spoonitnow's 5000th Post: Game Theory and Poker

Yeah so I started this thread the other day knowing that it would take a whole lot of posts before I finished everything I wanted to on the subject. A lot of people start into Mathematics of Poker wanting to learn some game theory to apply to poker, but without a background in mathematics it can be a pretty difficult read. It also doesn't cover some of the more basic games in-depth, opting to leave a lot of that work to the reader. My goal here is to present a series of toy games that are particularly instructive, some that are covered in that book, and some that aren't. I'm doing this for myself just as much as I am anyone else, but I promise that if you work through these games and the discussion of them that you'll learn a lot that you can apply directly to your game right now and see good improvements.

I think it's worth mentioning that I'm aware that the vast majority of people who open this thread won't work through more than 10-20% of it. That's because less than 10-20% of people who read this thread want to be successful.

Best of luck, and here's to 5000 more posts.

The Half-Street Heads-Up 13-Card Fixed-Limit Game

So here's how this works. I'm going to describe the rules of a poker game to you and ask you some questions. I'm not ready to go to bed yet but am bored as hell which is why I'm doing this. Before I even get started, I already know there's going to be some jackass who wants to chime in and say that this is useless in terms of learning "real" poker or some horse shit like that. You sir, are wrong.

Okay so here's the deal. We have a 13-card deck that comprises of all of the spades in a normal deck. So there's an Ace, King, Queen, Jack, Ten, Nine, Eight, Seven, Six, Five, Four, Three, and Two in the deck.

There are two players, Hero and Villain, and Hero is in position with Villain out of position. Each player posts an ante of P/2 before the cards are dealt. Each player receives one card, then Villain is forced to check in the dark. Hero then has the choice to bet 1 unit, or check to showdown. If Hero bets, then Villain has the choice to call and see a showdown or fold. At showdown, whoever has the highest card wins.

Here are some questions, and I'll probably come up with some more questions later.

1. How often does Villain have to fold for Hero to have a +EV bluff?
2. If Villain calls with a 7 or higher, what hands can Hero make a +EV value bet with?
3. If each ante is 0.25 units, what is the minimal number of hands Villain has to call a bet with so that Hero can't make a +EV bluff?
4. If Villain calls with an 8 or higher, what is the EV of betting a T for Hero? of checking a T for Hero?
5. In question 4, is betting ever better than checking for Hero?
6. If Villain calls with an 8 or higher, what is the EV of betting a J for Hero? of checking a J for Hero?
7. In question 6, compare betting and checking for Hero. What do you notice? Should Hero bet or check, and why?
8. If the antes are 1 unit each and Villain calls with J+, how should Hero play his entire range?
-- 8a. How does the EV of betting hands change as you look at worse and worse hands?
-- 8b. How does the EV of checking hands change as you look at worse and worse hands?

That should get you started studying this game.

A guide to finding the answers is in the next post of this thread, but you're highly advised to work it out on your own to begin with.