Their popularity can’t be questioned, but are ambrose events as fair as they could be?
Ambrose days are as common at most Australian golf clubs as duffed chip shots and three-putts. Whether two, three or four-player, we’ve all taken part in them and revelled in the team atmosphere and confidence-infusing format that wipes away the carnage of bad shots and instead focuses solely on the good ones. The game often referred to as a “scramble” is a great entry to golf for novice players, corporate events and once-a-year golfers.
Ever since Richard Ambrose at South Australia’s Victor Harbor Golf Club first coined the format, we’ve been using the same method for assigning ambrose team handicaps: one-eighth the combined handicap for a four-player team, one-sixth for three players and one-quarter for a duo. It’s simple, easy to calculate and familiar to most golfers. But it’s also flawed and unfair, according to one Adelaide golfer who has put in the hours to prove it.
Sensing the system’s unfairness, Denis Toohey applied his chemical engineering and mathematical background. When it comes to ambrose, he is golf’s ‘beautiful mind’ – our sport’s veritable John Nash. Toohey has spent countless hours experimenting with weightings for players in handicap order, charting results data, thinking through the logic and then using dozens of actual events to verify his findings: the way ambrose teams have been handicapped is flat-out wrong [see panel below].
His 2e Method weightings for a four-player ambrose total 75 (the sum of 40, 20, 10 and 5). Any individual competition allows for 100 percent of a golfer’s handicap. A two-player event conducted in the ambrose format necessitates a reduced percentage simply because of the multiple shot options available to the pair. Moving to three-player events, that percentage needs to be even lower due to the greater number of shot selection options, then lower again for four players. Toohey insists that’s a fundamental flaw of the traditional method, which uses half the team average regardless of team size. All teams should have proportionally increased handicaps, meaning higher handicapped teams gain more from this system than lower ones – which is why the latter win ambrose events more often.
Numbers and data can paint a pretty picture for any learned person trying to prove a statistical point, but sometimes you need to also see the evidence at work. And we’ve all seen it happen: an ambrose event where the handicaps of the players in some groups are more choreographed than your average ballet. Play in enough ambrose events and you come to know which combinations of handicaps can exploit the system. Usually, it’s one or two very low markers gelling with a couple of high handicappers. (Unlike foursomes, for example, pairing ‘like’ golfers is a statistical no-no in ambrose.) Yes, you’ve still got to play well, and putt well in particular, but the current method does provide an advantage to certain team compositions.
In some respects, changes have already been made to thwart teams who ‘stack’ their side. The Volkswagen Scramble is the largest and longest-running ambrose tournament in the country and one that attracts thousands of entrants nationally every year. When it was still known as the Holden Scramble, the event first began stipulating a minimum number of each player’s tee shots be used (now also commonplace at most club ambrose events) while mandating a limit to the total aggregate of handicaps for each team and requiring a certain composition of handicaps. Later, the elimination format was introduced, which prohibited the player whose shot had been selected from hitting the next shot until the green was reached. Both alterations added a distinct strategy to the format but neither did nearly enough to rectify the numerical imbalance, and Toohey says his results prove it.
His crusade began in 2017 when Toohey analysed club ambrose events. To verify his four-player method on a large sample, he obtained detailed results of 40 Holden Scramble events, comparing his method with the traditional (and also then USGA) versions. There might not be a better data sample available for such a number-crunch.
“Little bit by little bit, pieces fell into place,” Toohey says. “Once they did, the logic became undeniable. But it’s a case of arriving at that, questioning it and thinking it all through.”
Toohey’s findings led him to compile a 13,000-word submission replete with graphs, examples and recommendations. Titled “Much Fairer Handicapping Methods for Scramble Events” he sent it to the powers-that-be at the USGA in August 2018 for their consideration within the forthcoming World Handicap System (WHS). A copy was also provided to the Golf Australia organisation (GA). In February 2020, the WHS released handicapping allowances for ambrose events. These were a disappointment to Toohey – in more ways than one.
“After reading my submission, the WHS lifted their four-player model from 50 percent up to 70 percent, so all handicaps would increase by 40 percent,” Toohey says. “That was a distinct improvement but there is still a bias towards low-handicapped teams.
