Game Theory Problem Of The Week - Page 3

TNixon · #21 09-13-2007, 12:58 PM

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

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

[/ QUOTE ]

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.

[ QUOTE ]

Optimal is:

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

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

[/ QUOTE ]

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.

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.

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.

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.

*EDIT*

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

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

[/ QUOTE ]
"Fold" is not an option for P2.

mykey1961 · #22 09-13-2007, 01:22 PM

[ QUOTE ]
[ QUOTE ]
Ok here is what I just don't seem to understand.

You apply a formula, get an answer. But do you do anything else to verify it actually answers the question?

Your P1
[1,83] fold
[84,100] call

Your P2
[1,74] Check
[75,100] Bet

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

My P2
[1,11] Bet
[12,78] Check
[79,100] Bet

My P1 vs Your P2
EV = -0.094545

Your P1 vs My P2
EV = -0.118182

My Strategy wins 0.011815 Ante's per hand from yours while rotating positions.

Maximal against your P1 "P1MO"
[1,65] Bet
[66,92] Check
[93,100] Bet

Maximal against your P2 "P2MO"
[1,83] Fold
[84,100] Call

Your P1 vs Your P2
EV = +0.023636

Your P1 vs P1MO
EV = -0.447879

P2MO vs Your P2
EV = +0.023636

Maximal would win 0.2357575 Ante's per hand from your strategies while rotating positions

[/ QUOTE ]

Mykey , you got to be kidding me . Why on earth would you restrict player one to check call with only 57-100 ?

[/ QUOTE ]
If P2 is playing optimal
[1,11] bet
[12,78] check
[79,100] bet

then P1 is indifferent to calling with [1,78] and calls with [79,100]

To find optimal for P1, you need to find which [x,78] x to maximize P1's EV against any strategy for P2.

x happens to be 57.

[ QUOTE ]

If player two is betting [0,11] and [89,100], then player one "could" check-call with any number from [12,88] since he will always be able to beat 11 numbers and lose to 11 numbers .

[/ QUOTE ]

You can't fine tune a strategy to just one opponent (even if that opponent is optimal) and call it optimal against all.

[ QUOTE ]

Your strategy for player one is NOT optimal !!

Here is a better strategy for player one than the one you proposed .

Player one check calls with [12-88] given that player two bets with [0,11] and [89,100] .

[/ QUOTE ]

Your P1 [12,88] Check else fold vs my P2 [1,11] and [79,100] bet else fold gives:
P1 EV = -0.226263

My P1 [57,100] call else fold vs your P2 [1,11] and [89,100] bet else check gives:
P2 EV = +0.090909

My strategy for P1 and P2 vs your's for P1 and P2
Your total EV = -0.067677 per hand when rotating positions

Can you find a strategy against my P1 which gives P2 an EV > 1/9?
Can you find a strategy against my P2 which gives P1 an EV > -1/9?

TNixon · #23 09-13-2007, 01:31 PM

[ QUOTE ]
You can't fine tune a strategy to just one opponent (even if that opponent is optimal) and call it optimal against all.

[/ QUOTE ]
If you can fine-tune your strategy in a way that is +EV, then one of the two of you isn't playing optimally at all.

That's the whole point of trying to find these equilibrium strategies.

***NINJA EDIT ADD***

[ QUOTE ]
Can you find a strategy against my P1 which gives P2 an EV > 1/9?
Can you find a strategy against my P2 which gives P1 an EV > -1/9?

[/ QUOTE ]
I already gave examples of adjustments vs your P1 *and* your P2 that are quite clearly +EV.

mykey1961 · #24 09-13-2007, 01:55 PM

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

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

[/ QUOTE ]

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.

[ QUOTE ]

Optimal is:

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

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

[/ QUOTE ]

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.

[/ QUOTE ]

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

[ QUOTE ]

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.

[/ QUOTE ]

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.

[ QUOTE ]

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.

[/ QUOTE ]

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.

