Masters Pool Strategy: Double your Odds

I want to discuss Masters Pool strategy and see what we can say mathematically that will help you get the best chance at winning. If any of my family reads this, they’re going to laugh. I’ve only entered a Masters Pool a few times and have never won. So why am I giving you advice? The short answer is this: entering into any sports betting pool – a Masters Pool, a March Madness Pool, whatever! – takes a lot of luck to win. Nobody realistically picked Charl Schwartzel to win the 2011 Masters or for UMBC to upset Virginia in the 2018 March Madness. However, you can study the mathematics to get an idea for what the best strategy to win is. So, that is exactly what I am doing here: studying the mathematics. My techniques won’t give you a 60% chance to win the pool, but they can increase your chances from, say, 5% to 15% in a 20 entry pool.

At first it was surprising to me how interesting the mathematics behind this event were. I thought it would be fairly straightforward that the best technique is to simply pick the five best players. However, winning your Masters Pool depends heavily on how the other players in your pool pick their teams. That is, winning your Masters Pool is not necessarily about picking the best team, it is about picking the best team relative to your opponents. Because of this, the optimal masters pool strategy will need to rely heavily on an analysis of your opponents pick styles.

What is a Masters Pool?

A Masters Pool is a betting event centered on the Masters Golf Tournament. The idea is pretty simple: pick five golfers to be on your ‘team’. Then, we judge an entry by looking at the collective amount of money won by the golfers you picked to be on your team. Whichever entry ends up with the most collective money wins the pool.

For example this year if you pick the winning golfer and 3^rd place, your team scores $2,070,000 for the first place win and $782,000 for the third place golfer for a total of $2,852,000. If another team correctly picked the 2^nd, 4^th, 5^th, and 6^th golfers, they would win $1,242,000 + $552,000 + $460,000 + $414,000 = $2,668,000. Your team was better because you had a higher total payout.

We can already see from this small example that it is extremely important to get the winner. 1^st and 3^rd beats 2^nd, 4^th, 5^th, and 6^th. A typical winning Masters Team looks like the winner + 2 top 10 guys. But of course, what a winning team is comprised of depends heavily on the number of entries there are.

Why Masters Pool Strategies Must Involve Your Opponents

Events like this are extremely difficult to analyze because you have to take into account how everyone else plays. This turns finding the optimal Masters Pool strategy from a statistics and data science problem into a game theory problem. Let me explain what I mean. Your job isn’t necessarily to pick the best Masters Team. Your job is to beat everyone else. Let me explain what I mean with an admittedly contrived example. This example is meant to highlight a way that picking a team that doesn’t just have the best players on it could be the best strategy.

There are exactly 21 distinct combinations of five objects you can pick from a group of 7. To say that another way, there are exactly 21 different combinations of Masters Teams you could pick if you restricted yourself to selecting five of the best 7 golfers in the world.

Suppose you were in a Masters Pool with 20 other people where each of those 20 people picked a distinct team of 5 from the top 7 golfers in the world. Is it a better strategy for you to pick the remaining combination of 5 of the top 7 golfers or to pick golfers 8,9,10,11, and 12 to be on your team? I claim picking the tier two team is better.

In order to win picking from the top 7 golfers you need to pick exactly 1-5 on your team which is a really tough ask. However, if you pick 8-12, you are likely to win the pool if any one of your guys wins the entire tournament. Picking 1-5 exactly correct is extremely unlikely. However, having one of 8-12 win is not-so-unlikely at all.

All of this is to say, your Masters Pool strategy needs to rely on knowledge of what other people are likely to do.

My Approach to Finding the Optimal Masters Pool Strategy

My technique to try to answer this question is via random simulation. In particular, we’re going to use one of our favorite techniques: Monte Carlo Simulation. Monte Carlo simulation is the idea that simulation of a random event – or an event that has so much uncertainty that it can modelled as random – can yield insights. For instance, instead of directly computing the best strategy to win your Masters Pool we can simply test a few different strategies, simulate lots of different outcomes, and see which strategy does the best. This will help us find the optimal Masters Pool strategy.

