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New Wild Cards and the Playoff-Probability Curve

Some say the new wild card is a gimmick to artificially create drama. Some say it rewards teams for wining their divisions. No matter what you think about the new playoff format, it’s here — and the playoff probability curve we’re used to is suddenly out of date.

In March, Major League Baseball announced a major change to the playoffs. Each league would get a second wild card, and the two wild card teams would play a one-game playoff with the winner going to the division series. The change is effective for 2012.

This expansion is the first for baseball since the wild card was instituted in 1995. In the 17 seasons since that change, the probability that a team would make the playoffs — given a number of regular-season wins — is described by the following function:

There are two important concepts when analyzing the graphs in this post: inflection point and marginal value of a win. The point of inflection in the above function is between 89 and 90 wins. If a team wins 89 or fewer games, that team has less than a 50% chance to make the playoffs. If a team wins 90 or more, the team has greater than a 50% chance.

The marginal value of a win is the additional probability each win adds. For example, teams winning 96 games have about a 97% chance of making the playoffs. Teams that win 97 games have about a 98% chance. The marginal value of the 97th win is 1%. It should also be noted that, by definition, the marginal value of a win peaks at the point of inflection.

One more housekeeping note: These probabilities are based on averages; they are not specific to a league or to a division. In any given season, 90 wins probably isn’t enough to make the playoffs out of the American League East, but it would probably be enough to make it out of the National League Central.

Going back through the yearly standings and determining the teams that would have won the second wild-card spots — had they existed — allows for the estimation of a new function. An that yields playoff probabilities for the 2012 season:

The new inflection point occurs at about 87 wins. As expected, when you add playoff teams, it becomes easier to make the playoffs. The new wild card also increases the marginal win value around the inflection point. In the old system, the peak marginal win value was about 12.7% for win No. 90. In the new system, the peak marginal win value is 17.4% for win No. 87.

If all you care about is getting out of the regular season, then the story is over right here. Win 87 games and you’ll probably see Game 163. However, I suggest that the real story goes deeper. The new playoff format also created a new wrinkle: all playoff appearances are not created equal. The three division winners in each league basically get a first-round bye. The two wild-card teams have to play-in to the division series. How does this change the playoff probability function?

For all intents, a wild-card spot is now a 50-50 chance to join the division series. Given, the actual odds of the one-game playoff will be skewed toward the home team, but for our purposes we’re not trying to control for home field advantage or team quality. Using this method yields a much more realistic view of playoff probability:

The adjusted function looks similar to the pre-2012 function, but flatter. Technically speaking, they have the same point of inflection — 90 wins — but the marginal value of a win has changed:

Under the old rules, teams needed only to get into the playoffs to make the division round, so getting to the 90-win threshold had a very high marginal value. Under the new rules, teams can get into the postseason at about 87 wins, but they’re not guaranteed a spot in the division series unless they win their division. That takes about 93 wins. Therefore, wins No. 87 through 92 are less important than they used to be — but wins below 87 and above 92 are more important.

To sum up, teams that win 88 or fewer games are better off in 2012 because they have a better chance of making the division series than they used to. Teams that win 89 games or more are worse off  because there’s now the possibility they have to play-in to the division series — something they previously might not have had to do. This effect extends all the way into the 100-plus win teams. Remember the 2001 A’s? They won 102 games, but finished second in the American League West, behind the 116-win Mariners. In the new playoff system, Oakland would have had to play 85-win Minnesota in a one-game playoff.

You can argue how that game would have turned out, but there is no question that the A’s would have had one fewer chance making it to the divisional round playing the Twins than they would have by simply hopping on a plane.

Nothing in this analysis attempts to estimate the effect of player fatigue or using an ace to win the one-game playoff. It his highly likely that this analysis underrates the real-world negative effects on the wild card teams. Instead of a wild card being counted as half of a playoff spot, it might be more accurate for it to be 0.4 or even less.

The new wild cards will, without question, keep more teams in the race longer. A team that wins only 84 games now has a 12.6% chance of making the postseason, twice the odds as last year. However, there may be some unintended consequences, with teams in the high 80’s to low 90’s win range having less to gain from a marginal win, and the potential for 100+ win teams to be knocked out by 85-win teams in one game.

For reference, the below graph includes playoff probability curves and marginal win values for both versions of the expanded playoffs –the wild card as a full playoff spot and the wild card as a 50-50 play-in: