WS 12/13 Game Theory Exercise Set

Question 1 (GE with production)

In an economy, there are two consumers (i = 1, 2), two firms (j = 1, 2) and two goods (labor/leisure and rice). Each consumer is endowed with 1 unit of leisure but no rice. Denote xi as consumer i’s consumption of rice and li leisure. (Note that if consumer i consumes li amount of leisure, she has (1 − li) amount of labor available to sell.) The utility they derive from the consumption bundle (xi, li) is ui(xi, li) = ln(xi) + ln(li).

The two firms use labor to grow rice. Let yj denote firm j’s output of rice and Lj firm j’s input of labor. Firm 1’s production function is y1 = AL1 where A is a positive real. 1 Firm 2’s production function is y2 = (L2) 2 . Firm 1 is owned by consumer 1 and firm 2 is owned by consumer 2. The earnings of a firm are paid lump-sum to the owner. a) Define a competitive equilibrium for this economy. b) Assume both firms produce positive output. Calculate the competitive equilibrium price vector and allocation, as a function of the parameter A. (Hint: Normalize the price of rice to 1.) c) For what values of the parameter A will we have a corner solution, where one of the firms produces zero output?

Question 2 (GE with uncertainty)

In a village, there are 150 consumers and one physical commodity (rice) per state of nature. For i = 1, ..., 150, consumer i is a von Neumann-Morgenstern expected utility maximizer with (Bernoulli) utility of certain consumption given by ui(xi) = ln(xi).

For i = 1, ..., 100, consumer i lives on the highlands, and has an endowment of 1 in all states of nature.

For i = 101, ..., 150, consumer i lives on the flood plain. When a flood occurs, all con- 1 sumers living on the plain receive an endowment of 0. The probability of a flood is 10 . 9 When a flood does not occur (probability 10 ), all consumers living on the plain receive an endowment of 1. a) If the entire village trades state-contingent commodities, define a competitive equili- brium for this economy. b) Calculate the competitive equilibrium price vector and allocation. (Hint: Normalize p2 to 1.

(Hint: treat rice in the flood state and in the no flood state as two different goods.)

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Question 3 (Nash-Equilibrium in a three player game)

To prevent humans from being bitten, the government offers one unit of banked blood per day to every vampire for free. The blood bank of the capital has three units of banked blood of different conditions. Suppose that three vampires want to go out for lunch and are interested in these blood units. The manager of the blood bank makes a special offer: with a small ”contribution” of 20 Dollars a vampire can increase the probability of recei- ving a good unit. Dracula, one of the three vampires, just like his two colleagues, doesn’t want to pay the contribution. Because of different expected quality, the vampires value the best unit with 150 Dollars, the second best unit with 100 Dollars and the worst with 50 Dollars. The allocation of the units occurs according to the following:

If everyone or no one pays, the blood units will be randomly assigned, and Dracula assumes that the probabilities of getting any of the three blood units are the same. If two and only two vampires pay the contribution, these two will get the best two units with the same probability. The one who didn’t pay will get the worst unit. If one and only one pays, that vampire gets the best unit and the other two have equal probabilities receiving the worst unit.

Model the game in normal form. Assume that the vampires have common valuation of the banked blood, what can you recommend to Dracula?

Question 4 (Dividing Money)

Two people have 10 Dollars to divide between themselves. They use the following procedure. Each person names a number of Dollars (a nonnegative integer), at most equal to 10. If the sum of the amounts that the people name is at most 10, then each person receives the amount of money she named (and the remainder is destroyed). If the sum of the amounts that the people name exceeds 10 and the amounts named are different, then the person who named the smaller amount receives that amount and the other person receives the remaining money. If the sum of the amounts that the people name exceeds 10 and the amounts named are the same, then each person receives 5 Dollars. Determine the best response of each player to each of the other player’s actions; plot them in a diagram and thus find the Nash-Equilibria of the game.

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Question 5 (Tullock Contest)

Assume that there are only two producers on the market for refuelling aircrafts: Bobus and Airing. The Air Force of Tasmania is using old aircrafts made in the late 50’s. Accor- dingly, the minister of defence wants to buy new aircrafts, either from Airing or Bobus. The revenue of the contract is the same for both producers. Accordingly, it is denoted by V . But only one producer can get the contract. To convince the minister, both firms invest in marketing and bribery. The expenditures of Airing are given by xA and those of Bobus are given by xB. The probability that the minister chooses one firm is given by

( xi for xA + xB > 0 xA+xB pi = 1 i = A, B. 2 for xA + xB = 0 a) What is a strategy of a producer in the given game? What is the expected revenue for Airing and for Bobus? Is it possible that xA + xB = 0? (Suppose that there is no communicationn between Airing and Bobus.) b) Calculate the Nash-Equilibrium (xA and xB) as well as xA +xB! What does a producer earn in expectation?

