The given game is a two-period repeated game, where players observe the strategies played in period 0 before they play in period 1. Player 2 tells player 1 that if player 1 doesn't play strategy T in period 0, player 2 will punish player 1 by playing strategy R in period 1. Otherwise, player 2 will reward player 1 by playing strategy L in period 1.
To determine if this is a Subgame Perfect Nash Equilibrium (SPNE), we need to analyze the game played over two periods and solve the second period first.
In the second period, player 1 has two strategies: T (trigger) and N (no trigger). Player 2 has two strategies: R (punish) and L (reward). Player 1's payoffs are denoted as (a, b) where a is the payoff for player 1 and b is the payoff for player 2.
Let's consider the possible outcomes in the second period:
1. If player 1 plays T (trigger), player 2 will play L (reward). In this case, player 1 receives a payoff of a1 and player 2 receives a payoff of b1.
2. If player 1 plays N (no trigger), player 2 will play R (punish). In this case, player 1 receives a payoff of a2 and player 2 receives a payoff of b2.
To determine the best response for player 2 in the second period, we need to compare the payoffs of playing R (punish) and L (reward). If a2 > a1, player 2 will choose R (punish) as it gives a higher payoff. Otherwise, if a1 > a2, player 2 will choose L (reward) as it gives a higher payoff.
Now, let's consider the first period. Player 1 knows that player 2 will choose the best response in the second period based on player 1's strategy in the first period.
If player 1 plays T (trigger) in the first period, player 2 will play R (punish) in the second period. Therefore, player 1 receives a payoff of a2.
If player 1 plays N (no trigger) in the first period, player 2 will play L (reward) in the second period. Therefore, player 1 receives a payoff of a1.
To determine the best response for player 1 in the first period, we need to compare the payoffs of playing T (trigger) and N (no trigger). If a1 > a2, player 1 will choose T (trigger) as it gives a higher payoff. Otherwise, if a2 > a1, player 1 will choose N (no trigger) as it gives a higher payoff.
In this case, we have a unique SPNE:
- Player 1 plays T (trigger) in the first period.
- Player 2 plays R (punish) in the second period.
This SPNE is the outcome where both players maximize their payoffs based on the given strategies and punishments/rewards in the two periods.