In this way it is not especially insightful that the first team to a certain number of points is more likely to win. If the teams were equal from this point forward, the team with more points would win, and at any rate the better team is likely to score more points at any point in the game.

But I thought it would be interesting to see what the odds of winning were when you were first to each number of points, so I analysed every game since 2012 to see what the odds were.

I was also curious if you were ever likely to lose if you got to a certain number of points first. I do not think this will be the case, but we must remember that the team that kicked the first goal in the Grand Final went on to lose in every game from 2014-2019.

Here are the results (full table at the bottom):

This is as expected - at any point, if you are ahead, you are unsurprisingly more likely to win. Additionally, being the first to 100 points means you win 98.6% of the time. This is because you are a) scoring 100 points, which the other team may not and b) ahead late enough in the game for you to have scored 100 points.

You’ll notice the highest score (in my data) that a team got to first but still lost was 124 - this was of course by North Melbourne against Adelaide in Round 9 2013.

I think the most interesting part of this data are the probabilities at the lowest scores. There are clear jumps in win probability between 6-7 and 12-13. This is likely because each of these jumps requires another score (usually a goal) to be the first team there - in this way we are looking at a greater sample size. It is also clear why the 3-6 range is flat - teams normally kick a goal before scoring 3 points.

Here is all the data if you want to have a look through the numbers in the above graphs:

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2026 Women’s Asian Cup

In Asian Women’s football, there are a very clear top 5 teams, being Australia, China, Japan, North Korea and South Korea. These teams have dominated in terms of World Cup qualification for Asian teams and between them have not lost to a team outside this group in an Asian Cup game that I could find since 1999.

They are now all playing at the 2026 Women’s Asian Cup. Unfortunately, in a tournament with quarter-finals, this means that there will be at least one quarter-final with multiple top 5 teams. This isn’t awful - competitive matches are great - but could be seen as unfair for the top 5 teams that have to play each other.

To add to this, the teams that make the semis get direct qualification to the World Cup. There is still a repechage system that will allow quarter-final losers to qualify: the winner of each match between quarter-final losers also qualifies for the World Cup, and the losers of these matches go to further inter-confederation playoffs.

I’m not sure if the organisers were aware of this, but if they were they have expertly designed the tournament. Groups A and B each had two top 5 teams, where as Group C had one. The runner-ups in Groups A and B have to play each other in the quarter-finals (at the time of writing Australia and either North Korea or China), where as the (presumably not top 5) runner-up in Group C has to play the winner of Group B. The other two group winners get to play third-place teams. In this way all the Group winners should in theory avoid the other top 5 teams until the semis, creating a nice reward for topping a group.

As a comparison, this was done poorly in the structure of the 2018 tournament. With North Korea absent (being eliminated only on goal difference without losing a game after unluckily being drawn into the same qualification group as South Korea), there were only four top 4 teams. However due to the treatment of the host nation Jordan, we ended up with 3 top 4 teams in the one group!

South Korea were thereby eliminated without losing a game simply because they ended up with the two eventual finalists in their group.

2026 Men’s T20 World Cup

The 2026 Men’s T20 World Cup has:

• 4 groups of 5 teams, with the top 2 teams going through, then:

• 2 groups of 4 teams, with the top 2 teams going through, then:

• semi-finals and a final.

It’s worth noting that depending on how you count, there are generally 9 to 11 decent cricket teams - this doesn’t mean the others can’t beat these teams (as I’ll discuss), but they very rarely make it far in tournaments. Partly because of this, the ODI Cricket World Cup has only 10 teams, that all play each other in a round robin stage, before the top 4 teams go through to semi-finals.

After Bangladesh pulled out before the tournament started, these were the groups:

Groups A-C were predicted to be broadly uncompetitive with the top two teams advancing. Group D was predicted to be a tussle between New Zealand, South Africa and Afghanistan. In this way there were a ‘top 9’ countries who most people thought could plausibly win the tournament.

So confident was the ICC in this that the second round groups were ‘pre-seeded’. This means that, for example, if England qualified, they would end up in Group 2 regardless of where they finished in their first group. This had the ridiculous consequence that all the first-placed teams ended up in Group 1, and all the second placed teams ended up in Group 2.

As it turned out, Group B was not at all uncompetitive - in fact Australia surprisingly lost to Zimbabwe, and so were eliminated. In Group D, Afghanistan played well but lost to two other top 9 countries, and so were eliminated because of the group draw.

Interestingly, probably slightly due to the tournament structure, there was really not much noise in Australia about this tournament. I think this was because we were expected to waltz through the first group stage, so no one paid attention, and then we got eliminated before the second group stage, so no one paid attention.

The old T20 World Cup format actually quarantined the weaker countries and made them play an initial group stage before they could qualify to play the stronger countries. I think the new format is better than this - it is hard to argue that we should quarantine the non-top countries after Zimbabwe just beat Australia. But I feel as if Afghanistan were hard done by and there may be potential for a better tournament structure.

It’s also worth noting that some cricket fans think the ICC is not above ‘engineering’ the draw - India and Pakistan have managed to be drawn into the same group at the last 6 T20 World Cups (and the 2015 ODI World Cup, the last ODI World Cup with more than one group). Many suspect this is not a coincidence.

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2020 & 2024 Men’s Olympic Basketball

The USA have won 17 out of 20 Men’s Olympic Basketball tournaments, and very rarely lose. In this way being drawn against the USA in the quarter-finals is often a death sentence for teams’ medal hopes.

In 2020 this played out in the form of a very important final group game between Spain and Slovenia - the winner would top the group and definitely avoid the USA in the quarter-final, whereas the loser was at risk. Slovenia won and topped the group, and Spain, the #2 team in the world at the time, was unlucky enough to draw the USA in the next round - where they put up a good fight but lost. Interestingly, this partially only happened because the USA finished second in their group after losing a game to France.

In 2024 this was improved by further seeding within the quarter-finals as below (D plays G, E plays F). This meant that teams that performed better in the previous round avoided the USA until later in the tournament. Still, you could argue the silver medal was handed out partially by lottery - France ended up winning it by not playing the USA until the final, whereas maybe Serbia could have won silver if they were on the other side of the draw (they were instead beaten by the USA in the semis and took home bronze).

This only matters if medals are on the line of course - in many tournaments not winning is simply not winning.

This concept of a superpowered USA team gives credence to some challenge tournament systems, such as the America’s Cup, where the defending champions must be beaten to win the trophy, and old VFL finals systems, where the top team got to play the final again after being beaten.

2023 Men’s Rugby World Cup

The 2023 Rugby World Cup used a similar model to the 2026 T20 World Cup - 4 groups of 5 teams, but then followed by quarterfinals into the final. In most groups there were 3 competitive teams instead of the 2 in the T20 World Cup - simply for this reason this group format suited the competition a lot more.

Does tournament design need to consider the teams in the tournament?

I think it is agreed upon by most sports fans that the best team should win each tournament. However there are other factors at play that make tournaments great. We want tournaments with exciting, meaningful games. We want to both be able to follow our team through and also be able to enjoy watching matches as a neutral. We also want tournaments to be as fair as possible, especially where they impact other tournaments by functioning as qualifiers. All of these are impacted not just by the structure of the tournament, but by the relative strengths of the teams in it.

And so this discussion boils down to a single question:

Should the design of tournaments be agnostic as to the relative quality of the teams actually playing in the tournament?

I think the answer is no, and I think at some level most sports fans agree with this - no one argues that the Cricket World Cup should have the same format as the FIFA World Cup with 32 (or 48) teams and a long knockout stage, purely because the relative strengths of teams are different.

In this way I generally support the double group-stage model of the T20 World Cup (but not the pre-seeding).

The implication of this is that we should be engineering tournaments to get the big teams to play each other in the later stages, but avoid each other if possible until then - especially when medals or qualifications are on the line. This is of course how tennis tournaments are structured - but this is easier with no group stages that could throw draws into the air.

This was also a motivation of the Champions Trophy in cricket and the Confederations Cup in soccer - smaller tournaments between the top teams. I think tournaments like the UEFA Nations League do this well, and could be a useful model.

But what do you do when there is a big gap between the top teams and the rest, and the number of top teams isn’t a factor of 2?

One option is a heavy-handed approach: for example, the Women’s Asian Cup could instead say that the group winners and the top two second-place finishers are going into a second-round group of 5 teams. The top two teams in this group will then play the final against each other. You could even say that the winner of the round-robin is the champion, but people tend to like having a final (I think it would be bad to get rid of one).

The biggest problem here is that the top 5 is clearly contrived - the tournament is obviously engineered to get this outcome. But I think this is ok.

