Queen’s University
Department of Economics Econ 411, Prof. Christopher Cotton
Take-home midterm in Game Theory Instructions:
Please upload a PDF of your answers to onQ by 11:59pm on Monday, November 13, 2023. You may type up your answers (in a program that allows for math), or you may write your answers by hand and then scan them to upload.
Each question is worth 25% of the points. Show your work for full credit. However, partial credit should not be expected for an incorrect answer.
You may discuss questions in general terms with other students. However, to receive credit on a question, you must be able to explain it to the TA or professor if asked how you solved the problem.
1. Consider the one-shot ultimatum game. It is a non-repeated game. Shu and Ming divide $100 in the following way: First, Shu proposes a division of the dollar, with pm ∈ [0, 1] defining Ming’s share and ps ≡ 1 − pm defining Shu’s share, as proposed by Shu. Second, Ming observes pm and decides whether to accept or reject the proposal.
If Ming accepts the proposal, then their payoffs are us = 100ps and um = 100pm. If Ming rejects the proposal, then their payoffs are us = xs and um = xm, where the xi are the (discounted) values of the players’ outside options when they do not come to an agreement.
(a) In the version of the ultimatum game from lecture, xs = xm = 0. Suppose that is the case again here. What is the subgame perfect equilibrium of the ultimatum game? An appropriate answer requires defining the range of pm for which Ming accepts or rejects the proposal; and the proposal p∗m made by Shu.
(b) Now, consider an alternative version of the game where xs , xm ∈ [0, 100] and xs + xm ≤ 100. What is the subgame perfect equilibrium of the ultimatum game? An appropriate answer requires defining the range of pm for which Ming accepts or rejects the proposal; and the proposal p∗m made by Shu.
(c) Now, consider an alternative version of the game where xs, xm ∈ [50, 100] and xs +xm > 100. What is the subgame perfect equilibrium of the ultimatum game? An appropriate answer requires defining the range of pm for which Ming accepts or rejects the proposal; and the proposal p∗m made by Shu.
2. Using ChatGTP or a similar AI-driven Large Language Model, ask the following ques- tions:
• “What are some real-world examples of prisoners’ dilemmas?”
• “What are some examples of prisoners’ dilemmas that individuals might face in their
interactions with family, friends, and colleagues?”
• “What are some examples of prisoners’ dilemmas from pop culture, including books or movies?”
(a) Choose one ChatGTP generated strategic situation in response to each of the questions (3 in total, 1 per question). What are they? Construct 2x2 games (3 games in total) for each of these strategic situations. The payoff matrix should label the players and actions and list payoffs that are consistent with the chosen strategic situations.
(b) Choose one of the prisoners’ dilemma games that you created for part (a). Assume the simultaneous move game is played only once. Answer the following: (i) Do any pure strategy equilibria exist? What are they? (ii) Do any mixed strategy equilibria exist? What are they? (iii) Is the equilibrium payoff pareto efficient? (Or could both players be better off if a different feasible outcome occurred.)
(c) For the same game you chose in part (b), assume now that the game is played 5 times, with discount factor δ = 1 applying across periods (all 5 periods are equally important for expected payoffs). Write down all of the pure strategy equilibria of the 5-time repeated game.
(d) For the same game you chose in part (b), assume now that the game is played indef- initely, with discount factor 0 < δ < 1 applying across periods. (i) What would a grim trigger strategy look like in your game? (ii) For which range of δ is there an equilibrium in which both players ”cooperate” (or ”not confess” or whatever the cooperative strategy is labeled as in your game) in every period along the equilibrium path? (iii) For which range of δ is there an equilibrium in which both players ”don’t cooperate” (or ”confess” or what- ever the selfish strategy is labeled as in your game) in every period along the equilibrium path?
3. There are 100 disgruntled citizens. Each citizen simultaneously chooses whether to protest (peaceful or with looting and rioting; it doesn’t matter).
When n citizens engage in protest, there is a n/100 probability that the protest results in reform and a 1 − n/100 probability that no reform occurs. If reform occurs, all 100 citizens receive benefit B = 100, regardless of whether they engaged in protest. If a reform does not occur, all citizens receive protest benefit B = 0.
Additionally, all of the citizens who engage in protest might face arrest (and punish- ment). If a citizen is arrested, then they face a cost of C = 20.
The government wants to suppress protests. It doesn’t like reform. However, the government is broke (likely due to its corrupt leadership) and only has the policing capacity to arrest A protesters. If n citizens protest, then the probability of any given protester being arrested is A/n.
(a) Consider citizen i’s decision when the other 99 citizens choose to protest. What is citizen i’s expected utility from protesting? From not protesting?
(b) For what value of A is citizen i indifferent between protesting and not protesting? When A is less than this threshold value, does citizen i prefer to protest or not to protest?
(c) When A is less than the threshold value (answer to (b)), does there exist an equilibrium in which everyone protests? Does there exist an equilibrium in which no one protests?
(d) When A is greater than the threshold value (answer to (b)), does there exist an equilibrium in which everyone protests? Does there exist an equilibrium in which no one protests?
(e) Now suppose that A = 1. The government is so broke that it only has the policing capacity to arrest and punish one protester, which it selects from the protesting mob at random. What is the Nash equilibrium of the game?
(f) Continue to suppose that A = 1. But, now, instead of selecting whom to arrest at random, the police arrest the tallest protester. Suppose also that individuals know how their own height compares to other citizens and that no two people are exactly the same height. Is there an equilibrium in which protests take place? Why or why not?
(g) Arresting people based on height is a silly example. But, suppose instead that each player chooses when to show up for the protest, choosing any time before midnight. If no one shows up for the protest ahead of midnight, the protest is canceled and everyone goes to sleep. Suppose also that the police only arrest the first person to show up and that even if two people show up at about the same time, slow-motion video footage can be used to show who stepped into the street first (e.g., not everyone can show up at exactly the right time). Is there an equilibrium in which protests take place? Why or why not?
4. For each of the following lessons learned, list one examples of games from class or lecture notes that encompass that lesson. You don’t need to formally write out the entire game. But, do write a sentence or two describing how the games you name encompass the lesson learned.
(a) “People may be worse off when people act in their own self-interests”
(b) “Focal points, contracts and communication can help facilitate coordination by making
the outcome predictable”
(c) “Sometimes its better to be unpredictable”
(d) “Sometimes moving first gives you an advantage”
(e) “Sometimes letting the other player move first gives you an advantage” (f) “Sometimes letting the other player move first gives you an advantage”
