## Convexity in odds

Originally posted on 06/09/2021:

One thing that I think many struggle with, I know that I did at the beginning, is how to understand advantages and the triggers for different systems. Through messing around, and working off of probabilities instead of odds, I began to understand this concept, which I haven't seen written about, doesnt mean I came up with it, but ive never seen it mentioned, but in my mind I always though of the term "Convexity" for.

In the chart above, on the Y axis you have the change in implied probability for a one cent move, compared with the X Axis, which is the odds it is moving onto(for example, the very first item on the left is a move from -300 to -299, which is an implied probability of .75 (-300) to 0.74937, roughly 0.000627. We can see, compared to this, that a move from -101 to 100 is worth .0025, 4 times as much.

This information explains a few things I've heard throughout the years. The first being that money lines are almost always lower juice than spreads. In an NFL games, you are giving up the most valuable ten cents there are in vig. If you instead give up 10 cents in the -130 (-130 on a -120/120 vig free), your loss in implied probability is .019763 as compared to 0.02381, roughly a 17% savings.

Another point being that I always use difference in calculated vs offered probability to trigger systems. Earlier in my betting career, I would use cents. This was incorrect, as they are not all priced equally.

Hopefully, especially for newer bettors, they can use this information to save themselves some vig, and have a better idea of how to measure an edge.