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March Madness: Using Game Theory to Win Your Upset Picks


If you win more than half your upset picks, then you have an advantage over every other player in the pool. Of course, this still doesn't mean you'll win.

Two weeks ago, I wrote about the game theory of March Madness bracket picks (see March Madness: Using Game Theory to Win the Office Pool). Assuming you know the probabilities for each game, and everyone in the pool agrees with you, and everyone plays a game theory optimal strategy, and there is no collusion, the correct way to make picks is to simulate an outcome according to the actual game probabilities and pick it.

In practice, however, that's a terrible strategy because not everyone agrees on the probabilities and definitely not everyone will play a game theory optimal strategy. Also, there is frequent collusion, including people filling out more than one entry and people consulting each other on picks.

To pick the best "exploitive" strategy (a game theory strategy that makes assumptions about other people's play), you would also need to know the distribution of everyone else's picks. Then you would use a computer search algorithm to make your picks.

Fortunately, there is a much easier way that works pretty well. It works for lots of different scoring systems. We start from the assumption that you're going to have to beat someone who picks all favorites in order to win. For one thing, there are people who pick all favorites. For another, most people pick too many favorites, meaning the field will be crowded if there are fewer than the expected number of upsets. Therefore, you must win at least half of your upset picks.

In this article an "upset" pick is one in which the higher-number seed team wins. In some cases, the higher-number seed team is actually the better team, but it's my experience that most people filling out the bracket sheets don't know enough about basketball to know when that is true. For the first four games, I assume the favorite is the Las Vegas favorite, since the odds are easy to find on the Internet. For the final four, I assume the favorite is the higher-ranked team according to the RPI system.

When you make an upset pick, it's generally best to pick the most likely team to fill the slot, other than the lowest-number seed team that can fill it. Your goal is to make your sheet different from other people's, without picking too many underdogs. Picking anything but the favorite will get you a lot of differentiation. You don't get much more differentiation by picking a second-most likely team other than the favorite to fill the slot, and you have less chance of hitting.

If you win more than half your upset picks, then you have an advantage over every other player in the pool. Consider a random other player. She will have picked some games the same as you; these do not matter. She will have picked the favorites in some games in which you picked upsets; since you won more than half your upset picks, you have the advantage here. She will have picked some upsets where you picked favorites; here you have the advantage because the favorite wins more often than it loses. And in some games you both will have picked upsets, but different teams. Here you have an advantage both because you picked the best underdog so you have the better chance, and also because we know you won more than half your upset picks. This logic does not depend on the exact scoring method used; it's generally true for a variety of systems.

Of course, having the advantage over every other player doesn't mean you win, or even that you're likely to win. In a large pool, even with your advantage, some other player is likely to get lucky. On the other hand, in a reasonably large pool, you have very little chance of winning if you don't win at least half of your upsets.

My system is to set the number of upset picks such that your probability of winning at least half of them is one over the square root of the number of people in the pool. With 100 people, for example, that means you have a 10% chance of winning more than half your upsets. If you do that, in my experience, you will have something between a 30% to 90% chance of winning. The lower number is appropriate for a sophisticated pool where a lot of people know about the right number of upsets to pick, and pick shrewdly. The higher number is for a pool of people who pick foolishly. That means you have between three and nine times the chance of winning of the average player.

How many upsets is that? The table below gives my recommendations. If there are five or fewer, pick all favorites. With the sixth person, pick one upset. Add one upset per person up to 21, then you start adding them more slowly. Actually, this table is not very useful for pools smaller than 20 or larger than 1,000 as the assumptions break down. But I show you the whole thing for reference.

Upset picks People in pool
1 6
2 7
3 8
4 9
5 10
6 11
7 12
8 13
9 14
10 15
11 16
12 17
13 18
14 19
15 20
16 21
17 23
18 25
19 28
20 32
21 36
22 41
23 47
24 54
25 62
26 72
27 84
28 98
29 120
30 140
31 160
32 200
33 240
34 290
35 360
36 440
37 550
38 700
39 900
40 1,200
41 1,500
42 2,000
43 2,800
44 3,800
45 5,400
46 7,800
47 12,000
48 18,000
49 29,000
50 48,000
51 87,000
52 170,000
53 370,000
54 1,000,000
55 4,300,000
56 1,800,000,000

There are a few adjustments you can make. If there are runner-up prizes, I divide the number of people in the pool by the number of prizes. The numbers above assume you pick the most likely upsets (listed below). If you pick a lot of deep upsets, you should pick fewer of them. As a rule of thumb, for every upset you pick more than twice as far down the list as your total, pick one less upset. For example, with 100 people in the pool, you want 28 upsets. If you pick from the top 56 (twice 28), pick all 28. For every upset you pick deeper than 56, make one fewer upset pick.

