4349 Econ Midterm #3 solutions
1. (30 points) a. (10 points)
b. (10 points)
c. (10 points)
2. (30 points) In the strategic form below, boldface payoffs correspond to pure-strategy best responses. Player 2 Player 1 A B A B 1,1,1 0,9,0 9,0,0 1,1,1 Player 3: A Player 1 A B Player 2 A B 0,0,9 1,1,1 1,1,1 0,0,0 Player 3: B
Based on the best responses above, pure-strategy Nash equilibria are (B,B,A), (A,B,B), and (B,A,B). Since the game is symmetric, but none of these equilibria are, there should be a mixed-strategy equilibrium (and it should be symmetric). So, let (p,1-p), (q,1-q), and (r,1-r) be the three players' mixed strategies. For Player 1, the expected payoff to choosing A is 1qr+0(1-q)r+0q(1-r)+1(1-q)(1-r) = 1-q-r+2qr, and the expected payoff to choosing B is 9qr+1(1-q)r+1q(1-r)+0(1-q)(1-r) = q+r+7qr. Player 1 will be willing to choose a mixed strategy only if the payoffs to A and B are equal, which is true if 1-q-r+2qr = q+r+7qr. This equality simplifies to 5qr+2q+2r-1=0. Similar arguments for Player 2 and Player 3 give the conditions for their playing mixed strategies: 5pr+2p+2r-1=0 5qr+2q+2r-1=0 Since we are looking for a symmetric equilibrium, we can set p=q=r, so each of these equations simplifies to 5p2+4p-1=0. This quadratic has two solutions: p=-1 and p=1/5. Since probabilities must be between 0 and 1, we can eliminate p=-1. Then, since p=q=r, we have q=1/5 and r=1/5, so the mixed-strategy Nash equilibrium is ( (1/5,4/5), (1/5,4/5), (1/5,4/5) ). 3. (30 points) a. (8 points) At Player 1's second decision node, she will choose H (which gives her 8) instead of G (which gives her 7). Then, at Player 2's decision nodes, he will choose C (which gives him 3) over D (which gives him 1) and E (which gives him 0) over F (which gives him -1). Then, at Player 1's first decision node, she will choose B (which gives her 7) instead of A (which gives her 0). So, the rollback equilibrium is (BH,CE), and payoffs are (7,0). b. (12 points) A game matrix is shown here. (Other correct are answers possible.) Player 2 CE CF DE DF Player AG 0,3 0,3 7,6 7,6 1 AH 0,3 0,3 8,1 8,1 BG 7,0 -1,-1 7,0 -1,-1 BH 7,0 -1,-1 7,0 -1,-1 c. (10 points) Pure-strategy best responses correspond to boldface payoffs. Pure-strategy Nash equilibria are (AH,CF), (BG,CE), and (BH,CE); note that this last one is the rollback equilibrium.
4. (30 points) a. (10 points) A game tree is shown below.
b. (10 points) Player 1's available pure strategies are: Quit, Up, Top Continue, Up, Top Quit, Up, Bottom Continue, Up, Bottom Quit, Down, Top Continue, Down, Top Quit, Down, Bottom Continue, Down, Bottom Player 2's available pure strategies are: First, Left, Links Second, Left, Links First, Left, Rechts Second, Left, Rechts First, Right, Links Second, Right, Links First, Right, Rechts Second, Right, Rechts c. (10 points) This game has four subgames: the entire game, the portion of the game starting from Player 2's first move (after Player 1 has chosen Continue), and the portions starting from after Player 2 has chosen either First or Second. To find the subgame perfect equilibrium, we start from these last two subgames. Both correspond to the simultaneous-move games below. Player 2 Player 2 Lef...