The Shortstack Game, Part 1

A recent post by gnome got me thinking, in part because I think he made an error in his post, which I describe in my comment there. As you might know if you follow the blog closely, I am a grad student in Economics. One of the things I study is game theory. When game theorists "solve" a game, what they look for (or at least the first thing they look for) is a "Nash equilibrium". When two players's strategies are in Nash equilibrium, each one knows the other's strategy, and even with that knowledge, neither one would choose a different strategy. Here's another way of saying it. Take some game with two players. Fix a strategy for player 2. Call player 1's "best response" to that strategy as the strategy available to him that maximally exploits player 2's strategy. Two strategies strategy1 (player 1's strategy) and strategy2 (player 2's strategy) are in Nash equilibrium if and only if strategy1 is player 1's best response to strategy2, and strategy2 is player 2's best response to strategy1.

As a whole, poker is far too complicated a game to find equilibrium strategies. There is not any guarantee that there exists a unique equilibrium. But sometimes we can take toy games that mimic at least some situations in poker, with some simplifications, and find an equilibrium. You'll find many such games in The Mathematics of Poker by Ankenmann and Chen, which I highly recommend. In any case, here we will solve 'The Shortstack Game', which works as follows:

Player 1 and Player 2 receive two cards. Player 1 can either fold or raise. If player 1 folds, he gets a payoff of zero and player 2 gets a payoff of 1.5. If player 1 raises, player 2 can either fold or shove. If player 2 folds, player 1 gets a payoff of 1.5 and player 2 gets a payoff of zero. If player 2 shoves, player 1 can either call or fold. If player 1 folds, he gets a payoff of -3 and player 2 gets a payoff of 4.5. If player 1 calls, a Hold 'Em board is dealt out and the player with the best hand wins. The winner gets a payoff of 21.5, whereas the loser gets a payoff of -20.

This game preserves the important features of a Hold 'Em situation where a player raises on the cutoff with a shorty on the button. I have made some simplifying assumptions, some of them important, some not. For instance, I say that when player 1 folds, player 2 also gets a payoff of 1.5. In reality, of course, player 2 won't always win the blinds when player 1 folds, but insteads enters into a new game with the players in the blinds. But that is inconsequential to our analysis here (although I plan to come back to it later), as all the decisions we're interested in are 1) whether player 1 decides to raise, and 2) what happens after player 1 raises. Neither of these depend at all on what player 2's payoff is when player 1 folds (convince yourself this is true or ask for clarification in comments if it is unclear).

Another assumption I've made is that the blinds fold every time (if player 1 raises and player 2 folds, player 1 wins 1.5, the blinds, automatically, and if player 2 shoves, the action is immediately back on player 1). This is obviously consequential as the potential for blinds calling or re-raising affects the payoff of player 1 when he raises and player 2 folds, and therefore how good raising is relative to folding for player 1. I could eliminate this need for simplification by making player 1 the small blind and player 2 the big blind, but: 1) If I recall correctly, there is already analysis of some very similar game in Ankenmann and Chen, and 2) Often as a good deep-stacked player you're going to want to be raising in position vs. the blinds so you can take a flop in position. I'd venture that 1.5 (i.e., winning the blinds for sure) is actually much lower than player 1's true EV of player 2 folding on the button to his raise.

Anyway, this post is plenty long enough already, so I think I'll stop there for now and allow for digestion and questions of the game setup before proceeding to solving the game. But if you want to do some work on it before next post, note that a strategy for player 1 consists of choosing either FOLD, RAISE/CALL, or RAISE/FOLD for every hand possible. A strategy for player 2 consists of choosing FOLD or SHOVE for every hand possible. What is player 2's best response to player 1 choosing RAISE/FOLD for every hand? What is player 2's best response to player 1 choosing RAISE/CALL for every hand?

-BRUECHIPS