GAME THEORY: Analysis of Strategic Thinking Exercises on Repeated and Bargaining Games

Transcription

1 GAME THEORY: Analysis of Strategic Thinking Exercises on Repeated and Bargaining Games Pierpaolo Battigalli Università Bocconi A.Y Exercise 1. Consider the following Prisoner s Dilemma game. 1n2 C D C 3, 3 0, 4 D 4, 0 1, 1 Suppose rst we repeat this game twice (the actions chosen in the rst period are observed by all the players at the beginning of second period. 1) (*) What is the cardinality of the set of strategies of each player? 2) (*) Determine the unique subgame perfect equilibrium. Write down the reduced strategic form game and determine the Nash equilibria. 3) (**) Find the Nash equilibria in the non reduced strategic form. Is there any Nash equilibrium inducing a terminal history di erent from that of the perfect equilibrium? Suppose now the game is repeated more than twice, but still consider a nitely repeated game. 4) (***) Show that every Nash equilibrium of the nite repetition (with observable actions and, say, T stages) of this game induces the terminal history z = ((D; D); (D; D); :::; (D; D)). [As an alternative to the previous question, you can answer an easier one: Show that in the unique subgame perfect equilibrium s of the nitely repeated Prisoner Dilemma the players would always defect, that is, for all histories h 2 H and each player i, s i (h) = D.] Suppose now that the game is in nitely repeated. 5) (**) Does the result of point (4) still hold? Does the discount factor play any role? If yes, explain why and give an intuition of this result. [Hint: What does represent?] Now consider the nite repetition (T stages) with observable actions and no discounting of the following stage game (the upper-right corner of the game is 1

2 still a Prisoner s dilemma, but we have had an action for both players): 1n2 C D E C 3, 3 0, 4-1, 0 D 4, 0 1, 1-1, 0 E 0,-1 0,-1 0, 0 6) (***) Find the (pure) equilibria of the stage game. Show that (in the nite repetition of the game) there is a subgame perfect equilibrium inducing cooperation (C; C) in all stages except the last one. [Hint: the stage-game equilibrium played in the last stage may depend on the previous history!] Exercise 2. Consider the game obtained repeating twice the BoS game with the following payo s. 1n2 B S B 3,1 0,0 S 0,0 1,3 1) (*) How many strategies does each player have? 2) (*) How many di erent plans of actions (reduced form strategies) does each player have? [Hint: just collect all the identical rows and columns of the strategic form game] 3) (***) Can you nd a lower bound to the number of subgame perfect equilibria? Is there any subgame perfect equilibrium in which the pro le played in the rst period is not a Nash equilibrium of the stage-game? Is this result general (that is, it can be applied to any repeated game)? 4) (**) Is there any pair of subgame perfect equilibrium strategies such that the outcome of the second period depends on the actions chosen in the rst period? 2

3 Exercise 3. Consider the following game: rm 1 pays rm 2 w 0, rm 2 produces and delivers to rm 1 y 0 units of an intermediate good ( rm 1 chooses w and rm 2 chooses y). Payo s are as follows: u 1 (w; y) = 2y w; u 2 (w; y) = w y 2 : (2y is the value of y to rm 1 and y 2 is the cost of producing y for rm 2). Assume for simplicity that the stage game is simultaneous. Consider rst the stage-game. 1) (*) Find the equilibrium of the stage game. [Hint: just look for dominant actions] What would be the rst-best outcome, i.e. the outcome that maximizes the total surplus? 1 2) (*) Suppose that players could sign formal contracts, that is, contracts that specify a payment (w 0 ) and an amount of units (y 0 ) and that can be enforced by a court. Suppose further that both parties can ask the court to verify the agreement without costs and that the court is able to verify the wage actually paid and the amount of units actually delivered. Argue intuitively how the introduction of formal contracts can improve the situation described at point (1) Now consider the game originated by in nite repetition of the stage game. 2) (**) Specify a pro le of trigger strategies s implementing surplus maximization on the equilibrium path such that s is a SPE and for su ciently high values of the discount factor. (Prove that it is a SPE). [Hint: suppose that any deviation is punished with Nash reversion, that is a reversion to the Nash equilibrium of the stage game] 3) (***) Let (s) denote the minimum threshold for above which strategy pro le s is a SPE. Find the minimum value of (s) over the set of Pareto e cient SPEs. [Hint: Consider rst the trigger strategy SPEs with a constant w t on the equilibrium path and nd the minimum threshold over this set of equilibria. Then argue that (a) in this game Nash reversion is the harshest credible punishment, (b) non constant payments tend to hurt incentives.] 4) (**) Is there any SPE such that rm 2 overproduces (that is, rm 2 produces more than y = 1)? If yes, can you provide an example? [Hint: try to use the incentive constraints you should have found at point (3)]. 1 In this simple economy the total surplus S is given by the sum of the utilities, that is: S = u 1 + u 2 3

