S
sxm59
Rising Star
Bronze Level
I will use the following example in several posts to explain a few GTO concepts and their usefulness and limitations.
Example:
The river is a blank. Villain has air with probability 50% (say, he missed a flush) or a strong hand with probability 50% (say, he flopped a set). Hero has only a bluff catcher (say, she flopped top pair). [Technical note: I will assume that both players are rational, selfish, and risk-neutral. Furthermore, everything about this example is common knowledge of the two players. In future posts, I may discuss how the equilibrium changes if these assumptions are not satisfied.]
For simplicity, suppose there is $30 in the pot, Villain is first to act and has $10 left in his stack. Hero has a much larger stack. Suppose Villain will either check or shove (that is, bet all his remaining $10).
The Perfect Bayesian Nash Equilibrium strategies are as follows:
1) When Villain has a strong hand, he shoves; when he has air, he shoves 25% of the time and checks 75% of the time.
2) When Villain checks, Hero checks; when Villain shoves, Hero believes that Villain has air with probability 20%, and Hero calls 75% of the time and folds 25% of the time.
In my next post, I will prove that the above strategies and beliefs are the unique (Perfect Bayesian Nash) equilibrium. That will be a bit technical. Here, I will do something more fun. I will try to explain how the equilibrium changes if the pot is larger than $30.
There are two key facts to keep in mind. First, your opponent's action conveys information. In the above example, initially (just before Villain's action) Hero correctly believes that Villain has air 50% of the time. However, because in equilibrium Villain shoves less often when he has air than when he has a strong hand, as soon as Hero sees Villain shoving, Hero understands that it is now less likely (than before seeing the shove) that Villain has air. In the example, that likelihood or probability decreases from 50% to 20% because Villain only shoves 25% of the time when he has air (versus 100% of the time when he has a strong hand).
Second, a rational player will randomize between two actions--say, between calling and folding--only if he or she is indifferent between the two actions, that is, only if the two actions have the same expected value ("EV"). Indeed, if calling has a lower EV than folding (in a given particular spot), why would a rational player call with any positive probability? If you prefer beer over wine, why would you flip a coin to decide what to drink?
Now, consider again the above example but suppose that the pot is $40 instead of $30. If Villain does not change his strategy, then Hero will prefer to call Villain's shove all the time (since when Villain has a strong hand, Hero still loses the same $10 as before, but she now wins $50 instead of $40 when Villain has air). But if Hero calls all the time, then Villain should change his strategy: he should not shove when he has air. So, the equilibrium strategies must be different. Specifically, to prevent Hero from calling all the time and to keep her on the fence between calling and folding, Villain must bluff-shove less often (16.7% of the time instead of 25%) when the pot is larger ($40 instead of $30). And to prevent Villain from shoving all the time and keep him on the fence between shoving and checking, Hero must call more often (80% of the time instead of 75%) when the initial pot is larger ($40 instead of $30).
To be continued...
Example:
The river is a blank. Villain has air with probability 50% (say, he missed a flush) or a strong hand with probability 50% (say, he flopped a set). Hero has only a bluff catcher (say, she flopped top pair). [Technical note: I will assume that both players are rational, selfish, and risk-neutral. Furthermore, everything about this example is common knowledge of the two players. In future posts, I may discuss how the equilibrium changes if these assumptions are not satisfied.]
For simplicity, suppose there is $30 in the pot, Villain is first to act and has $10 left in his stack. Hero has a much larger stack. Suppose Villain will either check or shove (that is, bet all his remaining $10).
The Perfect Bayesian Nash Equilibrium strategies are as follows:
1) When Villain has a strong hand, he shoves; when he has air, he shoves 25% of the time and checks 75% of the time.
2) When Villain checks, Hero checks; when Villain shoves, Hero believes that Villain has air with probability 20%, and Hero calls 75% of the time and folds 25% of the time.
In my next post, I will prove that the above strategies and beliefs are the unique (Perfect Bayesian Nash) equilibrium. That will be a bit technical. Here, I will do something more fun. I will try to explain how the equilibrium changes if the pot is larger than $30.
There are two key facts to keep in mind. First, your opponent's action conveys information. In the above example, initially (just before Villain's action) Hero correctly believes that Villain has air 50% of the time. However, because in equilibrium Villain shoves less often when he has air than when he has a strong hand, as soon as Hero sees Villain shoving, Hero understands that it is now less likely (than before seeing the shove) that Villain has air. In the example, that likelihood or probability decreases from 50% to 20% because Villain only shoves 25% of the time when he has air (versus 100% of the time when he has a strong hand).
Second, a rational player will randomize between two actions--say, between calling and folding--only if he or she is indifferent between the two actions, that is, only if the two actions have the same expected value ("EV"). Indeed, if calling has a lower EV than folding (in a given particular spot), why would a rational player call with any positive probability? If you prefer beer over wine, why would you flip a coin to decide what to drink?
Now, consider again the above example but suppose that the pot is $40 instead of $30. If Villain does not change his strategy, then Hero will prefer to call Villain's shove all the time (since when Villain has a strong hand, Hero still loses the same $10 as before, but she now wins $50 instead of $40 when Villain has air). But if Hero calls all the time, then Villain should change his strategy: he should not shove when he has air. So, the equilibrium strategies must be different. Specifically, to prevent Hero from calling all the time and to keep her on the fence between calling and folding, Villain must bluff-shove less often (16.7% of the time instead of 25%) when the pot is larger ($40 instead of $30). And to prevent Villain from shoving all the time and keep him on the fence between shoving and checking, Hero must call more often (80% of the time instead of 75%) when the initial pot is larger ($40 instead of $30).
To be continued...
