The Green Bay Packers were defeated by the Tampa Bay Buccaneers last night. The Packers trailed 31-23 when it was fourth down and goal with 2:22 to go in the fourth quarter. The Packers decided to kick a field goal instead of trying for a touchdown. The decision was universally criticized. Without crunching the numbers, I knew it would be better to go for it and attempt to get a touchdown, even though either decision was a longshot. The Packers lost 31-26.

Since the game ended, I crunched the numbers.

Here is how I approached the decision. First, the Packers needed a series of events to occur, with all or nearly all events working in their favor to win. Computing the probability of the intersection of multiple events occurring is likely to be a small number. I examined the pathways to winning below. There were some fluke ways to win that I left out because those probabilities were negligible. My calculations are in this spreadsheet.

Decision #1: Go for it on fourth down. There are two ways to win in this scenario.

1. Score a touchdown.
2. Make the two point conversion to tie the game.
3. Stop the Buccaneers defensively (a TB field goal means the Packers lose).
4. Win by scoring within regulation or in overtime if time expires.

I estimate that the Packers had a probability of 0.6 of scoring a touchdown based on Aaron Rodgers’s pass completion numbers. Teams have a probability of 0.48 of getting the two point conversion. Teams have a probability of 0.68 of stopping their opponent from scoring on a possession. There was not much time on the clock, so this may have been an underestimate. However, both teams had multiple time out to stop the clock, and there had not yet been the two minute warning. Winning in overtime for two evenly matched teams is 50-50. Winning within regulation with very little time left has a small probability (say, 0.03). Putting this together, I estimate that the Packers had a win probability of 0.104.

Decision #2: Make a field goal attempt. There are also two ways to win in this scenario:

1. Make the field goal.
2. Stop the Buccaneers defensively while leaving enough time on the clock to score.
3. Win by scoring a touchdown within regulation.

or

1. Miss the field goal.
2. Stop the Buccaneers defensively while leaving enough time on the clock to score.
3. Score a touchdown within regulation, make the two point conversion to tie, and win in overtime (see Decision #1).

I estimate that the Packers had a probability of 0.96 of scoring a field goal. Teams normally have a probability of 0.68 of stopping their opponent from scoring, but I lowered that to 0.5 here because it needed to happen in such a way that the Packers had enough time for one last drive. That is likely an optimistic estimate. I estimate that the Packers could score a touchdown with a probability of 0.15 with the remaining time (Rodgers had an MVP worthy season). The second way to win involved missing the field goal and tying the game in regulation with a last second touchdown and later winning in overtime. Putting this together, I estimate that the Packers had a probability of 0.076. I believe this is optimistic.

Takeaways

1. Going for a touchdown increasing the win probability by about 3% compared to kicking a field goal. It’s not a huge different, but it’s also not insignificant.
2. Either way, the Packers were unlikely to win. So while the decision was bad, it wasn’t a decision that likely cost the Packers the game.
3. Kicking the field goal (Decision #2) could make sense with high confidence in a defensive stop or scoring a TD with time expiring. For the best defensive team in the NFL, decision #2 might be the better option. If Tampa Bay had, say, the worst defense in the country, especially if their secondary was weak, Decision #2 would be more attractive.
4. The Packers had two bad choices.

Last year, I tweeted about a win probability model I created for soccer (or football, depending on where you are from) and the 2019 FIFA Women’s World Cup case study. I promised to blog about this case study I developed for my probability models course. This is a long overdue blog post on this topic.

Here is a portion of the soccer analytics case study.

Soccer win probability model

Soccer (or football, which it is called outside of the the United States) is based on a 90 minute match. The data from FIFA women’s soccer indicate:

• Home teams score 2.34 goals/match (in regulation)
• Visiting away teams score 1.71 goals/match (in regulation)

We assume the goals are scored according to a Poisson process with exponentially distributed arrival times. Assume each team scores independently of one another and of the score. We can use the home team data for the team that is favored in the match.

Consider the situation when the home team is down by 1 goal with 4 minutes in regulation. Find the probability that the home team wins. This is a win probability. We do not consider pulling the goalie.

