### Is finishing 12th better than finishing 5th in Formula 3?

The rules dictating grid formation and points distribution in the Formula 3 2021 season are somewhat complicated. A race weekend consist of 1 qualifying session and 3 races. The winner of the qualifying session is awarded 4 points for this feat, and the result of the session dictate the starting grid in the third race. The third race is the most prestigious with 25 points for the winner and a gradually decreasing number of points for each of the remaining top 10 drivers.

The starting grid for the first race is also determined by the qualifying, but the top 12 drivers are reversed, so the winner of the qualifying get start position 12, number 2 gets position 11 and so forth, all the way to the driver who qualified 12th getting pole position. The winner of the race score 15 points, with points for the rest of top 10.

For the second race the grid is determined by the finishing positions in the first race, but again the top 12 positions are reversed. The scoring is the same as the first race.

For each race, 2 pints are also awarded to the driver who drove the fastest lap, these points will be disregarded for the remainder of this post as they are not particularly relevant to the problem.

Confusing? Indeed, but the whole grid reversal idea helps creating some exciting close races with lots of overtaking where the best drivers can't just put it on pole and cruse to the finish, at least that is the idea.

### Incentive graphs

For the purpose of this post I will introduce the concept of incentive graphs. An incentive graph is a graph detailing the point scoring incentive that drivers have for finishing a race or qualifying in each of the possible positions. For race 2 and 3 this is just a table detailing the points.

Fairly simple, but in the qualifying and race 1 drivers do not simply score points, they also score grid positions, that in turn help or hinder them scoring further points. In order to produce incentive graphs for these two events we have to quantify the value of grid positions in terms of points. One way of doing this is to run simple simulations with drivers overtaking each other, this way each grid position can be mapped to a probability distribution of all the possible finishing positions, which in turn can be mapped to an expected number of points.

Based loosely on observations of the season so far I have set up a model where in each race 60 overtakes happen, and then each driver has a 15% chance of retiring, sending them to the back of the field. This retirement chance is somewhat higher than the official retirement statistic indicate, but is supposed to also emulate spins, nose changes and other events that ruin someone's race.

With this model we can produce an incentive graph for qualifying.

The positions from 1st to approximately 8th seems fine, then a drop of less than 1 point from 9th to 12th seems a bit on the low side, only to be followed by a massive 6 point drop from 12th to 13th. Why this shape? Well, every position lost from 1st to 12th is rewarded with a grid slot improvement in race 1. While a better grid slot in the higher scoring race 3 is ultimately worth more, the difference is just not particularly big when one gets outside a realistic chance of scoring the big points. Qualifying 13th instead of 12th is no big difference in race 3, but a massive drop from 1st to 13th on the grid of race 1, hence the big incentive difference.

We can also make an incentive graph for race 1, combining the race 1 scores with the expected race 2 scores for the derived grid positions.

Again a big gap from 12th to 13th, to be expected as 12th gets a pole and 13th gets nothing much. Less expected perhaps is that 12th is rated as more valuable than 5th through 11th. That 12th is better than 11th is easy to explain, both gain 0 points, but 12th result in a better grid position for race 2. But when you move into the points surely that would offset the grid advantage? So it would be if points were distributed linearly.

In Formula 3 the points distribution is progressive, meaning that the score gap between two adjacent positions is greater in the top than in the bottom. A driver finishing 5th in race 1 will score 6 points and start in position 8 for race 2, if they don't manage to improve in race 2 that is just 3 points on top of the 6, making 9 in total. Meanwhile a driver finishing 12th in race 1 will start first in race 2, 15 points if they can hold the spot throughout the race, but still 12 or 10 points if they manage to only drop 1 or 2 spots.

### Real data

That was the theory, how does reality compare? This scoring system has so far only been used for 6 race weekends, which is not a lot for the purpose of drawing statistical conclusions. In any case, here is the average points scored in race 1 and 2 as a function of the position in race 1.

With lots of points for the first places, a plateau in the middle, and a sudden drop from 12th to 13th, this graph is certainly reminiscent of the its theoretical counterpart. The biggest difference is the relatively poor score for 10th and 11th place, it does not seem likely that these should be much worse than 9th place, so most likely the difference is either due to random variance in performance or the 10th and 11th place finishers have on average been worse drivers.

To test the last theory we can compare drivers finishing in various positions in race 1 to their season performance. Here is the average score per race for drivers finishing in various positions in race 1.

So indeed 10th and 11th place finishers are on average substantially worse than 9th place drivers, and a lot of other differences between the theory and the real data is neatly explained by difference in driver skill. If we subtract 60% of this table from the score table we get a rough correction for difference in driver skill.

With this correction the outliers then become position 6, 10 and 12. This makes the reverse pole seem very powerful, relative to the closest positions more so than the model predicts. This could be chance, or it may be that the model does not accurately reflect the value of pole. A pole position offers the driver a chance to run away, pull away from the beginning and never have to battle other drivers, offering both a lower chance of being overtaken, and a lower chance of retiring due to collisions. If this is true a 12th place may be more valuable than the model predicts.

### Yes or no?

So is a 12th place better than a 5th place in race 1? The model and data indicate that it may indeed be, but the data material is too shallow to allow a firm statement. In reality it probably depends a great deal on the track. A track where overtaking is hard makes grid position very important, and it seems very likely that 12th and resulting pole is better in this case. If overtaking is easy the pole is less important and a 5th place is probably a bit better.

In either case it seems silly that the rules are such that this question can be asked in a serious manner. That we haven't seen race strategists ask their drivers to drop positions is probably due to rules against intentionally losing, they would be punished if they did so in the open. On the other hand there is no rules against asking drivers to save the engine and stay out of trouble, which is boringly the very best advice anyone can give to a driver on their way to a 12th or better, with no hope of 4th, in a race 1.