Since the topic of equivalent EVs has popped up again, I thought to add where my thoughts ended up after the discussion of the
Equivalence of EV thread on the poker theory forum.
First of all I’d start with the assertion that against a GTO opponent there are in fact an infinite number of optimal solutions. But only one of those solutions is unexploitable, namely the GTO solution. This seems quite obvious and consistent with the observation that against a non-adaptive player it doesn’t matter which line is taken when the EVs are equivalent.
In the thread linked there is an excellent and simple example presented by whosnext which demonstrates this perfectly.
Player 1 can choose Top/Bottom Player 2 Left Right. There are specific rewards for each player depending on which of the four possible outcomes occurs. The Nash Equilibrium is 70:30 Top/Bottom for Player 1, 40:60 Left/Right for Player 2. The EVs of the two lines within each player’s strategies are equivalent (48/48 and 52/52 respectively). Of course if we freeze Player 2’s strategy, as done in GTO, then clearly Player 1 can choose any distribution and still be optimal. Against a non-adaptive opponent he could freeze at say 90:10 and still have an EV of 48.
As pointed out the EVs at this point are equivalent whichever line the player takes; however, Player 1 is exploitable if Player 2 is unfrozen, except of course at the Nash strategy.
Line frequencies: the simulators care about one metric only, EV. So if the line-frequencies are not arbitrary then they matter because they affect the EV. It is often inferred, it seems, that these frequencies are chosen to avoid exploitation as if this is perhaps some qualitative or meta-metric distinct from EV, but of course it must be translatable into the numbers on the bottom line. As such line-frequencies must determine the EV’s of other lines, otherwise what else is there to affect?
As we see in the above example and was stated in the other thread, the optimal unexploitable mixed strategies exist where lines of EV are equal: i.e. the frequencies that correspond to these line-distributions are requisite for the equality present in the mixed strategies (against an adaptive opponent). In short, line frequencies determine line-EVs.
As for mixed strategies it would be nice to see a proof demonstrate how a mixed strategy of unequal EVs can always be improved upon and so optimisation/unexploitability occurs only with equality of EVs (which would see to be the case). The position that a mixed strategy can always be improved upon by maxing the line with the greater EV holds of course only against a frozen strategy and doesn’t necessarily represent an improvement against an adaptive opponent where moving to a mixed strategy of equal EVs clearly does.
Anyhow, these were my concluding thoughts, until at least I spend some more time on this - I don’t wish to clog up this thread any further as I’m sure there are more interesting subjects from Matthews book to discuss.
Thanks.