Suppose that Bobus can cooperate with a local producer. If the local producer is contrac- ted by Bobus, then the local producer invests in convincing the minister. Bobus does only pay a share of V if Bobus can sell the aircrafts. If Airing can sell the aircrafts, then Bobus pays nothing, but the local partner has to bear his effort xL. c) What share ∗ does Bobus offer to the local partner? Assume that Bobus first contracts the local partner and, afterwards, Airing and the local partner invest in convincing the minister simultaneously. (Hint: Use backward induction.) d) Assume that an unit of effort made by the local partner is worth two units of effort made by Airing. Calculate the share ∗∗ that Bobus is offering his local partner, now.

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Question 6 (War of Attrition)

The game known as the War of Attrition elaborates on the ideas captured by the game Hawk-Dove. It was originally posed as a model of a conflict between two animals fighting over prey. Each animal chooses the time at which it intends to give up. When an animal gives up, its opponent obtains all the prey (and the time at which the winner intended to give up is irrelevant). If both animals give up at the same time then each has an equal chance of obtaining the prey. Fighting is costly: each animal prefers as short a fight as possible.

The game models not only such a conflict between animals, but also many other dis- putes. The ”prey” can be any indivisible object, and ”fighting” can be any costly activity - for example, simply waiting.

To define the game precisely, let time be a continous variable that starts at 0 and runs indefinitely. Assume that the value party i attaches to the object in dispute is vi > 0 and vi the value it attaches to a 50% chance of obtaining the object is 2 . Each unit of time that passes before the dispute is settled (i.e. one of the parties concedes) costs each party one unit of payoff. Thus if player i concedes first, at time ti, her payoff is −ti (she spends ti units of time and does not obtain the object). If the other player concedes first, at time tj, player i’s payoff is vi − tj (she obtains the object after tj units of time). If both players 1 concede at the same time, player i’s payoff is 2 vi − ti, where ti is the common concession time. a) Find the best response function Bi(ti), i ∕= j, for player i, j = 1, 2 and give a graphical description. b) Determine from a) all Nash-Equilibria of the game! What are the features common to all equilibria?

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Question 7 (Second-price sealed-bid auction)

Denote by vi the value player i attaches to the object; if she obtains the object at the price p her payoff is vi−p. Assume that the players’ valuations of the object are all different and all positive; number the players 1 through n in such a way that v1 > v2 > ... > vn > 0. Each player i submits a (sealed) bid bi. If player i’s bid is higher than every other bid, she obtains the object at a price equal to the second-highest bid, say bj, and hence receives the payoff vi − bj. If some other bid is higher than player i’s bid, player i does not obtain the object, and receives the payoff of zero. If player i is in a tie for the highest bid, her payoff depends on the way in which ties are broken. A simple (though arbitrary) assumption is that the winner is the player among those submitting the highest bid whose number is smallest (i.e. whose valuation of the object is highest). Under this assumption, player i’s payoff when she bids bi and is in a tie for the highest bid is vi − bi if her number is lower than that of any other player submitting the bid bi, and 0 otherwise. a) Show that bids (b1, b2, ..., bn) = (v1, v2, ..., vn) form a Nash-Equilibrium of the game! Who gets the object? b) Is (b1, b2, ..., bn) = (v1, 0, ..., 0) an equilibrium of the game? Who gets the object? c) Is (b1, b2, ..., bn) = (v2, v1, 0, ..., 0) a Nash-Equilibrium? Who gets the object? d) Find a Nash-Equilibrium in which player n obtains the object! e) Show that equilibrium a) is the only which uses (weakly) dominant strategies!

Question 8 (First-price sealed-bid auction)

Show that in a Nash-Equilibrium of a first-price sealed-bid auction the two highest bids are the same, one of these bids is submitted by player 1, and the highest bid is at least v2 and at most v1. Show also that any action profile satisfying these conditions is a Nash-Equilibrium.

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Question 9 (Mixed strategy equilibrium)

∗ ∗ Compute the mixed strategies equilibrium (1, 2) for the following game:

Player 2 L R Player 1 . T a,e b,f B c,g d,h

Question 10

Show that any strictly dominant strategy in game ΓN = (N, Δ(Si)i∈N , (ui)i∈N ) must be a pure strategy.

