5. [12 points] Two players, 1 and 2, play the following game repeatedly over time. The values in the table are the monetary rewards to the players for every possible combination of their actions. At any point in time, the actions taken by both players in the past (i.e. the history of play) are commonly known by both players. C D Player 2 l's Reward=2 l's Reward=0 C 2's Reward=2 2's Reward=3 l's Reward=3 l's Reward=l D 2's Reward=0 2's Reward=l Player 1 These players evaluate their (long-run) payoffs according to the discounting criterion, with discount factor 0.9. In other words, if Player 1 receives a stream of monetary rewards (x0, x1, K2, xi, ) over time, his payoff is U= x0 +x1(0.9) + X2(0.9)2+ + m(0.9)t +_... Similarly for Player 2. (I) First, suppose it is common knowledge that the game will be repeated only 100 times, and then end. Consider the following strategies for the two players. Which of these constitutes a Nash Equilibrium (NE) of the repeated game, and which ofthese is Subgame Perfect Equilibrium (SPE)? (Circle the right answers below.) (a) Both players adopt the strategy: 'Take action D regardless of the history'. This is a SPEJ This is a NE but not SPEJ This is not :1 NE at all. (h) Both players adopt the strategy: 'Take action C regardless of the history'. This is a SPEJ This is a NE but not SPEJ This is not a NE at all. (c) Both players adopt the strategy: 'Take action C as long as both players have always played C in the past, but take action D from now onward if any player (myself or the other player) has deviated from C (to D) even once. This is a SPEJ This is a NE but not SPEJ This is not a NE at all. ((1) Both players adopt the strategy: 'Takc action C in the rst period, and in any subsequent period take whatever action the other player has taken in the previous period.' This is a SPEJ This is a NE but not SPEJ This is not a NE at all. (e) Is there any Nash Equilibrium in which both players receive the payoff 2+ 2(0.9)+ 263.9)2 + + 2(0.9)99? No! Yes, and the equilibrium strategies for the players are: 4. (continued) (II) Now suppose it is common knowledge that the game will be repeated indefinitely without end. Consider the following strategies for the two players. Which of these constitutes a Nash Equilibrium (NE) of the repeated game, and which of these is Subgame Perfect Equilibrium (SPE)? (Circle the right answers below.) (f) Both players adopt the strategy: 'Take action D regardless of the history'. This is a SPE./ This is a NE but not SPE./ This is not a NE at all. (g) Both players adopt the strategy: 'Take action C regardless of the history' This is a SPE./ This is a NE but not SPE./ This is not a NE at all. (h) Both players adopt the strategy: 'Take action C as long as both players have always played C in the past, but take action D from now onward if the other player has deviated from C (to D) even once. However, I continue to take action C if I myself might have deviated to D but the other player has not deviated.' This is a SPE./ This is a NE but not SPE./ This is not a NE at all. (i) Both players adopt the strategy: 'Take action C in the first period, and in any subsequent period take whatever action the other player has taken in the previous period.' This is a SPE./ This is a NE but not SPE./ This is not a NE at all. (j) Is there any Subgame Perfect Equilibrium in which Player 1 gets the payoff 1 + 3(0.9)+ 2(0.9)2 + 1(0.9)3+ 3(0.9)4+ 2(0.9)5+ .. and Player 2 gets the payoff 1 + 0(0.9)+ 2(0.9)2 + 1(0.9)3+ 0(0.9)4+ 2(0.9)5+ ... ? No (explain why not) / Yes (find a SPE strategy profile) (k) Is there any Subgame Perfect Equilibrium in which both players receive the payoff 2+ 2(0.9)+ 2(0.9)2+ ...? No (explain why not) / Yes (find a SPE strategy profile)