The [0, 1] Half-Street Fixed-Limit Game

Each player, Hero and Villain, are given a random real number between 0 and 1, inclusive, and this serves as their hand. The player with the lowest number at showdown wins.

Hero is in position with Villain out of position. Each player posts an ante of P/2 before the hands are dealt. Each player receives their hand, 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.

This is similar to the n-card game, but is much easier to analyze, and is much easier to adapt to "real" poker river situations. We'll be using intervals and interval notation to represent ranges. For example, [0, 0.3) would mean that a range consists of all numbers from 0 to 0.3, including 0 but not including 0.3. We'll also be using U to denote a union of two intervals, so [0, 0.2] U [0.9, 1] would mean all numbers from 0 to 0.2 and 0.9 to 1, including 0, 0.2, 0.9, and 1. For more information, see Interval (mathematics) - Wikipedia, the free encyclopedia.

Here are some questions to get us started:

1. If Villain calls with a 0.4 or lower, what hands can Hero make a +EV value bet with?
2. If each ante is 0.25 units, what is the minimal range Villain has to call a bet with so that Hero can't make a +EV bluff?
3. If Villain calls with 0.3 or lower and P = 1, what is Hero's optimal exploitative betting range?
4. What is Villain's optimal exploitative calling range against Hero's range from question 3?
5. What is Villain's unexploitable calling range (in terms of P)?
6. What is Hero's unexploitable betting range (in terms of P)?