In one possible solution, we divide the match into small increments of, say, a half minute in length. We recursively solve for the home team’s win probability for any score differential and any time. This way, we want to be able to answer questions like this repeatedly for different score differentials and lengths of remaining time.

The derivation is next. You can skip the math if you want and jump to the figures below.

Let the random variable capture the event that the home team wins with a score differential of with increments to go.

We want to find , the probability that the home team wins if there is a score differential of with increments to go.  In our problem, the home team is down by 1 goal with 4 minutes in regulation, yielding for a win probability with a score differential of -1 with eight 30 second increments to go.

We use a recursive expression to compute the probability of scoring in small intervals of time and put the solution in a spreadsheet. The spreadsheet approach computes our solution, and it allows us to assess a variety of situations, including different score differentials with different amounts of time to go. The boundary conditions with 0 time increments to go are if (the home team is winning when time expires), if (the home team is losing when time expires), or if (the match ends in a tie).

There are two ways to compute the probability that a team scores in a small amount of time , which is the length of an increment: (1) we can use exponential interarrival times, or (2) we can use the Binomial approximation. I’ll illustrate the latter approach below. We have to make the time increments small enough such that having at most one goal scored during the time interval is a reasonable assumption.

We compute by conditioning on 𝑌, where 𝑌 captures what happened in increment 𝑖 with a home goal (+1), an away goal (-1), or no goals (0):

After taking advantage of independent increments, we can simplify this to . Here, we recursively solve for by conditioning on what happened last and formulating a new expression based on the win probability with time increments to go.

Here, , where home goals per match. Likewise, , where visiting team goals per match.

Let’s look at the answer

We can put this into the spreadsheet and estimate the probability of 0.048 that the home team wins when down by 1 goal with 4 minutes to go. We can see answers to other scenarios. A home team wins with a probability of 0.517 if the match is tied with 5 minutes to go.

What is more interesting is that we can move across these spreadsheet to estimate real-time win probabilities. Every time there is a goal, we jump to another row in the spreadsheet. We jump up a row if the visiting team scores and down a row if the home team scores.

I made two win probability charts for the USA v ENG and USA v FRA games in the 2019 World Cup. I set the USA Women’s National Team as the “home” team since they were favored to win in each of the matches, even though the matches were played in France. You can also see that the home team has a probability of 0.61 of winning when the game begins.

Earlier we assumed that goals are scored according to a Poisson Process. Is that a good assumption? Not exactly (see this post using Premier League data) but it’s not a bad approximation except for the end of game situation when a team pulls their goalie. The model we built above can be easily changed to have time-specific scoring rates. Pulling a goalie is trickier but doable. When pulling a goalie, we have to consider new Poisson scoring rates that depend on time and the score differential.

On a side note, the Poisson process assumption holds up better with National Hockey League data.

https://twitter.com/lauraalbertphd/status/1197640721437777924

NOTE: see the February 2024 update to this post and my two point conversion analysis here.

On Sunday November 10, 2019, the Carolina Panthers were down 14 against the Packers early in the 4th quarter. They scored a touchdown, putting them down by 8, and they went for a two point conversion.  The two point conversion did not succeed. This has been the subject of debate, with journalists both applauding and criticizing the decision.

I created a dynamic programming model to determine whether or not to go for a 2 point conversion. The dynamic programming model is based on Wayne Winston’s book Mathletics, which is a fantastic introduction to sports analytics. The state captures the team with possession, the score differential when they obtain possession, and the number of remaining possessions. When there is one remaining possession, it is the last possession. When there are three remaining possessions, the team with the possession has two scoring attempts. Each possession ends in a touchdown, a field goal, or no score. I assume half of all games end in a tie. The probabilities I used are based on average team statistics. I do not model other decisions, such as whether to go for it on fourth down, although these could further improve a team’s win probability.

The slides are below.

Bottom line: teams should go for two points when they score a touchdown and they are down 10, 8, 3, or 2 or up by 1, 2, 4, or 5 (including the points from scoring the touchdown) near the end of the game. These conclusions hold when there are at least two additional possessions in the game.