There are other factors that come into tournament design - one of the most significant is time. A direct elimination tournament is usually the fastest way to quickly eliminate lots of teams, whereas multiple group stages are quite slow. A 10-team round robin (as at the ODI Cricket World Cup) may be one of the slowest.

In this way I get why Olympic Basketball is in a rush to get to the elimination stage. However the Women’s Asian Cup and similar tournaments could perhaps be structured differently.

Time will tell whether the new FIFA World Cup format creates interesting games and whether it ends up giving some teams much easier paths to the final than others. I think because soccer is low-scoring with a good chance of upsets it will still be very entertaining.

Afghanistan and South Africa just played a match at the cricket world cup with not one, but two super overs. The match itself was full of swings and exciting moments, including a no-ball on a would-be match winning wicket, some classic tail ender running between wickets, and both teams squandering chances to win the game before South Africa eventually pulled through.

This is all a lot more fun, and satisfying from a cricket perspective, than a bowl-out would have been. Before some franchise league tries to make a whole tournament of one over games (would we call them O1s?), let’s have a look at what’s happened in Super Overs since they came into being.

Origins

Test cricket matches do not get tied very often (it’s happened twice). We are also very used to drawn test matches, such that in the context of a test match series it is perfectly fine for a game to be tied. It would also be a bit unfair to decide a 5 day contest off two overs of cricket, and so I don’t see a super over being used in test cricket any time soon.

We were also happy enough for ODIs to be tied. Famously the 1999 World Cup semi final between Australia and South Africa was tied, and no super over was played. Instead Australia got through as they finished higher in the group stage.

The rise of T20s necessitated an effective tiebreaker, as these shorter games were prone to being tied far more often. The first three tied T20Is were decided with bowl-outs - like penalty shootouts, except the bowlers had to hit the stumps with no keeper. A World T20 game between India and Pakistan was even decided in this way.

But bowl-outs were both random and bowler-focused, so in 2008 the ICC brought in Super Overs in T20Is. Eventually Super Overs were brought in for knockout ODIs, with the very first one happening in the 2019 World Cup Final. However at the time the rules only provided for one Super Over, and as the Super Over was tied, England won the world cup on boundary countback (somewhat controversially).

Since a December 2023 rule change, Super Overs are expected for all ODIs and T20Is. This does not mean they have always occurred - there was controversy around the lack of a Super Over following a tie between India and Sri Lanka in August 2024, after both the officials and teams did not realise that there was meant to be a Super Over.

Rules (international)

The current ICC Super Over rules can be found in the ICC Men’s T20 International Playing Conditions (at page 103 of this document) and the ICC Men’s One-Day International Playing Conditions. As far as I can tell these rules are identical. Domestic leagues may have slightly different rules however broadly the ICC rules predominate.

When to have a Super Over?

Firstly, when do the Playing Conditions provide for a Super Over? Nowadays, clause 16.3 of both rules says

If the teams’ scores are equal after both innings have been completed, then a Super Over shall be played. If the Super Over is a tie, then unless exceptional circumstances arise subsequent Super Overs shall be played until there is a winner.

So Super Overs should happen whenever there is a tie.

How does a Super Over work?

A Super Over, understandably, is an over of six balls with two wickets in hand for each team. Here are a few more rules that apply:

• The team batting second in the match will bat first in the Super Over

• Each team will be allowed one unsuccessful review

• The fielding side gets to choose what end it bowls from

• The fielding team shall choose the ball to be used in its over from the box of spare balls provided by the umpires. The team fielding second can pick a different ball from the box or use the same ball as the first team.

How does a repeat Super Over work?

A repeat Super Over works similarly to the first one:

• The team batting second in the previous super over will bat first

• The fielding side must swap ends

• The fielding side must use the same balls

However there are two rules to make the Super Overs seem like more of a normal innings:

• The bowler who bowled the previous Super Over cannot bowl the next one

• A batter who got out in a previous Super Over cannot bat in any subsequent super over.

There is an interesting implication to this last rule - if we ever were to have 6 Super Overs in a row, one team may find themselves ‘bowled out’ having picked up 10 wickets in the previous 5 Super Overs, and therefore unable to bat. As far as I can tell the rules do not actually provide for what would happen here. I assume that the team would either immediately lose or be taken to score 0 off their Super Over. This is very unlikely to occur of course, however perhaps the ICC will provide for this in their next rules update.

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All stats below are current as at 15 February 2026.

Amount of Super Overs (international)

There have been 6 Super Overs in ODIs, with one of these tied (the 2019 World Cup Final) - however a repeat Super Over was not played.

There have been 29 tied T20Is with Super Overs. Of these, there have been two games with two Super Overs (India-Afghanistan in January 2024 and the recent South Africa-Afghanistan game), and one game with three Super Overs!!! (between Netherlands and Nepal in June 2025).

Amount of Super Overs (domestic)

With the data I had access to, I found the following number of Super Overs for the following competitions:

• IPL - 15 Super Overs - 2 tied (1 resolved on boundary count, one resolved by repeat Super Over)

• BBL - 4 Super Overs

• BPL - 3 Super Overs

• CPL - 3 Super Overs

• SA20 - 1 Super Over

• The Hundred - 1 “Super Five”

Super Over Records

These are records that I could find from international and domestic games - please let me know if you are aware of anything that breaks these from other international competitions!

Highest Score

30 - Netherlands (Logan van Beek) vs West Indies, June 2023 - 4 6 4 6 6 4

Lowest Score

0 - Multiple, including Trinidad and Tobago Red Steel vs Guyana Amazon Warriors (Sunil Narine), July 2014 - 0 0 0 0 W 0

Highest Score Chased

17 - Multiple teams

19 (equalised) - Netherlands vs Nepal, June 2025 (first Super Over)

Lowest Score Defended

11 - Multiple teams, including Guyana Amazon Warriors (above) and Mumbai Indians (Jasprit Bumrah) vs Gujarat Lions, April 2017

Most Innings in a Cricket Game

There is an argument that due to Super Overs, the game with the most innings in cricket history is actually a T20 game! The aforementioned Netherlands-Nepal game could be argued to have 8 innings (or 4 each, being 1 regular and 3 Super Overs). This is far more than a Test match, which only has 4 innings.

Because it does not affect the outcome of the game, the ICC Playing Conditions do not comment on whether a Super Over is an innings or not. In all fairness it probably isn’t, however this could be a fun quirk where the game with the most innings ever was also complete in under 4 hours.

Affecting other records?

It is possible that a player could break the most runs or most wickets in a T20 game record due to getting access to more balls in a Super Over. However this is unlikely to occur as a game where a player is so dominant as to break the record is probably not going to go to a Super Over.

Players that have both batted and bowled in Super Overs in the one match

Here are all the players I could find that both batted and bowled in a Super Over in the one match in a T20I or ODI - there may be other domestic examples:

• Marlon Samuels (for WI v NZ, 2012 World T20) - went for 0/17 and scored 9* off 3

• Logan Van Beek (for Netherlands against WI, June 2023 as above) - scored 30* off 6 and then took 2/8

• David Wiese (for Namibia against Oman, 2024 T20 World Cup) - scored 13* off 4 then took 1/10

• Azmatullah Omarzai (for Afghanistan against South Africa, 2026 T20 World Cup as above) - hit 16* off 5 in the first Super Over, went for 0/23 in the second Super Over, then was 0* off 0 in the second Super Over.

A Final Comment - Duckworth Lewis for Super Overs

Because teams can be bowled out in Super Overs, we could in theory apply a modified version of the Duckworth Lewis Stern method to calculate a par score at each point in the Super Over. This would be very noisy, and I assume that batters do not actually get significantly less conservative after the fall of the first wicket - however they are more likely to be bowled out within the over. This could be a fun (but very silly) project if anyone is interested in doing it.

While watching the Australian Open Men’s Singles final, one of my friends remarked to me that Novak Djokovic had been especially clutch at saving break points over the last two matches.

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In the semi-final against Jannik Sinnner, he saved 16 of 18 break points (89%). In the final against Carlos Alcaraz, he saved 11 of 16 (69%).

This reminded me of my analysis of how tennis would play out if points were decided on probability alone. I thought that 89% was a very high amount of service points to win, but 69% might not actually be that good.

The ATP has thankfully gathered this data for me already - Novak has won 68% of service points throughout his career. In this way saving 69% of break points is expected - and saving 89% is indeed remarkable. The ATP data says that Novak has saved 66% of break points throughout his career.

In this way, my friend was (somewhat) wrong - we should expect a top player to save most of their break points, even if we might assume otherwise because of momentum or other factors.

Who’s the most clutch?

So let’s have a look at which players are the most and least clutch at saving break points - i.e. which players have the greatest difference between their service point win % and their break point save %.

