About a month ago, I wrote a post about two semi-fictional sports teams called the Phillies and the Qankees. I asked: what is the probability that the Phillies win a best-of-seven series of games, given that they win each game with probability p? And how does this compare to the probability of them winning each individual game? It turned out that if we had p = 1/2 + ε, then the probability of them winning the series was about 1/2 + (35/16)ε (assuming ε was small). In general, the probability of winning a best-of-(2n-1) series (that is, a series that ends when the best team wins seven games) was about 1/2 + (1.11 n1/2) ε; I conjectured that the constant 1.11 was actually 2/√π.
We can easily compute the average number of games that are actually played in a seven-game series. If we let p be the probability that the Phillies win any given game, and q be the probability that the Qankees (now do you see why I called them the Qankees?), then the number of games we expect to be played is just
4(q4 + p4) + 5(4q4p + 4p4q) + 6(10q4p2 + 10p4q2) + 7(20q4p3 + 20p4q3)
since q4 + p4 is the probability that the series goes 4 games, and so on. If, as in the previous post, we let p = 1/2 + ε and act as if ε2 = 0, it turns out that this is independent of ε and is 93/16, or 5.8125. (Of course, it's not actually independent of ε; for those who know what this means, if we expand the expected number of games as a power series in ε, there's no ε term but there is an ε2 term. (It turns out, though, that more World Series have gone the full seven games than would be expected, both because of which games are played at whose stadium and for reasons of baseball strategy.)
So in the "standard" system we have to play 5.8125 games, on average, to get an amplification of 35/16. Could we do better? Let's consider a playoff system that works as follows: the two teams play two best-of-three series. If the same team has won both series, they are declared the champions; if they've each won one series, then a third series is played to determine the champion. This is a best-of-three series of best-of-three series. The amplification is easy to find. If the Phillies have probability 1/2 + ε of winning each game, then we know from the previous post that they have probability 1/2 + (3/2)ε of winning a best-of-three series; thus they have probability 1/2 + (3/2)2ε of winning in this format. The amplification is a bit better than in a best-of-seven series, but only barely. It turns out, though, that on average you play more games in this format than in the best-of-seven format. On average, a best-of-three series consists of 2.5 games. (Half the time, the same team wins both games and it's over; the other half of the time, they split the first two games and the third has to be played.) So a best-of-three series of best-of-three series requires, on average, (2.5)2, or 6.25, games. It could take as few as four or as many as nine. (For evenly matched teams, though, the chances of going nine games are one-sixteenth -- a lot less than the probability of a best-of-seven series going its maximum length.)
At this point I started to wonder -- is it possible to design a system that gets substantially better amplification than something on the order of the square root of the number of games played? As I said before, it seems unlikely -- the whole situation reminds me of opinion polling, and if there were a way to do better than the square root there, I'd bet some pollster would have figured it out. Jordan Ellenberg suggested a solution in 2004, in which the World Series would end when one team is up 3-0, 4-1, 4-2, 5-3, or 5-4. I don't like this system for a best-of-seven series, because I was living in Boston when the 2004 American League Championship Series happened -- the Yankees (the real New York team, not the fictional Qankees) won the first three games but the Red Sox came back to win the last four. That could never have happened under this system, although it does open up possibilities for some other exciting series where a team repeatedly just barely avoids elimination and comes back to win.
How long is the average series now? The series can go three, five, six, eight, or nine games. The Phillies win in three with probability p3; the Qankees win in three with probability q3. For the Phillies to win in five, they have to lose the first, second, or third game and win the other four, with probability 3p4q; similarly, the Qankees win in five with probability 3q4p. For the Phillies to win in six, they must win the sixth game, and three of the first five games; there are ten ways to pick the games they win. But if they win the first three games, then the series would be over, so there are actually nine ways. Thus the Phillies win in six with probability 9p4q2; the Qankees win in six with probability 9q4p2. For the Phillies to win in eight, they must win the final two games, and three of the first six; but if the win the first three or lose the first three, it's over. There are thus 20-2 = 18 ways to pick the games they win, and their probability of winning in eight is 18p5q3; similarly, the Yankees win in eight with probability 18q5p3. Finally, we use a bit of trickery to determine the probability that the Phillies win in nine. This is some constant times p5q4. But if we let p=1/2, then we know the Phillies have probability 1/2 of winning the whole series; using this tells us the constant is 36.
The length of the average series turns out to be, then,
3(p3 + q3) + 5(3p4q< + 3q4p) + 6(9p4q2 + 9q4p2) + 8(18p5q3 + 18q5p3) + 9(36p5q4 + 36q5p4)
which, when p = 1/2, is 369/64, or 5.765625 -- very close to the length of the average World Series in our world. The Phillies' win probability is
p3 + 3p4q + 9p4q2 + 18p5q3 + 36p5q4
which, letting p = 1/2+ε where ε2 = 0, turns out to be 1/2 + (147/64)ε. 147/64 is about 2.30. It seems that in a series where about six games are played on average, the amplification will always be a bit greater than two.
But the purpose of the baseball playoffs isn't necessarily to find the objectively "best" team. In fact, a lot of people will argue that a good playoff team is different from a good regular-season team, because there are more off days in the playoffs than in the regular season. This means that once a team makes it to the playoffs they only need three starting pitchers, as opposed to the five they'd need during the regular season. The playoffs don't reward depth. If we wanted to find the objectively "best" team, we'd just award the championship to the team with the best regular-season record. The purpose of the playoffs is to provide entertainment. Since it seems like any playoff format with a "reasonable" average length is about equally good at finding the best team, it seems to me that baseball should stick with its current playoff system because it is familiar, and familiarity enables people to remember how things were in the past and compare those results to the present. Baseball is, of course, a game that is very aware of its own history.