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iRating System

Important

iRacing does not publicly release its exact iRating formula, but we can approximate the system with a similar approach based on the Elo rating model with adjustments for position-based rankings, field size, and Strength of Field (SoF). Here’s a version of the formula that should give us reasonable results for the project.

How to calculate it - Step by Step

Variables definition

Variable Description
$R_{before}$ Driver current iRating before the race
$SoF$ Strength of Field, calculated as the average iRating of all drivers in the race
$Position$ Driver final position in the race
$N$ Number of drivers in the race

Calculate Expected Finish Probability

Driver Expected Performance against the $SoF$ can be modeled with a probability using the logistic formula: $$E=\cfrac{1}{1 + 10^\cfrac{SoF - R_{before}}{400}}\$$ $E$ represents driver expected performance or "win" probability against the field based on his iRating relative to the $SoF$.

Determine the Scaling Factor ($K$-Factor)

This factor influences how much driver's iRating can change in a single race. For motorsports, a typical $K$-factor might be set between 30 and 100, depending on field size and competition level. Let’s define a scalable $K$-factor based on the field size: $$ K=30+\cfrac{70}{N} $$

Calculate Actual vs. Expected Performance

Define an Actual Performance Score ($S$) based on your position: $$ S=1−\cfrac{Position−1}{N−1} $$ This way, the winner gets $S=1$, and the last-place finisher gets $S=0$, with values in between for other positions.

Calculate iRating Change

Finally, use the Elo-based adjustment for the iRating change: $$ΔR=K*(S−E)$$ This $ΔR$ is your iRating gain (positive) or loss (negative).

Update iRating

New iRating: $$R_{after}=R_{before}+ΔR$$

Example Calculation

Let’s say:

  1. Driver initial iRating $R_{before}$ is 1500. The race $SoF$ is 1600. Driver finish in 5th place out of 20 drivers.

  2. Expected Finish Probability: $$E=\cfrac{1}{1+10\cfrac{(1600−1500)}{400}}=\cfrac{1}{1+10\cfrac{100}{400}}≈0.36$$

  3. Scaling Factor ($K$-Factor): $$K=30+\cfrac{70}{20}=30+3.5=33.5$$

  4. Actual Performance Score ($S$): $$S=1−\cfrac{5−1}{20−1}=1−\cfrac{4}{19}≈0.79$$

  5. Calculate iRating Change: $$ΔR=33.5×(0.79−0.36)=33.5×0.43≈14.4$$

  6. Updated iRating: $$R_{after}=1500+14.4=1514.4$$

This approximation captures the essence of iRating adjustments in iRacing. The parameters (such as $K$-factor) can be further tuned to match observed iRacing behavior more closely.

Taking into account the Theoretical Performance of the car

To account for the theoretical performance of each car in a formula like iRating, we can adjust the expected performance ($E$) to include the handicap due to machinery. Here's how we are going to integrate this adjustment:

Incorporating Car Performance Handicap

New Variables:

Variable Description
$CarPerf$ The theoretical performance of the car. The fastest car has $CarPerf=0$, and slower cars have positive values representing the time (in seconds) they are theoretically slower.
$AdjustedPerf$ An adjusted theoretical performance score for each driver, used to calculate their relative competitiveness.

Adjusting Expected Performance ($E$):

To account for car performance, we are going to modify the Expected Finish Probability formula: $$AdjustedPerf=R_{before}−(α×CarPerf)$$

Where $α$ is a scaling factor to convert the time deficit (in seconds) into an iRating penalty. For example, if $α=50$, a car that is 0.5 seconds slower would lose $0.5×50=250$ iRating points in its adjusted rating.

Then, use the adjusted iRating in the logistic formula: $$E=\cfrac{1}{1+10^\cfrac{SoF−AdjustedPerf}{400}}$$

This means slower cars will have a lower Expected Performance $E$, accounting for the theoretical disadvantage.

Example adjusted calculation:

Scenario:

  • Driver A: $R_{before}=1500$ $CarPerf=0$(fastest car).
  • Driver B: $R_{before}=1500$ $CarPerf=0.5$ (0.5 seconds slower).
  • $α=50$: Time penalty conversion factor.
  • $SoF=1550$
  • $N=20$.

Steps:

  1. Adjusted Performance:

    • Driver A: $$AdjustedPerf_A=1500−50×0=1500$$
    • Driver B: $$AdjustedPerf_B=1500−50×0.5=1475$$

  2. Expected Finish Probability:

    • Driver A: $$E_A=\cfrac{1}{1+10^\cfrac{(1550−1500)}{400}}≈0.36$$
    • Driver B: $$E_B=\cfrac{1}{1+10^\cfrac{1550−1475}{400}}≈0.29$$

  3. Actual Performance (SS):

    • Assume Driver A finishes 1st ($S_A=1$).
    • Assume Driver B finishes 10th ($S_B=1−10−120−1≈0.53$).

  4. iRating Change: $$K=30+\cfrac{70}{20}=33.5$$

    • Driver A: $$ ΔR_A=33.5×(1−0.36)≈21.5$$
    • Driver B: $$ΔR_B=33.5×(0.53−0.29)≈8.0$$

  5. New iRating:

    • Driver A: $$R_{A after}=1500+21.5=1521.5$$
    • Driver B: $$R_{B after}=1500+8.0=1508.0$$

Summary:

By incorporating the $CarPerf$ handicap into the formula:

Slower cars are given a lower expectation ($E$), so finishing above expectations rewards them more. Faster cars face a higher $E$, meaning they need to win or perform well to avoid losing iRating.

This approach balances performance differences due to machinery while preserving competitiveness.