In order to be able to simulate all these outcomes and test our various strategies, we need to be able to endow the Masters outcomes with some well-defined randomness. In particular, we need to be able to simulate random outcomes for:

• The finish order of the players in the field, and
• The Masters Teams that are selected by the various entries into your Masters Pool.

The next two sections will describe how we randomize these events in order to accurately evaluate real outcomes.

Randomization for the Order of Finish

For the next two sections I need one thing: win probability for each golfer in the field. We’ll then use these win probabilities to generate random outcomes that are fairly representative of reality.

The method we use to compute the finish order is actually fairly straight forward. First, pick the winner randomly according to the odds we have. Then, we adjust the odds based on the first selection before picking the second place finisher. To adjust the odds, set the odds for the already selected golfer to zero so that he is not selected again. Then, increase everyone else’s odds upwards by multiplying by some number larger than one so that the remaining odds add up to 1.

For example, suppose the winner we pick first is someone who had a 20% chance of winning. To update the odds before making our second selection we first change this 20% into 0%. Then, we multiply everyone else’s odds by 1.25 (so that the remaining 80% now adds up to 100%). We repeat this process until each golfer’s finish order has been determined. This randomization method generates random finish orders that still respect the fact that some golfers are better than others.

Now, in general we can’t know the probability of each golfer winning the tournament. These numbers aren’t computable or attainable in any way. Many models exist to estimate the probability of an individual winning a tournament including my own world golf rankings system. However, in general, it is hard to do much better than using Vegas’ implied odds as an estimate for win probabilities. How might one do this? We have to convert money-line odds into probabilities and then normalize the probabilities to 1. Let me give an example.

Suppose there are 5 golfers in a tournament with odds as follows:

• Golfer A: + 200
• Golfer B: +250
• Golfer C: + 300
• Golfer D: + 400
• Golfer E: + 500

These numbers can be converted into percentages with the formula: prob = \frac{100}{100+odds}. Therefore, each golfer can be said to have win probability:

• Golfer A: 33%
• Golfer B: 28%
• Golfer C: 25%
• Golfer D: 20%
• Golfer E: 16%

However, notice that these don’t add up to 100%. In fact, they add up to about 122%. This is because Vegas cooks the odds so that they make a guaranteed profit. So, if we want to adjust the odds so that they actually do add up to $100, we should divide each of these percentages by 1.22. Therefore, the implied odds are:

• Golfer A: 27%
• Golfer B: 23%
• Golfer C: 20.5%
• Golfer D: 16.5%
• Golfer E: 13%

In this way we can estimate each golfer’s chance of winning the title.

Randomizing the Submitted Masters Teams

This is the part that is a bit strange to randomize. It can be very difficult to understand how an individual chooses their team. Certainly there is a bit of favoritism in selecting your guys. So, how can we randomize the team selection aspect when we can’t know how different entries favor different guys?

My suggestion is that we should ignore any favoritism. In a reasonably large group, these will tend to cancel out. So, again, we can assume that the frequency a golfer is picked is in line with how likely that golfer is to win. The favorite to win the tournament will likely be picked the most often.

So, to assign a random five players for an individual entry, we randomize the top 5 finishers in the exact same way we did the previous section. So, each team picks 5 random guys where the better guys are more likely to be selected.

Before continuing we should verify that this is a reasonable way to simulate ‘the field’. That is, we should verify that the distribution of picks via this method actually matches the distribution of picks in a real Masters Pool. The blue dots in the figure below show the percentage of teams that picked a specific golfer ordered from highest to lowest. The most commonly picked golfer was on about 55% of teams, the second most commonly picked golfer was on 51% of the teams, and so on. The orange dots below are how frequently those golfers were picked by teams in my simulation. We can see that in my simulated, randomized Masters Pool the most commonly picked golfer was on about 53% of the teams, the second most commonly picked was on 49% of teams, and so on.

We can see that our method of randomly selecting teams tends to actually match what we see in a real life pool. That is, the randomly selected pool of entries is a very good representation of what a Master Pool actually looks like. Because of this, the conclusions we make in analyzing the optimal Masters Pool strategy will have a strong bearing on real life.

Perhaps the only difference is that our model slightly over-selects long shots while real entries place slightly more emphasis on the favorites. This could certainly have an effect on our analysis but it could also be an artifact from the specific Masters Pool I was in.

Various Masters Pool Strategies

I am going to try a few different strategies with a few different pool sizes to see what kind of outcomes we will see. For each team selection strategy I propose, I will test the effectiveness of the strategy on small pools (10 entries), medium pools (50 entries), and large pools (250 entries) to see if the strategy changes. I am writing this before running the experiments so my guess is that indeed the optimal strategy does depend on the pool size.

The strategies I am going to try are as follows:

• Chalk – take the best 5 guys and don’t look back
• All risk – take 5 under dogs.
• Studs and Duds – Take 3-4 of the favorites and 1-2 long shots
• Tier 2 Studs – Take all very good golfers but not 1-5.

So, let us test each Masters Pool strategy above individually. In each case, I simulate 10,000 pools and see what percentage of the time our team wins with a given strategy.

In the small pool, I assume there are 10 entries to the pool. If your strategy is average, we expect to win 10% of the time. In the medium sized pool, there are 50 entries – an average strategy should yield a win 2% of the time. In the large pool of 250 entries, an average strategy should win 0.4% of the time. A good strategy is one that out-performs these ‘average win rates’ relative to the pool size.

Chalk

I am guessing this is going to be the best Masters pool strategy in the small and middle sized pools. Picking the best five players gives you a reasonable shot at 1^st, a top 5 finisher, and another top 10 finisher which should be close to enough to win (again, depending on pool size). Here is what we found:

• Small Pool: 20.6% win rate (relative to 10%)
• Medium Pool: 5.3% win rate (relative to 2%)
• Large Pool: 1.1% win rate (relative to 0.4%)

My quick analysis is that the improvement in each case is about the same. Pretty much independent of pool size, picking the best 5 players roughly doubles your chances of winning. This is a very good outcome, but I should point out one thing: this technique can result in very large chances of ties occurring. If this ends up being the optimal strategy and other teams decide to use this exact same strategy, then you both lose a lot of value because you can’t beat each other.

From a game theory perspective, this can’t possibly a Nash Equilibrium. If there are 100 teams all using this strategy, then it is a good strategy for you to submit, say, the 1-4 best golfers and number 6 instead of number 5. This is because in the fairly likely case that 6 outperforms any of 1-5, you will win.

All Longshots

My initial expectation is that this Masters Pool strategy will not yield good results. For our simulations, I am picking the 12^th, 14^th, 16^th, 18^th, and 20^th best golfers in the world for my team. This idea is this: Very few people will have one of these guys on their team. So, if one of these longshots ends up winning (which should happen about 5% of the time), you give yourself a really good shot.

However, even as the pool size increases, the odds someone else has one of these guys also increases so this strategy probably doesn’t buy you anything. Enough discussion, let us see what the data says.

• Small Pool: 3.7% win rate (relative to 10%)
• Medium Pool: 0.7% win rate (relative to 2%)
• Large Pool: 0.1% win rate (relative to 0.4%)

I don’t need to talk very long about this. Don’t use this strategy, it isn’t going to win you anything.

Studs and Duds

This is the strategy that I think most people employ when entering these types of pools. This Masters Pool strategy is reasonable: pick a couple longshots as ‘lottery tickets’ but hedge and still take the big guys. Another way to justify this strategy is that ‘you want the best guys on your team but you need to differentiate yourself from the other teams who overwhelmingly pick the favorites’. It seems like sound logic, let’s see how it holds up in our simulations. First let’s see the strategy when we pick 3 studs, 1 medium longshot, and 1 longer shot. In particular, I picked the 3 favorites, the 10^th best, and the 20^th best player to be on my team.

