"Make it, take it" benefits the underdog

In a variation of basketball called "make it, take it", a team scoring a basket gets to keep the ball for another possession, rather than giving it to the other team.  So it can ring up 4, 6, 10, 20 ... points in a row, before the other team gets a shot. 

You would think that in this variation, a good team could easily run up the score on a worse team, and you'd get lots of blowouts.

And I think that's true.  But, unexpectedly, I think this variation also gives the underdog team a *better chance of winning the game* than regular basketball rules.

To verify, I created a simulation of a simplified game.  The good team scores on 52 percent of its possessions.  The bad team scores on 48 percent.  A game goes 200 possessions; if the game is tied, the next score wins.  There are no three-point attempts or free throws. 

In my simulation of ten million games:

"Regular" game: favorite won .7157682
"Make it" game: favorite won .7146367

Difference: .0011315

Not a huge change in competitive balance, but definitely real (about 8 SDs).

(Note: Geoff Buchan (who writes the RotoValue blog), e-mailed me with his own simulation.  The results were similar, although not exact.  Geoff noted that the difference is very small, and wonders whether simulating a "traditional" overtime, rather than sudden death, would eliminate it.  We're still working on that ... but, for purposes of this post, I'm going with sudden death.

In any case, I'm not advocating for this rule change, nor do I feel like thinking through the practical ramifications (how will teams change their play?).  I just think it's an interesting result.)

So why does this happen, that the underdog benefits from a "rich get richer" rule change?  I would never have thought it did.  I kind of stumbled on a similar situation last year, and wrote up a puzzle asking for a proof.  This time, I don't have an actual seamless proof, but I'm going to try explaining.  I'm trying to make you feel like you understand what's happening, rather than convince you with rigorous step-by-step logic.

It's a fairly long (but not complicated) explanation.  If you can think of a better or shorter one, let me know. 

I'm going to do this in two parts.

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PART I

1.  First, and perhaps obvious: the shorter the game, the better the chance of the underdog winning.  If a bad team loses 30% of its games, it may lose only 40% of its quarters.  A bad team might go on a 10-3 run in the short term, but it's not going to go on a 50-15 run in the longer term.  The Charlotte Bobcats may have a 20% chance of winning a given game against (say) the Lakers, but much less than a 20% chance of winning the *season* against the Lakers.

2.  Therefore, the underdog has a better chance of winning the first half, than it does of winning the whole game.  It would love to negotiate with the other team to have the game end after two quarters.  As it turns out, that would increase its chance of winning from 28% to 34%.

3.  Now, suppose we play a full game.  But we make one change: at the end of the first half, we take whichever team is in the lead, and we award it an extra 100 points.  Then we play a normal second half.

The first reaction is: well, the overdog is going to be in the lead 2/3 of the time, so, it's going to get most of the benefit of the extra 100 points.  You're just making the rich team richer!

But, it's the opposite.  If you give the leading team 100 bonus points, then the team that wins the first half *has to win the game*.  Right?  There's no way to come back from more than a 100 point deficit.  So, effectively, the "100 point bonus" rule is exactly like ending the game at halftime!  And, as we saw, that favors the underdog.

4.  What if the bonus is less than 100 points?  It still favors the underdog, because it still makes it less likely for the lead to change.  Not impossible, this time, but still less likely than otherwise, which still hurts the favorite.

Remember: the underdog leads 34% of the time after the half, but only 28% after the full game.  Therefore, when the lead changes in the second half, it's usually in the overdog's favor.  Therefore, fewer lead changes benefit the underdog. 

That's true even if it's a small bonus -- say, 4 points.  That still favors the underdog, to some extent.  The explanation isn't as obvious as when it's a 100 point lead, but the mechanism is the same: reducing lead changes.

5.  Another way to look at it is this: when the game is a blowout, 4 points isn't going to matter.  The 4 points is going to matter more in a close game.

Now: in our simulation, the better team is favored by 8 points over the game.  That means, after the first half, the average point differential is 4 points. 

Of course, it's not always 4 points.  The reality is something like:

-- Two-thirds of the time, the favorite is leading at the half, by an average of maybe 7 points.

-- One-third of the time, the underdog is leading at the half, by an average of maybe 2 points.

