17	0	8	6.68	67%	30%	3.39	5.90
IRL	22	12	1	9	-0.95	48%	11%	9.30	12.50
WAL	26	9	1	16	-4.73	38%	6%	18.12	14.50
AUS	34	8	2	24	-7.79	31%	3%	34.08	19.50
SA *	12	5	1	6	0.91	52%	14%	6.97	5.50

Table 1: Points Differences, Big 6 Head to Head since RWC2015 to present (* SA including only Rassie era)

I have converted the Points Differences to Win Probabilities based on a Normal Distribution. I have sense checked this PD -> P[W] conversion with some matches currently priced up on betfair and it seems a fair conversion. I assume that each team will need 3 victories in a row versus another Big 6 (QF/SF/F) to win outright.  That is the probability of winning a Big 6 Head to Head (as calculated using PD of previous Head to Head's) raised to the 3. So for example - South Africa have 52% chance of winning "an average" Big 6 encounter so 52% x 52% x 52% = 14% of wining 3-in-a-row and hence outright.

Although South Africa are closer to New Zealand now, it still reflects the fact that even under the new Head Coach, their average points difference against the Big 6 teams is still less than that of New Zealand. For example South Africa's average Points Difference versus Australia in the first 3 encounters of the "new era" is 8 points, whereas New Zealand are putting up an average of 17 Points Difference vs Australia in the same time-frame. 

This effort is still showing value on New Zealand, England and Ireland. And S Africa still appears to be over-rated by the markets (After giving S.Africa a Rassie-Boost the probabilities are summing to over 100 now so I would need to renormalise them, which should push each team out a little so this should also be considered). However its not really taking into account Strength of Schedule.  

The Strength of Schedule angle might be key to understanding South Africa's price. Things going as expected their QF match is tough enough versus Ireland, but they will be in the opposite side of the draw to NZ and ENG. Meaning their route is IRL - WAL/AUS - ENG/NZ/Other. They have about 2 points advantage on Ireland (which by my Normal conversion makes SA 55% favs), 5 or 6 points advantage on WAL/AUS (66% fav). This puts them about 36% chance to make the Final at which point NZ and ENG will have knocked one or the other out and it just takes one minor upset and neither are in the Final leaving South Africa favourites on the day. 

In conclusion by adding weight to the "new era" under the new Head Coach and unscientific analysis on the Strength of Schedule we have tried to understand the South Africa price. If you disagree with either of these points (and I think I disagree with both) then South Africa are still firmly to be opposed. I agree there is an improvement under Rassie Erasmus but not to the extent we weight those matches at 100% and previous at 0%! The improvement has also not been across the board. The fault in the Strength of Schedule analysis is that we know from the Pool B first match that S. Africa would be 3.1 underdog versus the most likely finalist (current price to beat NZ). You can make some adjustment to that price as the Pool pricing includes a draw outcome and also water it down a bit as its not 100% certain that they would play NZ in the Final. However I think it will still be a long way above the 2.0 that would be needed for the S. Africa Outright price to make sense.

Appendix:

My Pts Difference to Win Probability conversion table (based on a "Normal" distribution of points with standard deviation determined based on looking at the historical results). 

Pts. Diff.	P[W]	P[L]	Dec Odds
8.5	71%	29%	1.41	3.42
8.0	70%	30%	1.44	3.29
7.5	68%	32%	1.46	3.17
7.0	67%	33%	1.49	3.06
6.5	66%	34%	1.51	2.96
6.0	65%	35%	1.54	2.86
5.5	64%	36%	1.57	2.76
5.0	63%	37%	1.60	2.67
4.5	61%	39%	1.63	2.59
4.0	60%	40%	1.66	2.51
3.5	59%	41%	1.70	2.43
3.0	58%	42%	1.74	2.36
2.5	56%	44%	1.77	2.29
2.0	55%	45%	1.81	2.23
1.5	54%	46%	1.86	2.17
0.5	51%	49%	1.95	2.05
0.0	50%	50%	2.00	2.00

My actual Pts. Difference distribution compared to the Normal distribution of same mean/std dev. It breaks down badly at some points ie if you are trailing by 4 and have a penalty in the last minutes, you're always going for the corner and either winning by 3 or losing by 4 - its unlikely we get matches decided by a single point and even less likely by 2. The five peaks that ruin the smooth distribution are -6, -3, 0, +3, +6, ie these points differences occur in a single match more than expected.