To understand Relative Brier Scores and their calculation, it is essential to first understand basic Brier scores and how they are calculated. If you are not familiar with Brier scores, you should start by reading our article What is a Brier Score and How is it Calculated?
Note: in the past, we referred to Relative Brier Scores as Net Brier Points. They are functionally the same  different names for the same calculations.
In forecasting tournaments, the importance of fair and equitable scoring is
paramount. If forecasters can gain an unfair advantage in scoring, then it
undermines the goal of identifying the best forecasters. A traditional
Brier score can be effective for comparing forecasters if all participate equally. That is to say, they all forecast
every day on every question.
But what happens in a forecasting tournament where participants
choose the questions and days on which they forecast?
In that scenario, a forecaster may choose to wait until late in a question
before submitting a forecast, when picking the correct outcome is much easier.
To illustrate the Relative Brier Score concept, we can continue the example from
What is a Brier Score and How is it Calculated? In that post, we showed a
hypothetical forecast on the question "Will the Cubs win the World
Series?" and calculated the associated error/score for the scenario where the Cubs win.
n.b. the values in the 'Forecast' row are 'yes,no' forecasts. So
'0.9,0.1' would correspond to the forecaster saying there is a 90%
chance the Cubs will win the World series and a 10% chance they won't.
Forecaster #1

Day 1  Day 2  Day 3  Day 4  Day 5  Day 6  Day 7  Overall Brier Score 

Forecast  0.9, 0.1  0.9, 0.1  0.9, 0.1  0.95, 0.05  0.95, 0.05  0.95, 0.05  0.95, 0.05  
Score  0.02  0.02  0.02  0.005  0.005  0.005  0.005  0.0114 
Now, let's consider 2 other forecasters in the same question:
Forecaster #2

Day 1  Day 2  Day 3  Day 4  Day 5  Day 6  Day 7  Overall Brier Score 

Forecast  0.25, 0.75

0.25, 0.75

0.2, 0.8  0.2, 0.8  0.2, 0.8  0.2, 0.8  0.2, 0.8  
Score  1.125

1.125 
1.28 
1.28  1.28  1.28  1.28  1.2357 
And Forecaster #3 (a blank cell indicates no forecast, and thus no score, for that forecaster on that day):
Forecaster #3

Day 1  Day 2  Day 3  Day 4  Day 5  Day 6  Day 7  Overall Brier Score 

Forecast  0.99, 0.01  0.99, 0.01  
Score  0.0002  0.0002  0.0002 
So at 0.0002, Forecaster #3 ends up with the best Brier score (remember, 0.0 is the best possible Brier score and 2.0 is the worst). This seems a little unfair though  Forecaster #3 waits until Day 6 when the answer might already be obvious. Shouldn't Forecaster #1 be rewarded for making good forecasts early in the question?
Relative Brier Scores (also known as Net Brier Points) were created to correct this inequity  a more fair
scoring system that rewards early, accurate forecasts. Like traditional Brier Scores, lower is better with Relative Brier Scores (you can think of it like a golf score, where below par is better).
To calculate a Relative Brier Score:
To complete our example, let's calculate the median of the daily scores (step #1 above):
Forecaster #1

Day 1  Day 2  Day 3  Day 4  Day 5  Day 6  Day 7 

Forecaster #1 Score  0.02  0.02  0.02  0.005  0.005  0.005  0.005 
Forecaster #2 Score

1.125

1.125 
1.28 
1.28  1.28  1.28  1.28 
Forecaster #3 Score

0.0002  0.0002  
Median Daily Score

0.5725

0.5725

0.65

0.6425

0.6425

0.005  0.005 
Now we can subtract the median from the
forecaster's score, sum the forecaster's daily Relative Brier Scores, and divide by the total number of days for the question (7 days, in our example):
Forecaster #1

Day 1  Day 2  Day 3  Day 4  Day 5  Day 6  Day 7  Overall NBP 

Forecaster #1 NBP

0.5525  0.5525  0.63  0.6375  0.6375  0.0  0.0  0.43 
Forecaster #2 NBP

0.5525  0.5525  0.63  0.6375  0.6375  1.275  1.275  0.7943 
Forecaster #3 NBP

0.0048  0.0048  0.00137 
If you're interested in learning more about Brier scoring, Relative Brier Scores, or running a forecasting tournament, feel free to contact us.