Midterm Exam Economics 410-3 Spring 014 Professor Jeff Ely Instructions: There are three questions with equal weight. You have until 10:50 to complete the exam. This is a closed-book and closed-notebook exam. Please explain all of your answers carefully. 1. Consider the following two-person game. The game has k+1 stages for some positive integer k. The two players move in alternating stages, with player 1 moving in stage 1, player in stage, etc. At the beginning of the game there is a dollar on the table. In each stage the player to move can either grab the money on the table or pass. Every time a player passes, the amount of money is doubled. The game ends as soon as a player grabs the money. The grabbing player leaves with all of the money on the table and the other player leaves with nothing. If after k + 1 stages no player has grabbed then the two players share equally the money left on the table. Each player s payoff equals the amount of money she leaves with. This is a finite extensive-form game with perfect information. (a) Represent this game in strategic form. Formally describe the pure strategy sets of the two players and the strategic-form payoff function. A strategy for player i specifies each stage at which i moves whether he will grab or pass. One way to describe them is and S 1 = {grab, pass} k+1 S = {grab, pass} k The payoff to pure strategy profile σ is determined as follows. Find the earliest stage z at which some player grabs. If z k + 1 then the grabbing player earns z 1 and the other player earns zero. If no one ever grabs then both players earn k. (b) What are the sets of rationalizable (pure) strategies for players 1 and? Prove that every strategy in the set you have identified is rationalizable and every pure strategy outside of that set is not. Every strategy is rationalizable for both players. To prove this, note that every strategy for is a best-response to a strategy for 1 that grabs in the first stage. 1
Consider any strategy for 1 that grabs in some stage. Let z be the first stage in which it grabs. This strategy is a best-response to the strategy for that first grabs in stage z + 1. Finally consider the strategy for 1 that never grabs. It is a best-response to the strategy for that never grabs. To see this, if they both pass until the end then they both get k. This is the same payoff 1 would get by grabbing in the last stage and this is higher than 1 could get by grabbing earlier. (c) What is the set of all Nash equilibria of the game, both in pure and mixed strategies? i. There are two SPE which you can compute by backward induction. The first is always passing and the second always grabbing (for both players). ii. There are several other Nash equilibria in pure strategies which involve player 1 and player grabbing in their first chance and either grabbing or passing at each node in the future. iii. Mixed strategies in which player 1 grabs in the first period: Let ˆΣ 1 be the set of all such mixed strategies for 1. Consider any mixed strategy α for such that the maximum expected payoff player 1 can earn against it is less than or equal to 1 i.e., max ˆσ 1 u 1 (ˆσ 1, α ) 1 (There are many such strategies for.) Let ˆΣ be the set of all such mixed strategies for. This set of Nash equilibria is ˆΣ 1 ˆΣ. iv. Mixed strategies in which no one grabs: In order for someone to mix in equilibrium at some node reached with positive probability it better be the case that the expected payoff of passing is the same of grabbing. Inductively: think of the strategy where both players pass always except player 1 mixes in the last node choosing a probability greater than 1 of passing, this is a NE. Now think of the same strategies except player 1 mixes with probability exactly 1 and player also mixes in her last turn with any probability at least 1, this is also a NE. All of this mixed strategies will be of that form: pass up to a point, then at one node someone mixes with probability greater than or equal to 1 of passing, and both players mix with probability exactly 1 at every node after that. (d) Select one Nash equilibrium that is not Subgame-Perfect and find a profitable one-stage deviation for one of the players.
