On a remote island in a tropical sea there is an economy where silver orchids are grown by gentle giants on a hidden meadow in the middle of a lush rainforest. The orchids are translucent and delicate, every day each orchid grows one special silver petal. The gentle giants provide nutrition and minerals for the orchids to grow and be happy. The orchids concentrate all their cosmic energy in producing silver petals of the most precious qualities, each silver petal is a unique fine piece of filigree; the giants then sell the silver petals to the outside world as healing totems. The silver petals for the previous day are harvested at 730am the next morning, which takes approximately an hour, until 830am, and at 9am the sun rises and the new day
2 begins.
The orchids are very sensitive and each giant has to make an effort in order to
care for his orchids properly, especially when harvesting the silver petals; If the giant is not careful enough, he can hurt the orchid. The orchids are very special orchids and can sometimes be restless: during the night, when the giants are asleep, instead of calmly resting in the moist fertile earth on the edge of the meadow, the orchids are tempted by the sounds coming from the forest. If an orchid yields to temptation it draws out its roots, silently floats away from the meadow into the wild forest, and then just as silently returns to the meadow by 6am in the morning and replants its roots by 7am. The adventure may be fun, but it has its costs: lack of sleep disturbs the orchid's cosmic energy and the silver petal the next day is of an inferior totemic quality; but floating in the jungle at night makes an orchid happier.
On their part, the giants are gentle but they are also lazy: lazily lounging in their cave until 730am is for the giants more pleasurable than the effort of carefully preparing for the harvest of the silver petals, which takes about 30minutes and therefore requires the giants to get up by 7am. If a giant doesn't prepare well for the harvest, then he hurts his orchids during harvest, which doesn't affect the totemic quality of the silver petals but it makes the orchids less happy.
(a) Represent the interaction between a giant and one orchid during one full day cycle, from sunrise to sunrise, 9am-9am, as a game. Do players move simultaneously, or does one of the players move first? Does that matter at all? Under what circumstances is this game a prisoners' dilemma? Under such circumstances, what are the Nash equilibrium outcomes of this game? Are these outcomes Pareto efficient?
(b) Choose some numbers for the payoffs such that the interaction between a giant and an orchid during one full day is a Prisoners' dilemma (your answers to part (a) should be helpful). Suppose that both, the orchids and the giants live infinitely many days, but they all prefer present consumption to future consumption - there is some discounting: they both discount the future by the same δ. Again consider a game between one giant and one orchid, but now consider the full game throughout all the days. Consider the following strategy by the orchid:
"the first night I will not go out to the jungle; at any consecutive night, I will not go to the jungle if none of the previous nights I had gone to the jungle and none of the previous mornings the giant hurt me; if any previous morning at 730am the giant hurt me, or if any previous night I had gone to the jungle, I will from then on always float into the jungle."
Suppose also that the giant plays the following strategy:
"the first morning I will not be lazy and I will carefully prepare. On any consecutive morning, if all the silver petals harvested from the orchid until then were fine, and if on no previous morning I got up too late, then I will get up at 7am on that morning as well. Otherwise, I will get up at 730am."
Can this strategy profile be a Nash equilibrium strategy profile? For what δ, if any, will the outcome of this strategy profile lead to the orchid calmly resting at night, and the giant caring for the orchid?