[ QUOTE ]

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.

[/ QUOTE ]

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.

[ QUOTE ]

*EDIT*

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

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

[/ QUOTE ]
"Fold" is not an option for P2.

[/ QUOTE ]

[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

mykey1961 · #25 09-13-2007, 02:15 PM

[ QUOTE ]
Why not avoid betting 1-11 and bet with 67+ instead ?

I'm not sure the significance in why you're betting with 1-11 here .

[/ QUOTE ]

I believe it's because when P2 checks:
[12,78] vs [1,100] gives P2 an EV of -0.074444
[1,66] vs [1,100] gives P2 an EV of -0.226667

When P2 Bets with
[1,11] and [79,100] gives P2 an EV of +0.185556
[67,100] gives P2 an EV of +0.295354

+0.185555-0.074444 = +0.111111
+0.295354-0.226667 = +0.068687

TNixon · #26 09-13-2007, 02:17 PM

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

[/ QUOTE ]

Then you are calculating EV incorrectly, because against a P1 who plays this strategy:

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

It is clearly obvious that betting 67+ is better than betting 1-11,79+

Just think about what happens when you bet 67-78 instead of 1-11. If P1 calls, 67-78 has a chance to win, where 1-11 does not. If P1 folds, it doesn't matter what P2's number was, so there's no difference.

If there's extra value when P1 calls (because you win more often), and the same value when P1 folds, betting 67-78 instead of 1-11 is clearly +EV.

This:
[ QUOTE ]

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

[/ QUOTE ]

leads me to believe that you're running this through a simulator program. Is that correct?

If so..

I'm going to have to say there's clearly a bug in your code, because you're getting incorrect answers to *very* obvious problems.

[ QUOTE ]
The answer in both cases is No.

[/ QUOTE ]

Actually, the answer in both cases is Yes, and the fact that you disagree means that you're doing something wrong, probably in your EV calculation, because it is so blatantly obvious that betting 67+ is better than betting 1-11 and 79+ that I don't even need to run the math, because simple deduction shows that it's true.

But since other threads have shown that people who are confused simply don't accept the most simple logical chains, I'll go ahead and run the math.

In the meantime, why don't you show us how you're calculating EV, because you're obviously doing something wrong.

mykey1961 · #27 09-13-2007, 02:27 PM

[ QUOTE ]
[ QUOTE ]
You can't fine tune a strategy to just one opponent (even if that opponent is optimal) and call it optimal against all.

[/ QUOTE ]
If you can fine-tune your strategy in a way that is +EV, then one of the two of you isn't playing optimally at all.

That's the whole point of trying to find these equilibrium strategies.

[/ QUOTE ]

That's only true if the there is no positional advantage in the game.

You can find strategies for P2 that are +EV because this game has a 1/9 ante advantage for player P2.

[ QUOTE ]

***NINJA EDIT ADD***

[ QUOTE ]
Can you find a strategy against my P1 which gives P2 an EV > 1/9?
Can you find a strategy against my P2 which gives P1 an EV > -1/9?

[/ QUOTE ]
I already gave examples of adjustments vs your P1 *and* your P2 that are quite clearly +EV.

[/ QUOTE ]

That's not true.

Optimal EV for P1 against my P2 is -1/9
Optimal EV for P2 against my P1 is +1/9

None of your strategies reached or exceeded those EV's

mykey1961 · #28 09-13-2007, 02:37 PM

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

[/ QUOTE ]

Then you are calculating EV incorrectly, because against a P1 who plays this strategy:

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

It is clearly obvious that betting 67+ is better than betting 1-11,79+

Just think about what happens when you bet 67-78 instead of 1-11. If P1 calls, 67-78 has a chance to win, where 1-11 does not. If P1 folds, it doesn't matter what P2's number was, so there's no difference.

If there's extra value when P1 calls (because you win more often), and the same value when P1 folds, betting 67-78 instead of 1-11 is clearly +EV.

