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Is it correct when PIO shows the same EV if you bet or check? I thought EV will be diferent between betting and checking...
Sometimes PIO suggests to call a bet even when EV is negative... if we know EV is less than zero why should we call?
The way Pio works is that there are two strategies, one for IP one for OOP. Let's call them stratIP and stratOOP. When Pio starts running they are set to equal for all actions. In every step the solver moves them in direction of the highest EV. After a while this process gets the strategies closer and closer to equilibrium.
The way the solver calculates EV is to run current stratIP vs current stratOOP. Those strategies are not perfect equilibrium (just the best approximation till now) so there will be some places where action with not the highest EV is chosen some of the time. This will be less and less visible when strategies gets close to real equilibrium. The longer you run the solver and the better accuracy it reaches the less often not higher EV actions are chosen.
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if we know EV is less than zero why should we call?
The problem is that we don't know what EV is going to be in perfect equilibrium. We only know EV against current approximation. We can't make huge adjustment as always playing the best action against current opponent's strategy because it will result in being very exploitable (opponent can adjust and win vs us).
Imagine this happening in Rock, Paper, Scissors game. If you start with something like 50%-50%-0% strategy and your opponent is playing 0%-100%-0% (always paper) you want ot start playing scissors more often but you don't want to move to 0%-0%-100% instantly, you want to make small step first, let's say to 40%-40%-20% to see how opponent reacts. They will adjust as well and sooner or later you reach equilibrium (33%-33%-33%). If you keep going 100% Scissors, they will do 100% Rock, you do 100% Paper and you never reach the equilibrium.
I hope this is clear enough, feel free to ask follow up questions.