Analysing Football Finals with A'spreadsheet

Copyright AgencyIlnliled (CAL) licensed cop;..l'urth~coaing and Communication.prcfilbircd except onpayment of fee per~op'yor Commuication And otherwise iil~a~;&brd:1nce wnh the licence from CAf:.-fQ /\C~R.For more Information contatd::AL 0" (02)939·1-7600 or [email protected] Analysing Football Finals with a'Spreadsheet

Stephen R. Clarke Swinburne University of Technology

Using sporting applications to make mathematics more interesting to both secondary and tertiary students is often discussed in the literature, eg Clarke (1984,1991), de Mestre (1987). Australian Rules Football, being the winter sport followed by most people in the southern states of Australia, certainlyqualifies as being of interest to students, and several authors (Clarke (1984,1988),Weal (1987),Schwertman & Howard (1989),Watson (1991)), have suggested ways to incorporate football into the classroom. In this article, we show that a mathematical analysis capable of being performed by secondary students can shed light on the appropriateness of the current AFL finals system - a topic causing much discussion on radio and TV and in the daily press. To generate the 128 equally likelyresults the aid of a spreadsheet (or a word processor) is enlisted.

Introduction From 1931 until 1971, the Victorian Football League (VFL) finals were played underthe 'Page final four' system. Teams in third and fourth position played a knockout first semi-final, the loser being eliminated and the winner going into the preliminary final. The top two played a second semi-final- the winner going straight into the grand final and the loser getting a 'second chance' by playing in the preliminary final.. The winner of the preliminary final then progressed to the grandfinal. In 1972 the VFL introduced a final five, played under the 'Mclntyre Final Five' system. By1991, with the introduction of interstate clubs, the competition had grown to 15 teams, and the Australian Football League (AFL) which was by now the governing body, introduced a new finals system played between the top 6 teams. After some criticism, they adjusted the system again for 1992 and introduced the 'McIntyre Final Six' system. This system of matches includes 2 double chance matches and 5 knockout matches to produce an eventual premier. The first week of finals eliminates 2 teams and produces a top four, from where the finals are played exactly as they were under the Page final four system. In week 1, the top two teams play in the qualifying final, the winner retaining its double chance by going into the old position 1, the loser to position 3 (in1992 Geelong beat Footscray). In the other two elimination finals, 3 plays 6 (Collingwood lost to St Kilda) and 4 plays 5 (West Coast beat Hawthorn), the losers being eliminated. The highest ranked winner (West Coast) gains a double chance and goes into position 2, and the other winner (St Kilda) to position 4. The most controversial aspect of the system is that the path the winner of 4 and 5 takes is determined by who wins between 3 and 6. Thus in 1992, because St Kilda beat Collingwood, West Coast went straight into the second semi-final. Had Collingwood won, West Coast would have gone into the first semi-final and had no double chance. In week 2 of the finals in 1992, West Coast beat Geelong to go into the grand final, and Footscray defeated St Kilda. In week 3, Geelong qualified for the grand final by beating Footscray, and in the following week West Coast won the Grand Final. The order of the teams after each week of the finals is shown in table 1.

35 Stephen R. Clarke

Table 1 Order of teams after each week of 1992 finals.

Week1 Week2 Week 3 Week4 Geelong Geelong West Coast West Coast Footscray West Coast Geelong Geelong Collingwood Footscray Footscray Footscray West Coast St Kilda St Kilda St Kilda Hawthorn Collingwood Collingwood Collingwood St Kilda Hawthorn Hawthorn Hawthorn

