An Egyptian exporter, Zaytoon Zagazig (ZZ), supplies olive oil to Crypto Coop (CC), a blockchain-savvy gourmet food cooperative and coworking space in Tribeca. ZZ can supply either high-quality olive oil (sustainably made in small batches using cold-pressed single-origin heirloom olives) or low-quality olive oil (second-rate lamp oil that has been adulterated with chemical additives to conceal its oxidative rancidity). CC must decide whether to buy 1,000 or 2,000 blockchain-encrypted bottles from ZZ's current harvest. All bottles from a given harvest are of the same quality. CC cannot observe the quality of the olive oil when it decides to make an order, but it does discover and learn the quality once a shipment arrives and is opened. (The nonrefundable payment must be made before the oil is shipped.)

If CC buys 2,000 bottles, its profits are $30,000 if quality is high and $0 if quality is low. When it buys 1,000 bottles, its profits are $20,000 if quality is high and $10,000 if quality is low. If ZZ sells 2,000 bottles, then it earns a profit of $40,000 if it ships low-quality oil, but $15,000 if it supplies high-quality oil. If, instead, ZZ sells only 1,000 bottles, then its profits are $10,000 if it makes low-quality oil and $0 if it makes high-quality oil.

a. Suppose the two companies interact only once, and they make their decisions simultaneously (i.e., Zaytoon Zagazig decides on quality before knowing how large an order it will receive, and Crypto Coop must decide how many bottles to order before learning their quality). Describe the game in matrix form and find the Nash equilibrium.

b. What outcome is collectively preferred to the above-described equilibrium outcome? Explain why this better outcome cannot be achieved in a one-shot simultaneous move game. Who has an incentive to deviate from that outcome?

Next, suppose that ZZ and CC interact and play this game for exactly two harvest seasons (and then their relationship ends forever). In other words, in the first period, ZZ chooses the quality of oil they will deliver while (simultaneously) CC chooses the size of its order. Then, in the second period, the game is repeated (ZZ again chooses quality and CC again chooses how much to buy).

c. Suppose that in the first period, ZZ and CC achieve the outcome identified in part (b). What outcome do you expect in the second period? Why?

d. What equilibrium outcome should be anticipated in the first stage of the two-stage game? Justify your answer.

Now suppose that ZZ and CC enter a long-run ongoing business relationship. We can model such an ongoing relationship as an infinitely repeated game in which the two firms play the one-shot game of part (a) in every period. We aim to study the possibility of a cooperative equilibrium in which ZZ makes high-quality oil in every period, and CC buys 2 thousand units in every period.

e. Describe clearly (in words) possible strategies of each player that could sustain such a cooperative arrangement. (In particular, specify how the two companies should agree to play the game, and what each would do in case anyone cheats.)

f. Find the conditions on the annual discount rate