If you have the last possession: go for 2 when a touchdown on this last possession puts you down by 2.

If you just scored a touchdown but your opponent will have the last possession: go for 2 when a touchdown puts you down by 2 or up by 1, 4, or 5. You normally will want to go for 2 when a touchdown puts you up by 2 except in this situation, because missing the extra point means your opponent could win with a field goal.

Carolina went for two when down by 8 after scoring a touchdown. According to my math, Carolina made the right choice. However, the best strategy does not guarantee a win nor does it drastically improve the win probability.

We can examine the decision in more detail. When down by 8 with four possessions to go (which matches up with when Carolina went for a two point conversion), a team has one of two choices:

1. They could kick an extra point, which would give them a 11.3% win probability if successful (with probability 0.96) or a 7.9% win probability if not successful. Together, this yields a 11.2% win probability.
2. They could go for a two point conversion. If they succeed (with probability 0.48), they would have a 18.3% win probability. Otherwise, they would have a 7.9% probability of winning if not successful. Together, this yields a 12.9% win probability.

There are four things to keep in mind:

• Carolina improved their probability of winning by 1.7% by going for two.
• A good process does not guarantee a good outcome.
• Carolina was not likely to win using either approach.
• Carolina could have further improved their win probability by considering other decisions (who is playing, which plays are called, and whether to go for it on fourth down).

My conclusions are summarized in the chart below. For more reading: Benjamin Morris of 538 wrote an article about when to go for two here. My analysis is consistent with his, although we make different comparisons.

When to go for a two point conversion in NFL football

Last night the Washington Nationals beat the Houston Astros in the seventh game of the World Series. The visiting team won all seven games of the series. This has never happened before.

Two evenly matched teams should each win about half of the time. Home field advantage indicates that the home team has a slight advantage, with the home team in Major League Baseball winning approximately 55% of their games and the visiting team winning 45% of their games.

The probability that the visiting team wins all seven games in a seven game series is:

(0.45)^7 = 0.0037

This is less than 1% and while it is rare, we would expect the visiting team to win all seven games every 268 World Series or so. There have been 114 World Series so far.

For comparison, the probability that the home team wins all seven games in a seven game series is:

(0.55)^7 = 0.0152

To put this in context, the home team is four times as likely to win all seven games in a seven game series than the visiting team. The home team won all seven games of the World Series three times, which is about what we would expect based on the math above.

So far, either the home team or the visiting team winning all games in a seven game series would account for about 2% of all possible outcomes for a seven game series. The other 98% captures a mix of home and visiting team victories as well as the series ending in fewer than seven games.

I commonly hear others argue for expanding the four team College Football Playoff (CFBP) to an eight team playoff. I oppose expanding the playoff to eight teams because for all practical purposes we already have an 8+ team playoff.

Hear me out. There are five major conferences, each of which has a conference championship game, plus an additional five conference championship games in the non-major conferences. The five conference championship games from the major conferences help whittle down the field so the CFBP committee can select four teams for the playoff. These five conference championship games serve as a de facto first round of the playoff, with the losing teams being eliminated from advancing in the “playoff.” None of the losing teams have ever been selected for the College Football Playoff.

As further evidence of my claim, the conference championship games are so important for selecting teams for the College Football Playoff that the Big 12 added a conference championship game after their conference missed a berth in two of the first three playoffs.

The conference championship games serve as a first round of a playoff as follows. First, as noted earlier, the teams who lose the conference championship games are (for all practical purposes) eliminated from the playoff, since the CFBP committee has never selected a losing team for the College Football Playoff (e.g., 2017 Wisconsin). Second, note that not all five winning teams in the conference championship games have a reasonable case for making the playoff, since some may already have two losses (e.g., 2017 The Ohio State). As a result, some of the winning teams are also eliminated from the playoff. Who remains are the conference championship winners, other teams who do not have a conference championship game such as Notre Dame as well as Baylor and TCU in 2014, and other top teams who did not qualify for the conference championship game (e.g., Alabama in 2017). The College Football Playoff committee invites four of the teams who emerge from this process to participate in the College Football Playoff.