Schwertmanand Howard (1989, 1990) look at a probability model for the AFL Finals seriesas it was played from 1972until 1990 - a seriesof 6 games betweenthe top 5 teams. They list the 4 paths that result in the 4th team winning the grand final, and the 16 paths that result in the second team winning. Forthe top team they say"Directcomputation of the probability that team A wins the grand final is quite involved, with many different paths"andsuggestindirect methods. The current system,witha final 6playing aseriesof 7 matches has the extra complication that the position of a winningteam now depends not just on their match but on results of other matches. Forthe finalsixsystem,wewish to calculate not onlythe chance of eachteam winning the grand final, but also someother probabilities of interest such as the chance of pairsof teamsmakingthe grand finaland the chance of eachteamfinishing inanyposition. All the probabilities wouldfollow from the chance of allpossible finishing orders. Sothe problem is, for example, given the original order before the finals of Ceelong, Footscray, Collingwood, WestCoast, Hawthorn, St Kilda, what is the probability of a finalfinishing order suchas that whichultimately occurred, of WestCoast,Ceelong,Footscray, StKilda, Collingwood, Hawthorn?

Development of all possible outcomes Supposewe designateeachteam bytheir finishing position at the endof the homeand away matches. (In a classroom situation, teachers may wish to make the example more concretebyusingthe actualnamesof the teams,as we have donein table1).One possible sequence of match results (the one that actually occurredin 1992) could be: week 1: 1-2 (1 wins), 3-6 (6 wins), 4-5 (4 wins); this produces an order after week 1 of 142635. week2: 1-4 (4wins), 2-6 (2wins); this produces an order after week 2 of 412635. Preliminary final: 1-2(1 wins); this produces an order after the preliminary final of 412635 Grand final: 4-1 (4winslthlsproduces a final finishing order of 412635. There is no obvious pattern between the finalorder and the match results. Also there are other match results that also produce the same finishing order. For example, team 1 could loseits matchin week1 and the same final order could stillresult. It wouldbequite tediousto workout byhand allpossible 2' =128sequencesof matchresults.The factthat the positions of teams depend not just on the results of their matches but the results of

36 Analysing Football Results with a Spreadsheet others further complicates matters. Howeveraspreadsheet(orawordprocessor) comes to our aid. These packages allow the copying and movementof columns as wellas rows of text. Using this facility, the 128 sequences can begeneratedeasily ina normallessontime. The methodinvolves keeping not just the current order asabove but allpossible orders after eachmatch. Everymatchhas 2 possible results,eachofwhichproduces an associated changein the order. Before the final seriesthe order is 1, 2, 3, 4, 5, 6, so we have Order 1, 2, 3,4 , 5,6 Let us considerthe match between 1 and 2. This can have 2 results, so we copy the wholerow twice. Nowif1 beats2 the orderstaysthe same,soweleave the first rowalone, but if 2 beats 1 they interchange, so we do this to the second row. This gives us

I, 2, 3, 4, 5, 6 2, I, 3, 4, 5, 6

Werepeat the process for the match between4 and 5. This works in exactly the same way,with a duplication of the wholetableand aswapof twocolumnsin the second halfof the tableto give I, 2, 3, 4, 5, 6 2, I, 3, 4, 5, 6 1, 2, 3, 5, 4, 6 2, I, 3, 5, 4, 6 The next match, 3 versus6 is the mostcomplicated of the wholeprocedure. Againwe copythe wholeset of4 rows twice. If 3 wins,the positions stay the same,so weleavethe first 4 rows unchanged. If 3 loses, it moves to 5th while6 movesto 4th, so againwe move the relevant columns in the second half of the table. Finally, because the highest ranked winneractually jumps overthe loserofthematchbetween1 and2,weswapthe second and third columnin the table to give

I, 3, 2, 4, 5, 6 2, 3, I, 4, 5, 6

I, 31 21 5, 4, 6 2,3, I, 5, 4, 6

1, 41 21 6, 3, 5

2, 41 I, 6, 3, 5

1, 5, 2, 6, 31 4

2, 5, I, 6, 31 4 Thus we have all possible orderings after the first weekendof the finals. It might be argued that the table up to this point could be more easily built up from first principles. Howeverfrom here that method could no longerbe usedas the positions of teamsdepend on previousresults. Forexample there arenow2 possibilities for1st place, 3possibilities for 2nd place, 2 for 3rd and3 for 4th. The remaining 4 matchesare quite straight forward, as

