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Are Seven-game Baseball Playoffs Fairer Than Five-game Series? Dr. Brian Dean.

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Presentation on theme: "Are Seven-game Baseball Playoffs Fairer Than Five-game Series? Dr. Brian Dean."— Presentation transcript:

1 Are Seven-game Baseball Playoffs Fairer Than Five-game Series? Dr. Brian Dean

2 The Conventional Wisdom Teams that have earned home-field advantage over the course of a 162-game regular season prefer longer, seven- game playoff series to five-game series, feeling that the “better” team is more likely to win in a longer series. Question: Is the difference between seven-game and five- game series so great that baseball should consider changing the Division Series round to a best-of-seven format? Goal: Create a mathematical model to analyze this situation.

3 Dr. Lee May (1992): Seven-game series are not significantly fairer than five-game series (where significantly fairer is defined to mean that the better team has at least a four percent greater probability of winning a seven-game series than a five-game series.) May’s model: Let p denote the probability that the better team will win a given game. Since he’s looking at things from the point of view of the better team, p must lie in the interval [0.5, 1]. (For example, p = 0.7 if the better team has a 70% probability of winning a given game.) Note that May’s model treats each game of the series the same.

4 Probability that the better team will win a five-game series In May’s model, the probability of each W for the better team is p, so the probability of each L for the better team is 1-p. There are ten total scenarios of victory for the better team in a five-game series. The probability of each scenario is the product of the probabilities of each individual game. ResultProbability WWW p³ LWWW p³(1-p) WLWW p³(1-p) WWLW p³(1-p) LLWWW p³(1-p)² LWLWW p³(1-p)² LWWLW p³(1-p)² WLLWW p³(1-p)² WLWLW p³(1-p)² WWLLW p³(1-p)² Adding these, the total probability that the better team would win a five-game series is 6p⁵ - 15p⁴ + 10p³

5 Probability that the better team will win a seven-game series There are a total of 35 different scenarios in which the better team would win a seven-game series. Rather than listing each individually, we will summarize the probabilities of the different scenarios under May’s model: Series length# of ScenariosProbability of Each 4 games 1 p⁴ 5 games 4 p⁴(1-p) 6 games 10 p⁴(1-p)² 7 games 20 p⁴(1-p)³ Adding the probabilities of the 35 scenarios, the total probability that the better team would win a seven-game series is -20p⁷ + 70p⁶ - 84p⁵ + 35p⁴

6 Comparing five-game and seven-game series To compare five-game and seven-game series in May’s model, let f(p) denote the probability that the better team would win a seven-game series, minus the probability that it would win a five-game series: f(p) = (-20p⁷ + 70p⁶ - 84p⁵ + 35p⁴) – (6p⁵ - 15p⁴ + 10p³) = -20p⁷ + 70p⁶ - 90p⁵ + 50p⁴ - 10p³, 0.5 ≤ p ≤ 1 The maximum value of this function is ≈ (when p ≈ 0.689), and the minimum value is 0 (when p = 0.5). In other words, under May’s model, the better team is at most only about 3.72 % more likely to win a seven-game series than a five-game series. Therefore, a seven-game series is not significantly fairer than a five-game series.

7 What are some possible ways to modify May’s model? My model, the subject of the rest of this talk, will attempt to take home-field advantage into account. That is, the probabilities of victory/defeat in road games will be different from those in home games. Another possible modification, which we won’t discuss, would be to account for the effects of momentum/morale. That is, would the status of the series after each game affect the probabilities of victory/defeat in the next game? For example, would the probability of victory in game 2 differ depending on whether the team won or lost game 1?

8 Model taking home-field advantage into account Let Team H be the team with home-field advantage in the series, and let p be the probability that Team H will win a given home game. (Since Team H is not necessarily the “better” team, our model does not imply that p ≥ 0.5 like May’s did. Instead, we will allow p to be anything in the interval [0,1], though it seems unlikely that it would ever be much less than 0.5 in practice.) We will take the probability that Team H will win a given road game to be rp, where r is a parameter we will call the road multiplier.

9 Road Multiplier For a given team, we define the road multiplier as the ratio of a team’s road winning percentage to its home winning percentage. If a team’s road multiplier were 0.9, for example, we could say that they would be 90 % as likely to win a given road game as they are a given home game. For the 112 playoff teams of the first 14 years of the wildcard era ( ), the average road multiplier has been (to three decimal places) The three highest and three lowest road multipliers, rounded to three decimal places, have been: TeamHomeRoadRoad Multiplier ‘01 Braves ‘97 Orioles ‘01 Astros ‘03 Athletics ‘05 Astros ‘08 White Sox of the 112 road multipliers have been or higher, and 17 have been or lower.