The WTA has similar data as well, but unfortunately only by year. So I’m going to look at the current top 100 men, as well as every past #1 player (except for Nastase and Newcombe because there is no data), to see who is the most and least clutch of all of them.

Here’s the data:

So the most clutch break point defender is apparently Filip Misolic, who wins 4% more break point defences than he does regular serve. The most clutch current top 10 player is Ben Shelton.

It’s worth noting Misolic’s data comes from only 460 service games and 327 break point opportunities - this isn’t a very big sample size compared to Novak’s 17064 service games and 6862 break point opportunities. There’s every chance that Misolic has just got lucky so far and that his performance will experience some reversion to the mean as his career progresses.

Here are the least clutch players:

Further Analysis

For the average player, we would expect that their break points saved % is less than their service points won %. Looking at it on a game-by-game level, players would win a greater % of points in games where they do not face a break point (whether because they are serving better or they are facing different opposition). So we would expect that if they have already given up a break point, they are doing worse than the average game where they do not do that.

This is backed up by the data - the average Service Points Won % is 64.04%, and the average Break Points Saved % is 62.32%, a difference of 1.72%.

It’s also interesting to use this data to see which players were most dominant on their serve. Here are the players with the highest Service Points Won %:

Firstly, there are quite a few very good players on this list - unsurprisingly, if you win a lot of service points, you will win a lot of matches. But none of them stand out for their clutchness in saving break points.

Of course this data does not show us whether these points were saved in especially clutch situations such as saving a match point. In general however, the players that are best are saving break points are, as expected, simply the players that are best at winning points on serve.

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Gerrymandering is the practice of drawing electoral boundaries to advantage a particular party or candidate. This is typically possible because districts can be winner-take-all. Unlike many other sports, in tennis there are boundaries drawn around points that are winner-take-all - namely games and sets. This allows us to gerrymander a tennis game.

Similar to how CGP Grey calculated that you could win the Electoral College with only 22% of votes (noting that there is no active gerrymandering in the Electoral College as the boundaries are just the state borders), let’s work out the most efficient point allocation so that you could win a (complete - no retirements) best-of-5-sets tennis match despite losing an overwhelming number of points.

Losing the first two sets badly

Firstly, you’d have to lose every set that you lost by the highest amount possible (this is called packing in gerrymandering). This would be a 6-0 set loss, losing each game to love. This would be 6*4 = 24 points for the opposition. So let’s suppose you lose the first 48 points and find yourself 6-0 6-0 down.

Winning the next three sets narrowly

You then want to win the next set as narrowly as possible. Let’s test winning 6-4, 7-5 and 7-6 - the reason we are not jumping to testing the tiebreak option straight away is that in a tiebreak you have to win at least 7 points (and we want to win as few as possible) so it might be easier the other way.

Similar to before, we have to lose each game without scoring a point. Let’s now consider what the best way to win a game is.

Your initial instinct might be that we want a game that goes to deuce many times. Clearly, winning 5 points to 3 means that you won a lower percentage of points than a game that you won 4 points to 2 (62.5% to 66.7%).

However, supposing that you lost the other game 4 points to 0, this would actually mean that you had won a higher percentage of the total points. You would have won 5 points out of 12 (41.7%) rather than 4 points out of 6 (40%). This is because the game that you won is now a greater percentage of the total points.

This is similar to Simpson’s paradox.

In a similar vein, we want to win as few points as possible (not just a low percentage), because we want our opponent’s 48 points to mean as much as possible.

Therefore we must win each game by 4 points to 2 (i.e. game-30). Similarly, in a tiebreak, we want to win 7-5.

Another way of thinking about this is that either way to change this score would lead to us winning a higher percentage of points:

• a higher margin (e.g. 7-3) would just mean less points for our opponent

• a higher total (e.g. 9-7) would mean we are adding the same number of points each, which moves our total percentage towards 50% - which we don’t want to do if we can avoid it.

So our possible set results are:

Let’s now see what happens if we were to have each of these set results three times:

As we can see, winning each set 7-6 is the best way - replacing some of these with 6-4 wins does not improve the percentage. So the lowest percentage of points you can win and still win a match is 35.23%, if the game goes 0-6 0-6 7-6 7-6 7-6.

However this isn’t true in grand slams, because there final set tiebreaks are first to 10, not 7. This would mean that the match would actually be 96 points to 174, so you would win 32.56% of points. In this way the the lowest percentage of points you could win to win a match would actually be 35.25% - winning 0-6 0-6 7-6 7-6 6-4 (as in the table above).

This also holds for a best-of-3-set match - the lowest percentage of points needed to win (in a grand slam) is 37.16% (55 to 93 in an 0-6 7-6 6-4 win). If you didn’t have grand slam final set tiebreak rules, it would instead come in a 0-6 7-6 7-6 win, with 36.9% needed (62 points to 106).

Finally, this provides a hard limit to the number of points you need to win to never lose a match, which I explored here. While you will usually find it very hard to lose when winning 59% of points, if you are winning 65% of points in a match (100% minus 35%), it is truly impossible for you to lose.

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Tennis provides leverage to players with a small advantage, as this small advantage compounds through many separate games in separate sets. Eventually, we would expect the player that has won more points to win the match (although part of the fun of tennis is that there is no guarantee of this.

So, assuming that each point has the same probability of being won by a player (which of course it does)

If you want to skip the maths, you can go straight to the results below.

What % of points do you need to win to be unbeatable?

Let’s first assume that our player has the same probability of winning each point (irrespective of service - something we can correct later). We can very quickly calculate their total win probability by working out their odds of winning each game, then each set, then each match (assuming best-of-5-sets matches). It’s worth noting that throughout Federer’s career a lot of matches were best-of-3-sets matches.

To do this we’re going to use a lot of Binomial Distributions so acquaint or re-acquaint yourself with them here if you need.

Probability to win a game

We’ll assume we have probability p to win a point.

To win a game, you need to win at least 4 points and 2 more points than your opponent. If you win before deuce, your opponent has won 0-2 points, but you won the last point, and so the probability of winning a game before deuce is:

If deuce is reached, there are 3 possible outcomes for the next 2 points: you win both, you both win one and get back to deuce, or your opponent wins both. If you get back to deuce, we’re back at the beginning, so we only need to consider the other two options. The probability of winning a game after deuce is therefore:

So given probability p of winning a point, the probability g of winning a game is:

Probability to win a set

Let’s now work out the probability of winning a set from g. You can either win 6-something, 7-5, or 7-6. Using similar logic to before, the probability of either of the first two is:

The probability of getting to a tiebreak is:

But to work out the probability of winning a tiebreak we have to use points again. As it is first to 7 but you must win by 2 points, we can use our game probability model again, but instead imagine it as a 7 point game. The probability T of winning a tiebreak is therefore:

and so the probability s of winning a set is:

Except there’s a slight problem here - every win probability is partially inflated! This is because when we use the binomial distribution to work out all the ways to get to 6-6, this includes paths when one player got to 6-4 then the other player won the next w games. This can’t happen because the set would be over (and we would have already counted this case in the 6-something set).

To fix this, we can normalise our probabilities by comparing the probability of winning a set with point win probability of p to the probability of winning a set with point win probability of 1-p. I did this as follows:

For example, the unadjusted probabilities s for p=0.45 and p=0.55 are 0.209 and 0.861 respectively, but the adjusted probabilities are 0.196 and 0.804 (summing to 1 as we would expect).

Probability to win a match

A match is easier to work out, as there are no tiebreaks or deuces. Thus to win 3-0, 3-1 or 3-2, the probability m of winning a match is:

The results

So we can now graph all our results. For the 40%-60% range of point win probabilities, our graph looks like this:

As you can see, even if you were to win only 52% of points, we would expect you to win about 74% of your matches. If you could win 58-59% of points, you would probably never lose a match.

With a 54% point win rate, we would expect Federer to win 90% of matches (not 80%)! However this is of course because his point win rate varied by opponent (and so was of course lower than 54% against good opposition).

Zooming out, for all possible point win probabilities our graph looks like this:

This shows that once you win about 67% of points you will likely win all your sets, and that you would need to win 88% of points to win all your games.

So if Federer had won an extra 5% of his points (definitely easier said than done) he may have never lost.

Some notes on the model

We could of course factor in point win probability when serving and when returning. This would improve the model, but for simplicity I have not done this - it returns approximately the same outcome.

We could also consider how difficulty increases in a tournament - Federer probably won more than 54% of points in the first round of a grand slam for example.

For a better model have a look at this paper by Franc Klassen and Jan Magnus.

I thought we could use a common economic measure to measure ‘batting inequality’ in T20 cricket - i.e. when a few batters are scoring all the runs and the rest of the team is not.

If you want to skip the maths and just see the outcome then scroll down to ‘The Gilly Coefficient’.

The Gini Coefficient

The Gini Coefficient is used to measure income or wealth inequality. It is calculated by looking at the difference between the actual distribution and what the distribution would be.

In the above graph, we have a line at 45 degrees showing what the cumulative distribution of income would look like if everyone earnt the same. That is, the bottom 50% of people would have 50% of the income and the bottom 75% of people would have 75% of the income.

The blue area is the actual distribution of income. Here, the bottom 50% have something like 20% of the income, and the bottom 75% have something like 60% of the income.

The Gini Coefficient is just the area A divided by the total area of the triangle. This is a great indicator of inequality, because it captures the shape of the distribution. For example, here are some Gini Coefficients and distributions.

Calculating the Coefficient

The area A can be obtained through integration:

where L(x) is the actual distribution (the Lorenz curve). Because the triangle has area 1/2 (side length 1), the Gini Coefficient is therefore:

However you can alternatively calculate the Gini Coefficient as half of the average absolute difference between incomes, divided by the average income. This is useful for our purposes because it avoids integration when we’re just looking at a small number of batters. I won’t get into the proof of this but this comes out to be:

We will use this formula for our cricket analysis.

The Gilly Coefficient

I thought we could use the Gini Coefficient to measure inequality of runs for a cricket team’s top 7 batters. This lets us see where teams have had the biggest deviations in batting performance between their best and worst batters.

Just because we’re looking at cricket, I figured we could call this the Gilly Coefficient after Adam Gilchrist. He’s not especially relevant to the stats, however he did top score for the Deccan Chargers in the 2008 and 2009 IPL seasons so we’ll name it after him - he was also player of the series in 2009 and led the Chargers to the title.

I’ve decided to look at the top 7 because a T20 team will normally have recognised batsmen down to at least position 7. Often the 6th or 7th highest run-scoring player in a season for a team turns out to be a bowler because a batter underperforms, but in that case we’ll just classify that bowler as an all-rounder cause they’re clearly batting alright.

We’ll look at the IPL since 2008 (its whole history) and the BBL from 2015/16 to 2023/24 (because my data goes that far). I’ve picked the biggest league and my local one. I’ve only looked at franchise tournaments because international tournaments often aren’t long enough for a good sample size.

Some inequality is expected, because openers have more possible balls they can face and so will likely score more than the person batting 7.

As an example calculation, David Warner has done a great job this BBL season while his team, the Sydney Thunder, have struggled. The top 7 scores for the team wee:

These scores have the following inequality distribution:

From these scores, the Gini Coefficient is equal to 0.265.

The Results

Here are the Gini Coefficients across the IPL and BBL:

Amazingly, the top three Gilly Coefficients all go to RCB teams (and it’s not particularly close). In all these seasons, RCB had 2 of the top 5 run scorers (Kohli and either Gayle or Du Plessis), but quite a few players that didn’t make many runs. Perhaps this happens when your top players bat so well, but perhaps this is also why they couldn’t complete the chase in the 2016 Final when Gayle and Kohli got out.

The lowest Gilly Coefficient goes to 2022 CSK who had 7 players score between 230 and 368. They came second last. However the second lowest Gilly Coefficient goes to the 2017 Mumbai Indians, who won the league.

Finally, for a sense of scale, here are the Gini Coefficients for all Test-playing countries (with the UK for England, the Republic of Ireland for Ireland and a couple selected West Indies countries), according to the World Institute for Development Economics Research:

This makes it harder to tell exactly which votes a player got. While it is not possible to determine which votes came from which coach, I thought it would be useful to make a quick algorithm for decomposing the final votes into the two different votes received (as much as possible).

I should note that the fitzRoy R package already has a function for this, but I wanted to build one myself. It turns out I came up with the same algorithm the function used.

The Problem

Given a set of coaches votes for a game, we need to determine the 5-4-3-2-1 votes as much as possible.

Developing an Algorithm

We can solve this by iterating through the votes until we have worked out possible totals for all players.

There are three vote totals with only one possible combination:

• 10 votes - 5 & 5

• 9 votes - 5 & 4

• 1 vote - 1 (& 0)

Thus if we see a player with any of these votes we can ‘finalise’ their vote numbers and remove them from consideration. We could then use this information to finalise other players in another iteration.

• For example, if there was a player with 10 votes (5 & 5), we would know that the player with 8 votes got 4 & 4.

In a similar way, we can recursively work out the possible options after we pick a certain combination for a vote total. We will first need a function for working out the possible combinations for a vote total:

possible_votes <- function(n, max=5){
  possible <- c()
  for(i in 0:max){
    if((n-i)<=max&(n-i)>=i){
      possible <- append(possible,c(i,n-i))
    }
  }
  return(possible)
}

Then we can make a recursive tree to test all possible outcomes and prune them as they are unsuccessful. Our one condition is that a coach can only give each 5-4-3-2-1 vote once, so we need to make sure that any addition does not break this rule. Then we can print off final possible options.

find_votes <- function(votes,final_votes,count=1){
  #determining possibilities for highest vote
  vote = votes[1]
  votes = votes[2:length(votes)]
  possible <- possible_votes(vote)
  
  #iterating through possibilities
  for(i in 1:(length(possible)/2)){
    
    #if there is a plausible vote allocation then allocate votes - trying both arrangements of votes
    n = ifelse(count==1,0,1)
    for(j in 0:n){
      k = possible[2*i - j]
      m = possible[2*i - (1-j)]
      if((!(k %in% final_votes$vote1)|k==0) & (!(m %in% final_votes$vote2)|m==0)){
        final_votes[count,] = c(k,m)

        #if we have allocated all votes, publish the topic
        if(sum(rowSums(final_votes)>0)==nrow(final_votes)){
          print(final_votes)
        }else{
          #otherwise test the next vote
          find_votes(votes,final_votes,count+1)
        }
      }
    }
  }
}

This function now successfully decomposes votes into possible vote allocations.

As an example, if we decompose the below vote allocation:

We find that in every possible decomposition (an example below) Gryan Miers got 4 votes from one coach and no votes from the other. Maybe Sam Mitchell really doesn’t like him,

Similarly, for the above Brisbane-Collingwood prelim, it turns out that at least one of Will Ashcroft and Zac Bailey only got votes from one coach (a couple possible decompositions below):

Perhaps all of Jamie Elliot’s votes came from Craig McRae too.

I’m interested to run this algorithm on some votes during the season to find some more interesting outcomes.

A common (or at least commonly brought up, I’m not sure if it ever happens) fantasy football punishment is to complete 24 beers, donuts and miles (or kilometres), made up of any amount the punished player chooses. While this is only tangentially related to sports, it highlights the consideration of the marginal cost of multiple options to maximise a total.

I thought it would be fun to explore the probabilities surrounding similar ‘multiplication’ markets that exist for the AFL and NBA, specifically goals*kicks*handballs and points*assists*rebounds respectively.

Spread Betting

Most sports betting that occurs is either Fixed Odds (for most sports) or Tote/Parimutuel betting (for horse racing and similar). While there are differences between these two market types, in general they will result in a similar payout. Importantly, you cannot lose more than you bet, and you will not normally stand to gain more than the actual implied probability, that is:

Spread betting is different, and more resembles financial derivatives (fixed odds derivatives markets are something else - they are still fixed odds markets, simply the probability is derived from other odds rather than directly calculated). It provides far more leverage, can result in far higher wins and losses, and is related to accuracy in a way other betting types are not.

In a spread betting market, you will either bet above or below a certain number. However in contrast to Fixed Odds betting where your payout is the same as long as the market is successful or fails, the payout (or loss) from a spread bet changes depending on the scale of success.

For example, suppose that you bet $10 on Lebron James over 20 points. If he was to score 25 points, the market would have been beaten by 5 points, so you would win 5 times your stake (i.e. $50) - you would not have to pay the bookmaker the $10 in the first place so they would just pay you $50.

However if Lebron scored 10 points, you would have lost by 10 points, so you would lose 10 times your stake (i.e. $100). In this way there is a much greater risk to spread betting.

You may notice that this would lead to a situation where a person could be in debt to a gambling company. This is not permitted in Australia - betting operators cannot provide lines of credit to customers. In this way customers cannot make spread bets without sending all the money they potentially stand to lose to their betting account - seeing just how high this is usually makes people think twice about the bet, and in this way spread betting is not very common in Australia.

Common Multiplier Markets

Because spread betting is about numbers rather than probabilities, there are some fun markets that arise. Betting companies like markets with more leverage (as then they make more) - unfortunately for them (fortunately?) there aren’t many sports stats with high ranges of plausible numbers. Most player props in most sports are limited to two digits, and while basketball and cricket scores can be in the triple digits they still usually fall within a small range.

