The Hardy Tour

My last post investigated the merits of consistent and erratic playing styles using the simplified “Hardy Model” of the game of golf. It turns out I had so much fun diving into this model that the insights I found could not be contained in a single post! The major takeaway from the last post was that an asymmetry between two strategies can be responsible for favoring one over the other. The Hardy model (and my extended “smart” version) presented two asymmetries: consistent players have an edge when erratic players aren’t rewarded for good shots as much as they are punished for bad shots, and erratic players have an advantage in match play because they occasionally recover from bad shots.

The analysis in the last post considered averages over a single hole only. In reality tournaments are played against number of competitors for a finite number of rounds. Even if two players have the same scoring average, more value (both financial and historical) is placed on the player with more wins than the player that is consistent but never finishes on top.

This leads to an important question: How does consistent vs erratic play affect the chances of winning? Additionally, and importantly for PGA tour players, which strategy is more lucrative in the long run: lots of average shots or lots of good and bad ones?

I will use the Hardy model to attempt to answer these questions. It turns out the answer is amazingly straightforward.

A Regular Hardy Tournament

If you haven’t yet read the previous article, I recommend you do so now, as the rest of this post will make more sense with the details of the Hardy model fresh in your mind. To review, the Hardy model assumes that there are only three types of shots: excellent, normal, and bad. The equivalent of four normal shots are required to finish a hole, but excellent shots count as two normal shots and bad shots don’t count toward finishing the hole at all (they basically are like missing the ball entirely). Each player has a probability $p$ of hitting either an excellent shot or a bad one (so both are equally likely to occur). The last post described that the scoring average for a standard Hardy player on one hole is nearly $4+p$. That is, the higher the $p$ value, the more above par the average score will be.

An interesting thing happens if we imagine a tournament between a bunch of players with different $p$ values. As you might expect, if I pick a given low $p$ player and compare him to a high $p$ player, the low $p$ player will likely finish better in the tournament, since his average score is lower. However, if I were to predict whether the winner would be a low $p$ or high $p$ player, I would find it is more likely a high $p$ player will win. This is because, while I don’t know who it will be, it is likely that one of the high $p$ players will get hot. This poses the question of whether winning a tournament requires more risky play that could also lead to a poor finish if it goes wrong.

Thus if success means a consistent finish but few wins, the Hardy Model says conservative is the way to go. If however success is judged only by wins, then the erratic strategy is best.

Smart Hardy Tournaments

As interesting as this finding is, it is not particularly surprising. We know that the erratic player can swing up and down, but that the consistent player has a distinct advantage in the standard Hardy Model: she will not miss the “gimmie” putts the erratic player occasionally misses. What if we remove this advantage by using the Smart Hardy Model, where a player will revert to a normal shot when she knows it is all that is needed to finish the hole? In the last post we found that with this advantage removed, every player’s scoring average is exactly 4.0, regardless of the $p$ value. Now the difference between players is not in their scoring average but only in the variance of their scores.

This variance is depicted in the figure below, which was generated by simulating 1 million rounds of players with various $p$ values. As you can see, the median score plotted in black stays flat at 72, but the 25th and 75th percentile are also plotted, showing that the spread of potential scores grows with $p$. Another way to think of it: for a given $p$ value, half of the scores will be between the red and green lines, while half will be outside.

Where would you prefer to be on this chart? All the players average 72, but on the left side the scores will mostly be around even par, while on the right side the scores will fluctuate between the mid 60s and the high 70s. Your choice may depend on circumstance. If you are Johnny Miller before the final round of the 1973 US Open, you probably want to be on the right side of the graph, while if you are Tiger Woods at the same point in the 2000 championship, the consistent far left side is ideal. But supposing you had to base your entire tournament (or your entire career) on one of these $p$ values, which one would you choose?

There are three ways you might go about picking the best strategy: One is to attempt to maximize your wins, regardless of where your non-win finishes place you (the “second place is first loser” approach). Another is to pick the strategy that maximizes your earnings regardless of your number of wins (“I’m in it for the money”). A third strategy is to maximize the number of cuts made, giving yourself a consistent income and the most chances to play the weekend (“I just want to play more golf”).

We can use the Smart Hardy Model to find the optimal $p$ value for each of these measures of success, but first let us consider how PGA Tour events are setup to get a realistic picture of what can happen in a tournament.

A Typical PGA Tour Event

To avoid needless complications, let’s ignore special events like the WGC and assume every tournament is setup like a standard PGA Tour event. It turns out that even a standard event has more rules and technicalities than one might expect.

A standard tournament consists of 4 rounds of stroke play with a cut after the first two. Typically the field is initially 156 players and is cut to the top 70 and ties. Here is where it gets interesting. If more than 78 players would make the cut, the Tour has an additional rule to avoid making the weekend field too large. The Tour looks at how many players would make the cut if the line were one stroke lower. Whichever cut line puts the total number of players making the cut closer to 70 is chosen. If the lower number is chosen, the players who were in the top 70 and ties but still cut will still be paid despite not playing the weekend. If there are more than 78 players playing on Saturday, a 54 hole cut occurs using the same criteria as the Friday cut.

