Simple Game

[bxb]

You play this game against one opponent. Both of you are dealt a real number from 0 to 1. The game starts with $1 in the pot. There is one round of betting where you can either bet $1 or check. If you check, your opponent must check behind. If you bet, your opponent can either call or fold. Assume he will act rationally.

I think that our opponent will end up choosing some number y and folding all hands less than y to raises.

The solution will involve betting some hands at random, so we are looking for a function p(x) that tells us the probability we should bet when we are dealt x.

[PairTheBoard]

Seems to me we've seen this before. As I recall, you end up betting all the time when x is in the intervals [0,a] or [b,1] where a and b are fixed and optimum. I was suprised that you bluffed with your worst numbers.

PairTheBoard

[bxb]

This solution is easy to arrive at, but I think there are better ones. A simple check would be to calculate the EV when you use two sets of value a1,b1 and a2,b2 and switch between the strategies at random. Your opponent will have to choose a strategy best suited to combat an intermediate strategy, and thus your EV will end up greater. I'm pretty sure this ends up higher EV but I lost the paper i did this problem on initially, so I might be wrong.

The reason that it ends up better to bluff your low numbers is that by betting some of your higher numbers, you essentially turn some hands with good showdown value into bluffs.

[jay_shark]

The game is simple if you assume there are no raises and it's either check , call or fold with no raises .