1. (A finitely repeated game) Consider the two player game with the following payoff matrix: Player 2 b C A 3, 1 0,0 0, 0 5,0 B 0, 0 1,3 0,0 0,0 Player 1 C 0,0 0,0 2, 2 0, 0 D 0, 0 0, 5 0,0 4,4 (a) (5 points) Find all pure-strategy Nash equilibria of this game. (b) (15 points) Suppose this game is played twice (i.e. played and then repeated once). Construct a pure-strategy SPE in which (D, d) is played in the first stage. (c) (5 points) Argue that this SPE is more robust to the problem of renegotiation than is the equilibrium of the two-stage game discussed in class.2. (A hold-up problem) [This question introduces you to an important kind of problem in strategic settings. It might be helpful to look at section 6.1 of the textbook. ] Suppose that Renfe is choosing whether or not to build a new high-speed railroad in Spain. Building the railroad will cost an upfront sunk cost k. To keep accounting simple assume that the railway, if built, will run for exactly one year and that it will generate (new) revenues of (130,000,000. Operating the railroad for that year would cost 610,000,000 in fuel plus labour costs. The labour costs depend on wage. The railroad would need to employ 1000 workers all of whom would be unionised. The current going wage for union rail labour is 650,000. That is, without the new railroad, these workers would earn $50,000. (a) (5 points) Very briefly define what is meant by a sunk cost and what is meant by the 'sunk cost fallacy'? [Look it up if you do not know] (b) (5 points) Assume that labour can be hired at the going wage of 650,000. For what values of k should Renfe build the railroad? (Assume that Renfe aims to maximise profits without discounting) (c) (10 points) Suppose that if the railroad is built, after it is built, the rail union can make a take-it-or leave it wage demand w (a la Ultimatum Bargaining) for labour on the new line (assume that wage is continuous). What demand will the union make? Given this expected wage demand, for what values of k will Renfe build the new line? Why is your answer different from part (b)?Now suppose that the wage demand made after the railroad is built is not a 'take it or leave it' demand but rather part of negotiation. Suppose that, fearing strikes in the transport sector, the government has instituted compulsory arbitration in wage disputes. The arbitrator always follows a two-step approach. First, she disqualify any wage offers lower than the current going wage (that is, such that employees would rather walk away than accept the offer), and also any wage demand that would cause the employer to shut down (that is, such that the employee would rather walk away than accept the demand). Provided the offers and demands survive this test, she then "splits the difference". (d) (15 points) What wage demands and wage offers will be presented to the arbitrator after the railroad is built. Given this, if you were Renfe, for what values of k would you build the new line? Why is your answer different from that in parts (b) and (c)? Issues like this are sometimes called 'hold-up' problems. After Renfe makes their invest- ment, the union holds-up Renfe by renegotiating wage. One way to avoid the problem here (under-investment) is to give all the ex post bargaining power to the would-be ex ante investor. In general, a hold-up problem arises when one party makes a sunk relationship-specific investment and then the other party has an opportunity to expro- priate part or all of the return on that investment once it has been made. (e) (5 points) Give another example of a hold-up problem (not necessarily from Eco- nomics but it's fine either way) and potential solutions to it.3. (Another public good problem) There are two players: 1 & 2. Each player can choose a level to contribute to a public good: s. Each individual values the public good by "($1,$2) and contributing incurs a private cost for each individual, where c($) = 7. This means that the payoff for individual i is u,($1,$2) = ($1,$2)- 2 To keep things simple just consider: "($1, $2) = $1 + $2 + $152 2 (a) (5 points) What is the strategic form of this public good provision game? (b) (10 points) Solve for the Nash equilibrium of this game (Use best-response func- tions).(c) (5 points) Show that the Nash equilibrium is unique. (Easiest way would be to draw the best-response functions and show that there is a unique intersection) (d) (5 points) Now suppose that player 1 contributes first, and critically, that player 2 observes player 1's contribution prior to choosing how much to contribute. More specifically, player 2 observes s, before choosing $2. What is the set of terminal histories? What is the player function? (e) (10 points) Solve for the subgame perfect Nash equilibrium of this game