Question 11 (Neeskens-Effect)

Zack and Leo play football for rivalry local teams: Zack is the keeper of FC Farmers and Leo is the striker of Farmers United. In the last minute of Cup de Vila final played by these two teams, the United is awarded a penalty-kick. As the top goalscorer of the tournament thus far, Leo steps up, takes the ball from the referee and puts it on the 11-meter spot. Suppose there are two options for both Zack and Leo. Leo can either kick to the left or to the right. Accordingly, Zack can either jump to the left or to the right (from the perspective of the kicker). When different sides are chosen, Leo scores; Leo never misses the target. The payoffs are one for Leo and zero for Zack. When a same side is chosen, however, Zack cannot guarantee saving the ball for sure. It is known that Leo’s kick to the left is especially powerful; even if Zack jumps to the left 1 1 when Leo shoots to the left, Leo scores with probability of 50%. The payoffs are ( 2 , 2 ). When Leo shoots to the right and Zack jumps to the right, the probability that Zack 3 saves is 4 . a) Formalize the situation as a normal form game. Is there any Nash-Equilibrium in pure strategies? b) Find the mixed-strategy Nash-Equilibrium. What is the probability that Leo scores? Now suppose both Zack and Leo have three options to choose. Zack (Leo) can either jump (kick) to the left, the right or stay in (shoot to) the middle. Again, when different options 1 1 are chosen, Leo scores. Also as before, when both choose the left side, payoffs are ( 2 , 2 ); 3 both choose the right side, Zack saves with probability 4 . Zack, however, is good at saving balls shot to the middle. When Leo kicks to the middle and Zack stays in the middle as well, Zack saves for sure. c) Find the mixed-strategy Nash-Equilibrium in this game. What is the probability that Leo scores now? d) Compare the probability that Zack saves in b) and c). Does Zack’s ”advantage” in the middle help him? Why?

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Question 12 (First price all pay auction)

Consider the following imagined situation. Two cities 1 and 2 compete for becoming the location of the next Olympic games. City 1 puts a value of 10 million Euro on being chosen, whereas city 2’s value is 20 million Euro. Let this be known to both cities. The choice is made by some decision maker who, for some reason, may choose the city where the proposal for the organization of the games is more attractive. The cities may then spend money on architects, marketing agencies, lobbyists and other influence activities. All these types of expenditure improve the city’s proposal. Suppose that the cities are very similar ex ante and are capable of turning money into improvements of the proposal so that the city which spends more money wins the competition. If you are the mayor of city 1, how much money do you spend?

To model the situation, let us consider the following first price all pay auction. There are two contestants i = 1, 2 who attribute non-negative value to the prize that is allocated in the contest. Let these values be v1 and v2. The players know both their own valuation and the value their opponent attributes to winning the contest. By appropriate renumbe- ring of the town, v1 ≥ v2 > 0. Contestants choose their efforts xi ≥ 0 simultaneously, and the cost of effort is simply C(xi) = xi. Contestant 1 wins with probability ⎧ 1 if x1 > x2 ⎨ 1 p1(x1, x2) = 2 if x1 = x2  ⎩0 if x1 < x2

The probability that contestant 2 wins is p2 = 1 − p1. a) Model the auction as a strategic game. b) Are there pure strategy equilibria? Hint: By writing down the payoff functions, you should know that this game is different from ”war of attrition”. c) Find the mixed strategy equilibrium. d) What are the expected payoffs for the two players? What are the expected costs?

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Question 13 (Backward induction)

a) Find by backward induction the SPNE of the game above. b) Verify that the strategies identified through backward induction constitute a Nash- Equilibrium of the normal form game. c) Identify all other pure strategy Nash-Equilibria of this game. Argue that each of these other equilibria does not satisfy the principle of sequential rationality. d) What would Player 2 infer when he is called to move after Player 1 chose L?

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Question 14 (Extensive form)

Consider the following extensive form:

a) Explain the difference between a behavioral and a mixed strategy for player 2. Find an example for each type of a strategy. Under what circumstances are they payoff equivalent? b) Now, have a look at the the game tree. Specify the pure strategies of both players. How many strategies exist for each player? Give two mixed strategies of player 2 where 1 2 he plays F with probability 3 and R with probability 3 . c) Find a behavioral strategy for player 2 that always leads to the same lottery over 1 outcomes as the mixed strategy in which FR is used with probability 2 and FL with 1 probability 2 . Find a mixed strategy for player 2 that always leads to the same lottery over outcomes as the behavioral strategy in which he assigns equal probabilities to 1 each action at the first information set and assigns 3 to L at the second information set. (Player 1 chooses (x,(1-x)) over K and NK).

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Question 15 (A game in Edgeworth’s box)

A basket contains 4 liters of wine (W) and 5 liters of mineral water (M), which have to be shared among two players. He (H) has preferences

H H H H uH (xW , xM ) = xW xM , whereas she (S) has preferences

S S S S uS(xW , xM ) = min{xW , xM }.