This led companies to develop certain ‘multiplier’ markets with far more range and volatility (and therefore leverage). Two examples of this are:

and

Instead of varying across 40 possible outcomes, these markets could vary from 0-10,000 and 0-1000 respectively.

But we won’t look at these for the purpose of modelling when they are good to bet on. Instead we’ll simply look at when the highest score in these multipliers has been achieved.

Highest NBA Multiplier Scores

I calculated the highest multiplier scores in the NBA since the 2001-2002 season using the below code (my data source goes back that far).

library(dplyr)
library(hoopR)
data <- load_nba_player_box(seasons = 2002:2025)
datam <- data
datam <- datam %>% mutate(mult = points*rebounds*assists) 

datam <- datam %>% select(athlete_display_name,athlete_position_abbreviation,points,rebounds,assists,mult,game_date,team_name,opponent_team_name)

datam <- datam[order(-datam$mult), ]

There are a lot of mentions of Jokic, Doncic and Westbrook in this table, as is expected - this is pretty much just a list of impressive triple-doubles.

If you also thought it might be up here somewhere, in Wilt Chamberlain’s 100 point game, he scored 100 points, had 25 rebounds, but only 2 assists, meaning that he had a multiplier of 100*25*2 = 5,000.

I haven’t looked at all the data but amongst his five highest scoring game Wilt had 2, 1, 1, 0 and 1 assists respectively, so despite scoring a lot he probably didn’t beat any modern players at the Multiplier.

Highest AFL Multiplier Scores

I then calculated the highest AFL Multiplier Scores since the 2012 season using the below code (again, just working with R package limitations):

library(fitzRoy)
library(dplyr)
stats <- fetch_player_stats()
statsm <- stats
statsm <- statsm %>% mutate(mult = goals*handballs*kicks, opp = ifelse(team.name == home.team.name, away.team.name,home.team.name)) 

statsm <- statsm %>% select(player.surname, player.givenName, player.player.position, goals, handballs, kicks, mult, round.name, opp)

statsm <- statsm[order(-statsm$mult), ]

write.csv(statsm, "mult.csv",row.names=FALSE)

As expected, we see a lot of midfielders in here, dominated by Dane Swan and Gary Ablett Jr. (although Jeremy Cameron and Jarryd Roughead have done well to make it in).

However the all-time record (probably untouchable) is held by Greg Williams, who had a 6 goal, 25 kick, 28 handball game for Sydney against St Kilda in 1989. This is a multiplier of 4,200.

Optimisation

Clearly, the best way to increase your Multiplier is to score higher in the category you are scoring the highest in. A very simple formalisation of this is:

and so the fastest way to increase your score is to scale your best performing categories by multiplying them by a higher number.

For basketball, this almost always means increasing rebounds or assists. However, we must remember that points can go up by 2 or 3 at a time, whereas assists must go up by one at a time.

In this way if points are less than 2 times assists, it makes more sense to optimise for 2 pointers, or even 3 pointers if points are less than 3 times assists.

I.e., if

then

For AFL, it is almost always goals that are the lowest variable. To get a very high score requires a few goals, and more goals are always best (especially as they also count as a kick). Although using similar maths if you have already scored a couple goals and are not a typical goalscorer, it may be easier to amass kicks and handballs. Similarly to the above calculation, this may improve your score more than sacrificing that for one more goal.

Of course no players are thinking of this in games but if they were they now know how to score the highest.

At Euro 2016, Portugal only won one game in regulation time, and yet ended up winning the whole tournament. They got through their group (where they drew three times) by being the third-highest ranked third place team. It’s entirely possible a team could do the same at the 2026 World Cup…

Carrying on from this post, here I’m going to make a model of the 2026 World Cup group stage to work out every team’s probabilities of progressing. There are probably better models for the whole tournament out there, but my goals here are to make an intuitive model that can be simply explained, and one that I can use to show the differences in the probabilities of making it out of the group stage for the 2022 structure and the 2026 structure.

The Elo Rating System

The first thing we need is to be able to estimate the probabilities of a team winning each game. Thankfully we have the ready made FIFA World Rankings, which are an Elo rating system, similar to that used in Chess. The great feature of Elo systems is that the difference in ratings between two teams is intended to be directly related to the probability that one team will win a game. A team with an Elo rating 100 points higher than another team is expected to win 64% of the time, and a team with a rating 200 points higher is expected to win 76% of the time. The expected winning percentage for a team A against team B is:

where RA and RB signify the rating of each team.

(Note - there is a plus sign between the 1 and the 10 above, apparently it does not render correctly on some phones)

Here are the current FIFA World Rankings for the top 165 teams:

The FIFA World Rankings are not a strict Elo system - they do not weight all games equally, and instead assign an Importance coefficient, whereby World Cup games are weighted more than friendlies. This does slightly distort the Elo rankings however may be justified, as teams do tend to ‘try’ more in more important games - they are more likely to play new young players in friendlies than in World Cup qualifiers.

At any rate, the existing FIFA World Ranking system has been found to have decent, although not optimal, predictive power (Szczecinski and Roatis (2022)).

We’ll use the current rankings to predict how games will go throughout the group stage. It would probably be more correct to calculate new Elos after each game, to better capture movement and momentum, however this system will still provide decent predictions.

Draws

However we need to factor in draws. We can do that by calculating a draw probability for each game, and then assigning the remaining non-draw outcomes using the Elo rankings.

Draws are more common between evenly matched teams, and then become less common as teams’ skill levels get further apart. A good model for the probability of a draw is therefore a decay function such as the below function, which aligns well with empirical data:

So our code for predicting the winner of a game is:

p_draw <- function(Ra, Rb, d = 0.28, c = 0.003) {
  d * exp(-c * abs(Ra - Rb))
}

p_win <- function(Ra, Rb) {
  1 / (1 + 10^((Rb - Ra)/450))
}

match_probs <- function(Ra, Rb) {
  pd <- p_draw(Ra, Rb)
  pw_raw <- p_win(Ra, Rb)
  
  pw <- pw_raw * (1 - pd)
  pl <- (1 - pw_raw) * (1 - pd)
  
  c(win = pw, draw = pd, loss = pl)
}

Modelling the group stage

We can now model games. But the last thing we need to model the group stage is to know who will be in it - at the moment 6 spots are still up for grabs through qualifiers. To keep things simple, we will average the Elos of the teams still competing for the spots to create a substitute Elo that we can do our calculations with.

Now we can run the model, and keep track of which teams would qualify. We haven’t introduced goal difference in our simple model, so if teams are equal on points we will randomise the order they finish in.

There are a couple functions I haven’t shown here, but the majority of this model can be seen in the code below:

 actual_sim <- function(n=10000){
 
  qualify <- data.frame(teams=teams) %>% mutate(qual2=0,qual3=0)
  for (i in 1:n) {
    print(i)
    thirds <- c()
    tpoints <- c()
    for(i in 1:12){
      group_teams <- result %>% filter(group_letter==LETTERS[i])
      group_teams <- group_teams$team_name
      
      points <- data.frame(teams=group_teams) %>% mutate(points=0)
      
      matches <- combn(group_teams, 2, simplify = FALSE)
      
      for (m in matches) {
        A <- m[1]; B <- m[2]
        p <- match_probs_team(A,B)
        a <- which(points$teams == A)
        b <- which(points$teams == B)
        
        outcome <- sample(c("A", "D", "B"), 1, prob = p)
        
        if (outcome == "A") points$points[a] <- points$points[a] + 3
        if (outcome == "B") points$points[b] <- points$points[b] + 3
        if (outcome == "D") {
          points$points[a] <- points$points[a] + 1
          points$points[b] <- points$points[b] + 1
        }
      }
      
      # rank - GD randomised
      points <- points[order(-points$points, runif(nrow(points))), ]
      
      thirds <- append(thirds, points$teams[3])
      tpoints <- append(tpoints, points$points[3])
      
      
      #print(i)
      qualify$qual2 <- qualify$qual2 + qualify$teams %in% points$teams[1:2]
      
    }
    tdf <- data.frame(team = thirds, points = tpoints)
    tdf <- tdf[order(-tdf$points, runif(nrow(tdf))), ]
    qualify$qual3 <- qualify$qual3 + qualify$teams %in% tdf$team[1:8]
  }
  qualify <- qualify %>% mutate(qual = qual3+qual2)
  qualify <- qualify %>% mutate(prob2 = round(qual2/n,2))
  qualify <- qualify %>% mutate(prob3 = round(qual3/n,2))
  qualify <- qualify %>% mutate(prob = round(qual/n,2))
  qualify <- qualify[order(-qualify$prob),]
  qualify
}

After running this loop 10,000 times, I got the following results, where P2 is the probability of finishing first or second in a group, and P3 is the probability of being in the top 8 third placed teams:

While this is useful as a predictive model of the world cup, it also shows how allowing some third place teams to advance greatly advantages some teams. The teams with the largest increase in progression probability are:

Every single one of these teams was in Pot 3 of the draw - in this way allowing third placed teams to progress helps the third best team in each group the most (as expected).

