-Consider the pricing game depicted in Figure 5.5 in which payoffs are in millions of dollars. Suppose that this game is played repeatedly and the probability that the game will end in the next stage is 20 percent and the discount rate is 20 percent. What is the present value of the stream of payoffs by cooperating?
A) $8 million.
B) $12 million.
C) $18 million.
D) $30 million.
E) $108 million.

Question

-Consider the pricing game depicted in Figure 5.5 in which payoffs are in millions of dollars. Suppose that this game is played repeatedly and the probability that the game will end in the next stage is 20 percent and the discount rate is 20 percent. What is the present value of the stream of payoffs by cooperating?
A) $8 million. 
B) $12 million. 
C) $18 million. 
D) $30 million. 
E) $108 million.

Quizplus · Accepted Answer

The present value of the stream of payoffs by cooperating in an infinitely repeated game can be calculated using the formula for the present value of an annuity: $PV = \frac{C}{r}$, where $C$ is the cash flow per period and $r$ is the discount rate. Given that the probability the game ends in the next stage is 20%, the continuation probability is 80%, which corresponds to the discount factor (since the game is less likely to continue, future payoffs are discounted). However, in this context, the discount rate given is 20%, which directly influences the calculation. If the payoff for cooperating is $8 million per period, the present value of this infinite stream of payoffs, considering a discount rate of 20% (or 0.2), is $PV = \frac{8}{0.2} = 40$. However, the correct interpretation for the discount rate in the context of game theory often involves considering how future payoffs are valued relative to immediate payoffs, which might involve using the continuation probability or directly the discount rate provided. Given the options and typical confusion around the terms, the correct calculation aligns with the formula $PV = \frac{C}{r}$, but the exact match from the options provided seems to be misunderstood in the explanation process. The correct approach with a 20% discount rate and assuming $8 million as the annual cooperation payoff should indeed yield $PV = \frac{8}{0.2} = 40$, but no such option exists, indicating a misunderstanding in the application of the formula or the interpretation of the game's parameters. Given the options and typical scenarios in repeated games, a reevaluation of the calculation or the assumptions might be necessary to align with the provided choices.

-Consider the Pricing Game Depicted in Figure 5