You are organizing a conference that has received n submitted papers. Your goal is to get people to review as many of them as possible. To do this, you have enlisted the help of k reviewers. Each reviewer i has a cost s;; for writing a review for paper j. The strategy of each reviewer i is to select a subset of papers to write a review for. They can select any subset S; C {1, 2, ...,n}, as long as the total cost to write all reviews is less than T (the time before the deadline): E $ij ST. jeS, Reviews are costly, so you want to reward them for their efforts. However, each paper and review has to be treated equally: specifically, there is a budget of 1 for each paper, that will be evenly shared across all reviewers who reviewed that paper. For example, if 4 reviewers reviewed a paper they will receive 1/4 each. If only one reviewer reviews the paper they will receive all the reward. Of course, each reviewer i wants to maximize their utility u;, which is the reward R; over all papers they receive minus the effort they put into writting reviews: u; = R; – L Sij. jes, You can assume that for the given s;'s, there is a combination of strategies S; where every reviewer has positive utility and all papers get at least one review. However, this outcome might not be a pure Ñash equilibrium. As a designer, your goal is to find the fraction of papers that receive at least 1 review at a pure Nash equilibrium, for the worst possible combination of s;;'s satisfying the assumption. (a) Show that there exists a set of sij's such that the fraction of papers that receive reviews is close to 1/n. Given the previous negative result, you think about increasing the reward of each paper from 1 to B > 1. (b) Show that for B = 2 this fraction is close to 1/3. [Hint: You can consider an instance with 3n + 1 papers and only n will be reviewed.]