At the conclusion of the tournament, the payouts are awarded as a decreasing fraction of the total purse, with the winner getting 18% and the 70th place finisher receiving 0.2% (see chart). A tie in any position means the prize money is split among the positions covered (for instance, a three-way tie for 5th splits the 5th, 6th, and 7th place money equally). Of course if there is a tie for first, a hole-by-hole playoff is performed to determine the winner. If there are more than 70 places, the Tour pays each position beyond 70 with $100 less than the preceding position.

I have incorporated all of these rules into a model simulating a PGA Tour event in which every entrant is a Smart Hardy player, though I did not include the possibility of a 54 hole cut nor the $100 decreases in payments, as these complexities do not contribute significantly to the overall results.¹

The computational experiment

To investigate which type of player had an edge on Tour, I set up a tournament in which 156 Smart Hardy players were assigned random $p$ values ranging from 0 to 0.1 (this upper bound was chosen to keep the scores realistic – otherwise every round had scores in the 50s and scores in the 90s). Every player’s shot was computed across four rounds, following the Tour rules for cuts, playoffs, and payouts. This was all done for a million simulated tournaments. The number of wins, earnings, and cuts made were computed and averaged for each $p$ value.

Before looking at the results, it is important to recognize the assumptions and limitations of this model. Most importantly, this model assumes only three types of shot quality, with no distinction between shot type from tee to green. In reality there is a much wider spectrum of shot quality and the consequences of a bad shot vary across shot type. Additionally, this analysis assumes all players are at the same level, when in reality there is probably a range in what an “average” shot is worth across Tour players. This could affect the strategy of different players, as a poor ones may need to get really hot to win, while good players may only need their average play and choose consistency. Despite these two assumptions, I believe that this model still provides insight into how shot variability affects Tour performance. The simplicity of the shot types allows for an easy implementation and the fact that all players have the same scoring average allows us to isolate variability as the sole parameter affecting variations in performance.

Results

We begin with how consistency affects wins. The figure above plots the fraction of wins vs $p$ value. These results should not be too surprising – it makes sense that players with a higher $p$ value are more likely to get on a hot streak and win (they are also more likely to get on a cold streak and finish dead last, but if all we are counting is wins, this doesn’t matter). An interesting note is that the fraction of wins for any player is low, with even the most erratic player types winning at a rate of 2.5%. Another point is that the rate of change in wining also increases as $p$ increases (that is, increasing $p$ from 0 to 0.05 is less significant than going from 0.05 to 0.1). Finally, note that there are nearly no winners among players with $p$ below about 0.02.

It turns out the results for earnings are nearly the same. The figure above shows what one would expect to earn, in millions of dollars, over a typical PGA Tour season (which pays out around $342 Million each year), for various $p$ values. It seems that even though the erratic player is more likely than anyone else to finish dead last, the high chance of winning and the large fraction of the purse earned for doing so more than make up for these cold streaks. These results arise from another asymmetry in the consequences of good and bad scoring: finishing last doesn’t pay any less than barely missing the cut, while winning pays a lot more than barely making the cut.

Notice that the earnings plot is much more linear (straighter) than the win fraction plot. This tells us that generally becoming more erratic will increase your earnings at the same rate no matter what, while your chance of winning goes up faster the more erratic your play is.

The last comparison is the one that surprised me the most. I was expecting that although erratic players were more likely to win, consistent players would make more cuts. The plot above showing the fraction of cuts made vs $p$ has the most unique curve we have seen so far. Extremely consistent players have very little chance of making the cut – everyone else makes cuts at roughly the same rate, with a slight increase for larger $p$.

Why is it that extremely consistent players aren’t making cuts? Because the very consistent players will nearly always be around 78th place after 2 rounds (half of the 156 players better, half worse). Since the Tour cut is at top 70 and ties, a player must usually play slightly above average to make the cut. These score are relatively common for players with $p$ values above about 0.01, but below this it is just too unlikely for a consistent player to put together a good enough performance. This is also why these players basically never win – they almost never even make the cut!

In summary, the findings show that regardless of whether the goal is maximizing wins, earnings, or cuts made, the more erratic strategies are the most successful at achieving the results.

Summary

In general we tend to think that players who win more tournaments are better than those who don’t. This analysis shows that reality is a bit more complicated. Even in a world where we know every player has the same average results, those with more variation in their scores make more cuts, win more tournaments, and earn more money.