10 Probability that Team H will win a five-game series In the current five-game series format, Team H plays games one, two, and five at home, and games three and four on the road. Let W and L denote home wins and losses (with probabilities p and 1-p, respectively) and let w and l denote road wins and losses (with probabilities rp and 1-rp, respectively.) The ten scenarios for victory for Team H are as follows: ResultProbability WWw p²(rp) LWww p(rp)²(1-p) WLww p(rp)²(1-p) WWlw p²(rp)(1-rp) LLwwW p(rp)²(1-p)² LWlwW p²(rp)(1-p)(1-rp) LWwlW p²(rp)(1-p)(1-rp) WLlwW p²(rp)(1-p)(1-rp) WLwlW p²(rp)(1-p)(1-rp) WWllW p³(1-rp)² The total probability of victory for Team H in a five-game series would therefore be 6r²p⁵ - (9r²+6r)p⁴ + (3r²+6r+1)p³

11 Probability that Team H will win a seven-game series In a seven-game series format, Team H plays games 1, 2, 6, and 7 at home, and 3, 4, 5 on the road. Series Result# of ScenariosProbability of Each 2 W, 2 w 1 p²(rp)² 1 W, 3 w, 1 L 2 p(rp)³(1-p) 2 W, 2 w, 1 l 2 p²(rp)²(1-rp) 1 W, 3 w, 2 L 1 p(rp)³(1-p)² 2 W, 2 w, 1L, 1 l 6 p²(rp)²(1-p)(1-rp) 3 W, 1 w, 2 l 3 p³(rp)(1-rp)² 2 W, 2 w, 2 L, 1 l 9 p²(rp)²(1-p)²(1-rp) 3 W, 1 w, 1 L, 2 l 9 p³(rp)(1-p)(1-rp)² 1 W, 3 w, 3 L 1 p(rp)³(1-p)³ 4 W, 3 l 1 p⁴(1-rp)³ Adding the probabilities of the 35 scenarios, the total probability that the better team would win a seven-game series is -20r³p⁷ + (40r³+30r²)p⁶ - (24r³+48r²+12r)p⁵ + (4r³+18r²+12r+1)p⁴

12 Comparing five-game and seven-game series For each fixed value of r, let f(r,p) denote the probability that the better team would win a seven-game series, minus the probability that it would win a five-game series: f(r,p) = [-20r³p⁷ + (40r³+30r²)p⁶ - (24r³+48r²+12r)p⁵ + (4r³+18r²+12r+1)p⁴] - [6r²p⁵ - (9r²+6r)p⁴ + (3r²+6r+1)p³] = -20r³p⁷ + (40r³+30r²)p⁶ - (24r³+54r²+12r)p⁵ + (4r³+27r²+18r+1)p⁴ - (3r²+6r+1)p³, 0 ≤ p ≤ 1 Note that, if we take r = 1 (that is, treat road games to have the same probability of victory for Team H as home games), then we get f(1,p) = -20p⁷ + 70p⁶ - 90p⁵ + 50p⁴ - 10p³, the same function as in May’s model, with the only difference being that we’re no longer requiring p ≥ 0.5. Let’s first take r = 0.883, the average road multiplier of the 112 playoff teams from The maximum value of f(0.883,p) is ≈ (when p ≈ 0.728), and the minimum value is ≈ (when p ≈ 0.332). In other words, under our model, using this average road multiplier as our value of r, Team H is at most only about 3.39 % more likely to win a seven-game series than a five-game series (and is actually more likely to win the shorter series if its home-win probability p is low enough).

13 Maximum/Minimum Values of f(r,p) for different values of r We will consider values of r between and 1.200, since the road multipliers of all 112 playoff teams from have fallen in that interval. All of the max./min. values are rounded off to four decimal places: r Max. Min. r Max. Min In general, the maximum and minimum values of f(r,p) are increasing as r is increasing (and, though this is not shown in the table, the values of p at which the max./min. occur are decreasing as r is increasing.) The value of r for which the maximum value of f(r,p) is is r ≈ For r below this value, Team H is not significantly likelier to win a seven-game series than a five-game series. Of the 112 playoff teams from , 109 have had road multipliers below

14 Conclusion and Acknowledgements Though our model shows that, under certain circumstances, the team with home-field advantage may be significantly likelier to win a seven-game series than a five-game series, a general statement that seven-game series are significantly fairer than five-game series is incorrect. Thank you to all of the participating schools, students, and teachers in this year’s Eastern Shore High School Mathematics Competition, and to the contest’s co-chairs, Dr. Kurt Ludwick and Dr. Barbara Wainwright.


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