• Small Pool: 8.8% win rate (relative to 10%)
• Medium Pool: 1.7% win rate (relative to 2%)
• Large Pool: 0.44% win rate (relative to 0.4%)

Now, I’ll look at the performance when we pick 4 studs and 1 longshot. To me this feels a bit better.

• Small Pool: 15% win rate (relative to 10%)
• Medium Pool: 3.6% win rate (relative to 2%)
• Large Pool: 0.85% win rate (relative to 0.4%)

Commentary: It doesn’t seem like – in any case – this technique outperforms ‘picking the best 5 golfers’. This was a surprise to me, I guessed that this technique would be the best involved. My best guess for why this is the case is because the ‘differentiating yourself from the field’ effect is over rated.

Tier 2 Studs

This technique is pretty interesting to me. The idea is that we can differentiate ourselves from the field while not picking any ‘losers’. Again, I am going to test two versions of this strategy. In the first strategy, I am going to pick the golfers ranked 1, 3, 5, 7, and 9 in the world. In the second, I am going to take numbers 3, 4, 5, 6, and 7 in the world. In both cases, I am ‘differentiating’ myself without having to pick any losers.

First, the results for the 1, 3, 5, 7, and 9 team.

• Small Pool: 13% win rate (relative to 10%)
• Medium Pool: 3% win rate (relative to 2%)
• Large Pool: 0.7% win rate (relative to 0.4%)

Now, here are the results for the 3, 4, 5, 6, and 7 team.

• Small Pool: 13% win rate (relative to 10%)
• Medium Pool: 2.8% win rate (relative to 2%)
• Large Pool: 0.7% win rate (relative to 0.4%)

Obviously there are other realizations of this strategy. For instance, we could pick 1, 2, 3, 6, and 7. Or we could pick 1, 3, 4, 5, and 8. I think one could play with this idea and get results fairly close to the ‘chalk’ team. This is certainly a viable Masters Pool strategy.

Discussion and Further Work

It seems like the data suggests that the Masters Pool strategy of picking the five best golfers is indeed the best strategy and that fact still holds true no matter the pool size. However, I am likely to use a version of the Tier 2 studs strategy in my own pools. Not only are the results close to the chalk results, but it is also significantly more fun. Picking the five best guys is something a computer could do. Staring at golf results, reading articles, and watching a lot of golf in order to inform your choice of which 5 of the top 10 guys to take is significantly more enjoyable.

There are many further research and experimental topics I would like to do in the future (or that I would encourage anyone else to do!) I will highlight two of these other possible research directions.

First, how can one optimize their submissions if they are allowed to submit two different entries? Should we pick disjoint teams using different strategies? Should we pick two teams with a lot of overlap but some different guys near the bottom? How does the optimal Masters Pool strategy change here?

The second research question I am proposing is a bit more abstract. What if you suspect that the other entries to your pool will heavily over or under select a specific golfer? For instance, what if your family is all Irish and they tend to select players from the UK instead of Americans. Or, for instance, you have done some analysis and you think that Golfer X is severely underrated. How can you incorporate that information into your team selection? Succinctly: If players are selected in proportions not in line with their winning probabilities, how can one take advantage of this? I think this question is interesting but I suspect that we can’t convert this idea into one single coherent Masters Pool strategy. I think there will be too many cases to analyze, but I still encourage someone to try it!

I think this is an overwhelmingly interesting question that warrants more study and research by other analysts out there. I am not a game theory expert and I do not know the techniques they used. However, I believe my approach in analyzing the optimal Masters Pool strategy here is quite compelling. Some people have fun sports betting because they like watching sports. Me, on the other hand, I enjoy sports betting because I love having the opportunity to analyze things like this.