| 摘要 | Every March, millions of Americans fill out NCAA tournament brackets and enter competitions with friends, coworkers, and strangers. The term “March Madness” may just as aptly describe the mayhem associated with the typical office pool as the games themselves. Many of us devote a lot of time and effort to picking our winners. Much is at stake: experts estimate that $7 billion is wagered on the tournament each year. There is no perfect way to pick a bracket, but it is surprisingly easy to increase your odds of winning your pool if you study the math for a minute.
The optimal strategy for picking your bracket depends on the type of pool you are playing in. If you are competing only with yourself, and trying to get the highest number of games correct, it makes sense to use statistical models. These models create an equation to rank teams using complicated formulas that weigh record, strength of schedule, and a number of other factors. The best models in the literature, such as that of Jay Coleman and Allen K. Lynch, predict between 70 and 75 percent of the games correctly—a couple of percentage points better than what you would get by simply picking the higher-seeded team in each matchup. The models are designed to get the most games correct, so they pick very few upsets.
But what does this mean for the rest of us, whose goal is to win an office pool of 20, 50, 100, or 200 entries? The models help you pick a lot of games right, but, counterintuitively, the questions “how do I maximize the number of correct picks?” and “how do I maximize my odds of winning the office pool?” may yield different answers. When you are trying to win the office pool, following the statistical models will give rise to another problem: Your bracket will look like many others, making it hard for you to win. That is, the economics of office pools is as much about game theory as statistics.
A 2008 paper by Jaren B. Niemi, Bradley P. Carlin, and Jonathan M. Alexander argues that the optimal bracket is one that selects a contrarian champion and follows a statistical model for the remaining games. This strategy should get a lot of games right while simultaneously being unique, which is exactly what you want. Your bracket should not be completely off-the-wall, but, statistically speaking, you can increase your chances by finding ways to differentiate yourself from the pack.
The key to a successful bracket is picking the right champion. In all likelihood, your scoring system is such that you need to pick the winner in order to win the pool, so this is the best place to differentiate your bracket. In an online pool I participated in last year, with a fairly common scoring system (double the points for a correct pick each round), the three people who picked Duke to win it all finished first, second, and third of 44.
Of the 44 participants, 32, or 73 percent, picked Kansas as the champion. Forty-one people, or 93 percent, picked one of the four No. 1 seeds. But the historical data doesn’t justify this popularity. Since the field was expanded in 1985, only 16 of 26 champions (62 percent) have come from the ranks of the No. 1 seeds. In a well-functioning market, the share of brackets that picked each team would align with the odds of that team winning the tournament. If the favorite had 20 percent odds, that team would be picked in 20 percent of the brackets. But NCAA tournament pools are not well-functioning markets—a reality you can exploit to increase your chances.
Let’s look at it mathematically. If you need to pick the champion in order to win the pool, then, for any champion you select, your odds of winning the pool are (1/n), where n is the number of people who picked that team. Multiplying (1/n) by the odds that a team wins the tournament will give you an equation for the odds of winning the pool conditional on picking that team: (1/n)*(odds). To estimate the odds, you should use Vegas betting odds, which you can acquire with a quick Google search once the brackets are released.
We can demonstrate this using an example. It looks as though Ohio State will be the favorite this year. If your office pool has 50 participants, and 20 of them pick Ohio State, then n=20. If Ohio State’s odds of winning the tournament are 25 percent, then your odds of winning the pool are 1.25 percent ((1/20)*(.25)). The team fourth most commonly picked as champion, perhaps Pittsburgh, might have 10 percent odds and will probably be picked by about 4 people out of 50. In this scenario, Pittsburgh would give you (1/4)*(.1) = 2.5 percent chance of winning the pool, doubling your odds compared to picking Ohio State.
Rather than focusing on what team has the best chance to win, think about what everyone else is doing and weigh that with the logic above. In a perfect world, you would know who everyone’s champion was and pick accordingly. But if you are in an imperfect world, assume your opponents display common tendencies and use them to your advantage. A few guidelines may help:
1) If there is a local team in your pool, avoid it. It will be over-bet.
2) Avoid the favorite(s). But don’t go too far down the line either. No team seeded lower than three has won the tournament since 1997.
3) Consider Duke. There tends to be a bias against them, so they are frequently a smart pick.
4) When (2) and (3) conflict, go with (2). Especially if Duke is the defending champion.
5) Although we’ve focused on the champion, the rest of the bracket is important too. You’re going to pick upsets; everyone does. But go easy on them. Statistically, they are not a great idea. Besides, you’ve already differentiated yourself with your contrarian champion.
Every year the pool is different, and even the optimal bracket might not win. But tournament pools are simply a game of chance, statistics, and game theory. It makes sense to recognize this—and use it to increase your odds.
Chad Hill is a Jacobs Associate at the American Enterprise Institute.
Image by Rob Green/Bergman Group. |