The allocation is determined as follows: In the first stage, H proposes an allocation, i.e. two ’sub-baskets’ y = (yM , yW ) and z = (zW , zM ) such that yW +zW = 4 and yM +zM = 4 (E.g. if y = (1.5, 2) then z = (2.5, 2)). In the second stage, S chooses one of the proposed subbaskets ,y or z. The sub-basket not chosen is the share of H. a) Formulate this allocation mechanism as a game in extensive form (two stage game)! Give a precise definition of strategies and strategy space for both players! b) Explain the notion of a subgame-perfect equilibrium in the context of this game and find such an equilibrium! c) Are there further equilibria whose payoffs are different from those in b)? Explain your answer! d) Modify the game as follows: S now gets the additional option of destroying the alloca- tion (i.e. both sub-baskets) proposed by H (if she does so, all wine and water is spilled and useless). How does this change your answer to a), b) and c)? Can S gain from this strategy in equilibrium?

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Question 16 (warm glow of giving)

Suppose that a specific society consists of 2 individuals. One is rich and the other one is poor. The income of the rich individual is given by IR and is constant. This is due to the fact that the rich individual only gets paid capital interest. The poor individual has to work to earn income. Leisure and consumption determine the utility of the poor. ¯l denotes the maximal amount of hours that the poor can work. l denotes the hours worked, and the wage rate is normalized to one. Accordingly, leisure is given by ¯l − l. It is assumed ¯ that IR > l. The utility function of the poor is given by ¯ UP = ln(l + S) + ln(l − l).

S denotes a donation from the rich to the poor. The rich individual is altruistic and wants to help the poor. But the rich is also interested in his own consumption. His utility function is given by

UR = ln(IR − S) + UP . a) Suppose that S = 0. How many hours does the poor work? Compute the utility of the rich and the poor individual! b) Assume that the poor individual decides first how many hours to work. Afterwards, the rich becomes aware of the income of the poor and uses this information to decide how much to donate. Find the subgame perfect equilibrium of the game. Compute S, l, uP and uR. Have the hours worked declined? Give a verbal explanation! How has the utility of the poor changed compared to a)? c) Assume that there is a benevolent social planer that can decide how much hours to work and how much money to donate. If the social planer maximizes the sum of both utilities, how much money is donated and how many hours does the poor work? Does the poor work less compared to b)? d) Show that the utility of both increased compared to b) after the social planer decided! e) Suppose that the rich is not altruistic anymore and that there is no social planer. If the rich donates, he gets a warm glow of giving. That means he feels better after he donates. His utility function is now given by

UfR = ln(IR − S) + ln(S).

Explain verbally how this changes the Leisure decision of the poor in b).

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Question 17 (A bargaining procedure))

Two potential partners, A and B, negotiate about the division of a 1000 Euro profit from a feasible joint project. They agree to the following procedure: A has the right to suggest any division (z, 1000 − z) between herself (the 1st component) and B (2nd component). B has the right to accept or reject the offer. In case of rejection the joint venture does not materialize and both players get the payoff 0. A wants to maximize her absolute monetary payoff from the project; B, however, wants to maximize his relative monetary payoff compared to A. (If e.g. A suggested (300,700), then A would evaluate this as payoff 300, while B would evaluate it as payoff 700-300=400). These preferences are common knowledge. a) What are the payoffs of the two from (accepted) divisions (700,300), (295, 705) and (100,900)? b) Describe the extensive form of this game! Are there subgames? What is the strategy set of player A, what is the strategy set of player B? c) Determine the best reply correspondence of player B! I.e. which divisions should he accept, which divisions should he reject? What is A’s best response to this behavior? Comment on the nature of subgame perfect equilibrium in general and in this game in particular! d) Find the subgame perfect equilibrium of the game, if the roles of the two players are interchanged! I.e. B is proposing now and A responds. Comment on the difference of the solution compared to the one of c)! e) Suppose now the game has two rounds: after one player proposed and the other rejected the division; it is now the responders turn to propose, which the previous proposer can accept or reject. If he rejects, payoffs of (0,0) result. Describe the subgame perfect equilibrium of this game by using your results from c) and d)! Compare the equilibria obtained from having A or B proposing first! Would you prefer to be partner A or B?