Expanding the College Football Playoff to eight teams seems redundant to me given that there are conference championship games that serve as a mechanism for selecting the teams for a four team playoff. I might support an eight team playoff if it replaced and eliminated the conference championship games from the five major conferences. I do not see a need to eliminate the other conference championship games. I hope a team from a non-major conference is someday selected for the College Football Playoff, and conference championship games from those conferences help make a case to include teams from more conferences in the College Football Playoff.

A secondary reason for why I oppose expanding the field to include additional teams in the College Football Playoff is for the impact on the students’ educational plans. The football players are student-athletes, and as a professor, I bristle when the topic of education does not enter the conversation. And by and large it has not.

In the meantime, I will continue to rank college football teams and forecast the College Football Playoff every week on Badger Bracketology.

I introduced a Markov chain to model volleyball scoring schemes in my course on probability models. I am old enough to remember side out scoring, where a team could score a point only if they served the ball. If the team did not score the point, there would be a side out, and the other team would serve the ball in recognition of winning the play. Games were played to 15 points. Under rally scoring, a team scores a point for every play they win regardless of who served. Games are generally played to 25 points. USA volleyball switched from side out scoring to rally scoring in 1999, and college and high school volleyball switched soon thereafter. More here.

Years after the switch, I met Phil Pfeifer from Darden. We discovered that we both played a lot of volleyball in graduate school. There was a push to rally scoring (then called Fin-30) when Phil was in graduate school well before the switch. He wrote a paper* in 1981 that modeled the two scoring schemes as a Markov chain to identify the situations under which rally scoring would leader to shorter or longer games. Because rec sports could be so lop-sided, it was hypothesized by some that rally scoring would lead to longer games. This is counter-intuitive. The Markov chain analysis confirms that this is true in some circumstances. The games tend to be longer under rally scoring when the serving team tends to hold the serve or when the teams are lopsided.

My slides from class are below. In my slides, I also examine the benefit of serving first under side out scoring. Since the serving team is the only team that can score points, the team that serves first is one step closer to winning. That is an enormous advantage if there is a high probability that the team that starts with the ball holds the serve.

* P. E. Pfeifer and S. J. Deutsch, 1981. “A probabilistic model for evaluation of volleyball scoring systems, Research Quarterly for Exercise and Sport 52(3), 330 – 338.

Mike Trick of Carnegie Mellon University came to the Industrial and Systems Engineering department at UW-Madison to give a colloquium entitled “Sports scheduling meets business analytics.”

How hard is it to schedule 162 game seasons for the 30 MLB teams? It’s really, really hard.

Mike Trick stepped up through what makes for a “good” schedule? Schedules obey many constraints, some of which include:

• Half of each team’s games are home, half are away.
• Teams cannot have more than three series away or home.
• Teams cannot have three home weekends in a row.
• Teams in the same division play six series: two early on, two in the middle of the season, and two late, with one home and one away each time.
• Teams play all other teams in at least two series.
• Schedules should have a good flow, with about one week home followed by one week away.
• Teams that fly from the west coast to the east coast have a day off in between series.

Teams can make additional scheduling requests. Every team, for example, asks for a home game on Father’s Day, and this can only be achieved for half of the teams in any given year. Mike addresses this by ensuring that no team has more than two away games in a row on Father’s Day.

Mike illustrated how hard it was to create a feasible solution from scratch. You cannot complete a feasible schedule if you try something intuitive like schedule the weekends first and fill out the rest of the schedule later. This leads to infeasible schedules 99% of the time. One of the challenges is that integer programming algorithms do not quickly identify when infeasibility is reached and instead branch and bound for a long while.

