I enjoy probability puzzles almost as much as I enjoy gaming (and in fact gaming was what inspired me to learn some probability) so I got curious: What're the odds of winning an opposed die roll, and how do those odds change as my bonus (relative to yours) changes?
I worked it out and then wrote up how I approached it in case others here also enjoy the intersection of gaming and math. (I'm linking to it as a Google doc rather than pasting it in here because it involves equations that don't display well here). My goal in writing it up was to make the math understandable, and questions on it are welcome. Likewise, there may be other ways to model the same problem. If you would model it differently (especially in a simpler way) I'd love to learn from you.
Finally, I linked to it at the end of the doc, above, but here's a spreadsheet that I made which calculates the odds of winning for any relative bonus (and how much of an advantage an additional +1 gets you at each step). Feel free to skip straight to this if all you want to know is the "answer" for various rolls.