You are on an island with two tribes. One tribe always tells the truth, and the other tribe always lies. You meet three individuals from the island: A, B and C. Each individual belongs to one of the tribes. You ask each of them the same question: "Is B a truth-teller?"
A says, "Yes, B is a truth-teller."
B says, "No, I am not a truth-teller."
C says, "B is a liar."
Step 1: Analyze B's statement.
B says, "No, I am not a truth-teller." If B were telling the truth, then he wouldn't be a truth-teller, which is a contradiction.
Hence, B is lying, so B is a liar.
Step 2: Analyze A's statement.
A says B is a truth-teller. But we just proved B is a liar, so A must be lying too.
Step 3: Analyze C's statement.
C says B is a liar. That’s a true statement, so C must be a truth-teller.
Conclusion: C is telling the truth; A and B are lying.

Two players \( A \) and \( B \) are playing a game. Player \( A \) has two available actions \( a_1 \) and \( a_2 \). Player \( B \) has two available actions \( b_1 \) and \( b_2 \). The payoff matrix arising from their actions is presented below:

Let \( p \) be the probability that player \( A \) plays action \( a_1 \) in the mixed strategy Nash equilibrium of the game.
Then the value of p is (round off to one decimal place).
Three friends, P, Q, and R, are solving a puzzle with statements:
(i) If P is a knight, Q is a knave.
(ii) If Q is a knight, R is a spy.
(iii) If R is a knight, P is a knave. Knights always tell the truth, knaves always lie, and spies sometimes tell the truth. If each friend is either a knight, knave, or spy, who is the knight?
Consider the following statements followed by two conclusions.
Statements: 1. Some men are great. 2. Some men are wise.
Conclusions: 1. Men are either great or wise. 2. Some men are neither great nor wise. Choose the correct option: