Game Theory Problem Of The Week

[mykey1961]

[ QUOTE ]
If player 1 checks with [12-78] , then there is nothing player 2 can do to increase his EV . So I don't see a point in player 1 check-calling with [57-100] since he should be indifferent to betting or checking with [12-78] .

[/ QUOTE ]

The point is P1 [57,100] is optimal against the whole range of P2 opponents that are, or are not optimal.

Optimal does not mean maximal.

The question is how to prove it.

[mykey1961]

[ QUOTE ]
If player 1 checks with [12-78] , then there is nothing player 2 can do to increase his EV . So I don't see a point in player 1 check-calling with [57-100] since he should be indifferent to betting or checking with [12-78] .

[/ QUOTE ]

I assume you mean:

If player 2 checks with [12-78] , then there is nothing player 1 can do to increase his EV. So I don't see a point in player 1 check-calling with [57-100] since he should be indifferent to betting or checking with [12-78].


If you did, then that is true.

Remember with these two strategies:

P1 calling [57,100] P1's EV = -1/9
P2 betting [1,11],[79,100] P2's EV = +1/9

P1 isn't trying to improve his EV, just maintain it against all possible strategies for P2.

By P1 calling with [57,100]:

P2 has a better EV by betting compared to checking with each of 1 thru 11.
P2 has the same EV by betting compared to checking with 12.
P2 has a better EV by checking compared to betting with each of 13 thru 78.
P2 has a better EV by betting compared to checking with each of 79 thru 100.

So against P1's calling with [57,100]:

if P2 chooses to bet with either strategy ([1,11],[79,100]) or ([1,12],[79,100]) P2's EV is +1/9
If P2 chooses to check anything in [1,11], P2's EV goes down.
If P2 chooses to bet anything in [13,78], P2's EV goes down.
If P2 chooses to check anything in [79,100], P2's EV goes down.

As reference

If P1 had choosen to call with [56,100] instead:

P2 has a better EV by betting compared to checking with each of 1 thru 9.
P2 has the same EV by betting compared to checking with 10.
P2 has a better EV by checking compared to betting with each of 11 thru 78.
P2 has a better EV by betting compared to checking with each of 79 thru 100.

Since we agree P2's betting with [1,11],[79,100] is optimal, and makes P1 indifferent to calling or folding with the range [12,78] then when P1 changes to [56,100] the P2's EV stays the same.

If you notice above, with P1 playing [56,100] P2's best choice for 11 has changed from bet to check, and since betting with 11 gave P2 and EV +1/9, P2 increases his EV by checking 11 instead, and an increase for P2, is a decrease for P1.

The same idea is true for P1 calling with [58,100]

If P1 has choosen to call with [58,100] instead:

P2 has a better EV by betting compared to checking with each of 1 thru 13.
P2 has the same EV by betting compared to checking with 14.
P2 has a better EV by checking compared to betting with each of 15 thru 78.
P2 has the same EV vy betting compared to checking with 79.
P2 has a better EV by betting compared to checking with each of 80 thru 100.

P2 changing from checking to betting with 12, and 13 increases P2's EV beyond +1/9 which reduces P1's EV.

P1 has to call with exactly [57,100] to be able to guarentee P1's EV of -1/9 at minimum against all possible strategies for P2.

[mykey1961]

Let my try that again..
I assume you mean:

If player 2 checks with [12-78] , then there is nothing player 1 can do to increase his EV. So I don't see a point in player 1 check-calling with [57-100] since player 1 should be indifferent to calling or folding with [12-78].