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Question 18 (The importance of sunk cost)

Consider a product market with 2 competing firms (Firm 1 and Firm 2). Each firm simultaneously decides to compete either strongly (s) or weakly (w). If both firms choose to compete strongly, because of the high cost, they both realize a profit of 0. If both compete weakly, because of low consumer demand, they also both end up with a profit of 0. If, however, one of them competes strongly while the other weakly, the strongly competing firm enjoys a profit of 3 due to its larger share of high consumer demand and the other realizes a profit of 1 from the residue demand. a) Formulate this situation as a strategic game and find the three Nash-Equilibria.

Now let’s assume that, before playing this game, Firm 1 has the opportunity to run an advertisement of a cost x which has no effect whatsoever on the following competition. (money burning) b) Represent this game using a game tree. c) Formulate this extensive game also in normal form. d) Using the arguement of Forward Induction, what is the lower bound of x that Firm 1 by playing burning x can guarantee the outcome of (s, w) in the subgame following x? e) Intuitively, if Firm 1 can guarantee the outcome of (s, w) by burning the money and the amount of money is not too large, playing not burning and competing weakly afterwards for Firm 1 becomes unattractive. What is the upper bound of x such that Firm 1 can convince Firm 2 that it will compete strongly even without burning the money? (Hint: In this case, it’s an arguement of rationalizability of iterated elimination of dominated strategies so you may also work on the payoff matrix you identified in c).) f) Intuitively also, when x is too low, Firm 1 will never burn money. What is the highest value such that the choice of burning money becomes irrelevant?

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Question 19 (Overlapping generations (Rasmussen 1994 p. 140))

There is a long sequence of players. One player is born in each period t, and he lives throughout periods t and t + 1. Thus, two players are alive in any one period, a youngster and an oldster. Each player is born with one unit of chocolate, which cannot be stored. Utility is increasing in chocolate consumption, and a player is very unhappy if he consumes less than 0.3 units of chocolate in a period: the per-period utility functions are U(C) = −1 for C < 0.3 and U(C) = C for C ≥ 0.3, where C is consumption. Players can give away their chocolate, but, since chocolate is the only good, they cannot sell it. A player’s action is to consume X units of chocolate as a youngster and give away 1 − X to some oldster. a) If there is a finite number of generations, what is the unique Nash-Equilibrium? b) If there are an infinite number of generations, what are two Pareto-ranked perfect equilibria? c) If there is a probability  at the end of each period (after consumption takes place) that barbarians will invade and steal all the chocolate (leaving the civilized people with payoffs -1 for any X), what is the highest value of  that still allows for an equilibrium with X = 0.5?

Question 20 (Repeated prisoners’ dilemma (Rasmussen 1994 p. 142))

Column Deny Confess Row Deny R,R S,T Confess T,S 0,0

Assume that 2R > S + T . a) Show that the grim strategy, when played by both players, is a perfect equilibrium for the infinitely repeated game. What is the maximum discount rate for which the grim strategy remains an equilibrium? b) Show that tit-for-tat is not a perfect equilibrium in the infinitely repeated prisoners’ dilemma with no discounting.

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Question 21 (Trade wars)

Country A exports widgets to Country B. There is a perfectly competitive market for both consumers and producers of widgets in the two countries. In trade theory, the ”optimal tariff” argument states that a government of a large economy may be able to improve domestic welfare by exploiting the country’s market power in international trade by imposing a trade tax (that is, a tariff if the country imports the good and an export tax if it exports the good) which improves the country’s terms of trade. Of course if both countries impose trade taxes, the volume of trade relative to the free trade outcome falls and if the absolute sizes of the price elasticities of both the export demand and import supply are similar the result is worse for both countries than the free-trade outcome.

To simplify the analysis suppose the optimal tariff for Country B is independent of the level of export tax set by Country A and similarly, the optimal export tax for Country A is independent of the level of tariff set by Country B. For both countries let 0 and T denote the zero trade tax and optimal trade tax respectively. Suppose the payoff matrix is as follows: Country B 0 T Country A 0 3,3 1,4 T 4,1 2,2 a) What is the equilibrium of this trade-tax setting game if played once? b) Given that the one period discount factor of each country’s government is  ∈ (0, 1), what can be sustained as a subgame-perfect equilibrium if the trade-tax setting stage game is played five times? Make sure you explain the reasoning behind your answer. c) How large must  be in order that free-trade can be sustained as a subgame-perfect equilibrium of the infinitely repeated game? In particular write down a subgame-perfect equilibrium strategy profile that sustains free-trade for a sufficiently large  and show that it is indeed subgame perfect. d) Consider the ”limited punishment” strategy in this infinitely repeated game. ∙ Cooperative phase: Both players play 0 at the first period and play 0 if – every player has always played 0, – or k periods have passed since some player has played T ; ∙ Punishment phase: Play T for k periods if – some player played T in the cooperative phase. Find the condition on  that needs to be satisfied for this strategy profile to constitute a subgame-perfect Nash-Equilibrium for k ≥ 2. Show that if k = 1 then no value  less than 1 can make this profile a Nash-Equilibrium in this infinitely repeated game. Now suppose that each government’s horizon is only two periods and the (common) discount factor is 1. Moreover suppose that in either of these two periods a government has the option of setting its trade tax so high that the two countries revert to autarky for that period (that is, there is no trade between the two countries). Suppose the payoff to each country in autarky is zero.