Any of the strategies in bullet in the previous answer which are not grabbing always has a profitable one-stage deviation (just pass when the other player is going to pass after this stage).. In the two-player game matching pennies, each player chooses from {H, T } and the matcher tries to choose the same action as his opponent while the opponent, the mismatcher, tries to choose the opposite action of the matcher. The player who achieves his objective receives one dollar from the other player. This question is about an incomplete-information version of matching pennies. It is common knowledge that one player is the matcher and the other player is the mis-matcher but the incomplete information is about which player is the matcher. In particular, it is common knowledge that Each player is equally likely to be the matcher Player 1 knows who is the matcher (and knows this at the time the players simultaneously choose actions.) Player has no additional information beyond what is given above. (a) Represent this game of incomplete information as a Bayesian game with a common prior. What is the set of types for each player? What is the common prior? What is the Bayesian-game payoff function? Player 1 has two types {t 1, t 1}, player has only 1 type and the common prior assigns equal probability to the two types of player 1. The payoff function is represented by the following matrices. (b) Find a Bayesian Nash equilibrium in which player 1 plays a pure strategy. Player 1 plays H regardless of his type. Player randomizes with equal probability on {H, T }. 3. This question is about price competition between two firms under complete information. (a) Each firm can produce a single unit of output at cost c > 0. There is a single consumer who views the two firms products as perfect substitutes. The consumer has a maximum willingness to pay of v > c for a single unit of output sold by either firm. The consumer has no additional value for any additional units. The two firms will simultaneously make price offers to the consumer. The prices can be any non-negative real number. The consumer observes the price offers p 1 3
and p and then decides whether to purchase from firm 1 at price p 1, purchase from firm at price p or make no purchase at all. The chosen firm produces its unit and incurs cost c, the other firm does not produce and incurs no costs. Then the game ends. The consumer s payoff is v p if he purchases a unit at price p and the consumer s payoff is zero if he does not purchase any unit. Firm i s payoff is p i c if the consumer purchases from firm i and zero otherwise. This is an extensive-form game with perfect information and simultaneous moves. Prove that all subgame-perfect equilibria yield the same payoffs for all players: 0 for each of the two firms and v c for the consumer. (b) Now there are n distinct buyers who arrive in sequence (buyer 1 arrives in period 1, buyer arrives in period, etc.) and the same two firms will compete in each period to sell to the buyer arriving in that period. All buyers are identical to the buyer described above: maximum willingness to pay v > c and a demand for at most a single unit. Each of the two firms has a total production capacity (over the whole duration of the game) of n meaning that each unit the firm produces up to the nth can be produced at unit cost c. Competition in each period works exactly as in the single-period version above: the firms simultaneously quote prices and the consumer active in that period chooses which firm to purchase from (or to make no purchase). That consumer then exits the game and the next stage begins with the arrival of the next consumer. At the beginning of each stage, all players observe the prices quoted and purchase decisions in all previous stages. Each firm s payoff in the overall game is the sum of payoffs it earns in the n stages. This is a n + -player finite-horizon extensive-form game with perfect information and simultaneous moves. Prove that all subgame-perfect equilibria yield the same payoffs for all players: 0 for each of the two firms and v c for each consumer. (c) Now there are again n stages but firm 1 has capacity n 1 while firm has capacity n. Each firm can produce any number of units up to its capacity at unit cost c. Once a firm has produced to capacity, that firm cannot produce additional units and exits the game leaving the other firm in a monopoly position. (A monopoly firm quotes a price and the consumer chooses whether to purchase from the monopoly firm at that price or not purchase at all.) Again the firm s payoffs are the sum of payoffs earned in the n stages. All of the subgame perfect equilibria of this game yield the same payoffs for the firms. What are the firms subgame perfect equilibrium payoffs? The idea is that if Firm 1 spends all its capacity in the first n 1 rounds then in the last round Firm will be able to extract monopoly profit v c. But if 4
Firm 1 does not make a sale in any of the first n 1 rounds, capacity constraint stops being binding so firms compete as in 3(b) again and both make zero profits thereafter. Thus, Firm wants to ensure that Firm 1 makes sales in all n 1 first rounds. If the buyers are not strategic (randomize between firms when indifferent) the only SPNE are: Firm 1: In the round k = 1... n 1 if Firm 1 has been making sales in all previous rounds make an offer v. If Firm 1 did not make a sale in any of the previous rounds, make an offer c. In the last round, if there is capacity left, make an offer c. Firm : In the round k = 1... n 1 if Firm 1 has been making sales in all previous rounds make an offer > v. If Firm 1 did not make a sale in any of the previous rounds, make an offer c. In the last round, if Firm 1 has capacity left, make an offer c, and v otherwise. All buyers: buy from the firm offering the lowest price (if the price is v). If offers are the same, randomize between firms (choosing Firm with positive probability). To see why this is a subgame-perfect equilibrium note that in any of the first n 1 stages, Firm can earn at most strictly less than v c by a one-stage deviation. (He must price below v to make a sale and if he does so we will reach a subgame in which the continuation strategies earn both firms zero in all remaining stages.) In addition, if the buyers are strategic (don t randomize when indifferent), the following are also SPNE: Firm 1: In the round k = 1... n 1 if Firm 1 has been making sales in all previous rounds make an offer p k [c, v]. If Firm 1 did not make a sale in any of the previous rounds, make an offer c. In the last round, if there is capacity left, make an offer c. Firm : In the round k = 1... n 1 if Firm 1 has been making sales in all previous rounds also make an offer p k. If Firm 1 did not make a sale in any of 5
the previous rounds, make an offer c. In the last round, if Firm 1 has capacity left, make an offer c, and v otherwise. All buyers: buy from the firm offering the lowest price (if the price is v). If offers are the same, buy from Firm 1. Thus, if buyers are not strategic Firm 1 gets (n 1)(v c), Firm gets v c. If buyers are strategic Firm 1 s payoff is n 1 k=1 (p k c), p k [c, v], and Firm s payoff is v c. In particular, if all p k = c Firm 1 gets 0. 6