"What are the odds?" LeBron and AD are Evenly Matched in a Sudden-Death One-on-One Game

I've answered FiveThirtyEight's Riddler question before here, and aim to answer this week's Riddler Classic via simulation:

Now that LeBron James and Anthony Davis have restored the Los Angeles Lakers to glory with their recent victory in the NBA Finals, suppose they decide to play a game of sudden-death, one-on-one basketball. They’ll flip a coin to see which of them has first possession, and whoever makes the first basket wins the game.

Both players have a 50 percent chance of making any shot they take. However, Davis is the superior rebounder and will always rebound any shot that either of them misses. Every time Davis rebounds the ball, he dribbles back to the three-point line before attempting another shot.

Before each of Davis’s shot attempts, James has a probability p of stealing the ball and regaining possession before Davis can get the shot off. What value of p makes this an evenly matched game of one-on-one, so that both players have an equal chance of winning before the coin is flipped?

I pared down my one-on-one basketball simulator to answer this question, randomly assigned a value to the probability of LeBron stealing in each game set, simulated each game 1,000 times, and then logged only simulation runs that ended in a 50/50 tie between LeBron and AD until I had 1,000 such instances.

Over those 1,000 evenly matched games, LeBron's steal percentage is 33.325% - which I'm guessing is due to variance over the simulation runs, so my answer is 33.33..%, or 1/3.

I went ahead and ran all steal outcomes 10,000 times again, to graph the relationship between LeBron's steal success rate and his win percentage. Once again, his win % (purple line) touches 50% at about 33% steal %: