ECON6001/6701 Microeconomic Analysis 1, S2 2022 Problem Set 7

Strategic Behavior / Game Theory

Note 1. In each of the following exercises, take the verbal description of some eco- nomic/social environment and describe the associated strategic form game. These problems will later be used to illustrate different techniques for calculating equilibria and also to discuss the economics. That should give you enough practice on being able to solve for equilibria in games and derive the economic implications for any questions that you may get on the Exam (and hopefully  beyond).  Until week  11, you may not be  able to solve for Nash equilibrium. What I expect you to be able to do at this stage is create the strategic form game from the verbal description. Of course, you should be able to complete a problem in full if asked to show a dominant strategy or solve it by IEDS.

1. (Bertrand  Competition  with  differentiated   Products).   Firm  1  and  Firm 2 compete in two related markets. If the two firms set a price of p₁ > 0 andp₂ > 0 respectively, the respective quantity they can sell is given by the equations

q₁ = 10 ¡ 2p₁ + p₂ and q₂ = 10 ¡ 2p₂ + p₁

Assume that firms compete by setting prices simultaneously and operate with a constant marginal cost of c₁ and c₂.

2. (Second Price Auction) There aer n bidders in an auction for a single item. Bidders simultaneoulsy submit bids. If b = (b₁; :::; b_n) is the squence of observed bids, then the player with the highest bid, say player i  wins the auction.  In this  case, his payoff is v_i, which is his benefit from the item, minus the second highest bid. The payoff of the remaining bidders is zero. (Assume ”ties“ are broken with equal probability.)

Set b_i^× = v_i. In class we illustrated how any strategy b_i such that b_i > b_i^× is weakly dominated by b_i^×. Reproduce an analogous argument for the case where b_i < b_i^×.

Note. Once you are done with this, you  may  want  to  jump  to  Q10,  especially Part (b), which has a very similar structure to the second price auction.

3. (Lobbying as a strategic form game) There are n firms that are lobbying the government to enact their respective favorite policy. The vNM utility of firm i if she spends t_i 0 amount of time on lobbying activity is v_i t_i if her favorite policy is implemented and t_i otherwise.  Each firm simultaneously chooses how much  time to allocate for the lobbying activity. Express  this as a strategic form game under  the assumption that the firm that lobbies the longest wins. (Break ”ties“ at with equal probability.)

4. (Bidding for Telecom licenses) Two firms are bidding for two telecom licences. Firm A has a budget of 3 dollars and Firm B has 2 dollars. Each has to divide its budget to bid in multiple of dollars for two objects (telecom licences?). Firm A will win a license provided its bid on that license is at least as high as Firm B's bid.

Firm A needs both licences to be profitable. Firm B needs only one licence to force Firm A out of the market. Assume that winning both licenses (and becoming profitable) gives Firm A a payoff of +1 and Firm B a corresponding payoff of -1. If Firm A does fails to win both licenses, Firm B gets a payoff of +1 and Firm A gets a payoff of -1.

Unlike in some of the other questions, this is  a  two  player  game  with  finitely many strategies. You may therefore, simply write down the matrix of payoffs that describes the above strategic situation instead of using functional forms.

5. (King Solomon's problem) A man (M) and his estranged wife (W) seeking sole custody of their child. The man ”values“ the child at v_m > 0 and the wife ”values“ the child at v_w > 0. The family court would like to assign custody to the parent that places the highest value. That is, the child must be given to M if v_m > v_w and to W if v_m < v_w. If the court knew these values, then the matter is trivially resolved. But the problem is that the Family Court only knows that one parent values custody at at least 100 whereas the other values no more than 50 – the court does not know which parent values the custody more. To ensure that its desired outcome is reached, the following scheme is set up.

M is given two strategies – ”y“ or ”n“ which state whether he really wants custody. W is given two strategies ”c“ – which denotes ”I want to challenge M's claim if she were to say play 'y“' or ”n - I do not really want custody“. M is given the  custody  only if he plays ”y“ and and W plays  ”n“.  In all other cases W is given custody.  However,    if M plays ”y“ and W plays ”c“, both of them will have to pay a fine of  f dollars.

Question: What do you think will happen in this game, if v_m = 100, v_w = 50 and

f = 75?

Question: What do you think will happen in this game, if v_m = 50, v_w = 100 and

f = 75?

6. (Team Production) P1 and P2 are mangers of two divisions of a firm in which the former will receive a share t of the profits and the latter receives (1  t).  Suppose  the total profit that will be generated if they choose effort levels e₁ and e₂ is given by N(e₁; e₂) = a e₁ + Ø e₂. Assume that players choose simultaneously as to how much how  much  effort to put in.  But each  player  has a  cost of exerting an effort, an effort level e_i costs e² for player i.

1. Write down the payoff functions.

2. Use the first order conditions to determine a Nash equilibrium.

3.  As you see, the Nash equilibrium effort depends on how the profits are shared, i.e. t matters! Now suppose you are a designer who is seeking to maximize the total profits that the firm generates in equilibrium. How would you share the profits (i.e. choose the value of t) so that profits are maximized.

7. (Moral Hazard in Teams) See Lec 10 slides. Show that for any differentiable sharing rule, the Nash equlibrium effort choices are necessarily inefficient.

8. (Final Offer Arbitration) See Lec 10 slides.

9. (Bilateral Bargaining) P1 and P2 make  simultaneous demands for a share of  a pie in a certain bargaining game. If their respective demands are 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, then they will receive those demands as utility with probability p(x; y). With the remaining probability of (1 ¡ p(x; y)), they get a zero payoff. In fact, assume that p(x; y) = 1 ¡ x ¡ y if x + y ≤ 1 and p(x; y) = 0 otherwise.

10. (Funding a public project) Two framers live farm on one of  the banks of a  river and would like a bridge to be built to transport produce to the market more efficiently. The personal value of the bridge to Player i (farmer i) is v_i 0. It costs

$ 10K to build the bridge. As in the case of the auction problem, assume that if Player i were to end up paying $ b, her utility is v_i b if the bridge is built and b otherwise.

1. Suppose the government determines whether the bridge is built as per the following scheme: Each player must simultaneously report her valuation. There is no reason to expect the farmers to report their valuations truthfully. If the reported valuations are  say  s₁ and  s₂,  the  bridge  is  built  provided s₁ + s₂ 10, i.e.  in which case each players share the cost and each pays $ 5  in taxes. Write down the payoff functions.

2. Now consider the following revised rule for building the bridge and the asso- ciated tax scheme: Players are asked to to report their valuations. The bridge is built if and only if the reported valuations are such that s₁ + s₂ 10.

The taxes paid depend explicitly on the announced valuations (unlike in the previous cases where they pay an average cost) and only if a player's report has an effect on the decision. In particular, Player 1 is required to pay only in the event that s₁ + s₂ ≤ 10 but s₂ < 10 and the amount she pays is 10 ¡ s₂. (Thus, Player 1 only pays in cases where her announced value makes an otherwise inefficient project efficient.) Player 2's payment is 10 ¡ s₁ if s₁ + s₂ ≤ 10 but s₁ < 10.