37 Stephen R. Clarke from here on the result of any match either retains or swaps the order of the two teams involved. Eachmatch requires only 2 operations - a complete duplication of the whole table and a move in the second half of the table of the column for the lower placed team. The various stages are shown below - although for brevity we have only shown the first few and last few rows. Note each step doubles the number of rows. First semi-final: current 3 plays current 4 (16 rows) I, 3, 2, 4, 5, 6 2, 3, I, 4, 5, 6 1,3, 2, 5, 4, 6 2, 3, I, 5, 4, 6 I, 4, 6, 2, 3, 5 2,4,6, I, 3/ 5 1, 5, 6, 2, 3,4 2,5,6, I, 3, 4 Second semi-final: current 1 plays current 2 (32 rows) 1,3,2, 4, 5, 6 2,3, 1, 4, 5/ 6 1,3,2,5,4,6 2, 3, I, 5, 4, 6 4, I, 6, 2, 3, 5 4, 2, 6, 1, 3, 5 5/ 1, 6, 2, 3, 4 5, 2, 6, I, 3, 4 Preliminary final: current 2 plays current 3 (64 rows) 1, 3, 2, 4, 5, 6

2, 3/ 1/ 41 5, 6

11 3, 2, 5, 4, 6 2, 3, I, 5, 4, 6 4/ 6, 1,2, 3, 5 4,6, 2, I, 3, 5 5/6, I, 2, 3, 4 5,6,2,1,3,4 Grand final: current 1 plays current 2 (128 rows) 1,3,2,4,5,6 2, 3, 1, 4, 5, 6 1,3,2, 5, 4, 6 2, 3, 1/ 5, 4, 6 6,4, I, 2, 3, 5 6, 4, 2} I, 3, 5 6, 5, I} 2} 3, 4 6, Sf 2, 1, 3, 4

38 Analysing Football Results with a Spreadsheet

We now have a list of the 128 possible outcomes, Because of the systematic method used to derive the above there are some interesting patterns. The top half represents the matches in which thecurrently higher ranked team won the grand final (as was the case in 1992 - table 1 shows West Coast above Geelong after week 3), the bottom half whenthe currently lower ranked team won the grand final. The first and third quarters when the currently higher ranked team won the preliminary final etc. Thus the first line represents the result when the currently higher ranked team always wins. Note this does not produce the order 123456 because of the way.3 'Ieapfrogsz on winning the first elimination final. The order 123456 is produced twice - for either result of the qualifying final, provided the original higher ranked team wins all the other matches, (This finishing order was in fact impossible under the originalfinal six system as played in 1991).The last line represents the final orderif the currently higher ranked team always loses/and is the most different order possible from the original positions/ as 1 and 2 cannot finish lower than 4th position. Schwertman & Howard (1990) suggest several probability models that students could use to investigate finals. Here/ to test the fairness of the new system we assume that all teams in the finals are equal. In 1992 this was probably reasonable. If we use the ladder positions before the finals as a ranking/ of the 29 times the finals teams met during the season/ the higher ranked won on 14 occasions, the lower ranked on 15. This assumption implies the outcomes are all equally likely/ and it is now a simple matter to count the elementary outcomes that produce given orders/ or any other required compound events. Allspreadsheets and many word processors have the ability to order rows and this could be used to simplify the task. Although there are 128 different possibilitiesfor the results of the 7 matches/ because of double chances often 2 or even .3 of these individual results produce the same finishing order. There are 'only/ 72 different possible finishing orders of the 6 teams/ as can be shown by the following argument. There are 4 possible results for the matches between .3 & 6 and 4 & 5 and these determine 5th and 6th position. Since the highest ranked winner can finish no lower than .3rd there are 3 possibilities for this team/ leaving 3 x 2 x 1 for the remaining 3 teams. This gives 4 x .3 x .3 x 2 =72 orderings in totaL

Howfair is the newfinal 6 system? There are many events that students could investigate to check on the fairness of the system. It is one advantage of using an application area with which students have familiarity (or in this case even some degree of passion) that class discussion can be generated on suitable questions to be asked/ rather than students just answering preset questions. It is also important that students interpret theiranswers/ and write a discussion on their findings. Below are some of the results the author thought important/ but there are certainly others.

