There are n players. The game proceeds as follows:

In the first round, a player is chosen at random, and required to choose whether to transgress, or not. Transgressing gets/costs nothing, while avoid transgressing costs C > 0.

In the second round and each subsequent rounds, all the players are randomly paired, and each chooses whether to punish the other player, or not. Punishing costs c > 0 and harms the other by h > 0. Not punishing costs nothing.

The probability of proceeding from each round to the subsequent round is > 0. That is, the probability from proceeding from the first round to the second round is . If round two is reached, the probability of proceeding from it to round three is , and so on

(a) We will begin by considering the strategy we discussed in class, which is defined as follows. It is helpful to first define players as being in good standing:

In the first round, all players are considered to be in good standing.

In the second round, a player is considered to be in good standing unless he was chosen in the first round and transgressed. In third and all subsequent rounds, a player is considered to be in good standing unless, in the previous round: he was paired with a player who was not in good standing, and did not punish, or he was paired with a player who was in good standing, and did punish. Notice that standing is determined based entirely on one's behavior in the previous round. This known as 'memory-1'. Now, we define the strategy of interest as:

Don't transgress Punish if and only if your partner is not in good standing

i. Show that this strategy profile is a Nash equilibrium when C/h.

ii. Show that, for this equilibrium to be subgame perfect, it must also be the case that c/h.

Now consider the following, 'memory-2' strategy. In it, standing is defined as follows:

In the first round, all players are considered to be in good standing.

In the second round, a player is considered to be in good standing unless he was chosen in the first round and transgressed. In the third round, a player is considered to be in good standing unless he was chosen in the first round and transgressed, or he was paired with a player who was not in good standing in round 2 and did not punish, or he was paired with a player who was in good standing in round 2, and did punish.

3 In fourth and all subsequent rounds, a player is considered to be in good standing unless, in one of the two previous rounds, he was paired with a player who was not in good standing, and did not punish, or he was paired with a player who was in good standing, and did punish.

Notice that now standing is determined based on one's behavior in the past two rounds. The strategy of interest is otherwise the same as before: Don't transgress Punish if and only if your partner is not in good standing

(10 points) Find conditions under which this strategy is a subgame perfect Nash equilibrium. Describe the 'comparative static' on . That is, when is it easier for these conditions to be met-when is large or small? Does this comparative static differ meaningfully to the one you found for the above 'memory-1' strategy? If so, how?