“But unbelievably, they ignored my ‘always unfair’ findings of the USGA two-player version and it remains at 50 percent, which doesn’t make sense. It would have to be a number between 70 and 100. In mine, it’s 85 after meticulous optimisation. That was some of my chemical engineering expertise at play – you do experiments and gradually move in on where the ideal point is.”
The WHS revision also lacks any method for three-player teams. How many times has a four-person ambrose taken place at your club and a team or two has lost a player so three golfers battle on instead. Now there is no formula to apply. In contrast, Toohey devised a three-player method by logically interpolating between his four and two-player versions, so his system is internally coherent.
Toohey forwarded a response to GA in the hope of forestalling implementation of what he saw as a flawed, incomplete and unfair WHS system. GA’s reply focused on maintaining global uniformity rather than having a different Australian version. While a fan of uniformity, Toohey did not want to see the promised ‘fairness’ of the WHS being compromised so blatantly. He was also concerned that those who proposed the system would not now change it because that would be an admission of their mistake in the first place.
Passing on a message
Golf Australia agreed to forward Toohey’s findings to the R&A and USGA. Yet the governing body here sees no need to reinvent the ambrose wheel. In early March, Simon Magdulski, GA’s long-time senior manager of play management and regulations, e-mailed Toohey to say their stance matched that held by the larger governing bodies.
Within his reply was one particularly eye-catching statement: “Golf Australia has received almost no feedback over the past decade to suggest that Australian clubs are concerned with scramble/ambrose handicapping regulations. Should Australia be moving in a different direction to the rest of the world on scramble handicapping? Golf Australia has not seen any evidence to suggest there is any material difference between the way scramble events are held in this country compared to what happens overseas, as a result we do not see any argument for heading counter to the direction of increased global consistency.”
Using his spare time during the COVID-19 restrictions, Toohey pondered the issue further and was able to enunciate more clearly exactly why the WHS four-player version was flawed and unfair. While it did improve fairness across the range of handicaps by increasing the sum of player weightings from 50 percent to 70 percent, the WHS weightings themselves were too close together (5 percent constant separation), leading to unfairness among teams within the same handicap.
The 2e method weightings are quite different. “It’s logarithmic,” Toohey says. “It’s a halving as you go. And that’s where the 2e comes from: each factor is 10 times 2 to the power of ‘e’ where ‘e’ ranges from 2 down to -1. But there is another little twist… ‘2e’ also happens to be my family’s shorthand version of our surname.
“I could see where the problem in the WHS formula lay. But there was no direct mathematical way to prove it, as is the case with gradients for fairness across the range of handicaps. I then hit upon a novel way of demonstrating this using some clever logic and the common nous of other golfers.”
Toohey turned back to the Holden/Volkswagen Scramble results and its 1,006 teams across 40 years. He calculated all team handicaps by both the WHS and 2e methods and sorted them in handicap order. For each method, he selected five pairs of teams with the same handicap and the greatest difference by the other. For example, a team with daily handicaps of plus-2, 16, 21 and 36 has the same WHS handicap (9.5) as a team with 10, 15, 15 and 17, but their 2e handicaps are 6.3 and 9.4, respectively – a massive 3.1 strokes difference.
“Of the 40 Holden/Volkswagen Scramble events, 21 were decided by one stroke or less, so even one shot can make a big difference,” Toohey says. “And it’s not only winning on the day, it’s whether you go off to the regional final and eventually to Royal Pines. For an event like the Volkswagen Scramble to go to all the trouble that it does – there’s a lot of organising, expense and sponsorship involved – and be operating with a now proven unfair method doesn’t make sense.”
These 10 pairs were shuffled into a random order and compiled into a survey sent out to fellow golfers with the instruction to pick which team from each pair would beat the other if there was no handicap difference between them. From 26 responses, 75 percent of replies on average picked all five teams with same WHS handicap but lower 2e handicap, affirming the 2e method handicap difference. Correspondingly, a clear majority in all five cases of same 2e handicap picked the team with the higher WHS handicap. In other words, that team would be expected to win even without a handicap advantage. Toohey calls that a 10-out-of-10 win for 2e over WHS for fairness.
