Three people A, B, and C live in a village, and they want to build a road to facilitate commute. To make this happen, it needs contribution from at least two of them. Each person can individually decide whether to contribute or not, and the cost of contribution is 1. (Suppose this cost occurs no matter whether or not the road gets built eventually.) If the road gets built, each person, regardless of whether they have contributed or not, will receive a benefit 2. For example, if A contributes but B and C do not, then their payoffs will be -1, 0, and 0, respectively; if A and B contribute but C does not, then the road will get built and their payoffs will be 1, 1, and 2, respectively.

Suppose the three people only care about their own payoffs and do NOT consider fairness etc. You are consulted about what game rule will maximize the chance that the road will get built. Here are the three options under consideration:

(i) Suppose the three people make their decisions simultaneously at the same day (without talking to each other). Find the symmetric mixed-strategy Nash equilibrium (in which each person chooses to contribute with the same probability x between 0 and 1). What is the probability that the road will get built in this equilibrium?

(ii) Suppose A and B decide simultaneously in day 1, and then C decides in day 2 after seeing A and B's choices. What is the (sub-game perfect) Nash equilibrium? What is the probability that the road will get built? (In the day-1 game, focus on the symmetric Nash equilibrium where A and B adopt the same pure or mixed strategy.)

(iii) Suppose the three people decide sequentially whether or not to contribute in three consecutive days (in the order of A, B and C). What is the (sub-game perfect) Nash equilibrium? What is the probability that the road will get built?

(iv) Which game rule(s) are the best in terms of the chance that the road will get built?