This is very much in keeping with what seems to be FIFA’s strategy of allowing more countries to have more opportunities at the world cup.

This is of course all downstream of having a round of 32 rather than a round of 16 - of course more countries will make the knockout stage.

I was 3 when the Socceroos qualified for the 2006 FIFA World Cup, so I didn’t really grasp how meaningful it was (although I did enjoy the footage of John Aloisi sprinting around waving his shirt everywhere).

The next year, Australia won the Cricket World Cup for the third consecutive time (making it 29 World Cup games in a row without a loss). Melbourne had also just hosted the Commonwealth Games - my mum took me to watch the badminton - where Australia also did very well.

I was understandably convinced that Australia was the best at every sport. I got my hands on a book about the history of the Socceroos from my mum (a very big sports fan who has *technically* played soccer for a country internationally).1

This book (written in 2005) had a chapter with the history of the Socceroos’ World Cup qualifying campaigns. However, with discussion of games against Israel, Iran (still devastating) and Uruguay, 4 year old me was under the impression that these were Australia’s World Cup finals results, not qualification results. My mum was quite confused when I was telling her about how Australia actually won the 1974 World Cup (at the time of the book’s publication the only World Cup Australia had qualified for).

With its long and drawn-out group stages, the qualification process for the World Cup eliminates teams more slowly than the actual competition. Through a series of play-offs and repechages, teams have multiple chances to qualify (even more so now in a 48 team competition).

The new World Cup format raises several issues, such as the effects of fatigue from one extra game, whether a round of 32 creates too much randomness, or if there will be consequences to the quality of the tournament by letting worse teams in.

However I’m going to do a series of posts simply exploring the new group stage format. As an Australian fan, the group stage has mostly been the part of the tournament with the most excitement anyway, and I think the old group stage format was excellent for a global competition, so I’m curious to see if the new format will be as good.

Here I’ll look at the ideal group size and the excitement of the group stage - in the next two posts I’ll consider whether 4th place teams should be allowed to progress and potential collusion or manipulation, and then take a data-driven look at progression probabilities in the new format.

I should note that this tournament analysis doesn’t actually provide any guidance for teams beyond ‘try to win or, failing that, try to draw’ - same as always.

Group Size

The 2022 World Cup had eight 4-team groups, where the top two teams progressed. The 2026 World Cup will instead have twelve 4-team groups, where the top two teams in each group and the top 8 third place teams will progress.

The old 4-team groups were very popular as teams were rarely completely eliminated after the second game, however all three games were meaningful. There was a slight design flaw that was put on full display during the Disgrace of Gijon in 1982 - the teams playing in the last game could sometimes manipulate the results so they both qualified. In this case, with West Germany playing Austria, a 1-0 win for West Germany was enough for both teams to qualify for the next round, so after this score was achieved both teams pretty much stopped trying - much to the fury of Algeria, who were eliminated as a result. But since this flaw has been ironed out with simultaneous final games, the groups have functioned well.

Groups do not strictly have to have 4 teams. With 48 teams in the new World Cup, 3 or 6 (or even 8) team groups were possibilities. FIFA even initially planned for 3-team groups. However 3-team groups are quite punishing and mean some teams will only get to play two games, and 6-team groups just take too long.

Perhaps FIFA could have expanded the world cup to 40, 45 or 50 teams and had 5-team groups - one option is to have eight 5-team groups where the top 2 from each group make the round of 16.

However a common issue with groups with an odd number of teams is that not every team can play consecutively at the same time, which creates the potential for the same issue as before (and also just isn’t as exciting for the team not playing). Thus it makes sense that FIFA went with 4-team groups.

Excitement of new groups

One of the best examples of the old group system was Group E in the 2022 World Cup. Here, each of Japan, Spain, Germany and Costa Rica could theoretically qualify or be eliminated, and as the below image shows, the positions changed rapidly during the games.

Does this excitement still exist with some third place teams qualifying? Firstly, only a top 2 finish can guarantee advancement, and so these spots are still prized.

We should consider what is needed to qualify from third. We look at the recent Euro 2024 and AFCON 2023 tournaments as examples - in both these tournaments, 4 of 6 third place teams made the next round, the same percentage as the new World Cup format (8 of 12 teams).

As can be seen, a single win was enough to get close to qualification, and a win and a draw ensured qualification in both tournaments. In this way very few teams are now completely eliminated after the second match of the group stage.

Each group of 4 teams is independent of all the others. Therefore it is easy to calculate the expected required points for progression. We will consider this both empirically and with a model.

Looking at the 3 most recent Euro, AFCON and World Cup tournaments, the third place teams have finished on the following points:

If the top 2/3 of teams qualify, then the average result needed for qualification would be to be in the top 40 of these 60 teams:

In this way a single win without too bad a goal difference should allow teams to qualify more often than not.

We can also take a more theoretical look with a crude model. If we assume that each game between team A and team B is randomly either a team A win, team B win or a draw, there are

possible group stages.

This gives the following points for the third place team in each of these 729 scenarios:

Again, the marginal 2/3 position (position 486 would be required to qualify in a world cup of this size) is a team with 3 points (either obtained through one win or three draws).

An implication of this is that three draws would almost always be enough to qualify for the next round, as a team would have 0 goal difference. I like this outcome, as it means that a team now needs to actually lose a game (or at least a penalty shootout) to be eliminated. This does however mean that now it is a lot easier for a team to win the World Cup without ever scoring a goal outside of a penalty shootout.

Another interesting feature to note is that when qualification is dependent on other groups, it is impossible to completely eliminate Disgrace of Gijon situations, as later groups will know what is needed to finish in the correct position amongst third place teams.

However this also creates more excitement for earlier teams, as they may have to wait several days after their last game to know if they have progressed or not.

Overall, FIFA are working well towards their goal of getting more eyes on the World Cup - more teams will still be mathematically in the tournament for longer, and countries will now care about games outside of their group even more. Whether this is good for the competition as a whole remains to be seen, as more teams progressing may remove some excitement from the group stage.

Next, I will consider how the chance of progressing from a group changes between the 2022 and 2026 World Cup formats, both using a simple ELO model and looking at past empirical data. I am currently ironing out the model for the current World Cup groups that were drawn recently.

In my local futsal league, the clock doesn’t stop under any circumstances. Because of this, teams with narrow leads will happily give away yellow cards for time wasting to make sure they win the game. This got me thinking about another sacrifice a player made…

Late in the second period of extra time of the 2010 FIFA World Cup Quarter Final between Uruguay and Ghana, the score was locked at 1-1. Just as the game entered time-on, Ghana had a free kick on the edge of the penalty box. If don’t know what comes next, you can watch what happened here.

After a goalline scramble, a Ghanaian header directed the ball towards goal, only for it to be saved by the outstretched hand of… Luis Suarez. Suarez had already scored 3 goals in this World Cup, however was definitely not a goalkeeper, and so was shown a red card without much argument and a penalty was awarded to Ghana.

Asamoah Gyan, who had already scored 3 goals in the World Cup including 2 penalties, stepped up to take the penalty. Gyan beat the keeper, but hit the crossbar and the ball went out of play. Suarez could be seen celebrating ecstatically on his way to the changerooms. The game immediately went to penalties.

Ironically, Gyan converted his penalty in the shootout, however Uruguay won the shootout 4-2 and progressed to the Semi Finals.

Suarez was of course suspended and did not get to play in the Semi Final - however he definitely would not have got to play in the Semi Final if Ghana won the Quarter Final.

Ghana’s coach labelled Suarez a ‘villain’ and a ‘cheat’. Uruguay’s coach said that ‘we have to go by the rules’. As the World Cup was in South Africa and Ghana failed to be the first African team to make the Semi Final of a World Cup, the New York Times reported that “A Continent’s Hopes are Swatted Away”.

It would take until 2022 for Morocco became the first African country to make the Semi Finals of a World Cup.

xG (Expected Good Deeds?)

Morally, Suarez’s decision is unclear. If there is a Spirit of Football anything like the Spirit of Cricket, it’s probably against that but I’ll leave that for others to discuss.

However in terms of winning, the decision is incredibly clear cut. Without the handball, Uruguay win 0% of the time. With the handball, Uruguay win if Gyan misses the penalty and then Uruguay win the penalty shootout. If we assume that penalties are scored 75% of the time, and that shootouts are roughly random at 50/50, this means Uruguay wins 25% * 50% = 12.5% of the time. But this number isn’t particularly important. The important part is that it’s not 0%.

