Level 41: The Aggression Advantage
There’s a simple version of poker that is frequently used in GTO analysis of poker. It illustrates many concepts in poker, from polarization to the power of balance between bluffs and value, but it also seemed to me to show the basic power of aggression, and how it can in theory enable an over realization of equity.
The “test” game is described in many ways in different texts. For simplicity I’ll just make it a game with 3 cards, A, K and Q, and no streets and no community cards. Each player has $1 behind, and there is always a pot of $1, and players are allowed to bet $1 only, or check, and if facing a bet, call or fold. Player 1 will always be dealt either an ace or a queen, while player 2 just always has a king.
It’s pretty easy to see that the equity of player 2’s holding (K) is 50%, as half the time player 1 will be dealt an ace, and the other half a queen. If both players always check, that equity will be realized.
It is also pretty easy to see that player 2 should never bet. Player 1 knows he always has a king, and knows whether or not he is ahead or behind, and will always fold to the $1 bet if he has a queen, and always call if he has an ace. So a bet by player 2 can never win money, but can lose money.
What happens if player one always value bets the ace, and always checks the queen? Well, initially that will allow player 1 to improve his equity above 50% from the times player 2 calls and loses, but if player 2 sees player 1 bet 100 times, and every time has an ace, then eventually player 2 will fold and equity will be back at 50%.
Conversely, if player 1 never bets A, and always bets Q, if player 2 always folds, that will drive player 1’s equity to 100%, but if player 2 shifts to always calling, that would then drive player 1 equity down, as $1 winnings with the ace are offset by $2 losses with the queen.
So both always value betting and never bluffing, and always bluffing but never value betting seem limited in value. While always value betting can lift equity above 50%, especially initially, to the extent your opponent knows you never bluff, your equity again approaches 50%. Always bluffing is considerably worse; while it can again drive your equity initially over 50%, to the extent that your opponent begins to understand that all your bets are bluffs, it can actually drive your equity below 50%.
But what happens if player 1 always bets the ace, and sometimes bets the queen? Well, it turns out that if he bets the queen 50% of the time, there is actually nothing player 2 can do to prevent player 1 from capturing 75% of the pot, and that player 1 is actually indifferent to the calling percentage that player 2 adopts: at every point between 0% calls to 100% calls player 1 still captures 75% of the pot. Player 2 actually has an ideal calling frequency (the MDF from level 30), in that this is the only frequency that restricts player 1 to 75%. Any other frequency would allow a counter exploit (for example, 100% folds from player 2 would allow player 1 to always bet and capture 100% of the pot).
So by betting with a carefully crafted mix of value and bluffs, you can achieve a larger share of the pot than the equity share of your actual holdings that cannot be denied even with perfect play.