14.122 Problem Set #2 1. Find all of the Nash equilibria of the following games. 2.Consider a game in which two students simultaneously choose effort levels e1,e2∈[0, 1] in studying for an exam. The exam is graded on a curve and effort directly determinesperformance, so the student who works harder will get an A and the other student will geta B. If the students exert identical effort assume that they both get B’s. Assume that theeach student receives one extra util from getting an A instead of a B and a disutility of 2e2utils from expending effort e. Find a mixed strategy Nash equilibrium of this game.If you have time it would also be educational to try to prove that there are no otherNash equilibria.3.Consider a version of the Bertrand price competition game. Firms 1 and 2 simul-taneously choose p1and p2from the set of positive real numbers. Assume that there areno costs of production. If pi<pjassume that firm i gets demand D (pi) and firm j gets1demand 0, with D(p)=√1+p. If the firms charge equal prices assume that demand is splitevenly.(a) Give a case-by-case argument to show that the only pure strategy Nash equilibriumis p∗= p∗=0.12(b) While it only has one pure strategy Nash equilibrium, the model above actually hasan infinite number of mixed strategy Nash equilibria. Show by direct construction that themodel has a symmetric mixed strategy Nash equilibrium where the players choose strategiesfrom a distribution with full support on [8, ∞). What is funny about the set up of thismodel that lets these equilibria exist?Hint: What must the payoff in such an equilibrium be?1 3,04.When I give an exam in 14.122, my utility is decreasing in the effort which it takesme to make up new questions and increasing in the amount of time which students spendlearning the material the course covers. I have two options to choose from: to make upnew questions or to reuse questions from old exams. In preparing for the exam studentsalso have two choices: to spend their time available for studying trying to learn the coursematerial, or to try to find out what questions were asked in past years and memorize theanswers. While I don’t necessarily have these sentiments in real life, assume for the purposesof this question that having students only memorize old questions is sufficiently annoyingto me that I derive utility from having them fail an exam containing new questions if thisis all they’ve done. Assume that students care mostly about their score on the exam (butperhaps also about learning) so that if I ask new questions students are best off learningthe material, while if I ask old questions they are best off memorizing old answers.Write down what you would think are reasonable utility functions for a typical studentand for me (consistent with the description of the game above). Find the Nash equilibriumof the game and comment on how the probability with which I put old questions on theexam is affected by the parameters of the utility