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Game Theory: Rochambeau to Poker (Part 1)

The first time I heard the term, Rochambeau, it was in reference to a one-on-one pain contest between two men. Competitors take turns kicking each other in the balls, and as in all pain contests, the winner is the player who doesn’t concede. As much as I love testing my mettle, I’ve never engaged in such a game of Rochambeau (and I have no plans to do so in the future). However, I’ve engaged in another game called Rochambeau that many of you are probably familiar with: Rock-Paper-Scissors. An in-depth look at this seemingly simple game can teach an extremely valuable poker lesson.


(If you don’t know how Rock-Paper-Scissors is played, check out the Wikipedia entry before reading further)


In Rock-Paper-Scissors, each player’s strategy can be completely specified as follows (P(X) represents the probability that X is thrown):


{P(Rock),P(Paper),P(Scissor)}


Suppose you’re playing Rock-Paper-Scissors against me (of course, as a pain contest where the winner gets to execute a two-finger slap on the loser’s wrist, you wuss!). If I win a throw, the payout is that I slap you (+1 for me; -1 for you); if you win a throw, you slap me (-1 for me; +1 for you); if we tie a throw, we throw again (0 for me; 0 for you).


I tell you that my strategy for the first throw is {1,0,0} (i.e. I’m going to throw rock 100% of the time). Can you devise a strategy to beat me? Yes, you can use {0,1,0} for your strategy, and you’ll get to slap me 100% of the time. Ouch! The net payouts for the throw would be -1 for me and +1 for you.


For all subsequent throws, I tell you that my strategy is {1/3,1/3,1/3}. Can you devise a strategy to beat me on all subsequent throws? No; in fact, a little algebra shows that our expected payouts on each throw are 0 regardless of what strategy you choose. Therefore, the {1/3,1/3,1/3} strategy is simultaneously really weak and really powerful! It’s really weak, because by implementing it, you’re ensuring that you’ll never win. It’s really powerful, because by implementing it, you’re ensuring that you’ll never lose…it’s non-exploitable in the sense that your opponent can only break-even against it even if he knows exactly what your strategy is!


What does this discussion have to do with poker? Everything! There are international Rock-Paper-Scissors matches, and top players claim the ability to exploit their opposition by identifying how their opposition deviates from the {1/3,1/3,1/3} equilibrium strategy and then employing the proper exploitative strategy. When poker players read their opponents, they are doing the same thing that top Rock-Paper-Scissors players do; poker players simply do it in a game that’s much more complicated and where non-exploitable play is unknown in many situations.


Nonetheless, there are situations in poker where close approximations to non-exploitable play are known. Jam/fold play with respect to chip EV in no-limit hold’em is one of them. And unlike in Rock-Paper-Scissors, opponents who deviate from perfectly countering non-exploitable jam/fold play will lose chips in the long run. For example, you’re at a 9-handed table, action folds to you, and you’re in the small blind. Blinds are T100-T200 with a T25 ante. Prior to the posting of antes and blinds, you and your opponent both have T2,025. The approximate non-exploitable strategy with respect to chip EV here is to jam {22+, Ax, Kx, Qx, J2s+, J3o+, T2s+, T6o+, 92s+, 96o+, 84s+, 86o+, 73s+, 75o+, 63s+, 65o+, 53s+, 43s} (75.3% of hands), and your opponent’s proper response is to call with {22+, Ax, Kx, Qx, J2s+, J6o+, T5s+, T7o+, 96s+, 98o, 86s+} (59.6% of hands)…according to the ICM Nash Jam/Fold Calculator at HoldemResources.net.


To prove that this equilibrium is different from the one in Rock-Paper-Scissors (where payouts are zero as long as only one player plays non-exploitably), we can consider a simple counter-example: the big blind will fold 100% of the time. If the big blind folds 100% of the time, you’ll pick up the blinds and the antes (T525) 100% of the time that you shove, meaning the expected payout of your strategy is about (.753)(+T525) + (.247)(T0) = +T395.325 (i.e. you profit with respect to chips…even after accounting for the small blind and the ante). The big blind’s expected payout is about (.247)(+T525) = +T129.675 (i.e. the big blind also profits, but not after accounting for the big blind and the ante).


To this day, poker is mainly thought of as a game involving reads and adaptation. However, an increasing number of players are beginning to understand the theory behind non-exploitable play (even if non-exploitable play for numerous situations has yet to be solved for). Achieving an understanding of this topic will ensure that you’ll know more than your competition as poker continues to evolve.


Look for a second part to this blog post soon, where I’ll continue to analyze this jam/fold situation. Also, be sure to tune into Killer Poker Analysis this Friday at 5:00PM PT on Rounder’s Radio. I’ll be covering the following topics:


1.) For those with small bankrolls and who aren’t ready to shell out the cash for Poker Tracker 3 or Hold’em Manager: Ways to gain an edge in online cash games without using player tracking software
2.) A seemingly popular topic these days: preflop 3-betting in no-limit hold’em cash games
3.) Shorthanded no-limit hold’em and pocket pairs


May Your EV Always be Positive!


Tony Guerrera




  



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