4 Exercise 4. (**). Consider the following game of perfect information (Nuisance Suit) between a plainti (pl. 1) and a defendant (pl. 2): 2 1. The plainti decides whether to bring suit against the defendant at cost c. 2. If he brings suit, the plainti makes a take-it-or-leave-it settlement o er of s 0 [Note: s is a choice variable of the plainti that can take any nonnegative value.] 3. The defendant either accepts to give s to the plainti, or rejects the settlement o er. 4. If the defendant rejects the o er, the plainti decides whether to give up or go to trial at a (lawyer) cost p > 0 to himself and d > 0 to the defendant. 5. If the case goes to trial, the plainti wins and gets amount x ( damages ) from the defendant with probability (0 < 1), and gets nothing with probability (1 ). 1) Assume that x + d > c. Find the subgame perfect equilibrium as a function of the parameters (that is, c, p, d, x and ). 2) How does the solution change if the plainti can pay his lawyer the amount p in advance (for example, at stage 1, if he decides to bring suit)? To avoid the boring discussions of trivial cases, you may focus you attention on the SPE where the plainti does not sue if indi erent, and would not go to trial if indi erent. 2 The example is borrowed from Rasmusen Games and Information (2001), pp

5 Exercise 5 (***) There are unexploited gains from trade between Ann and Bob; this potential surplus is worth $1. Ann and Bob have to draw-up a contract in order to achieve the 1$ surplus and bargain over its distribution. But in order to draw-up a contract Ann and Bob have to pay a xed transaction cost c A and c B respectively; if either of them does not pay the contract is not drawn-up and no surplus is realized. We assume that c A + c B < 1; so that it is e cient to pay the transaction costs. To be more speci c, Ann and Bob play the following multistage game with observable actions: Round A: (1) Ann and Bob simultaneously and independently decide whether to pay their xed cost or not. If either one does not pay, the game moves to Round B. (2) If both have paid the xed cost, Ann proposes a distribution (; 1 ) of the surplus ( is the share of Ann), where 0 1: (3) Bob says Yes or No. If he says Yes, each player i s payo is her or his proposed share of the surplus minus her or his cost c i. If he says No, the game moves to Round B. Round B is similar to Round A, except that (i) the roles of Ann and Bob in (2) and (3) are reversed (Bob proposes a distribution of the surplus and Ann responds) and (ii) the game ends if either Ann or Bob does not pay the xed cost or if Ann rejects Bob s proposal. In both cases each player loses the sum of the costs paid in Rounds A and B. Of course, if both players pay the xed cost in Round B and Ann accepts Bob s proposal, each player s payo is her or his proposed share of the surplus minus the sum of the costs paid in Rounds A and B. Find the (unique) subgame perfect equilibrium of this game. Is the contract drawn-up? [Hint: once the xed transaction costs are paid, they are sunk. Does Ann s response to Bob in round B depend on whether she has been o ered more or less than c A?] 5

6 Exercise 6 (***) Ann and Bob are still trying to split the surplus equal to 1 of the previous exercise. This time they play a three-period split-the-pie bargaining game with alternating o ers and discount factor. In particular Ann o ers a split division in periods 1 and 3 (if the game does not end before t = 3), while Bob makes his proposal in period 2 (if the game does not end in stage 1). Compared to standard bargaining games, this one has the following additional feature: in each period, besides saying Yes or No, the respondent can choose an outside option, Out, that is worth a fraction f of the pie to her/him (0 < f < 1); if the respondent chooses the outside option, the proposer gets nothing and the game terminates. The parameter f is commonly known by players. The game ends when a proposal is accepted, or the outside option is chosen, or at the end of period 3: In particular, if Ann and Bob fail to agree and the outside option is not chosen, nobody gets anything. Find the subgame perfect equilibrium as a function of the parameters f and. 6