The Value of an Extra Home Game in the Six Nations.

Often the format under which a tournament is played can have a considerable influence on the eventual outcome. Tennis provides a prime example. Five set matches enable the superior quality of a higher seeded player shine through compared to a best of three set sprint and, unlike the FA Cup, the best players are often seeded to meet in the final, rather than the earlier rounds.

A Seattle/Denver Super Bowl wouldn't have been possible prior to realignment in 2002, when the Seahawks moved from the AFC West to the NFC West. And their return to the NFL's showcase game also owes something to a playoff format that rewards the highest seeded teams with fewer post season matches and home field advantage all the way to the Super Bowl.

Another epic sporting occasion begins on Saturday with the first round of the Six Nation rugby and it also has a format that potentially aids certain sides in particular years. The four home nations, plus France and Italy play each other over five weekends. Therefore, each country alternates between playing two and three home matches from their total of five fixtures. This arrangement, not only creates alternate feast and famine markets for six nations tickets, but it also should logically give the teams with three home ties an advantage compared to seasons when they only host twice.

The home field advantage, in keeping with the NFL and rugby league, is worth around three points in union. So a switch in venue accounts for a swing of six points or very nearly a converted try.

At the start of the 2013 Six Nations, Wales hosted Ireland in Cardiff in the first match. Overall, Wales was considered the fourth most likely side to win the tournament with around an 18% chance of success, their visitors were slightly more favoured with a 22% chance. So Ireland was rated as a slightly superior side compared to the Welsh and that was reflected in the matchday predicted margin of victory. With the help of home field advantage, Wales were expected to win by an average of two points in repeated trials.

Had the game taken place at the Aviva Stadium, Ireland would have been four point favourites.

Expected margin of victory or defeat can be readily converted to win or loss probabilities. Wales as 2 point favourites in Cardiff are likely to win around 53% of the time, compared to just 36% of the time as 4 point underdogs in Dublin.

Similarly, England's expected 16 point margin of victory when hosting Scotland, equated to a near 90% chance of victory compared to a ten point average winning margin and a 78% chance of bringing the spoils back from north of the border.

Therefore, points spreads (assuming their accuracy) can be easily used to simulate not only recently completed tournaments, to gauge how likely was the actual outcome, but the venues for each match can also be reversed to test the impact of the unbalanced five game format.

Simulation of Six Nations 2013 Based on Points Spread for each Game.

Country.	Number of Home Games in 2013.	Simulated % of Outright Title Wins.	Overall % of Title Wins (inc decided by points differential).
Wales	2	8.4	13.2
England	3	33.7	43.8
Scotland	3	0.7	1.6
Italy	3	0.4	0.7
Ireland	2	10.8	16.1
France	2	16.7	24.6

With the title decided over just 15 matches, it isn't uncommon for the top sides to end the championship level on points. Around 30% of the simulations for 2013 finished in at least ties and 4 out of the 14 tables (29%) since the competition was extended from five to six nations has seen the top sides level in terms of league points. So, points difference, akin to goal difference in football, now comes into play to decide the champions.Therefore, the second column denotes how frequently a side "won" the simulation without recourse to a tie breaking, points reckoning and the final column includes "wins" such a Wales' in 2013, where points differential across all five games was required to split teams at the top.

We can now reverse the venues and repeat "2013" using alternative probabilities that reflect the altered chances of each side winning or losing these virtual matches to see if the presumed advantage of playing three of the five matches at home is a significant factor.

Simulation of Six Nations 2013 Based on Points Spread for each Game, but with Venue Reversed.

Country.	Number of Virtual Home Games in Alternative 2013.	Simulated % of Outright Title Wins.	Overall % of Title Wins (inc decided by points differential).
Wales	3	7.4	12.4
England	2	33.2	44.0
Scotland	2	0.7	1.4
Italy	2	0.1	0.2
Ireland	3	8.8	14.8
France	3	18.7	27.4

Initially it appears that both simulations yield very similar results. The order in which each side was most likely to finish remains the same, whether a team had three or just two home matches. The chances of particularly England and Scotland are almost identical over 10,000's of iterations despite the reversal of venue.

Most interesting is that while the chances of France lifting the title improved for an extra home game, those of both Ireland and Wales declined slightly. There could be countless reasons for this apparent quirk, not least a faulty simulation or insufficient iterations.

An alternative explanation is that it is not just the number of home matches that impacts on  the likely number of points a Six Nations side may get, but also the relative ability of those opponents.

Based on the relative merits of the countries in 2013, Wales gain around an average of 18 extra wins per 100 matches when playing England at the Millenium compared to visiting Twickenham. As an extreme comparision, England's predicted 95 wins over Italy in the Stadio Olimpico has a very limited upside when the match takes place in TW2. Home field advantage in the latter case adds very few extra wins to England's cause.

Might home field advantage be a larger boon to teams facing similarly matched opponents, rather than taking on markedly inferior ones.

Both the Celtic nations each entertained two (on paper at least) strong rivals in their 2013 season with only two home matches. Wales hosted Ireland (3rd favs) and England (favs) and Ireland entertained France (2nd favs) and England (favs). France, by contrast had visits from Wales (4th favs) and Scotland (33/1 fifth favs).

So the interplay of two weakish (on paper) home opponents in one season for France, produces inferior expected returns compared to the reverse fixtures when they have three home matches. But the relative strength of the two home opponents for Wales and Ireland appears more advantageous than an extra home tie, but against markedly inferior countries.

In short, playing two very good teams at home and three,on average, less able ones away, may give you a better chance to win enough games to lift the title compared to the alternative of three home matches against overall weaker opposition and two really tough games on the road....Even though the latter alternative involving an extra home game superficially appears more attractive.

 Will Three Home Games Prove a Help or Hindrance to Wales in 2014?

Whatever the speculation, the way the unbalanced fixture list has evolved doesn't appear to unduly handicap any one side in the Six Nations and there may even be reasons to think that the complex interaction of matchups may actually slightly assist some "two home game" teams.

What the simulations do show is the scope for shock results to influence a competition decided over a relatively small numbers of matches. 2013's reality of Italy defeating both France and Ireland, Scotland finishing 3rd, Ireland on the opening weekend beating Wales, the eventual champions, before themselves nearly finishing bottom of the table, probably appears somewhere in the 10,000's of simulated championships, but it is a fairly unlikely one.