Test

Let's make this more specific: say the second Villain has 5c4s and also has a $5 stack. We make our $5 bet, and assume Villain 1 acts first while Villain 2 acts second.

From highest EV to lowest EV:

$12.00 - Both fold
$10.93 - AhKh calls, 5c4s folds
$10.09 - AhKh folds, 5c4s calls
$7.23 - Both call

Here the fundamental theorem holds since we benefit by their mistakes no matter what happens. Can you come up with hands for the two Villains here where the fundamental theorem doesn't hold?


Math Shit Below Showing The EV Of Each Possible Outcome

If both call:

Code:

Board: 7d 6c 2h
Dead:  

	equity 	win 	tie 	      pots won 	pots tied	
Hand 0: 	45.293%  	45.29% 	00.00% 	           409 	        0.00   { 9h9s }
Hand 1: 	23.588%  	23.59% 	00.00% 	           213 	        0.00   { AhKh }
Hand 2: 	31.118%  	31.12% 	00.00% 	           281 	        0.00   { 5c4s }

Our EV is (0.45293)(12+5+5) + (1-0.45293)(-5) = $7.23.

If AhKh calls and 5c4s folds:

Code:

Board: 7d 6c 2h
Dead:  

	equity 	win 	tie 	      pots won 	pots tied	
Hand 0: 	72.424%  	72.42% 	00.00% 	           717 	        0.00   { 9h9s }
Hand 1: 	27.576%  	27.58% 	00.00% 	           273 	        0.00   { AhKh }

Our EV is (0.72424)(12+5) + (1-0.72424)(-5) = $10.93.

If AhKh folds and 5c4s calls:

Code:

Board: 7d 6c 2h
Dead:  

	equity 	win 	tie 	      pots won 	pots tied	
Hand 0: 	68.586%  	68.59% 	00.00% 	           679 	        0.00   { 9h9s }
Hand 1: 	31.414%  	31.41% 	00.00% 	           311 	        0.00   { 5c4s }

Our EV is (0.68586)(12+5) + (1-0.68586)(-5) = $10.09.

If both fold, our EV is $12 since we just win the pot.