W24_MATH_MTHE_339_Assignment_1

Department of Mathematics and Statistics, Queen’s University MATH/MTHE 339: Evolutionary Game Theory Homework Assignment 1 Due January 31st, 2024 25 marks total 1. Cattle Battle (5 marks total) There are n cattle herders that share a common pasture in which they allow their cattle to graze. Each herder has two available strategies: they can choose to graze m cattle, or they can choose to graze m + 1 cattle. Each herder chooses simultaneously, without being able to see what the other herders have chosen. The grass in the pasture is limited, and it can only support a total of mn cattle before it begins to degrade. If the pasture is not degraded at all, each cow that grazes will bring a profit p to the herder that owns it. For each additional cow that grazes above the critical mn threshold, the quality of the pasture degrades by an amount c > 0. The effect of this degredation is to reduce the profit of every herder by c/n . Note that this reduction is applied to each herder (whether they graze m or m + 1) and it applies for each additional cow above the mn threshold. (a) (1 mark) We assume that c n < p < c . In a couple sentences, explain what these inequalities signify within the context of this game. (b) (2 marks) Calculate the payoff to each herder in the following cases: • Every herder grazes m cattle. • Every herder grazes m + 1 cattle. (c) (2 marks) If the herders are perfectly rational, will each herder graze m or m + 1 cattle? Explain and show your work. 2. Ad Attack (5 marks) Two politicians, Abby and Brooke, compete against one another in a heated election. As part of their election campaign, each candidate will choose (simultaneously) to run their campaign advertisements in one of three ways. The candidates can (i) focus ads on positive aspects of their own platform (call this strategy P ), (ii) focus ads on negative aspects of their opponents’ platform ( N ), or (iii) choose a balanced approach that combines their own positives and their opponent’s negatives ( B ). The probability that a candidate wins depends on her choice of ads as well as her opponent’s choice. The probabilities of winning for each combination of outcomes are given as follows: • If both choose the same type of campaign then each wins the election with probability 0.5. • If candidate i uses a positive campaign while j uses a balanced campaign, then j wins with probability 1. • If candidate i uses a positive campaign while j uses a negative campaign, then i wins with probability 0.3. • If candidate i uses a balance campaign while j uses a negative campaign, then i wins with probability 0.4. (a) (1 mark) Can we set payoff to each player under each outcome to be the probability that they win the election? Why or why not? (b) (2 marks) Set up a payoff matrix for the game. (c) (2 marks) Reduce the game by eliminating dominated strategies. Is there a dominant strategy equilibrium? 3. Best Picture (5 marks total) Every year, on the Saturday before the Academy Awards, film enthusiasts Larry, Curly and Moe have a movie marathon and mini awards show to decide on the ‘Best Picture’ of the previous year, fueled by a mutual love of movies and a lot of popcorn. This year, they decide on a shortlist of three movies that seem to be getting the most buzz: an action-adventure film packed with twists and turns, Adrenaline 1

Rush (which we’ll denote by A ); a dramatic, character-driven biopic, Born and Raised ( B ); and the dark, gritty crime thriller, Cops and Robbers ( C ). After watching and discussing all three movies, each person’s preferences are as follows: • Larry prefers A to B and prefers B to C . • Curly prefers C to A and prefers A to B . • Moe prefers B to C and prefers C to A . Since they seem to be stuck, they agree to have a vote. Here’s how it will work. Each person writes their selection for best picture on a ballot. If one movie receives more votes than the others, it wins the award. If there is a three way tie in which each movie gets a single vote, then Larry (as the host) breaks the tie. ( Note: Larry doesn’t get to vote a second time in the case of a three way tie, but rather whatever film he had voted for already would be the winner.) We assume that all three movie lovers are perfectly rational, and that each knows the preferences of the other two; we would like to determine how each of them would vote, and which movie will take home the coveted prize. (a) (3 marks) For each individual, determine which of their strategies are strictly or weakly dominated. (b) (2 marks) After eliminating the strategy (or strategies) found in part a), which strategy/strategies can you now eliminate? What will be the result of the vote? (“...and the Oscar goes to .... ”) 4. Corporate Division (5 marks total) There are five executives on the board of directors of a major financial corporation: a CEO and four vice-presidents. After a very profitable year, there is a surplus of 100 million dollars that is to be shared among the board members as bonuses. The CEO calls a meeting to decide how to split up the money. After some deliberation, they agree on the following procedure: • Each board member will submit a proposal about how to divide the money, one after the other. Each proposal will outline a potential division of the 100 million dollars among the five board members. • After each proposal, there is a vote. In order for a proposal to pass, it needs to be agreed on by 50% or more of the board members that are voting (members are permitted to vote for their own proposal). • If the proposal passes, then the money will be split up accordingly and there are no more proposals; if the proposal is rejected, then the board member who made the rejected proposal will lose their right to vote in any future votes. If the proposal is rejected, the next board member in line will then make a new proposal and the process repeats. • The order for proposals is as follows: CEO → VP1 → VP2 → VP3 → VP4 Assuming that board members are perfectly rational, how will the money be divided? Explain and justify your answer. 5. Survivor Island (5 marks total) Four players (Alice, Bob, Carlos, and Diane) are contestants on a three-day long reality television game show, Survivor Island . Here’s how it works: • Each player is assigned a value as follows: Alice → 4, Bob → 3, Carlos → 2, Diane → 1. • At the start of each day (before any of the votes), the total value of all remaining ‘survivors’ on the island is added up. That number, multiplied by 10000 dollars, is added to a pot of prize money that will divided among the two survivors remaining at the end of the final day as outlined below. (Note that all four contestants are considered to be survivors on the first day, so a total of (4 + 3 + 2 + 1) × $10000 = $100000 is added to the pot at the start of day 1). • At the end of the first and second day there is a vote among survivors to expel one of the survivors from the island. Each survivor votes privately (without seeing any other contestant’s vote). We’ll assume that each player must vote to expel a player other than themselves. Whoever receives the 2