3. Game trees with chance moves Consider the following game involving two real players, Ann and Bob and a chance move. We can model this as \"Nature\" being a player: Think of nature as an impartial person tossing a fair coin so that with probability % Ann gets to be the first-mover of the game while Bob moves second, and with probability %, Bob moves first while Ann moves second. The player who moves first decides whether to attack or wait. If the first-mover attacks, the game ends, and the player who attacked gets a payoff of 1, while the other player gets 4. If the first mover waits, then the second mover gets to move. The second-mover too must decide to attack or wait. If the second-mover attacks, then the game ends, and the second-mover gets 1 and the other player gets 4. If the second-mover waits then both players get 0. Assume that each players maximizes his/her expected payoff. a. First assume that Ann and Bob observe nature's choice so they know whether they are first or second. Find any pure-strategy subgame-perfect equilibria (SPE) of this game. Are there any Nash equilibria which are not SPE? b. Now suppose that neither Ann nor Bob observes the move by nature, or each other's move. That is, if a player is asked to move they do not know whether they are the first-mover, or whether they are the second mover and the other person has just chosen to wait. Draw the game-tree. Find two symmetric pure-strategy SPEs in this game that have very different outcomes. c. Now suppose that Ann can observe the move by nature and can also observe Bob's move (when she is the second mover). Bob knows what Ann can observe, but, Bob himself can observe neither nature's move nor Ann's move. Draw the game-tree, and compare the SPE of this game with the SPE of the game of parts (a) and (b)