[mykey1961]

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Why do you assume P2 bets with [x,1]?

why not [0,x] and [1-2x,1]?

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Because in this game, where lower hands *never* beat higher ones, there is no reason to bluff with the absolute lowest cards, as opposed to the cards just below the range where you're betting for value.

For example, lets say P2 is betting 50% of the cards, and that P1 is calling with 75%. (Yes, I realize this is not anywhere near optimal for either player, it is an example to illustrate a point)

It simply *does not matter* if P2's actual betting range is 1-25,75-100 or 50-100, or even 1-10,21-30,41-45,75-100 for that matter. P2's EV is exactly the same in all three of these situations, because P1 *never* calls with a card lower than 75. So P2 can only win when P1 folds, in which case it doesn't matter if P2's card was a 25 or a 74.

This seems to be a common mistake through these problems, based on the fact that in holdem it is often better to bluff with the bottom of the range and fold the middle, because you're far less likely to be dominated when you do call, giving you an extra 10 or 15% chance to win.

With that in mind, let's consider your "optimal" for a moment.

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Optimal is:

P1
[1,56] Fold
[57,100] Call

P2
[1,11] Bet
[12,78] Fold
[79,100] Bet


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This is *clearly* not optimal for P2, because P2 could gain value without changing the total number of cards he's betting simply by betting 67-100 instead of 1-11,79-100. You have P1 calling with many numbers lower than 67, so 1-11 has absolutely no chance of winning when P1 calls, but 67-78 definitely do have a decent chance of winning.


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If P2 changes his strategy to betting [67,100] then his EV goes from +0.111111 to +0.068687 against my P1

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And if P2 knows P1's strategy, he can increase his EV even further by betting a much wider range, somewhere in the neighborhood 30-something to 100.



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if P2 changes his strategy to:

betting [30,100] P2's EV = -0.169091
betting [31,100] P2's EV = -0.165455
betting [32,100] P2's EV = -0.161616
betting [33,100] P2's EV = -0.157576
betting [34,100] P2's EV = -0.153333
betting [35,100] P2's EV = -0.148889
betting [36,100] P2's EV = -0.144242
betting [37,100] P2's EV = -0.139394
betting [38,100] P2's EV = -0.134343
betting [39,100] P2's EV = -0.129091

That takes P2 from winning +0.111111 (my way) to losing instead.


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Your strategy is also clearly sub-optimal for P1. P2 is betting 33 numbers, so it is absolutely silly for P1 to call with anything lower than 67, because it has absolutely no chance of winning.


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against my P2, P1 is indifferent to calling or folding with 12 thru 78.

The thing is, P1 also has to defend against non-optimal P2's as well, so it can't always fold when it's indifferent to the optimal.

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An equilibrium (or "optimal", take your pick) strategy is one where neither player can adjust their strategy to gain EV, *even if they know the other player's strategy*.

In the "optimal" strategy you've given, if P2 told P1 what his strategy was, P1 could clearly adjust to increase his EV, and vice-versa.

That alone should be enough to show that your strategy is not optimal.


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That's exactly what I've done here

I'm telling you my P1, and P2.

Can you do better than +1/9 against my P1?
Can you do better than -1/9 against my P2?

The answer in both cases is No.

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*EDIT*

You also seem to be solving the wrong problem (and making a mistake that I also made in reading the problem).

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P2
[1,11] Bet
[12,78] Fold
[79,100] Bet

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"Fold" is not an option for P2.

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[12,78] Fold is a typo, should be check
Bouncing back and forth between P1 and P2 can get confusing.

P1's options are call and fold
P2's options are bet and check