# How does a tournament's structure affect the likelihood that the best player will win?

## Background

The origin of this question is a conversation I had with some friends a few years ago. At the time, Roger Federer and Tiger Woods were dominating professional tennis and golf, respectively, and we were comparing and contrasting the two. It occurred to me that there was a mathematical question that was relevant to our discussion; namely, the structure of golf tournaments vs. tennis tournaments.

For example, the Masters is a four-round tournament. After two rounds, roughly the bottom half of the field is sent home. The winner is the person with the lowest total score after four rounds. On the other hand, Wimbledon is a single-elimination tournament; the winner must defeat seven other players in head-to-head competition. From a structural standpoint, if you are the best player in the field, is it harder to win a tournament like the Master's or harder to win a tournament like Wimbledon?

## The mathematics

More generally, how does a tournament's structure affect the likelihood that the best player will win?

This question is a bit fuzzy because there are some modeling issues involved that will affect the answer. Something like the following seems a reasonable place to start.

Let $F_1, F_2, \ldots F_n$ be independent normal distributions such that a random variable drawn from distribution $F_i$ gives the performance by the $i$th best player in a particular round of competition. So we would have $\mu_{F_i} > \mu_{F_j}$ when $i < j$. Assuming that the $F_i$'s have the same variance seems a reasonable starting point, too, as does the assumption that a given player's performances from round to round are independent.

First question: What is the best way to model the $\mu_{F_i}$'s vs. the $\sigma_{F_i}$'s? The difference $\mu_{F_1} - \mu_{F_2}$ ought to be much larger than $\mu_{F_{99}} - \mu_{F_{100}}$, so maybe something like $\mu_{F_i} = 1/i$ would work, but I'm not sure what makes for a reasonable variance to go with this function.

For specificity's sake, let's assume three types of tournament structure: 1) that of the Masters, 2) that of Wimbledon, and 3) that of the World Cup (which has a round-robin stage before moving to a single-elimination stage).

Second question: Given a satisfactory answer to the first question, what is the probability that Player 1 will win each of these three tournaments?

## My reasoning so far

It seems to me that the two most important factors involved that would prevent the best player from winning the tournament are

1. an unusually poor performance from the best player in a particular round, and
2. an unusually good performance from someone else in the field in a particular round.

There's not much that the tournament's structure could do to mitigate factor (1), although a single-elimination tournament would seem to be the most unforgiving. On the other hand, the structure of the tournament probably has a large effect on the impact of factor (2). For instance, an incredible performance from someone in two separate rounds of the Masters raises the bar quite a bit for the best player. On the other hand, in a tournament like Wimbledon two great performances might lead to two upsets of major players but doesn't provide any advantage in later rounds, and, for the best player to be negatively affected, he/she would have to be playing directly against the overperforming player. Also, if there are enough players around (like the early rounds of Wimbledon and all the way through the Masters) there is a high probability that someone in the field will turn in two great performances in two different rounds.

So, if you are the best player in the field it seems to me that contests in which you are essentially playing most of the field simultaneously, like the Masters, would be more difficult to win than single-elimination tournaments like Wimbledon, which in turn would be more difficult to win than those with a round-robin format in the early rounds and single-elimination in the later rounds, like the World Cup.

Third question: Are there any known results that address this problem of the effect of tournament structure on the best player's chances of winning?

I would be happy to see critiques/comments on my modeling and my reasoning as well.

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A fun book called Calculated Bets recounts a computer scientist leveraging the tournament structure in the game jai-alai to build an automated wagering bot. I suppose that the analysis will be very different from situation to situation, but in the book I recall that they go through the math in this one case. cs.sunysb.edu/~skiena/jaialai –  R Hahn Oct 17 '10 at 6:19
Thanks for the reference! –  Mike Spivey Oct 17 '10 at 13:31

It turns out this problem has been studied extensively in the economics literature. The motivation is to create some sort of competition that will maximize the likelihood of the best candidate for a job or the best application for a grant actually being awarded the job or grant.

For example, "The Predictive Power of Noisy Elimination Tournaments," by Dmitry Ryvkin, examines the effects of seeding under different numbers of players and some different performance probability distributions.

The paper "Three Prominent Tournament Formats: Predictive Power and Costs," (apparently published in Management Science under the title "The Predictive Power of Three Prominent Tournament Formats"), by Ryvkin and Andreas Ortmann, addresses my question exactly, though. They calculate the exact probability (under uniform, normal, and Pareto distributions for player performance) that the best player wins a round robin tournament, a binary elimination tournament, and a contest. (The last involves all players performing simultaneously at once; the winner is the player with the best performance.) By calculating these probabilities for specific values they show numerically that for all but small numbers of players in a tournament, the best player in the tournament has a higher probability of winning a round robin tournament than a binary elimination tournament and a higher probability of winning a binary elimination tournament than a contest.

Given these results, it does appear that (all other things being equal) the structure of golf tournaments makes them more difficult to win than tennis tournaments, and, consequently, that dominating professional golf is even more impressive than dominating professional tennis.

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The question

"Which tournament is hard to win for the best player?"

seems to depend on the number of rounds, where a round is a minimal unit of competition such that any two rounds are independent.

If a single golf tournament stretches over a week, with say 18 holes each day, and each hole consisting of approximately 4 shots of the golf ball, do we divide up by shots, holes, days, or weeks? Is there so much rest between shots that it is essentially like starting over?

If the number of rounds is $\approx\infty$ then the best player will almost surely win in the golf tournament, whereas in the tennis tournament, the opposite may be true.

So maybe a possible fourth question...

what kind of statistical experiment could you do so as to determine the realistic length of a round?

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That there is some amount of rest between rounds suggests that the distributions of performance should also depend on time, i.e. how many rounds one has already played. Certainly, stamina is sometimes decisive in tennis, for instance. So, I don't think performance being independent between rounds is a good assumption, at least if one wants to have an accurate model (though that isn't really the problem yet, I guess). –  Ketil Tveiten Oct 17 '10 at 12:00
Thank you. Of course, one of the problems with a question like this is the modeling aspect. I'm hoping that the collective minds on Math Overflow will help me improve my model as well as answer the probability questions. –  Mike Spivey Oct 17 '10 at 13:35