To highlight this point, suppose Sam is an erratic Smart Hardy player, while Ben is a consistent Smart Hardy player. Because Ben and Sam have the same scoring average, we know that on any given stroke play round Sam is just as likely to beat Ben as he is to lose to him (moreover, if they kept a running total across a lifetime of rounds, their scores would be nearly the same). However, because Ben is so consistently average, he rarely plays well enough to make the cut, and when he does, he stays near the bottom of the leader board and takes home a modest check. Sam sometimes has terrible rounds, but these rounds don’t hurt him any more than Ben’s average rounds. Meanwhile, Sam’s good rounds nearly always lead to him making the cut – guaranteeing him a paycheck and giving him a good chance of winning. Sam goes on to be considered one of the greatest players on Tour, while Ben’s earnings are not enough to keep his Tour card. This all despite the fact that Ben finishes ahead of same in half of the tournaments they play.

This analysis raises the question of whether we judge greatness fairly. Generally greatness is judged by number of wins, and it is assumed that a player who wins more is better (or at least more clutch) than a player who never wins. It may be true that in the real world some people are psychologically more apt to play well in big events, but this analysis says that a streaky player and random chance alone may be all that is necessary to describe why some players do well while others do poorly.

It is also worth considering whether the PGA Tour payout scheme is fair. The analysis clearly shows that a streaky player will make more money than a steady player with the same scoring average. From a marketing perspective, this may be exactly what the Tour wants, as streaky players will provide more tournament drama, with potential for epic comebacks and devastating meltdowns.

It is important to mention again that this model has a number of features that do not align with reality. Particularly important in this discussion of valuing greatness is the fact that some players truly are better than others. The consistent success of players like Nicklaus, Woods, Hogan, and others makes clear that the assumption of a uniform average ability is not accurate.² The assumption was useful, however, in showing that consistency does have an effect.

This leads to a question that perhaps is more realistic: In order to maximize wins or earnings, is it more important to have a low scoring average or an erratic strategy? We might expect that erratic play is especially important for players with high averages, as they need some luck to win or make cuts. On the other hand, it may be that the world’s best player needs only an average day to win, and therefore plays conservatively. It is also possible that an above average player needs an erratic strategy to win, yet a conservative strategy will mean making more cuts and earning more money. Furthermore, if a player has a high scoring average, is it better to work on a more erratic strategy to promote hot streaks or to lower his scoring average so that good rounds are more common?

This is a situation physicists love to find: when two effects are present and can enhance or counteract each other. It is in these areas where we find lots of rich and unexpected behavior and where deep new insights are often waiting to be uncovered.

For now, the interactions between scoring average and erratic play remain a mystery. As always, you can download the code used in this article from GitHub and use it to make your own discoveries.³

UPDATE 9/6/18: Check out my bonus post to see how going back to the normal Hardy model provides a good way to explore the relationship between scoring average and consistency in tournament performance

Applying it: Do you want to play great or avoid playing poorly?

I will admit, some of these results were especially surprising to me. I expected that streaky play would improve a player’s chance of winning, but I did not expect it to be the preferred strategy for all three metrics, including making the cut! Now it is clear that because making the cut requires playing above average, a strategy that leads to more above average rounds will allow you to do so more frequently.

This inherently means that the optimal strategy increases your chances of below average rounds as well. Playing erratically means occasionally finishing in last place. In order to have the best chance of success, therefore, one must be willing to psychologically accept occasionally posting a really high score or losing the tournament by a lot, rather than by a little.

Consider Tiger Woods’s final holes at the Open Championship this year. On the 11th hole of the final round, Tiger lost his lead in the tournament after he attempted to get a short-sided pitch close. Here is what one writer had to say:

Woods had acres of green to work with behind the flag but he tried to get too cute with his lob wedge, fluffing the chip short and having it roll back toward him, mockingly. It was not the kind of mistake a master tactician like Woods makes.

But was it such a bad decision? We know that Tiger’s only goal is winning the tournament. The winner is most likely going to be someone who succeeds at taking risks. This notion that Tiger made a tactical error is also short-sighted because it ignores what Tiger did on the previous hole. On the 10th hole, Tiger hits his tee shot into a pot bunker. With the burn in front of the green and the bunker’s steep walls, the conservative play would be to hit a recovery shot back into the fairway and accept a likely bogey. Instead, Tiger takes a Herculean swing at the ball with a wedge, manages to make clean contact, and puts the ball on the green with a chance at birdie and an easy par.

On the 10th and 11th holes, Tiger chose to play aggressively. On the 10th hole he succeeds, and on the 11th hole he fails, playing those two holes at 2 over par. Had he played conservatively on both holes, laying up out of the bunker on 10 and pitching to the middle of the green on 11, he probably makes bogey on both, resulting in the same score of 2 over par. If he succeeds at both, he plays both holes at even par, maintains his lead, and has a good chance of winning the tournament.

I think Tiger made the right choice considering his goals. To win the tournament, you have to pull off some excellent shots. Sometimes the risk goes bad, but as long as you are comfortable with just as many poor performances, and as long as success is worth the cost of failure, the Hardy model says risk is for you.