And so Suarez’s decision (if it was a decision) was the right one for winning the game. It is interesting to note that the red card did not disadvantage Uruguay at all - they did not play a second of open play with only 10 men, and of course they did not get less penalties in the shootout.

When should strikers become goalkeepers?

Of course this is easy to see in the last 10 seconds of the game. I think a more interesting question is if it would have been correct for Suarez to make his save in the 85th minute of the game, or even in the first minute.

For this analysis, let’s assume that both teams are evenly matched, at a neutral venue, and in a knockout game. In this way they would both score 1.5 goals in the average 90-minute game.

We can model goals scored as a Poisson distribution. This gives the probability of a certain number of events occurring, assuming that events occur independently and the rate of events occurring is constant. In this case, this would mean that a goal is equally likely to occur at any stage of the game. This is of course not 100% accurate, but is a good model for goal probabilities in this sport.

In a Poisson distribution, the probability of getting k events when there are an average of λ events is:

In this way if there are n minutes remaining,

and so the distribution of goals would be:

We now need to consider what would happen to scoring rates when a team is down to 10 players. This can be considered more empirically: Badiella et al (2022) looked at data from the top five European leagues and unsurprisingly found that it is easier to score against 10 players and harder to score with 10 players.

To simplify our analysis, let’s assume that the team with 11 players will score 2 goals per game on average, and the team with 10 players will score 1 goal.

The Poisson distribution for the team with 11 players is therefore:

and for the team with 10 players it is:

We can set up a matrix of the probabilities of all goal combinations using the below R code, and use this to work out the chances of a comeback.

joint_poisson_matrix <- function(lambda_team, lambda_opp, max_goals = 5) {
  P <- matrix(0, nrow = max_goals + 1, ncol = max_goals + 1)
  for (i in 0:max_goals) {
    for (j in 0:max_goals) {
      P[i+1, j+1] <- dpois(i, lambda_team) * dpois(j, lambda_opp)
    }
  }
  rownames(P) <- paste0(”Team_”, 0:max_goals)
  colnames(P) <- paste0(”Opp_”, 0:max_goals)
  return(P)
}

For example, with 30 minutes to go with both teams with 11 players, the probabilitiy of score combinations looks like this:

Using this method and the above Poisson distributions, we can analyse the probability of winning in different game scenarios.

If a game goals to a penalty shootout, we will assume each side has a 50% chance of winning the shootout.

Scenarios

Firstly, let’s assume the score is 1-1 (or otherwise tied), there are n minutes to go and a player has the opportunity to stop a certain goal by giving away a red card and a penalty.

If the player does not do this, the score would be 2-1. With enough time remaining, there could be a good chance of a comeback. But with only a few minutes, it might be a better bet to concede the penalty and hope it gets saved.

As seen above, using our assumptions as soon as there are less than 23 minutes left in regular time, you will have a higher chance of winning by having a player save the ball and get a red card. Of course this is all dependent on the penalty actually being saved - if the penalty is scored, a team’s chances plummet now that they only have 10 men. You can also see above that under this model it would have even been optimal for Suarez to make his save at the start of extra time.

This 30 minutes figure is of course dependent on what each team’s probability of scoring is, and what the effect of going down a player is - further empirical work is needing to arrive at a more precise and accurate value.

Now let’s consider if the score was 1-0 in your team’s favour.

Similar to before, it is a good idea to handball if there are less than 10 minutes remaining of regular time or 24 minutes remaining of extra time.

The biggest dampener on these intentional red card strategies is that it’s quite likely that the other team will score the penalty. But goal line saves aren’t the only intentional red cards that stop goals. For example, take this tackle from Paulo Otavio for Wolfsburg against Hoffenheim.

If it wasn’t clear, Otavio has done this despite Wolfsburg being 2-1 down. It still saved them a goal on goal difference so maybe it was worth it.

To analyse this, let’s use a conservative estimate of a 10% conversion rate for a free kick on the edge of the box.

Our assumptions suggest that it is therefore almost a good idea to hack a player on the edge of the box to stop a goal (and get a red card and concede a free kick).

Under our model, it can be seen that at the start of the game going a man down is slightly worse than conceding a goal, but that otherwise it may be best to take the red card.

As before, perhaps empirical data will provide different results. It certainly seems like a bad idea to go a man down in the first few minutes of the game, however goals are often so hard to come by that perhaps this is a trade worth taking to stop a certain goal. There are also other complexities to consider, such as that edge of box tackles are often not stopping certain goals in the same way that goal line saves are.

Double Keepers?

So, if it is a good idea to make a goal line save in the last few minutes of a game, should a team just play with two goalkeepers? One of these ‘goalkeepers’ being an outfield player of course.

Quite clearly the answer is no. The lesson of the above results is that it is optimal to get a red card to stop a goal only when the goal is imminent. Suarez would not have handled the ball had it not been about to go into the net, and the odds are (as always) that within any specific 10 minute period it is not likely that there will be a goal. It is better for teams to focus on their own attacking opportunities and not confuse their own defensive setup.

I will try to report back on some more empirical data to determine more precise values for the above results. I will also see if I can find some more examples (other than Suarez and Otavio) of intentional red cards - please share if you have any.

I used R’s cricketdata library for my data - will upload code shortly.

One of the first stats mentioned when considering how good a batter is in cricket is their average. A batting average is calculated as:

On a basic level, the team with the highest averages should win more games in the long run. If a team has a mean batting average of 40, on average they would score 400 runs in an innings, whereas if a team has a mean betting average of 30, on average they would score 300 runs in an innings. This applies to test matches, not limited overs games - someone should have told Sunil Gavaskar before his marvellous 36* (174) in the 1975 World Cup.

This assumes that all batters’ scores are distributed similarly. This is of course not the case, and we will now explore some further considerations when evaluating batting averages.

It is useful to consider ‘hazard functions’, which represent the risk of getting out at any particular score, once that score has been achieved, i.e. :

Batting average vs mean

If a player had equal chance of getting out at any point in an innings, then the batting average would be equal to the runs per wicket. We can consider this to be a constant hazard function, in that the risk of getting out at any time is the same. This is the same as a constant probability of ‘success’ of a wicket, and so we would expect innings length (and thereby runs) to be distributed roughly geometrically (a geometric distribution is the probability of the first success when all trials have the same chance of success).

Of course, every cricketer believes that they bat better once they ‘get their eye in’ - in this way they are more likely to get out at the start of an innings. As a quick example, Travis Head’s distribution does not fit the geometric distribution.

Kimber and Hansford (1993) demonstrated that the hazard function is indeed greatest at the start of an innings. It decreases to a roughly constant amount as batters get their eye in, then increases slightly at very high scores as they get tired.

In this way the batting average could be improved to better capture when we expect not out innings would have ended. Kimber and Hansford developed their own ‘Alternative Batting Average’ using a product-limit estimator. This was quite similar to the classical batting average for most players, for example Bradman had an Alternative Batting Average of 98.98 instead of 99.94.

Is it fair for a number 11 who scores 5 20 times in a row and only gets out once to have a batting average of 100? In all likelihood if this happened they would not find themselves a number 11 for very long. Given that the hazard function is highest when a player starts their innings, this would actually be quite impressive for the batter to continue to survive to reach 5 runs every time.

Modelling individual distributions

However, aside from increased risk of getting out early in an innings, the analysis does suggest that eventually the hazard function becomes roughly uniform. After this point, the batter’s scores could potentially be modelled as a geometric function.

In this way, perhaps there are only three key features of a batter’s hazard function:

• Starting Risk: How likely they are to get out at the start of their innings

• Settling Time: How long it takes for them to become settled at the crease

• Continual Risk: How likely they are to get out once they have settled

We can model this as an exponential decay function,

where:

• h0 = starting risk,

• h_inf = continual risk, and

• tau is set to get an appropriate settling time

The probability of surviving to a score is the product of the hazard functions for every preceding score. We can then calculating a batting average by summing the probability of surviving to each score, as shown below.

This model works ok - for example with Ricky Ponting (51.85 average), setting h0 = 0.07, h_inf = 0.015, and tau = 4 the estimated hazard function roughly fits his empirical hazard function, and returns an expected batting average of 51.47. It doesn’t perfectly fit his smoothed empirical hazard function - more work is required to get these to fit better, and perhaps this is a limitation of this parameterisation.

Spinners (the other kind)

It is possible for one player to have a better batting average than a different player, but for the other player to be more likely to score more than them.

For example, take the following two spinners:

The average value of the first spinner is 5. The average value of the second spinner is 4.67. And yet if you were to spin the spinners, the second spinner would win 2/3 of the time. This is a contrived example, however shows that one option can have a higher average while another option has a higher probability of superiority.

