Question

Mary and Raj are the only two growers who provide organically grown corn to a local grocery store. They know that if they cooperated and produced less corn, they could raise the price of the corn. If they work independently, they will each earn \(100\)dollar. If they decide to work together and both lower their output, they can each earn \(150\). If one person lowers output and the other does not, the person who lowers output will earn \(0\)dollar and the other person will capture the entire market and will earn \(200\)dollar Table 10.6 represents the choices available to Mary and Raj. What is the best choice for Raj if he is sure that Mary will cooperate? If Mary thinks Raj will cheat, what should Mary do and why? What is the prisoner's dilemma result? What is the preferred choice if they could ensure cooperation? \(A=\) Work independently; \(\mathrm{B}=\) Cooperate and Lower Output. (Each results entry lists Raj's earnings first, and Mary's earnings second.)

Short answer

If Raj is sure that Mary will cooperate, his best choice is to work independently (choose strategy A) and earn $200. If Mary thinks Raj will cheat, she should also work independently (choose strategy A) and earn $100. The prisoner's dilemma result is both working independently and earning $100 each. If they could ensure cooperation and both lower their output (choose strategy B), the preferred choice would earn them $150 each.

Step-by-step solution

Create a Payoff Matrix

First, we will create a payoff matrix to easily visualize the possible outcomes. | | Mary A | Mary B | |-----------|------------------|-----------------| | Raj A | (100, 100) | (200, 0) | | Raj B | (0, 200) | (150, 150) | Here, the rows represent Raj's choices and the columns represent Mary's choices.

Raj's Best Choice (Mary Cooperates)

Let's find Raj's best choice if he is sure that Mary will cooperate (Mary plays B). Since we know Mary's choice, we need to compare the payoffs corresponding to the strategies of Raj in the second column of the matrix: | Strategy | Payoff (Raj,Mary) | |----------|-------------------| | A | (200, 0) | | B | (150, 150) | Raj will choose the strategy that maximizes his payoff. In this case, he will choose strategy A, i.e., to work independently, in which he will earn $200 dollars.

Mary's Best Choice (Raj Cheats)

If Mary thinks that Raj will cheat (plays A), then she needs to compare the payoffs for her different strategies in the first column of the matrix: | Strategy | Payoff (Raj,Mary) | |----------|-------------------| | A | (100, 100) | | B | (0, 200) | To maximize her payoff, Mary should choose strategy A, i.e., work independently, in which she will earn $100 dollars.

Prisoner's Dilemma Result

The prisoner's dilemma result is the outcome where both players choose to work independently (strategy A for both): | | Mary A | Mary B | |-----------|------------------|-----------------| | Raj A | (100, 100) | (200, 0) | In this situation, both Raj and Mary will earn $100 dollars each.

Preferred Choice if Cooperation is Ensured

If Mary and Raj can ensure cooperation, meaning both of them choose strategy B, the preferred choice will be: | | Mary A | Mary B | |-----------|------------------|-----------------| | Raj B | (0, 200) | (150, 150) | In this case, both Raj and Mary will earn $150 dollars each, which is a better outcome than the prisoner's dilemma result.

Key concepts

Prisoner's Dilemma

The Prisoner's Dilemma is a classic concept in game theory, illustrating why individuals may not cooperate even if it seems to be in their best interest. In this problem, Raj and Mary face a similar dilemma with their organically grown corn. If both work independently, they each earn $100. However, if they cooperate and lower their output, they can each earn $150. The dilemma arises because if one cheats while the other cooperates, the cheater earns significantly more, while the cooperator earns nothing. This creates a situation where both acting independently (and hence, non-cooperatively) is a stable but sub-optimal outcome.

Payoff Matrix

A payoff matrix is a tool used in game theory to visualize the different outcomes and payoffs of a strategic interaction. For Raj and Mary, the matrix shows their potential earnings based on their choices:

• If both choose to work independently, they each earn $100.
• If one works independently while the other cooperates, the independent one earns $200, whereas the cooperator earns $0.
• If both cooperate, they each earn $150.

The matrix helps to clearly outline how choices affect each player's earnings and aids in determining the best strategic decision for each.

Strategy Choice

Strategy choice involves deciding the best course of action considering what another player might do. For Raj, if he knows Mary will cooperate, his best strategy is to work independently (cheat), earning him $200. However, this advantage deteriorates if Mary also decides to work independently in future rounds. For Mary, if she thinks Raj will cheat by working independently, her best strategy is also to work independently, ensuring she at least earns $100. This highlights the importance of anticipating the other player's moves in strategy selection.

Cooperation in Economics

Cooperation in economics refers to parties working together to achieve better overall outcomes. In the context of Raj and Mary, cooperation would mean jointly lowering their output to raise their individual profits to $150 each. This requires trust and possibly agreements to prevent one from breaching the cooperative strategy. Cooperation often leads to Pareto efficiency, where no one can be made better off without making someone else worse off, but achieving this in practice can be challenging due to incentives to act independently.