Wizard
Joined: Oct 14, 2009
• Posts: 21930
August 13th, 2012 at 2:28:54 PM permalink
Here is your next problem. The same as problem 1, except of X checks then Y can bet. Here are the full rules.

• Player X and Y each ante \$1.
• Both are given a random number uniformly distributed from 0 to 1. The higher number wins.
• Player X may bet \$1 or check.
• If player X checks then Y may check or bet.
• If player X bets then Y may call or fold.
• If player X checks, then Y bets, then X may call or fold.

What is the optimal strategy for both players? What is the expected value for X under mutual optimal strategy?

If sufficient interest is show in this exercise, then problem 3 will be the same, but allow betting up to a pot size of \$6.
It's not whether you win or lose; it's whether or not you had a good bet.
JTate503
Joined: Mar 3, 2012
• Posts: 7
August 13th, 2012 at 3:12:59 PM permalink
my initial guess (probably wrong)

X has the same initial strategy, raises with 0.1 or less, raises with 0.7 or more.

Y calls a raise with anywhere from 0.1 to 0.7 or higher (doesn't matter EV) wise.

If X checks, Y raises with 0.16 or less, raises with 0.52 or more.

X calls with anything from 0.16 to 0.52 or higher.

Expected value for X = 0.068

Wizard
Joined: Oct 14, 2009
• Posts: 21930
August 13th, 2012 at 3:23:31 PM permalink
Wouldn't it make sense that X should call less often than problem 1? In problem 2 Y has more power if X checks.

It's not whether you win or lose; it's whether or not you had a good bet.
JTate503
Joined: Mar 3, 2012
• Posts: 7
August 13th, 2012 at 3:29:02 PM permalink
Yes, of course. But this is the only answer I could do in 2 minutes :)
MangoJ
Joined: Mar 12, 2011
• Posts: 905
August 13th, 2012 at 3:35:00 PM permalink

Left side: Player X. Right side: Player Y.
dots are probability to bet, circles are probabiltiy to call.
The scattered data is (as before) almost undecidable - i.e. it oszillates in the simulation. Probably doesn't matter what you pick there.
[Edit^2: As X never bets between ~0.18 and 0.5, Y' call results between ~0.18 and 0.5 does scatter heavily. Possibly Wizards "doesn't matter" decision point. Also valid for the other side (X's call results between ~0.18 and 0.5)

I leave the analytical analysis to the experts around here.

Edit: Summary of the results (in case the graphic won't load):
X should never bet between ~0.18 and 0.5, and never call below ~0.18.
X should call above 0.5.

Y should never bet between ~0.18 and 0.5, and never call below ~0.18
Y should bluff below 0.18, and bet/call anything above 0.5

Edit^2 (see spoiler)
Wizard
Joined: Oct 14, 2009
• Posts: 21930
August 13th, 2012 at 3:36:06 PM permalink
Mango, I couldn't see the graphic.
It's not whether you win or lose; it's whether or not you had a good bet.
MangoJ
Joined: Mar 12, 2011
• Posts: 905
August 13th, 2012 at 3:41:34 PM permalink
Could you try again ? On my machine it works: http://img401.imageshack.us/img401/1475/strategies2.png
I hope this is not a cookie-related thing, I'm not very familiar with imageshack.
CrystalMath
Joined: May 10, 2011
• Posts: 1826
August 14th, 2012 at 1:08:30 PM permalink
I get that x should bluff on 19/112 or less and bet on 55/112 or higher. This is very close to MangoJ.
If x bets, then y should call on 40/112 or higher.
If x checks, then y should bluff on 15/112 or less and bet on 19/112 or higher.
If x checks, then y bets, x should fold on 31/112 or less, otherwise call.
I heart Crystal Math.
Wizard
Joined: Oct 14, 2009
• Posts: 21930
August 14th, 2012 at 1:52:58 PM permalink
I still can't see Mango's graphic. It may be a similar situation to problem 1 where multiple answers result in the same expected values. May I ask what you guys get? Here is my EV...

EV(player X) = -1/18
It's not whether you win or lose; it's whether or not you had a good bet.
MangoJ
Joined: Mar 12, 2011
• Posts: 905
August 14th, 2012 at 2:30:34 PM permalink
Quote: Wizard

I still can't see Mango's graphic.

Maybe try one of these links. Please let me know if the server / 404 or the PNG format is the problem.
http://s7.postimage.org/5f62aa9gr/strategies2.png
http://s14.postimage.org/c9q3lkmr5/strategies2.gif
http://s9.postimage.org/pz89t184v/strategies2.jpg

-0.055 for Player X