Table 2 - Probability of teams finishing in any position Original Final position Mean Final position 1 2 3 4 5 6 Position One/two .250 .250 .250 .250 .000 .000 2.50 Three .188 .188 .125 .000 .500 .000 3.44 Four/Five .125 .125 .125 .125 .250 .250 4.00 Six .063 .063 .125 .250 ,000 .500 4.56

39 8tephen R. Clarke

Table 3 • Probability of teams finishing in any position or higher Original final position position 1 2 3 4 5 6 One/two .250 .500 .750 1.000 Three .188 .375 .500 .500 1.000 Four/Five .125 .250 .375 .500 .750 1.000 Six .063 .125 .250 .500 .500 1.000

Table 4 • Probability of team i finishing above team j team 1 team 2 team 3 team 4 team 5 team 6 team 1 .500 .672 .750 .750 .828 team 2 .500 .672 .750 .750 .828 team 3 .328 .328 .703 .703 .500 team 4 .250 .250 .297 .500 .703 team 5 .250 .250 .297 .500 .703 team 6 .172 .172 .500 .297 .297

Table 5 • Chance of pairs of teams playing in grand final Original positions sum of ranks probability One & Two 3 .125 One & Three 4 .156 Two & Three 5 .156 One & Four 5 .094 Two & Four 6 .094 One & Five 6 .094 Two & Five 7 .094 Three & Four 7 .031 One & Six 7 .031 Two & Six 8 .031 Three & Five 8 .031 Four & Six 10 .031 Five & Six 11 .031

Table2showsthe top5 teamsanhavea reasonable chanceof winninga flag. The top2 teamshavean equalchance (0.25) of finishing 1,2,3,or 4. Fourand five havea 1in 8chance offinishing in 1,2/3,or4 anda 1 in4chanceoffinishing 5th or 6th.The tablealsoshowsthe mean or averagefinal position, whichis foundby adding the finishing positions weighted by the probability. Forexample the team originally in sixth position has a mean finishing position of 0.063xl +0.063x2 + 0.125x,3 +0.250x4 +0.500x6 = 4.56. The chance of winning the premiership and the average final position are both in monotonic order of original finishing order. In fact Table ,3 shows the probability of finishing in position jar higherisinmonotonic orderoforiginal finishing orderforeveryj. Perhapsaslightanomaly isthat team6,havingmadethe finals, haslesschance of winningthe flag than theyactually had at the start of the season (1 in 15 if an teams are equal). Table 4 produces some anomalies. It would be preferable if the chance of team i finishing higher than team j increased as j increased. This is true for all teams except for teams3 and6.Thus forexample team ,3 hasa0.70chance of finishmg above s or 51 but only a 0.50chanceof finishing above6. The probability of grand finals between the teamsin

40 Analysing Football Results with a Spreadsheet variouspositions is shown in Table5 and isgenerallyin the sameorder as the sum of the teams' rankings. One anomalyof the system is that a grand finalbetween1&3(or2&3) is morelikely than between1&2. In factagrand finalbetween the top 2teamsonlyhasa 1in 8 chance of occurring, compared with 1 in 2 under the old final four system. The results after the first week (equivalent to the old final four system) can be calculated ina similar manner or canalternatively becalculated byfirst principles. It iseasily shown that for teams who win through to the matches belowthe subsequent chances of winning the premiership are: Fromgrand final .500 From preliminary final .250 Fromfirst semi-final .125 Fromsecond semi-final .375