None of this stuff is set in stone; use your judgment. There are a wide variety of bracket selections that have close to equal chance of winning, closer than we can distinguish with this method.

The table below shows my ranking of the upsets from most likely to least likely. However, you don't want to just pick the most likely ones. For one thing, other people may read this column. For another, you know your pool. Don't pick local favorites as underdogs and randomize things a bit. My recommendation is to look at games with upset numbers twice the number of games you want to pick or less; for example, 56 or less if you are picking 28 upsets. Cross off the local favorites or teams you think might be popular for other reasons.

Next, start from the championship. Most people do the opposite and start with the first games (the "First Four" or FF below). You see I have picked North Carolina in the final as upset pick No. 25. That means I think there are 24 other slots more likely to have an underdog win them. If you are going to make 12 or fewer upset picks (meaning your pool is 22 or fewer people), you shouldn't look at upset picks 25 or higher. That means you fill in Kentucky for the champion. But with 13 or more upset picks, you can pick North Carolina if you like.

Once you make your final pick, you can fill it in all the way back to round one. That takes care of one semifinal pick, but you need to make another one. The semifinal picks are upset numbers 7 and 20. If you picked North Carolina to win the championship, you can pick Michigan State in the other semifinal if you want. If you picked Kentucky, you have the option of picking North Carolina in the other semifinal game. Once, again, when you make your pick, fill that team in all the way to the beginning.

Keep working backward filling in all empty slots at each round before moving on to the earlier round. Generally, you'll pick the upset most of the time (but not all of the time; remember to randomize) if its number is twice the number of picks you want to make or less, and it is not a local favorite or otherwise likely to be picked by many other people. Once you fill up your number of upset picks, pick the team with the lowest-seed number than can fill the slot at each point. If you get to the end and don't have enough upsets, switch some earlier picks.

Upset # Round Underdog
1 1 Texas
2 1 North Carolina State
3 1 Colorado State
4 2 Wichita State
5 1 Alabama
6 1 Saint Louis
7 Semi-Final North Carolina
8 4 Ohio State
9 FF Lamar
10 FF Loyola
11 1 California
12 FF Western Kentucky
13 4 Kansas
14 2 Texas+
15 1 Purdue
16 1 Long Beach State
17 FF South Florida
18 2 Colorado State+
19 2 New Mexico
20 Semi-Final Michigan State
21 3 Indiana
22 1 UConn
23 2 Vanderbilt
24 4 Missouri
25 Final North Carolina
26 1 Virginia
27 2 Temple
28 3 Louisville
29 2 Florida
30 2 UNLV
31 1 Virginia Commonwealth
32 4 Duke
33 1 Belmont
34 1 West Virginia
35 1 St. Bonaventure
36 2 North Carolina State+
37 1 Ohio
38 1 Harvard
39 3 Florida State
40 1 Colorado
41 1 Southern Mississippi
42 3 Georgetown
43 3 Michigan
44 1 South Dakota State
45 1 Xavier
46 1 Davidson
47 3 Marquette
48 1 BYU
49 2 Notre Dame
50 1 New Mexico State
51 2 Kansas State
52 2 Saint Louis+
53 2 Gonzaga
54 1 Montana
55 3 Wisconsin
56 3 Baylor
57 2 Alabama+
58 2 St. Mary's
59 1 Lehigh
60 2 Iowa State
61 1 Norfolk State
62 1 UNC-Asheville
63 1 Detroit
64 1 Loyola (Maryland)
65 1 Vermont
66 1 LIU-Brooklyn
67 1 Mississippi Valley State

You should make an effort to work in all of the top eight upsets, because in these games I believe the underdog actually has a better than even chance of winning. Of course, if you disagree, don't pick them. And if you see any other situation where the team with the higher seed-number is actually a favorite in your opinion, pick that regardless of the number of upsets you need. Also, I have added a "+" to teams where by seed order they are not even the second-favorite team to fill the slot. These give you a little bonus.

This is by no means an optimal strategy. I developed it with some math and some experience, and tried to simplify it enough to fit in a column without a lot of formulae. If you use it and win, do something good with the money. If you use it and lose, you should know better than to trust free advice on the Internet.
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