## Pickleball win percentage serving first

I had a discussion recently where it was proposed that starting on defense (in doubles) is preferable to serving first because the defenders have an advantage. While the defense *does* have an advantage because they are able to wall off the kitchen first, I would like to prove that the choice to serve or receive makes no difference in outcome. I wound up using two methods:

**Monte Carlo method**

I wrote a program that simulates games in order to find the win percentages. You can get the code here. By default, the program assumes the two teams are evenly matched and runs 100,000 simulations:

`./bin/console pickleball:serve`

The program reports that both teams end up with a 50% win percentage, meaning that it doesn’t matter whether you serve first or second.

We can incorporate a defensive advantage of a few percentage points like this:

`./bin/console pickleball:serve --defPct=5`

In this scenario, each team will win 55% of points on defense and 45% on offense. The end result is still 50/50 in terms of wins.

Note that when running only 100,000 simulations, you might get a little fluctuation in the output.

I didn’t really understand why the odds would be even, so I wound up proving the result in a different way:

**Mathematical “proof”**

“Proof” is in quotes because this is not as rigorous as a true mathematical proof, but I think it gets the point across.

First I assume that the only time you would realize an advantage due to order of serves is if the order results in one team having more scoring opportunities than the other.

So consider a simple case of singles with Player A serving first against Player B. View the serving possessions like so:

A B A B A B A B A ... 1 2 3 4 5 6 7 8 9 ...

If you count the number of serving opportunities that A and B have had at any given point, you’ll see that the number will be even when the total number of possessions in even. But when the total is odd, Player A will always have one extra possession. So in this case, Player A does have an advantage due to serving first because half the time (any odd number) she will have had one more opportunity to score points than Player B.

Now let’s consider the more complex case of doubles, where Team A’s first server, A1, serves on the first possession before handing off to Team B. Here are the possessions:

A1 B1 B2 A1 A2 B1 B2 A1 A2 B1 B2 A1 A2 ... 1 2 3 4 5 6 7 8 9 10 11 12 13 ...

You’ll see that again the number of possessions per team is even when the total number is even. But what about the odd numbers? On 1, Team A has one additional possession; on 3, Team B has one additional possession; on 5, Team A is back to having one additional possession; on 7, Team B is up one, and we continue to alternate to infinity.

So 50% of the time (all even numbers), the teams are even. 25% of the time (half of all odd numbers), Team A has one additional possession. And 25% of the time (the other half of all odd numbers), Team B has one additional possession. Given these probabilities, both teams have an even chance of having one additional scoring possession at any random point in the game – therefore, neither team has an advantage due to serving first.

As to defensive advantage, I think that is irrelevant because just as the teams are even in serving possessions, they are also even in defensive possessions.

**Conclusion**

Assuming I haven’t botched the code and logic, the Pickleball method of giving only one serving possession to the starting team seems to perfectly balance what would otherwise be an advantage to the first servers. And just because the receiving team has an advantage on any given point doesn’t mean you are more likely to win the game if you start on defense.