Note another anomaly- the chancesof team 4 or 5 winning dependas much on their matchas on the result of the match between3 & 6. Beforetheir elimination final, both 4 & 5 have a 0.125 chance of winning. After winning, the winner may still only have a 0.125 chance(if3 beats 6)or mayhave increasedit three foldto 0.375 if6has beaten 3. Thisisin factwhat happenedin1992,with WestCoastcontinuingon to the premiershipafter going straight into the second semi-final after St Kilda beat Collingwood. That a team'schances couldalter so dramatically dependingon the result of a third party maybe considered by someaflaw in the system. Students couldinvestigate further the effects the changesfrom the final four, final five, and two versionsof a final sixhavehad on the chances of competing teams. In mostcases, further calculation canbe reduced ifstudents seesimilarities between the oldsystemsand the current one.Forexample, in the previous final sixsystem,the chances of both 1 and 2 were the sameas under the current system,the chancesof both 3and 4were the sameas 3 under the current system, and the chances of both 5 and 6 were the same as 6 under the current system.

Importanceof matches.

We all know the grand final is the most important match of the year. It would be desirable if finals matchesbuilt up in importance, but how canwe quantify this notion of importance. Morris (1977) definesthe importance of a point in tennis as the probability of winningthe matchifa playerwinsthe pointminusthe probability ofwinningthe matchifa playerlosesthe point.Applying this definition to the final series, the grand finalisthe most important game with an importanceof 1.00, followed by the preliminaryfinalat 0.50. In tennis,pointsare equally important to bothplayers, but applying the definition to the finals sometimesresults inmatchesbeingofdifferentimportanceto the two teams.Inadditiona match can be important to a team not participating. The calculations are shown below.

41 Stephen R. Clarke

Grand final 1.00 - 0.00 1.00 Preliminary final 0.50 - 0.00 0.50 Second semi-final 0.50 - 0.25 0.25 First semi-final 0.25 - 0.00 == 0.25 Qualifying final between 1 & 2 0.375 - 0.125 == 0.25 Elimination final between 3 & 6 For 3 0.375 - 0.00 0.375 For6 0.125 - 0.00 = 0.125 For winner of match between 4 & 5 0.375 - 0.125 == 0.250 Elimination final between 4 & 5 depends on the result of the final between 3 & 6. If 3 wins 0.3125 - 0.00 == 0.125 but if 6 wins 0.1375 - 0.00 0.375 so on average it is 0.25 In general finals are in increasing order of importance, although there are some exceptions. In 1992 the match between 3 & 6 was played first and resulted in 6 winning. This meant the second match of the final series between4 & 5 was actually moreimportant than either of the two semi-finals on the second weekend. It could also be argued that the match between 3 & 6 is more important than the first and second semi finals, because of its importance to the non-participating winner of 4 & 5. The importance of othermatches can also be found. In the final match of the home and away round Carlton was playing West Coast. Had Carlton won they would have finished in 5thposition, Hawthorn dropped to 6th and St Kilda dropped out of the finals. So in fact the importance of this match to Carlton was 0.125, to Hawthorn 0.065 and to St Kilda 0.065. Again it could be argued that this match was at least as important as the final match between 4 & 5 would have been had 3 beaten 6.

Bookmakers' odds It is also interesting to see if the bookmakers' odds reflect these chances. In The Sun newspaper, September 5, the odds on winning the flag were quoted as Geelong 2-1, West Coast 3-1, Collingwood 4-1, Hawthorn 4-1, Footscray 5-1 and St Kilda 10-1. Translating these to probabilities, and multiplying by the required factor to produce a total probability of 1 we get: Table 6: Bookmaker's odds Team Odds to 1 Probability Adjusted Actual probability probability Geelong 2 0.333 .266 .2500 Footscray 5 0.166 .133 .2500 Collingwood 4 0.200 .162 .1875 West Coast 3 0.250 .205 .1250 Hawthorn 4 0.200 .162 .1250 St Kilda 10 0.090 .073 .0625