That does, in fact, give an overall average of +4: two-thirds of +7, added to one-third of -2. 

Now, in the second half, the better team will add on another 4 points of differential.  So you can add 4 points to both of those:

Two-thirds of the time, the favorite is leading at the half, and the final differential will average +11 points.
One-third of the time, the underdog is leading at the half, but the final differential will average only +2 points (for the favorite).

Now, what if you give a 4-point bonus to the team leading at the half?  Then, two thirds of the time, it turns a +11 into a +15.  Doesn't matter much; the favorite is overwhelmingly likely to win either way.  But, one-third of the time, it turns the worse team from a 2 point underdog, to a 2 point favorite!  That's a big deal.  And that's why the bonus favors the underdog team.

The bonus makes a much bigger impact when the game is close, and when the underdog has a chance, the game is more likely to be close than a blowout. 

6.  Now, suppose that instead of giving points to the team that's leading at the half, you give them *extra possessions*.  Same thing, right?  The extra possessions will just translate to points.  Even more so if you give them extra possessions, and simultaneously take extra possessions away from the trailing team!  Even though it's the favorite that gets the extra possessions twice as often as the underdog does, the rule change still helps that underdog win more games.

7.  In summary: anything that gives a benefit to the team that's leading *favors the underdog*.  That's because the underdog is more likely to be leading partway through the game than at the end of the game, and the benefit has the effect of increasing that lead, thus partially cementing it.

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PART II

Now, what does this have to do with "Make it, take it"?  Let me show you.

Here's a possible play-by-play of a "make it, take it" game.  I'll use "U" for underdog and "F" for favorite, and follow each possession by the number of points scored (either 2 or 0).  I'll put each "run" of possessions on its own line. 

U2 U0
F0
U0
F2 F2 F2 F0
U0
F2 F0
U2 U2 U0
F0
...

You'd read the game horizontally.  So, "U2 U0 [new line] F0" means, "the underdog scored 2 points (U2).  So they kept possession, but this time missed (U0).  So the favorite then took possession, and missed (F0)."

The chart will continue on until the game hits 200 possessions (which we assume to correspond to 48 minutes).

How many rows will the chart have?  About 100.  There will be 100 "runs" on average.  (Why?  Because the average length of a "run" is around 2 possessions.  The teams are both around a 50 percent success rate.  Therefore, there's roughly one make for every miss.  Since every run corresponds to exactly one miss, the average run has 1 make.  That gives a run length of two.)

Now, the key to it all.  Turn the above play-by-play on its side:

U2 F0 U0 F2 U0 F2 U2 F0 ...
U0 -- -- F2 -- F0 U2 -- ...
-- -- -- F2 -- -- U0 -- ...
-- -- -- F0 -- -- -- -- ...


Now, it doesn't matter what order we consider the possessions in, since they're independent, and since the total score is going to be the same either way.  So, imagine that the game actually unfolded this new way.  In that case, the top row, being 100 possessions wide, can be considered the first half.  The rest of the rows, combined, can represent the second half.

See where we're going with this?  The first half has alternating possessions by team.  So it's like normal basketball. 

Therefore, we can rewrite the rules of "Make it, take it" as follows: Play a normal first half.  Then, spend the second half awarding the extra "make it" possessions. 

How do we do the second half?  We start by awarding the second row.  Each team gets one extra possession for every basket they made. 

That means that the team leading after half the game gets more extra possessions than the team trailing after half the game.  That is -- we're giving a bonus for the team leading.  And as we already saw, that favors the underdog.

And the same for the third row: we give an extra possession for every basket made in the second row.  Again, the more successful team gets the bonus.  In this case, it's not necessarily the leading team -- it's the team that had a better second row.  But it's *probably* the team in the lead, so it still *probably* benefits the underdog.

And the same for the fourth, and fifth, and sixth rows, until there are no more bonuses to give.  Each row, on average, benefits the underdog a little bit.

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Get it?  The "Make it, take it" rule gives the underdog a better chance to win the game, because it has the effect of helping cement a lead at several points during the game.  Since the underdog is more likely to be leading partway through the game than after it, the cementing helps them disproportionately.

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There you go!  As I said, I'm open to better explanations; this was the simplest I could think of.




UPDATE: Geoff Buchan's results are here.