1. The stage game is shown in Table 1.
   \begin{tabular}{c|c|c|}
   \hline & $H$ & $L$ \
   \hline \hline $\mathrm{H}$ & $(3,1)$ & $(0,0)$ \
   \hline $\mathrm{L}$ & $(1,2)$ & $(5,3)$ \
   \hline
   \end{tabular}
   Table 1: Stage game
   Consider the infinite repetition of the game in Table 1 with discounted criterion to evaluate payoffs. Find a subgame perfect equilibrium of this game such that
   (a) the equilibrium payoff of Players approach $(4,2)$ as $\delta \rightarrow 1$.
   (b) the equilibrium payoff of Players approach $(3,2)$ as $\delta \rightarrow 1$.

If we repeat prisoner’s dilemma game for two periods, how many strategies does each player have in this repeated game?

3. Consider the stage game $G$ shown in Table 2 .
   \begin{tabular}{c|c|c|c|}
   \hline & $a$ & $b$ & $c$ \
   \hline$A$ & $(4,4)$ & $(-1,5)$ & $(2,2)$ \
   \hline$B$ & $(5,-1)$ & $(1,1)$ & $(2,2)$ \
   \hline$C$ & $(2,2)$ & $(2,2)$ & $(3.5,3.5)$ \
   \hline
   \end{tabular}
   Table 2: Stage game
   (a) Find the worst Nash equilibrium (pure action) for each player in $G$ and the corresponding payoffs.
   (b) Consider $G^2$ : the finitely repeated game, where $G$ is repeated for two periods.
   (i) Is there a subgame perfect equilibrium of $G^2$ where $(A, a)$ is played in the first period? Explain your answer.
   (ii) Is there a Nash equilibrium of $G^2$ where $(A, a)$ is played in the first period? Explain your answer.

4. Suppose instead of discounting criterion for evaluating payoffs, we evaluate payoff of Player $i$ from a stream of payoffs $\left{v_i^t\right}_1^{\infty}$ as
   $$
   \lim {T \rightarrow \infty} \frac{1}{T} \sum{t=1}^T v_i^t .
   $$
   \begin{tabular}{c|c|c|}
   \hline & C & D \
   \hline \hline $\mathrm{C}$ & $(2,2)$ & $(0,3)$ \
   \hline $\mathrm{D}$ & $(3,0)$ & $(1,1)$ \
   \hline
   \end{tabular}
   Table 3: Prisoner’s dilemma
   Verify if the grim-trigger strategy continues to be the Nash and subgame perfect equilibrium of the Prisoner’s Dilemma game of Table 3 using this criterion for evaluating payoffs.