His detailed ‘Review of the WHS Handicapping Recommendations for Scramble Events’ was 4,500 words longer than his original submission and lambasted the recommendations. In June this year, Toohey sent his review to GA with the invitation for the body to validate his results by conducting his survey and to forward his review to the bodies running the WHS.
Chemical engineers have a proud reputation of being practical rather than theoretical, sifting through real data and deriving equations to bring order out of chaos. Toohey says his conclusion that the WHS four-player method is unfair is not reliant on complex mathematical computations. Instead it comes from using actual team handicaps and events, logical reasoning and basic golf nous.
“I am not seeking adoption of an Australian version, but I hoped GA could see the exposed flaws of the WHS recommendations and advocate for their change. I’m fighting for fairness and I was hoping that Golf Australia would too,” Toohey insists. “The WHS was to come up with the fairest methods for all formats of golf. In a way, they’re breaking their own aim by being blinkered. It seems they did not want to adopt a system devised by some mug golfer from Adelaide, so they tried their own quick fix on the unfair USGA four-player version and then did nothing about the other two. I’m totally satisfied the integrity of my system is watertight, but the WHS recommendations have holes right through them. We now have golf’s equivalent of ‘The Emperor’s New Clothes’.”
Scrambled signals
Amid the stalemate, one question worth asking is: where on the totem pole of importance does fairer ambrose handicapping sit? Surely there are more pressing issues at hand for the game. Plus, it is a fun, novelty event in most instances – and part of the enjoyment can be in trying to ‘game the system’ with a shrewd bit of tactical recruitment of golfers with certain handicaps within the parameters given.
“When you look at all of the data, it’s impossible to come up with a robust solution that accounts for all of the different ways that ambrose is played and the different mixes of teams,” Magdulski says today. “Is it the biggest issue on our radar? No. Is it an important issue? It would become important if we were getting a theme emerging from clubs that they want us to do something, but at the moment that’s not there.
“We’ve moved to a World Handicap System, and a part of that is taking the view that: rather than having a whole lot of different solutions for the one problem, that there be one uniform message to clubs, to golfers, on what the best approach is. In us looking at this, it’s as much a case of saying, ‘We don’t really want to be coming up with a specific Australian direction on this.’ If there’s going to be a discussion, it should be at a global level. And the WHS has a recommendation on ambrose/scramble and we’re loath to be looking at it in isolation from them.
“At a global level, the view would be: it’s very difficult to come up with a perfectly equitable solution that accounts for all the different ways that ambroses are played. Handicapping is complex enough when you’re just talking about individual play; when you start getting teams of four in an environment where players aren’t playing their own ball… it becomes very challenging.”
Challenging indeed. And placing restrictions on team composition, for example, isn’t a one-size-fits-all remedy. As Magdulski says, requiring every ambrose side to include at least one single-figure handicapper, for instance, can be problematic at small country golf clubs where there might not be many single-figure markers among the membership.
“Is Denis right in saying that some of these existing pieces of formal guidance aren’t bulletproof? I’d totally agree with that and the R&A and USGA would agree with that. But is Denis’s way forward also going to tick all of the boxes across the landscape? Almost certainly not, either.”
The PGA of Australia, which runs the Volkswagen Scramble, began the current season in August by using the old system, having not yet taken on board the WHS version. “The 2020/2021 season of the Volkswagen Scramble has launched and will use its existing ambrose handicapping format as this has proven to be fair and successful for many years,” the PGA said in a statement.
That’s a stance Toohey agrees with… for now. He says next year would make far more sense for the event to make a change – by adopting his 2e method.
Toohey remains adamant his system is much fairer than both the previous version and the revised WHS offering. And while the term “2e” has it roots in mathematics, it does provide a nice synergy with its creator’s surname. Much like Ken McIntyre gave our football codes the McIntyre System for finals series and former Australian Golf Digest editor Geoff “Pinky” Prenter gave cricket competitions the “Pinky (bonus) Point”, Adelaide man Denis Toohey’s 2e Method could position him alongside Dr Frank Stableford as format visionaries in golf.
Ambrose has its origins in South Australia. It seems appropriate for any overhaul to emanate from the same place.