15 WS 12/13 Game Theory e) Modify the payoff matrix of the stage game to include the option of a country setting a trade tax that induces autarky (label this action A). Analyze whether it is possible to sustain free-trade in either period as part of a subgame-perfect equilibrium of the once repeated (that is, two period) game.

Question 22 (Greedy (or not) pharmaceuticals)

Consider a pharmaceutical firm (player 1) that produces a unique kind of drug that is used by a consumer (player 2). This drug is regulated by the government so that the price of the drug is p = 6. This price is fixed, but the quality of the drug depends on the manufacturing procedure. The ”good” (G) manufacturing procedure costs 4 to the firm, and yields a value of 7 to the consumer. The ”bad” (B) manufacturing procedure costs 0 to the firm, and yields a value of 4 to the consumer. The consumer can choose whether to buy or not at the price p (the payoff of the consumer when she not buys is 0) and this decision must be made before the actual manufacturing procedure is revealed. However, after consumption, the true quality is revealed to the consumer. The choice of manufacturing procedure, and the cost of production, is made before the firm knows whether the consumer will buy or not. a) Is this a game of perfect or imperfect information? Is this a game of complete or incomplete information? Explain using clear definitions. b) Draw the game tree and give the normal form representation of this game. Find all the Nash-Equilibria of this game. c) Now assume that the game described above is repeated twice. (The consumer learns the quality of the product in each period only if he consumes.) Assume that each player tries to maximize the (non-discounted) sum of his stage payoffs. Find all the subgame-perfect equilibria of this game. d) Now assume that the game is repeated infinitely many times. Assume that each player tries to maximize the discounted sum of his or her stage payoffs, where the discount rate is  ∈ (0, 1). What is the range of discount factors for which the good manufacturing procedure will be used as part of a SPE? e) Consumer advocates are pushing for a lower price of the drug, say 5. The firm wants to approach the regulator and argue that if the regulated price is decreased to 5 then this may have dreadful consequences for both consumers and the firm. Can you make a formal arguement using the parameters above to support the firm? What about the consumers?

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Question 23 (The Monty Hall Problem (Rasmusen p.63))

You are a contestant on the TV show, ”Let’s Make a Deal.” You face three curtains, labelled A, B and C. Behind two of them are toasters, and behind the third is a Mazda Miata car. You choose A, and the TV showmaster says, pulling curtain B aside to reveal a toaster, ”You’re lucky you didn’t choose B, but before I show you what is behind the other two curtains, would you like to change from curtain A to curtain C?” Should you switch? What is the exact probability that curtain C hides the Miata?

Question 24 (First and Second Price Auction with incomplete information)

In a single object auction there are two bidders, i ∈ {1, 2}. Bidder i has valuation vi for the object, that is, if a bidder gets the object and pays price p then her payoff is vi −p, otherwise her payoff is zero. The two bidders’ valuations are independently and uniformly distributed at [0,1]. Bids are submitted simultaneously and should be nonnegative. The higher bidder wins the good and pays the bid she made in a first price auction or pays the other player’s bid in a second price auction. a) Model the auctions as games of incomplete information: formulate players’ strategies and the payoff functions.

Denote player i’s strategy as fi(vi) and assume f(0) = 0 and f is increasing and differen- ′ tiable, so f (vi) > 0. b) Find the symmetric Bayesian Nash Equilibrium in the case of First price auction. c) Use your solution procedure in b) to find the symmetric Bayesian Nash Equilibrium in the case of Second price auction. (Your answer should confirm that biding own valuation is a weakly dominant strategy in a second price sealed bid auction, thus a Nash-Equilibrium.) d) Show that the average price received by the seller is the same in both cases.

In experimental simulations of the first-price auction we typically observe huge overva- luation in bidding behavior. This finding is often explained by risk-aversion of the parti- cipants. Suppose now that the utility of any bidder who gets the object at the price p, is

(vi − p) , where ∈ [0, 1] describes the risk-aversion of the participant and the utility of bidder with lower bid is zero.

1 e) Show that the bidding strategy xi = 1+ vi is a Bayesian Nash Equilibrium of this auction. How does this solution explain the observation of over-valuation?