Worse average, but more likely to score more

One (marginal) example of this occurring is with AB De Villiers (50.66 average) and Kane Williamson (54.88 average). If you take a random innings from each of them, AB scores higher 49.6% of the time, whereas Kane scores higher 49.2% of the time.

Similarly, Travis Head (42.34 average) scores more than Adam Voges (61.87 average) 50.9% of the time, whereas Voges scores higher 47.7% of the time. This happens with quite a few players and Voges - likely because Voges’s average is so high because of some very high not out scores.

However, Harry Brook (56.31 average) scores more than Head 54.3% of the time, whereas Head scores higher 44.4% of the time.

So even trying to find an alternative position, Harry Brook’s stats are still better than Travis Head’s. This is not to say expected or average batting performance is the only metric - it could fail to capture a matchwinner like limited-overs Glenn Maxwell. I think many teams would be happy to wear a few ducks from Maxwell in exchange for his 201* against Afghanistan.

Despite all these stats, there’s no real ‘Moneyball’ advice here for playing better cricket - as always, score more runs and don’t get out . However there may be some team dynamics where you want a certain amount of matchwinners and a certain amount of solid scorers, which could be something to explore later. Across 11 players, you probably still want to be the team with the better average overall.

There’s plenty more exploration to be done here - in future I may look at how averages work within team innings, including if there are certain players that just feast when their team is already doing well. It could also be interesting to explore how batting averages work within partnerships.

I’ve been threatening to write more for quite a while and now that I’ve finally finished uni I thought I’d start here…

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All Victorian law students interested in getting a grad role at a major commercial law firm have to do an internship (called a clerkship) at that firm the previous year. These clerkships are handed out through an arduous process of questions, video interviews and in-person interviews. After spending far too long preparing and with all sorts of nerves, you can imagine my shock when the first question I was asked in my first in-person interview was

If the AFL finals were going to have 10 teams, how would you structure it?

Melbourne’s great.

After some brief reflection, I gave some options but said the most likely would be a 7v10 and 8v9 wildcard round (with reference to the NBA) that would feed into elimination finals. For a variety of reasons, this was always the proposal for a 10 team finals series, and so of course was the model used by the AFL when the 10 team finals was recently announced.

The clerkship process raises all sorts of interesting questions around selection procedures and randomness, and so it does in a way make sense that our conversation turned to sports and playoff systems.

A Brief History

In the VFL’s early years, a variety of finals series were trialed for their 8 teams. These included:

• A top 4 round-robin.

  • Problem: No guarantee of a Grand Final.

• Two groups of 4 teams (with 8 clubs this meant that all teams made the ‘Major Round’), then a playoff between group winners, then potentially a grand final if the Minor Premiers won enough games in the ‘Major Round’ but didn’t win the Major Round.

  • Problem: Melbourne won the 1900 Premiership after finishing 6th out of 8.

  • There was an additional structural problem in that 2nd place got to play weaker teams (4,6,8) than 1st place (3,5,7).

• Two semi-finals, then a final, then potentially a grand final if the Minor Premiers didn’t win the final.

  • Problem 1: Massively advantages the Minor Premiers.

  • Problem 2: Unclear if final will be the last game of the season.

The VFL was clearly looking to provide a structural advantage to the Minor Premiers.

In 1931, the VFL shifted to the McIntyre system, variants of which are still used to this day. The key features of this system are:

• Double chances for high-finishing teams

• Weeks off for successful high-finishing teams

• Dynamic fixturing based on rankings of winners and losers

The 4 team McIntyre system is commonly used in many sports, including many local basketball leagues:

• Match 1 (QF): 1v2

• Match 2 (EF): 3v4

• Match 3 (SF/PF): Loser of Match 1 vs Winner of Match 2

• Match 4 (GF) Winner of Match 1 vs Winner of Match 3

The original 8 team version has since been replaced with the current AFL finals system as it has more rigid fixturing that makes it easier to schedule and for teams to know who they are playing next week.

The double-chance is distinctively Australian (the NRL now uses it too) - it is not used in any major American sports league (where direct elimination is preferred).

Purpose

So after all this, what is the point of a playoff system? A better question is that of the point of a sporting competition. Beyond fun and community and other similar benefits, a competition is designed to produce a winner. A winner should be the best team, but should also be able to beat the best teams (perhaps these are the same?).

From this we arrive at a tension between league systems and playoff systems. Leagues have more data points, whereas playoffs introduce uncertainty. Uncertainty is probably a net good for a sport (excitement, underdogs) however can create issues.

This is not to say leagues cannot have uncertainty - Agueroooooo!!! - however they will necessarily have less than a competition with a single grand final.

We can see this tension everywhere. A couple examples:

• MLB - 162 regular season games (winning even 60% of these is impressive), followed by a playoff series that puts 1-seeds in a first-to-3 series to advance

  • Not a massive reward for performing well throughout the whole season

• NBA - 82 regular season games, followed by a 16-team direct elimination bracket of best-of-7 series

  • Top teams have no structural advantage over lower teams, but do get a theoretically easier draw

Overall, we want a playoff winner to at least be close to the best team, and to have had to prove themselves. We also want competitive, exciting finals.

Part of the problem is that we want the league stage to still mean something. One good way to measure this is to look at the percentage of teams that make finals. We can see how this has evolved over the history of the VFL/AFL below:

The 1994 AFL season is the only time in the past 100 years that over half of the teams played finals. This will now happen in 2026. Perhaps this new system is more appropriate given the upcoming expansion of the league with Tasmania and potentially a 20th team.

The home and away season, aside from the fun of all the games, has two main purposes - align the teams for finals, and decide who gets the top draft picks the next year. In this way we want meaningful differences in Premiership probabilities for each position in finals.

We can approximate the probability each position has of making the Grand Final with some crude assumptions:

• Each team has a 50% chance of winning each game, or

• The home team wins 70% of the time, or

• The higher ranked team wins 70% of the time

Because the new Top 10 system is just a play-in, all it does in these models is redistributes the chances of teams 7-8 to teams 7-10. In this way there is now a key incentive to finish 5/6 instead of 7/8 beyond the home final it currently gives.

Some more examples

The AFL’s wildcard system most closely resembles the NBA, however the NFL and MLB have also introduced wildcards. These are moreso genuine ‘wildcards’ than in the AFL and NBA - instead of just being the next best team, these are the teams that performed best without winning their division. However, the NFL’s system highlights the potential creep of play-in rounds. The NFL had a 12-team playoff where both 1 & 2 seeds (there are four because there are two conferences) got a bye past the Wild Card round. This is now a 14-team playoff, where only the 1 seeds get byes. The Wild Card round has even been renamed to Super Wild Card round.

Athletics normally has direct elimination ‘playoffs’ - in which more than one runner can advance, but with very few double chances (there are sometimes repechages). In events such as sprinting, competitors cannot normally directly affect each other’s performances, and so even with natural variation you will typically still end up with the best runners making the final. Of course in competitions where competitors can interfere with each other this can lead to chaos (see Steven Bradbury). While AFL teams can’t trip each other over, this does highlight the benefits of having a double-chance reward for the top teams.

My View

The 8 team system has been excellent and creates an exciting finals system. As a Hawthorn supporter, I loved being able to watch our underdog run from 8th this year.

The new system will make underdog runs harder. However, it was already quite rare for a 7/8 seed to make the grand final. In this way while it may be harder for the 8th place team to make the prelim, this system probably will not significantly affect who the Premier actually is. The finals situation of teams 1-6 is unchanged, and the Premier will almost always come from these top 6 teams.

Outside of watching the games themselves, fans stay engaged because of the excitement and hope of the tournament. The finals race in 2025 was exciting, but you can also end up with years like 2016 where the Top 8 does not change from round 6. One of the most fun weeks in footy that I can remember was the lead up to the 2019 Grand Final, with Big Big Sound memes galore. It was great to watch everyone get behind the Giants (no one but Richmond fans liked the Tigers) even though we all kind of knew that Richmond was going to win. And despite Richmond winning by 89 points, the hope of the week was still lots of fun.

And in essence the change from 8 to 10 teams extends the hope of multiple teams. This firstly occurs because teams 9 and 10 get to play a final, but also because the team in 13th in round 22 may still be in with a shot. It felt like Hawthorn always managed to string a few wins together late in the season from 2019-2022, and so they were still mathematically involved - even though you knew they weren’t going to make finals, still being in control of their own destiny was great.

Now at some point teams need to be eliminated, but the play in system makes it so hard to win the Premiership (you need 5 wins in a row) that I am ok with teams not being fully eliminated but needing to run a very impressive gauntlet.

There are of course other factors I haven’t discussed like teams getting a week off, tanking and alternative finals systems - I may get around to writing something up about these.