This:
[ QUOTE ]

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

[/ QUOTE ]

Lead me to believe that you're running this through a simulator program. Is that correct?

If so..

I'm going to have to say there's clearly a bug in your code, because you're getting incorrect answers to *very* obvious problems.

[ QUOTE ]
The answer in both cases is No.

[/ QUOTE ]

Actually, the answer in both cases is Yes, and the fact that you disagree means that you're doing something wrong, probably in your EV calculation, because it is so blatantly obvious that betting 67+ is better than betting 1-11 and 79+ that I don't even need to run the math, because simple deduction shows that it's true.

But since other threads have shown that people who are confused simply don't accept the most simple logical chains, I'll go ahead and run the math.

In the meantime, why don't you show us how you're calculating EV, because you're obviously doing something wrong.

[/ QUOTE ]

P1Fold := [1..56];
P2Bet := [67..100];
Sum := 0;
for v1 := 1 to 100 do
for p2 := 1 to 100 do
if (v1 <> p2) then
begin
..p1 := v1;
..Net := 1;
..if p2 in P2Bet then
..begin
....Net := 3;
....if p1 in P1Fold then
....begin
......p1 := 0; Net := 1;
......// this makes P1 always lose when it folds and makes P2 only net 1 unit
....end;
..end;
..if p1 > p2 then inc(Sum,Net) else dec(Sum,Net);
end;
writeln(Sum/9900.0:9:6);

mykey1961 · #29 09-13-2007, 02:57 PM

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

[/ QUOTE ]

Then you are calculating EV incorrectly, because against a P1 who plays this strategy:

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

It is clearly obvious that betting 67+ is better than betting 1-11,79+

[/ QUOTE ]
That's your opinion

[/ QUOTE ]

Just think about what happens when you bet 67-78 instead of 1-11. If P1 calls, 67-78 has a chance to win, where 1-11 does not. If P1 folds, it doesn't matter what P2's number was, so there's no difference.

If there's extra value when P1 calls (because you win more often), and the same value when P1 folds, betting 67-78 instead of 1-11 is clearly +EV.

[/ QUOTE ]

As I've shown elsewhere, the bigger difference is when P2 Checks

[ QUOTE ]

This:
[ QUOTE ]

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

[/ QUOTE ]

leads me to believe that you're running this through a simulator program. Is that correct?

[/ QUOTE ]

Calculate instead of simulate

[ QUOTE ]

If so..

I'm going to have to say there's clearly a bug in your code, because you're getting incorrect answers to *very* obvious problems.

[/ QUOTE ]

Again, that's your opinion

[ QUOTE ]

[ QUOTE ]
The answer in both cases is No.

[/ QUOTE ]

Actually, the answer in both cases is Yes, and the fact that you disagree means that you're doing something wrong, probably in your EV calculation, because it is so blatantly obvious that betting 67+ is better than betting 1-11 and 79+ that I don't even need to run the math, because simple deduction shows that it's true.

[/ QUOTE ]

I think it's time you run the math, and give your deductionator a checkup.

[ QUOTE ]

But since other threads have shown that people who are confused simply don't accept the most simple logical chains, I'll go ahead and run the math.

In the meantime, why don't you show us how you're calculating EV, because you're obviously doing something wrong.

[/ QUOTE ]

Again for the 3rd time, that's your opinion.

jay_shark · #30 09-13-2007, 03:42 PM

Mykey here is where you're mistaken .

If player two bets with [0,11] and [79,100] but checks everything else , then player two is indifferent to calling or folding with [12,78] .

ie, 12 beats 11 numbers but loses to 22 numbers . Since he is getting 2:1 odds , any call he makes is neutral EV, even with the number 12 . Similarly , if player one checks calls with 78 , then he can only beat precisely 11 numbers and STILL lose to 22 numbers ;namely , [79,100] . In both situations , any call he makes from [12,78]is neutral EV .
So saying that player one should check call with [57,100]is wrong !!