We don't like this graph very much. It could be wrong, and even if it's right, the parameter could be off by ~0.192. But fortunately, we have other evidence for the multiplier which comes from looking at the communication between client and server.

Here is a sample MITM catch log. What can be seen in the client-server communication is that a variable named normalized_reticle_size (NRS) is sent to the server when a ball is thrown. Experimental evidence shows that

$$NRS = 2 - r$$

where r is the radius of the inner circle as a fraction of the outer circle. In other words, for a full-sized circle, NRS is equal to 1, and for a circle of minimal radius, NRS approaches 2. We also have some lines from the game's code that shows:

"excellentThrowThreshold": 1.7, "greatThrowThreshold": 1.3, "niceThrowThreshold": 1,

NRS Range Avg Fit Nice 1.0 to 1.3 1.15 Great 1.3 to 1.7 1.5 1.518+/-0.192 Excellent 1.7 to 2.0 1.85 1.902+/-0.073

This means that a NRS from 1.0 to 1.3 gives a Nice throw, 1.3 to 1.7 gives a Great throw and 1.7 to 2.0 gives an Excellent throw. Summarizing the NRS range, the average of this range and our estimated values gives this table:

While the numbers in our fit aren’t exactly the average of the numbers within the range, they are sufficiently close to those values. Our data did not rule out the possibility for there to be flat multipliers for different bonuses, but the fact that the client sends the server a parameter which fits very well for both Great and Excellent bonuses suggests that this parameter is the multiplier.

To add to the evidence, there would be no reason for a value of 2-r to even be calculated if all that mattered was if a throw was Nice, Great or Excellent. Niantic could easily make the thresholds the exact ratio of the inner circle radius to the outer circle radius and spare all our phones some battery life. The fact that they don't do this suggests that the value is "normalized" so that it can be used as a throw multiplier.

Thus, we hypothesize that the Throw multiplier is equal to 2-r and varies continuously as the inner circle size decreases.