This is a fun problem
btw, the reason why I asked earlier about an unexploitable strategy being the same thing as the game theory optimal is this: I understand what was said above about the NE strategy being where neither opponent can improve their play despite knowing the other person's strategy. However, I don't care about my opponent adjusting to my play - or rather, I will assume that he in fact plays extremely +EV against me (to envision a worst case scenario for me), so that I can make adjustments from there based on the ways I think my opponent will fail to play perfectly against me.
If that's confusing (probably is), let's go back to the model I posted above:
When I say I want an unexploitable shoving strategy in BB's spot, what I want is to know what hands I can profitably shove even if I turn my hand face up and the SB plays perfectly against my shove. In algorithmic form, it would be like:
Code:
for (hand X : all hands bb can have) {
// so here we iterate through each hand - 32o, 42o, ..., AA
range Y = compute hands with which SB can profitably call vs X
EV = compute EV of BB's shove against a calling range of Y
if EV > 0, add X to the list of hands BB can profitably shove, else don't
}
I don't write pseudocode for a living so hopefully that gets the point across. In the situation I'm looking at, each hand is discrete and its EV is solely determined by its hand strength, not other hands in my range. As such, in this situation, I would never shove a -EV hand.
When talking about NE strategies, though, I do see what scylla says about the need to include -EV hands. Even if I can't explain it, all I can say is that I've experienced the same thing he has when I try not to include -EV hands. In the model above, say I give SB a calling range of "all hands" as a starting point. I then compute BB's optimal shoving range and set it. In response, I then set SB's optimal calling range of that shoving range and set it. I now have to recompute BB's optimal shoving range, and what happens (like scylla said) is that these ranges get tighter and tighter until SB's calling range is so tight that BB's optimal shoving range opens back up to almost ATC - and then we're right back where we started. The ranges don't converge. Since this is what happens when you only include +EV hands in each player's range, I can only conclude that a NE solution will have -EV hands in player's ranges.