COU 3: Distributing the Burden

The lesson just used that case in which N=2 to illustrate the claim that the burden of producing any particular likelihood \hat{L} of the favorable outcome can be distributed in an infinite number of ways across the members of the group. In this check of understanding, you’ll develop your understanding of the model by illustrating the same thing for the case where N=3.

Instructions

Respond to the following prompts:

  1. Write three different distributions of the burden of achieving a particular likelihood \hat{L} of the favorable outcomes in the case where N = 3, i.e. the case when the group consists of 3 persons.
  2. For each of the three distributions of the burden you offered in response to prompt 1, show that that distribution achieves the likelihood \hat{L} of the favorable outcome by showing that when the levels of effort you specified in that distribution are substituted into the expression \frac{e_1 + e_2 + e_3}{e_1 + e_2 + e_3 + 10}, the expression is equal to \hat{L}.
  3. For each of the three distributions of the burden you offered in response to prompt 1, write the utility level each of the three persons would experience if their effort levels were given by that distribution.
  4. For each of the three persons, rank the three distributions you wrote down in response to prompt 1 with respect to the utility level that person would experience if the effort levels were given by that distribution.

Example

This example is here to help you understand what you’re being asked to write in response to the prompts. Here are answers that would work for this COU if you were asked to work with the case of N=2 instead of N=3:

Response to Prompt 1

  1. Person 1 exerts effort level 0 and person 2 exerts effort level \frac{10\hat{L}}{1-\hat{L}}.
  2. Person 1 exerts effort level \frac{10\hat{L}}{1-\hat{L}} and person 2 exerts effort level 0.
  3. Person 1 exerts effort level \frac{1}{2}\frac{10\hat{L}}{1-\hat{L}} and person 2 exerts effort level \frac{1}{2}\frac{10\hat{L}}{1-\hat{L}}.

Response to Prompt 2

  1. \frac{0 + \frac{10\hat{L}}{1-\hat{L}}}{0 + \frac{10\hat{L}}{1-\hat{L}} + 10} = \frac{\frac{10\hat{L}}{1-\hat{L}}}{\frac{10}{1-\hat{L}}} = \hat{L}
  2. \frac{\frac{10\hat{L}}{1-\hat{L}} + 0}{\frac{10\hat{L}}{1-\hat{L}} + 0 + 10} = \frac{\frac{10\hat{L}}{1-\hat{L}}}{\frac{10}{1-\hat{L}}} = \hat{L}
  3. \frac{\frac{1}{2}\frac{10\hat{L}}{1-\hat{L}} + \frac{1}{2}\frac{10\hat{L}}{1-\hat{L}}}{\frac{1}{2}\frac{10\hat{L}}{1-\hat{L}} + \frac{1}{2}\frac{10\hat{L}}{1-\hat{L}} + 10} = \frac{\frac{10\hat{L}}{1-\hat{L}}}{\frac{10\hat{L}}{1-\hat{L}} + 10} = \frac{\frac{10\hat{L}}{1-\hat{L}}}{\frac{10}{1-\hat{L}}} = \hat{L}

Response to Prompt 3

  1. Person 1’s utility level: 1000\times \hat{L}. Person 2’s utility level: 1000\times \hat{L} - \frac{10\hat{L}}{1-\hat{L}}.
  2. Person 1’s utility level: 1000\times \hat{L} - \frac{10\hat{L}}{1-\hat{L}}. Person 2’s utility level: 1000\times \hat{L}.
  3. Person 1’s utility level: 1000\times \hat{L} - \frac{1}{2}\frac{10\hat{L}}{1-\hat{L}}. Person 2’s utility level: 1000\times \hat{L} - \frac{1}{2}\frac{10\hat{L}}{1-\hat{L}}.

Response to Prompt 4

Person 1’s Ranking
Distribution (a) gives Person 1 the highest utility level, followed by distribution (c), followed by distribution (b).
Person 2’s Ranking
Distribution (b) gives Person 2 the highest utility level, followed by distribution (c), followed by distribution (a).

Rubric

You can earn up to four points on this COU, one point for each of the four prompts. However, you can only give responses to prompts 2, 3 and 4 that can be evaluated if you give a correct response to prompt 1! So, getting more than one of the four points requires giving a fully correct response to prompt 1!

You don’t answer prompt 1 correctly. A correct response to prompt 1 amounts to three distributions, each one specifying an effort level for each of the four persons, where the effort levels specified sum to \frac{10L}{1-L}, with all three distributions distinct from one another. If you specify two distributions, you get \frac{2}{3}rds of a point. If you specify one distribution you get \frac{1}{3}rd of a point. If you specify zero distributions, you get 0 points.

You correctly answer prompt 1. This sets your base score to 1 point. We then add points to that based on your responses to prompts 2, 3 and 4 as follows:

  • For prompt 2, you get \frac{1}{3}rd of a point for each expression you write that corresponds to one of the distributions you wrote in response to prompt 1 and that correctly shows that the effort levels result in a likelihood of the favorable outcome of L.
  • For prompt 3, you get \frac{1}{3}rd of a point for each expression you write that corresponds to one of the distributions you wrote in response to prompt 1 and that correctly calculates the corresponding utility levels for all three persons.
  • For prompt 4, you get \frac{1}{4}rd of a point for writing a correct ranking for each of the three persons. Note that a correct ranking for any person requires having written an answer to prompt 1 that got full credit. So if you didn’t write an answer to prompt 1 that got full credit, you get 0 points on this prompt automatically.