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I've seen a number of solutions that claim to be optimal, but they don't all agree. It seems to be time for a public discussion of methods of solution so that the peer-review process can take place.
I'm more than happy to talk about how I arrived at my (very good) approximate solution.
For those worried about losing EV, don't. The willfully ignorant will remain blissfully so.
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I got my solution by using ficticious play (as did Streib). I checked my solution against one that was constructed independently using the same method and they agreed. My solution agrees with the sample solution posted by eastbay, who used the simplex method to solve the problem directly. If I remember correctly, both methods produce the same optimal solution. The simplex method produces it directly, whereas ficticious play slowly converges to the solution.
Streib used ficticious play, but didn't allow for mixed strategies, which meant that his final solutions probably oscillated around the optimal solution, but never converged. My guess is that he just picked one of the solutions when it started to oscillate.
I certainly didn't mean to impune Streib's honesty. I was just looking for clarity in my heavy handed manner. And to be fair, the main point of the article isn't the solution per se, but how deviating from the solution affects your EV.
In truth, because it's the solution to a toy game, it's not vital that it be 100% correct, because it doesn't map directly to true play.
- Andrew
www.pokerstove.com