[0,1] game: Methods

[Aisthesis]

After one of the supposedly "simple" exercises led to an inability (on my part at least) to figure out where exactly my attemt at solving the problem by logic went wrong, it seems like a good idea to consider various methods of solving these things. We've basically seen three of them (including Jerrod's "primer material" link):

1) Partial derivatives
2) Indifference equations (elaborated by Jerrod in the link)
3) The logic of pot odds

Applied correctly, all 3 of these methods have to lead to the same solution. The logic of pot odds (method 3) has the big advantage that it is typically much simpler, hence much less likelihood for clerical errors--as long as the logic is applied correctly every step of the way (which I must not have been doing). Jerrod, whom I'm assuming is the person here probably most versed in this type of problem, apparently solved David's #4 primarily by the logic method.

So, anyhow, while for more complicated [0,1] games some kind of combined approach may be necessary (as Jerrod also seems to do on occasion), I'd like to explore a (simple) game or 2 trying all 3 methods in the hope that that might illuminate how they fit together.

What seems to me to be the case is that there are at least certain peculiarities in games where both players have put money in the pot and have the option of folding. These peculiarities are what I think is giving me some problems with making the correct logical steps all the way through.

I'll start off with a seemingly quite simple game, which I'll call game #5:

A and B both ante $1. B is first to act and can check or raise one additional dollar. A only has the option of calling or folding. He can't raise.

I'll try to solve this by all 3 of the methods given in subsequent posts. Hopefully they'll all be the same.

First, however, I think I'll try proving a "lemma" for games of this kind.