By Ben Blatt
Before starting a pick-up basketball game the question usually arises: “Are we playing winners’ take or losers’ take?”. The difference in rules is that with winners’ take the team that scores gets to keep possession of the ball. The opposite is the case for losers’ take. One would think that based on the difference in skill between two teams, one set of rules would be beneficial to a particular team. Specifically in my case I sought to answer the question: “Which strategy will be most beneficial given that I am considerably worse than my opponent?”
To tackle this problem you have to start with a few assumptions. Firstly, I assumed that there is a probability ‘p’ that your team will score when you start with the ball and a different probability ‘q’ that your team will score when you don’t start with the ball. I assumed the values were constant over the duration of the game. I broke the problem down into the different probabilities of reaching n points on your own or your opponent’s kth possession. This was done using combinatorics. Below are the equations used to calculated the chances of winning and a chart showing chances of winning in a game to 15 points with different values of p and q.
Probability of player one winning under Winners’ take:
|P(W) with Winners’ Take||P(W) with Losers’ Take|
The result: there is no difference in winning between winners’ take or losers’ take. This is true for all values of p and q, not just the ones listed above. I had intended to explain further mathematically how I obtained the formulas and why they are equal, but after doing my analysis I found a paper from the journal Operations Research titled ‘A Probabilistic Model of Winners’ Outs versus Losers’ Outs Rules in Basketball’ . The paper, by Christian Albright and Wayne Winston, approached the same problem with the same methods and got the same answers. However, they beat me to the punch by 32 years.
In order to make this post have some originality I decided to consider a few other alterations. First let’s consider a new set of rules, that I will call ‘Catch-Up’ rules, in which the team that is losing always gets possession of the ball after a basket. If the score is tied, the team that most recently was behind maintains possession.
I was unable to directly use combinatorics to determine the probability of winning under these rules. Instead, I made a 16×16 grid (since I decided to model a 15 point game) with each row corresponding to the number of points player one has and each column corresponding to the number of points player two has. Each box therefore represented a score of the game and the value within the box indicates the probability that the score will happen. Each box’s probability was defined recursively. For instance, the box with score 2-2 was defined as P(2-2)= p*P(1-2)+(1-q)*P(2-1). The p and q correspond to the p and q defined earlier and P(1-2) is simply the probability that the score 1-2 would occur. Below is a chart showing the differences between the ‘Catch-Up’ rules and the winners’ take or losers’ take rules (the chart says Winners’ but remember those likelihoods are identical).
|P(W) with Winners’ Take||P(W) with ‘Catch-Up’ Rules|
As you can see, the ‘Catchup’ rules bring the values closer to 50%. This is intuitive as it gives teams that are behind a way of catching up. However, these rules do not help the worse team as much as I had expected. This is because these rules help the better team in the rare case that they fall behind.
Winston and Albright had an interesting proof in their paper which showed that if teams were given the option to switch strategies midgame, the team behind would always want winners’ take and the team ahead would want losers’ take. We already showed the strategies are identical when starting at 0, but it can easily be seen why switching strategies would be beneficial. Consider the extreme case where a team is winning 14-0 in a game to 15. Clearly it would be better for the team behind to choose winners’ take since they require a streak to catch up and winners’ take gives them the best chances to do that (since we assume p>q).
However, they did not quantify how much of an advantage switching strategies would have. Below is a chart showing the benefit of switching to Winners’ at a certain score given you were playing losers. The values below were generated with two different 16x16x2 grids. The reason this grids needed to be three dimensional with length 2 is that there is a difference in probability, in terms of reaching a score and what score will occur next, depending on whether or not the previous point was scored by player one or by player two. For instance under winners’ rules, P(2-2)= p*P(1-2)w+(1-p)*P(2-1)w+ q*P(1-2)L+(1-q)*P(2-1)L. In this case, P(1-2)w is the probability of reaching the score 1-2 by player one winning the previous point and P(1-2)L is the probability of reaching the score1-2 by player one losing the previous point. One 16x16x2 grid was the winners’ take grid and the other was the losers’ take. The value of switching to winners is simply the difference between the two grids at any particular score.I used a first to 15 point game with p=.6 and q=.5. Player one’s score (before the switch in rules) is in the left column and player two’s is in the top row.
As you can see, it is beneficial to switch to Winners’ take if and only if you are losing. When the score is tied this is the same as a game to n points. We have already shown that winners’ and losers’ are equal so it is logical that 0% is the advantage at these points.
This model is definitely not perfect, but any sport which has different teams start with possession dependent on a previous score (such as soccer, football, or volleyball) could hypothetically be modeled this way. The result that the probability of winning with winners’ take and losers’ take is equal means, unsurprisingly to all but me, that spending my time doing math will not help me at all on the basketball court.
Ben Blatt can be contacted at firstname.lastname@example.org.