Additionally, it is equally hard to change a small piece of a feasible schedule based on a new requirement and easily get another feasible schedule. For example, let’s say the pope decides to visit the United States and wants to use the baseball stadium on a day scheduled for a game. You cannot simply swap that game out with another. Changing the schedule to free up the stadium on that one day leads to a ripple of changes across the entire schedule for the other teams, because changing that one game affects the other visiting team’s schedule and leads to violations in the above constraints (e.g., half of each team’s games are at home, etc). This led to Mike’s development of a large neighborhood search algorithm that efficiently reschedules large parts of the schedule (say, a month) during the schedule generation process.

Mike found that how he structured his integer programming models made a big difference. He did not use the standard approach to defining variables. Instead he used an idea from Branch and Price and embedded more structure in the variables (which ultimately introduced many more variables) to solve the problem more efficiently using commercial integer programming solvers. This led to 6 million variables that allowed him to embed his objectives such as travel costs.

In most real-world problems, Mike noted that there is no natural objective function. MLB schedules are a function of travel distance and “flow,” where flow reflects the goal of alternating home and away weeks. The objective reflects the distance teams travel. He cannot require each team to travel the same amount. Seattle travels a minimum of 48,000 miles per season no matter the schedule because Seattle is far away from most cities. Requiring other teams to travel 48,000 miles in the season leads to schedules where teams often travel from coast to coast on adjacent series to equal Seattle’s distance traveled. That is bad.

Mike ultimately included revenue in his objective, where revenue reflects attendance. He used linear regression to model attendance. He acknowledged that this is a weakness, because attendance does not equal profit. For example, teams can sell out afternoon games when they discount ticket prices. Children come and do not purchase beer at the stadiums, which ultimately fills the stands but does not generate the most revenue.

Mike summarized the keys to his success, which included:

1. Computing power improved over time
2. Commercial solvers improved
3. He solved the right problem
4. He structured the problem in an effective way
5. He identified a way to get quick solutions for part of the schedule (useful for when something came up and a game had to change).
6. He developed a large neighborhood search algorithm that efficiently retools large parts of the schedule.

Three years ago I wrote a blog post about Mike Trick’s keynote talk on Major League Baseball (MLB) scheduling at the German Operations Research Conference (blog post here). that post contains some background information.

I post weekly NCAA men’s basketball rankings over at Badger Bracketology. Every week I also post the rankings of the Big Ten conference teams. Here are the rankings right now. They differ from the rankings on the site because I made a small change to how I score games in overtime–I now count those as having been decided by a single point to capture the closeness of the games. Here are where the B1G teams end up in the overall rankings:

5 Michigan St
11 Indiana
14 Purdue
19 Maryland
21 Iowa
32 Wisconsin
39 Michigan
52 Ohio St
63 Northwestern
90 Nebraska
93 Illinois
102 Penn St
167 Minnesota
210 Rutgers

These rankings reflect all games across the season without discounting games earlier in the season.

The committee will look at how the teams did down the stretch, and I wanted to get a sense for how I would rank teams based on only the conference games played in the second half of the season. This doesn’t discount the games and it completely ignores the early, non-conference schedule, but it gives a sense as to how the teams should be ranked based on the quality of the wins and losses in the conference schedule.

The results are below using my Modified Logistic Regression Markov Chain method. I only rank the 14 B1G teams because I am only considering B1G games. My B1G rankings are really close to the official ranking based on the standings (in parentheses). I am able to rank order the four teams tied for third place.

Ranking just B1G conference games (official ranking based on the standings in parentheses):

1. Indiana (1)
2. Michigan St (2)
3. Iowa (3)
4. Maryland (3)
5. Wisconsin (3)
6. Purdue (3)
7. Michigan (8)
8. Ohio St (7)
9. Northwestern (9)
10. Nebraska (11)
11. Penn St (10)
12. Illinois (12)
13. Minnesota (13)
14. Rutgers (14)

We can see a few differences between the win/loss (standings) rankings and my rankings. While Michigan State is the top ranked team in the B1G when considering all games, Michigan State finished second when considering just the conference games. Wisconsin, for example, played poorly earlier in the season and finished 5th when considering only conference games.

I’ll post my final B1G conference rankings after the B1G conference tournament.

Check out Badger Bracketology for more updates.