Question

Use the following normal-form game to answer the questions below.

(a). Identify the one-shot Nash equilibrium.

(b). Suppose the players know this game will be repeated exactly three times. Can they achieve payoffs that are better than the one-shot Nash equilibrium? Explain.

(c). Suppose this game is infinitely repeated and the interest rate is 6 percent. Can the players achieve payoffs that are better than the one-shot Nash equilibrium? Explain.

(d). Suppose the players do not know exactly how many times this game will be repeated, but they do know that the probability the game will end after a given play is . If $\theta$is sufficiently low, can players earn more than they could in the one-shot Nash equilibrium?

Short answer

a. The one shot Nash equilibrium strategy is A, C(30,30).

b. No, no better payoff is found.

c. Yes, the better payoff is found in this condition.

d. Yes, the players can earn higher benefits in this condition..

Step-by-step solution

Identifying the one-shot Nash equilibrium

a.

If the player 1 decides to choose option A, the player 2 seeking to obtain the highest payoff will choose option C. If player 1 chooses option B, player 2 will find option C more beneficial than option D.

n the other hand, if player 2 chooses C, player 1 will choose option A since B leaves him 0 payoff. Similarly, if player 2 chooses option D, player 1 will choose option A.

Therefore, the dominant strategy for player 1 will be option A while for player 2 it will be option C. We can then obtain the Nash equilibrium with the combination of strategy A, C (30,30).

 Finding how a player can achieve payoffs at game repeated three times.

b.

The fact that the game is repeated exactly three times does not affect the decisions made under the one-shot Nash equilibrium decision.

This is due to the fact that both players can only change their decision a maximum of three times. So, if in this last play each player does not seek to maximize their payoff they will not have a new opportunity and may be left in a disadvantageous situation that in the balance initial Nash equilibrium.

For example, it may happen that on the third move, player 2 chooses option D, counting that player 1 will be able to choose option B and both get a payoff of 60 , however player 1 can make the decision to play A and leave player 2 with 0.

Therefore, there are no incentives that games can continue to be played indefinitely, so in the third play both players will make the decision to stay in the dominant strategy for each one that as we analyzed in exercise (a) is A, C ( 30.30).

Find to player can achieve payoffs at game infinitely repeated in 6 percent.

c.

If both players decide not to cheat on the collusive agreement, then each player will be able to get the maximum payoff which is the combination B,D(60,60) in each period. Assuming player 1 will not cheat player 2, the payoff will be 60 each period forever. Also taking into account that there is an interest rate of 6 % we can calculate the present value of player 1 if he decides to cooperate in the following way:

$\begin{array}{c}{\text{PV}}_{\text{coopplayer1}}=60+\frac{60}{1+i}+\frac{60}{1+i}+\frac{60}{1+i}+\dots +\frac{60(1+i)}{i}\\ =\frac{60\left(1+0.06\right)}{0.06}\\ =1060\end{array}$

Now, we can obtain the present value if player 1 decides not to cooperate and cheat on player 2. Here, player 1 will get the maximum payoff of 70, and in successive games he will get a payoff of 30. The player 2 will observe that uncooperative will choose the dominant strategy just like player 1 in the following games. Keeping the same interest rate, we can substitute the previous formula:

$\begin{array}{c}{\text{PV}}_{\text{cheatplayer1}}=70+\frac{30}{1+i}+{\frac{30}{1+i}}^2+{\frac{30}{1+i}}^3+\dots +\frac{30\left(1+i\right)}{i}\\ =70+\frac{30\left[\frac{1}{1+0.06}\times \left(1+0.06\right)\right]}{0.06}\\ =70+500\\ 570\end{array}$

Therefore, the present value of both players cooperating is greater than Player 1's decision not to cooperate (1060 > 570).Thus, both players will be able to obtain a higher payoff cooperating than under Nash equilibrium and under the strategy of non-cooperation.

Step 4: If   is sufficiently low, can players earn more than they could in the one-shot Nash equilibrium

d.

If the probability of the game ends after a given play, we have $\theta$$0<\theta <1$

Based on the result of exercise (c) we know that the decision to cooperate will bring a higher payoff than not to cooperate, therefore taking into account the probability that the game will end after a given play is:

$\begin{array}{l}\frac{60}{\theta }\ge 70+\frac{30}{\theta }\\ \frac{60}{\theta }-\frac{30}{\theta }\ge 70\\ \theta \ge \frac{30}{70}\\ \theta \ge \frac{3}{7}\end{array}$

Since the probability of the game ending after the next move is 3 / 7(>0) but still low enough, the payoff players will get will be higher than in the Nash equilibrium with one-shot move scenario.