## Thursday, October 9, 2014

### GTO Brainteaser #8 -- Solving the Turn, Theory vs Practice

The internal alpha version GTORB is now capable of solving turn scenarios for GTO turn and river play so this brainteaser is going to focus on multi-street theory.  In the solution (probably a week from today) I'll post the first fully browse-able GTORB turn solution to the model game below for those of you who are excited play around with a GTO turn strategy.  Note that the version of GTORB that can solve the turn won't be released commercially for a month or two as there are some performance / scalability issues that I need to solve before it is ready for mass use.  It will likely cost extra.

### The problem

In GTO Brainteaser #6 I looked at a model scenario where the hero had a range of 50% nuts, 50% air while the villain had a range of 100% medium strength hands.  There was a 100 chip pot, 150 chip stacks and two streets of betting.  The hero could either bet 50 chips on the turn and then have the option to bet 100 chips on the river or he could shove the turn for 150 chips, and the question was which option is higher EV and what are GTO strategies for both players in this game.

The key simplification that made this scenario quite different from real world poker is that it was assumed that no river card was actually dealt, there were just two rounds of betting.

For those who are curious you can check out the full solution to brainteaser #6 here.  It turns out that it is optimal for the hero to bet 50 chips on the turn with all of his nut hands and 7/9ths of his air and then to bet 100 chips on the river when he is called with all of his nut hands and 3/7ths of his air hands.  The villain calls each of these bets 2/3rds of the time and folds 1/3rd.  The hero wins 8/9ths of the 100 chip pot in EV in this game.  Furthermore, it turns out that betting 50 chips on the turn and 100 chips on the river is the exact optimal bet sizing for the hero to maximize his EV, all other bet sizes are lower EV.

Lets now look at a very similar game.  Imagine the following (completely made up) scenario.

1. You are on the turn and the board is  AsAhKsKh
2. The hero has a hand range of AcAd and 3c2d
3. The villain has a hand range of KcKd
4. The pot is 100 chips and stacks are 150 chips
5. The hero can either bet 50 chips on the turn and 100 on the river or he can shove for 150 on the turn.

Clearly no matter what river card comes, the relative strengths of the hands in both players ranges will not change so in that respect this game seems identical to the model game from brainteaser #6.  AcAd will beat KcKd on every possible river and 3c2d will lose to KcKd on every possible river.

However, it turns out that GTO play in this game is different from GTO play in GTO Brainteaser #6.  Why?

1. What is the EV of the game for the hero, is it higher or lower?
2. What are the optimal strategies in this game and what is the hero's EV when both players play optimally?

Bonus:  Is betting half pot on the turn and the river still optimal or is there a higher EV bet size?