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Question 25 (Equilibrium bidding in a frst-price auction)

Assume that there are n bidders competing for an ancient coin, where n ≥ 2, in a first price auction. Each bidder i draws her true value vi independently and uniformly at random from the interval between 0 and 1. Show that it is optimal for any bidder i to bid n−1 n vi.

Question 26 (Productivity signaling)

A researcher is preparing a proposal document for a research project that he wishes a government funding agency to finance. The productivity of the researcher, denoted by p, is either 2, (that is high) or 1 (that is low). The researcher who knows what her productivity is, chooses the level of ”quality”, q ≥ 0, for the proposal document and the amount of funding , g ≥ 0, that he says is needed for the project to go ahead. The agency after observing the quality of the proposal document, decides whether to fund the project or not. Assume that the agency either provides the full amount of funding requested or rejects the project outright. If the proposal is successful (that is, the project is funded q by the agency) then the payoff to the researcher is g − p and the payoff to the research q agency is p − g. If the proposal is unsuccessful then the payoff to the researcher is − p and the payoff to the agency is zero. a) Suppose that there is complete information so that the agency does observe the resear- cher’s productivity at the same time as observing the quality of the research proposal. What level of quality will be chosen by each type of researcher? b) Suppose now that the agency does not observe the researcher’s productivity so that there is asymmetric information. In particular, suppose that the agency has no specific information at all about the researcher except that it is equally likely that his produc- tivity may be high or may be low, and that the researcher is aware of the agency’s lack of knowledge about him. How can this situation be analysed using game theore- tic techniques and concepts? Provide an example of a ”pooling” equilibrium and an example of a ”separating” equilibrium for this game.

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Question 27 (Chain-Store game)

Consider the following variant of a Chain-Store game. A single incumbent firm faces potential entry by two potential entrants (A and B) at two markets (A and B) in two periods. In the first period, a potential entrant A decides whether to enter or to stay out of a market A, in the second period entrant B decides about entering the market B observing the resulting outcome at market A.

If the entrant stays out at the particular market, the incumbent enjoys a monopoly payoff in that market; if the entrant enters the incumbent must choose whether to fight or to accomodate. Incumbent has two possible types: tough and weak. With probability , the incumbent is tough, meaning that its payoffs are such that it will fight in every market along any equilibrium path (alternatively, one can suppose that a tough incum- bent is simply unable to accommodate.) The payoff of the weak incumbent is > 0 if the entrant stays out, 0 if she accommodates and -1 if she fights. The incumbent’s objective is to maximize the discounted sum of its two-period payoff;  denotes the incumbent’s discount factor. The type of incumbent is private information.

Also each entrant has two possible types: tough and weak. Tough enrants always enter. A weak entrant has payoff 0 if she stays out, -1 if she enters and is fought, and > 0 if she enters and the incumbent accommodates. Each entrant’s type is private information, and each is tough with probability  independent of the other.

Find the perfect Bayesian equilibria of this game with respect to different values of parameters , , ,  and .

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Question 28 (A centipede game with incomplete information)

Consider a centipede game, except with a modification that player 1 may be behavioral with probability p > 0 and normal with probability 1 − p. The following game shows the payoffs of player 2 and of the normal type of player 1:

A behavioral player always chooses C. For the following questions, let us number the nodes from 1 to 5 from left to right. a) Is there a PBE where the normal type of player 1 always plays S in node 1? b) Suppose that in node 4 player 2 believes that player 1 is behavioral with probability p4. For what values of p4 will player 2 play S? For what values will player 2 play C? For what values is player 2 indifferent? Denote the value that makes player 2 indifferent ∗ by p4. c) Denote by p3 the probability with which player 1 is behavioral when node 3 is reached. ∗ Suppose that p3 < p4. Show that the normal type of player 1 must mix in a PBE. Conditional on player 1 playing C, what is the probability that he is behavioral? With what probability will player 1 play C? d) Suppose that in node 2 player 2 believes that player 1 is behavioral with probability p2. For what values of p2 will player 2 play S? For what values of p2 will player 2 play C? Hint: If player 2 plays C, then p3 = p2. In part (c) you found the probabilities with which player 1 mixes in node 3 as a function of p2.

1 e) Suppose p = 2 . How will player 1 act in node 1?

