Described in words, the game involves the following sequence of steps:
- Player 1 chooses either to hurt the chance of player 2 being strong-type or do nothing. If player 1 decides to hurt, then he spends \$1 in exchange for decreasing the chance of player 2 being strong-type from $2/3$ to $1/3$. Player 2 does not know whether player 1 has chosen to hurt or do nothing.
- The game randomly decides whether player 2 is strong-type or weak-type with appropriate probabilities. If player 1 has decided to hurt in step 1, then player 2 is strong-type with probability $1/3$. Otherwise, player 2 is strong-type with probability $2/3$. Player 1 does not know the type of player 2, while player 2 knows his own type.
- Player 2 chooses either to stay in the game or quit the game early. If he decides to quit, then he walks away free if he is strong-type, but he has to spend \$1 to quit if he is weak-type. If he decides to stay, then the game proceeds to step 4.
- Player 1 chooses either to fight or run away. If player 1 runs away, then player 2 walks away free regardless of the type. If player 1 decides to fight, then the outcomes depend on the type of player 2. If player 2 is strong-type, then player 1 loses the fight and has to pay \$3 to player 2. If player 2 is weak-type, then player 1 wins the fight and he can get a payment of \$3 from player 2.
- In any case, the final payoff of player 1 takes into account whether he has spent the \$1 to hurt in step 1.
This is quite a complicated game. I don't even know how to start to tackle it. How should I analyze such a game?