0 to 1 Game -- Problem 3

Same as problem 2, except Y may raise after a bet. X may not check raise. 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 Y bets, then X may then call or fold.
• If player X bets then Y may raise, call or fold. If Y raises, then X may call or fold.

What is the optimal strategy for each player?

This one is about twice as complicated as problem 2, just to warn you.

Method of solution:
Call x and y the random numbers received at beginning.
b(x) = P (X bets when she holds value x).
c(y) = P (Y calls if X bets). r(y) = P (Y raises if X bets). Constraint: c+r<1
k(x) = P (X calls if Y raises).
d(y) = P (Y bets if X checks), g(x) = P(X then calls).

Writing I(E) for the indicator function (0 or 1) of the event E, we write
SH(x,y) = [I(x>y) - I(x<y)] (SH for "showdown")

Then
EV(x,y) = b { r [k 3 SH + (1-k) 2] + c 2 SH + (1-c-r) } + (1-b) { d [g 2 SH + (1-g)(-1)] + (1-d) SH }

X maximizes EEV(x) = int(y=0 to 1) EV
= b { k (6R(x)+2R(1)-3) + 4c - 3R(1) -C(1) - 1 } + (1-b) { (2g-1) [2D(x)-1+D(1)/2] + (1-D(1))/2 }

where an uppercase letter is the integral of the correponding lowercase.

Do the same for Y, integrating over x.

You get a set of six maximizing conditions, fiendishly difficult to solve...
Using information from the previous problems, one can probably ease some by assuming the betting strategy is deterministic (a threshold below which you bluff, another above which you bet). But I see no hint about the optimal form of strategy for raising.

Work in progress.

For those interested, here is a solution. There may be others.


The Expected Value for X is - 0.49/18 = -0.0274 .

Surprisingly, alternative solutions include X calling the raise even though she was bluffing. That is because Y might be bluffing in his raise.
Other alternatives have Y folding for small values and bluffing (raise) for higher values.