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Question 29 (Bilateral trading (M-W-G, Question 23.E.6))

Consider a bilateral trading setting in which both agents initially own one unit of a good. Each agent i’s (i=1,2) valuation per unit consumed of the good is i. Assume that i is independently drawn from a uniform distribution on [0,1]. a) Characterize the trading rule in an ex post efficient social choice function. b) Consider the following mechanism: Each agent submits a bid; the highest bidder buys the other agent’s unit of the good and pays him the amount of his bid. Derive a symmetric Bayesian Nash equilibrium of this mechanism. Hint: Look for one in which an agent’s bid is a linear function of his type. c) What is the social choice function that is implemented by this mechanism? Verify that it is Bayesian incentive compatible. Is it ex post efficient? Is it individually rational (which requires that Ui(i) ≥ i for all i and i = 1, 2)?

Question 30 (Venture capital)

An Entrepreneur (E) has a project which needs an initial investment of k. The project’s random output, x ≥ 0 depends on E’s choice of effort as follows: x = ey where e ≥ 0 is E’s choice of effort and y is a random variable distributed uniformly on [0,2]. E’s private 1 2 cost of effort is g(e) = 2 e , and this effort is unobservable, while the output is observable and verifiable. E makes a take-it-or-leave-it offer to an investor (I). Assume that E has no starting capital, so that I will have to pay the start-up cost k. The contract also specifies the sharing rule of output, so that when output is x, E leaves w(x) to himself and the rest goes to I. E has limited liability which constrains w(x) ≥ 0 for all x. Both parties are risk neutral, and the market interest rate is normalized to zero. a) Suppose that E’s effort is observable and verifiable. Solve for the first-best level of effort. For what values of k would the project be worth undertaking if there were no moral hazard? b) From now on assume that the project is worth undertaking. Show that despite the unobservability of effort by E, an optimal contract implements the first-best level of effort. Hint: Consider contracts of the form ( 0 if x < a w(x) = w¯ if x ≥ a.

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Question 31 (Moral Hazard)

Consider a standard moral-hazard problem with the following features:

∙ The principal (p) and the agent (a) are both risk neutral. Let x be the verifiable output, e be the agent’s unobserved effort, and w(x) the payment to the agent. The two parties’ final utility levels are given by,

up = x − w(x),

ua = w(x) − v(e),

where v(e) is a strictly increasing function of effort.

∙ The agent has finite wealth, which constrains the principal to offer incentive schemes w(x) ≥ 0 for all x. Assume that this constraint guarantees that the agent is willing to work for the principal. (That is, no additional participation constraint is needed.)

∙ Output can take three values: x1 = 1, x2 = 2 and x3 = 3. Effort can take two values: e0 = 0 and e1 = 1. Normalize v(0) = 0. ∙ The probability of x given e, denoted (x∣e), satisfies the Monotone Likelihood Ratio Property given by,

(x ∣e = 1) (x ∣e = 1) j > j−1 for j = 2, 3. (xj∣e = 0) (xj−1∣e = 0)

Assume that a second best solution to this problem induces the agent to choose e1. a) Formulate the principal’s problem. b) Show that in such a solution w(x1) = w(x2) = 0, and w(x3) > 0.

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Question 32 a) Consider the following symmetric normal form game:

Player 2 L H Player 1 L 1,1 2,0 H 0,2 2,2

Derive the Nash-Equilibria of this game! Discuss the idea of evolutionary game theory in contrast to non cooperative game theory and define the notion of evolutionary stable strategies (ESS) and the relatedness to Nash-Equilibria! Is there an evolutionary stable strategy in the above-mentioned game? b) The following game is known as chicken or ’hawk and dove’ game:

Player 2 T W Player 1 T -1,-1 2,1 W 1,2 0,0

It exhibits two asymmetric and one symmetric equilibrium. Explain why in a symmetric recurring (anonymous) game only symmetric equilibria may emerge! Is the symmetric equilibrium an ESS?

ESS is a static concept which has an underlying dynamic conception. Is it likely that a dynamic adaptation process will find the symmetric equilibrium?

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Question 33 a) Consider the following symmetric normal form game:

Player 2 L M R Player 1 L 1,1 2,0 1,0 M 0,2 2,2 3,1 R 0,1 1,3 3,3

Derive the Nash-Equilibria of this game! Discuss the idea of evolutionary game theory in contrast to non cooperative game theory and define the notion of evolutionary stable strategies (ESS)! Is there an evolutionary stable strategy in the above-mentioned game? What should we expect in an evolutionary dynamic of this game? b) The following game is known as ’Rock-paper-scissors’ game:

Player 2 R P S Player 1 R 0,0 -1,1 1,-1 P 1,-1 0,0 -1,1 S -1,1 1,-1 0,0

It exhibits one symmetric Nash-Equilibrium in completely mixed strategies. Is this equilibrium an ESS? c) Discuss shortly the relation between ESS, dynamic stability (asymptotically stable equilibrium) and Nash-Equilibrium!

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