42 Analysing Football Results with a Spreadsheet

The biggest difference in the bookmakers' probabilities and the actual probabilities, assuming all teams are equally likely to win any match, are for Footscray and West Coast. This may reflect the weight of money, or alternative estimates of the two teams' relative strengths, rather than a lack of knowledge by the bookmakers on the advantages of different ladder positions. However St Kilda is slightly favoured by the bookmakers, whereas Collingwood is given less chance of winning than our assumption of even teams. This is surprising, given that Collingwood was clearly the favourite in the match against St Kilda(8-11favourites in The Sun, as against St Kildaevens), and probably reflects errors in the subjective estimates of the relative advantages of lying 3rd and 6th.

Conclusion Although we have here considered allteams to be equal, the method can be modifiedfor alternative models. For several years the author has been supplying computer generated tips for a daily newspaper (Clarke, 1988, 1992). The above method was used on a word processor when the probabilities of allpossiblefinishing orders were required, allowing for a different win-loss probability for every possible pair of teams. This probability even changed from week to week, because of the effect of changing grounds. To allow for this, another column was used to keep the probability of the current order. Each time the rows were duplicated, this column was multiplied by the probability of the particular match result that was being considered. In football, subjective judgements are often used to rate team chances of winning the premiership. These often tend to reflect the relative strengths of the teams, and ignore the current ladder position. With the complicated structure now in place for the Australian FootballLeague finals series, a mathematical analysis using a simple modelcan shed light on the chance of teams winning, or of finishing in any position, given their initial or current ranking in the final series. A large part of mathematics is about recognition of patterns. In this case, there was no obvious pattern between the results and the probabilities that produced that order. However there was a pattern in the way these orders were built up when individual matches were considered and the functions of a spreadsheet or word processor could be used to exploit this pattern to generate the elementary outcomes. Investigating such examples in sport provides not only an interesting area for the application of probability and logic, but demonstrates to students that mathematics can provide insights into everyday problems. When students are intimately familiar with the problem, and have a stake in the answer, they willoftengo that extra step and question the assumptions of the model. To generate some heated controversy in a mathematics lesson can provide a pleasant change.

References Clarke, S. R. (1984) Mathematics in Sport. In: Maurer, A. (Ed.) Conflicts in Education. Mathematics Association of Victoria, Melb, 130-135. Clarke, S. R. (1988) Tinhead the Tipster OR Insight, 1 (1), 18-20.

43 Step hen R. Clarke

Clarke, S. R. (1988) Computer Tipping of Football (and other Sports). In: Firth, D. (Ed.) Maths Counts - Who Cares? Mathematics Association of Victoria, Melb, 243-247. Clarke, S. R. (1991) Australian Student Projects on sport result in a USA Research Connection. ASOR Bulletin. 10 (2), 2-5. Clarke, S. R. (1992) Computer and human tipping of AFLfootball- a comparison of 1991 results. In: de Mestre, N. (Ed), Mathematics and Computers in Sport. Bond University, 81-93. de Mestre, N. (1987)Mathematics and Sport. Australian Mathematics Teacher, 43 (4), 2-5. Morris, C. (1977) The Most Important Points in Tennis. In: Ladany, S.P. & Macho!' R. E. (Eds.) Optimal Strategies in Sports. Amsterdam. North Holland, 131-140. Schwertman, N. C. & Howard, L. (1989) A Probability model for the Victorian Football League finals series. Australian Mathematics Teacher. 45 (2), 2-3. Schwertman, N. C. & Howard, L. (1990) Probability models for the Australian Football League finals series. Australian Mathematical Society Gazette. 17 (4), 89-94. Watson, j.M. (1991) Exploring Data from the AFL Grand Final. Australian Senior Mathematics Journal, 5 (1), 23-38. Weal, S. (1987)Mathematics in Australian Rules Football. In: Caughey, W (Ed.) From Now to the Future. Mathematics Association of Victoria, Melb, 183-189.