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THE 2e METHOD VERSUS OTHERS
The 2e method recommendations are:
4 players: 40%A + 20%B + 10%C + 5%D (total of 75%)
3 players: 45%A + 20%B + 15%C (total of 80%)
2 players: 50%A + 35%B (total of 85%)
where A to D represent low to high handicaps in order.
The traditional handicapping methods in Australia all have a total of 50%. The USGA recommendations were:
4 players: 20%A + 15%B + 10%C + 5%D (total of 50%) – now superseded by WHS (total of 70%)
2 players: 35%A + 15%B (total of 50%) – retained in WHS
As well as being simple in its formulation, any method for ambrose handicapping must be:
- Fair across the range of team handicaps, and also
- Fair for the range of possible combinations within a particular team handicap.
For the first condition, there is a clear measure. There should be no bias favouring either a high or low handicapped golfer (or team). Mathematically, a chart of results versus handicaps for any given event would be expected to have a trendline gradient (‘line of best fit’) of 0% to be ‘perfectly fair’. In practice, some small variation either way would still be seen as ‘very fair’. This gradient is also a measure of by how much handicaps would need to be adjusted to be ‘perfectly fair’.
For ambrose events, which have higher volatility, coloured bands of fairness were developed from ‘very fair’ to ‘very unfair’ according to the trendline gradient from each event. Tables of results show the relative fairness of the various methods for four and two-player events:
| Table 1: ‘Fairness’ from 40 Holden Scramble events | ||||
| 4-player Method: | GA | USGA | 2e | |
| Very Fair | +/-10% | 5 | 5 | 15 |
| Fair | +/- 11-20% | 7 | 3 | 11 |
| In Between | +/- 21-40% | 9 | 10 | 11 |
| Unfair | +/- 41-60% | 9 | 12 | 2 |
| Very Unfair | +/- 61+% | 10 | 10 | 1 |
| Average Gradient | +36% | +43% | -9% | |
GA was fair only 30% of the time and unfair 48%.
USGA was fair only 20% of the time and unfair 55%.
2e was fair 65% of the time and unfair only 8%.
The results for two-player events were even more emphatic.
| Table 2: ‘Fairness’ from 20 events of two-player ambrose | ||||
| 2-player Method: | GA | USGA | 2e | |
| Very Fair | +/-10% | 0 | 0 | 13 |
| Fair | +/- 11-20% | 0 | 0 | 7 |
| In Between | +/- 21-40% | 2 | 0 | 0 |
| Unfair | +/- 41-60% | 6 | 4 | 0 |
| Very Unfair | +/- 61+% | 12 | 16 | 0 |
| Average Gradient | 64% | 72% | 0% | |
GA was never fair and unfair 90%. USGA was always unfair. 2e was always fair. With these results at hand, WHS inexplicably chose USGA as its recommendation.
The WHS four-player recommendation, whether intended or not, is a hybrid of USGA + 40%GA, with a 5% weighting increase for all players. It reduced the average gradient from 43% to 3% (‘very fair’). But fairness across the handicaps is a necessary but not a sufficient condition; there must also be fairness within any given team handicap.
For the second condition, there is no direct measure, such as gradient. Let’s compare the handicaps of two teams by the different methods:
% Weighting (A to D) Team 1 (0, 0, 40, 40) Team 2 (20, 20, 20, 20) Difference
GA All 12.5 10 10 0
USGA 20/15/10/5 6 10 4
WHS 25/20/15/10 10 14 4
2e 40/20/10/5 6 15 9
A fatal flaw of GA is that there is no handicap recognition for the variations in team composition for a given team total. Few would argue that, under it, Team 1 would thrash Team 2. Already proven unfair under the first condition, USGA cannot exceed GA for Team 2 while WHS does not reduce below GA for Team 1. However, 2e can range from the low of USGA and to (just above) the high of WHS. The handicap difference of 9 is large, but equates to one stroke per two holes – and that seems about right in this example.
Furthermore, a survey was devised based on clever logic and the common golf nous of others to show how unfair WHS can be within a particular handicap, as described in the accompanying story.
– Denis Toohey
