Game theory

I've done a lot of algebra again on the game theory where 2 players ante $1 and get a real number between 0 and 1.

Seeing as some people are posting a solution, eg where player A bets with .86 or more and with .07 or less; I modelled 2 variables, ah and al, being the high point and low point respectively where A bets.


There is one other variable there, called b, which is where player B calls.


If b >= ah then...


EVA = ah squared - al squared - 2.al + 4.al.b + 3.(b squared) - 2.b - 4.ah.b + 2.ah + 1


EVB = 2 - EVA (because they both anted $1, and I didn't take it into account as -ve in the betting)


To work out b, you differentiate, and see when that = 0. It comes out to... b = 1/3 + 2/3(ah-al)


I have more formulae if b evaluates to less than ah, so be careful, this is a restriction on the above formulae.


So if A bets at .86 or above, and .07 and below; then I calculate b to be .86 (same as ah, but that is ok).


I haven't worked out whether A's EV is negative, yet, but I will be very surprised if it isn't. I think A's EV will always be negative if B calls according to his calculated maximum value.


I'll be back later and I will look at this further.

ah= 7/9 . al= 1/9 . b= 5/9 ... assuming A is betting the pot ($2).

Well I plugged the formula and values into my spreadsheet; and I have to admit that all those people who said betting is correct were correct.

A's EV comes to 1.0959; which is more than the 1 if they both always checked.


All I can say is that I did all this stuff for the draw game I usually play, which is 5, 10 blinds and 20 to play, and raising turned out to be bad.


Well I've learned something.

If anyone is interested

the formula for al 1; then b = al (the lower bound)

if (2.al + ah)

* A's EV comes to 1.0959 ... *


EVA= 1,111111111

**********

EVA= 1,111111111


Try again.

@#$#@ All my text has gone.


I'll put it up again later.

For those that are interested what the formulae are if b 1; then b = al (the lower bound)

If (2.al + ah)

For those that are interested, I have the formulae for when b lte ah. lte means "less than or equal to", and for those that haven't worked it out yet, this noticeboard won't let me write that in the way that would be synonymous with writing >=. So not only do I have to be smart and work all this stuff out, I also have to be smart and get it through this parser.


EVA = -2.(ah squared) - (al squared) - 2.al + 4.al.b - 2.b + 2.ah.b + 2.ah + 1

EVB = 2-EVA (as before)


Differentiating EVB with respect to b to find a maximum for b reveals...

If (2.al + ah) > 1; then b = al (the lower bound)

If (2.al + ah) lt 1; then b =ah (the upper bound)

If (2.al + ah) = 1; then b can be anything between al and ah inclusive.


So in our example with al = 0.07, and ah = 0.86, the values fit option 3, and b can be anything in between, so 0.5 is acceptable.

EVA = 1.0959 regardless of b


Next is to differentiate EVA with respect to ah and al and to see if we can come up with the optimal values for those, now that we know how b will play.


What fun I'm having.

And would you 2+2 people be so kind as to change the parser to accept signs like