EC9410 Game Theory 2021/22

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EC9410

May Examinations 2021/22

Game Theory

Section A: Answer the ONE question

 

1.

(a) Two co-workers, player 1 and 2, must independently decide whether to tackle a        simple project (S) or a hard project (H). In making this decision, both players               maximize the expected value of their own monetary remuneration. Anyone                attempting the simple project will accomplish it, while the hard project will be a        success only if both of them decide to work at it. If both players attempt the same     project, then they equally divide among themselves a payment of 4 if the project is   the simple one and a payment of 12 if the project is the hard one. If the two workers attempt different projects, then only the simple project gets completed and the         worker who executed it obtains 4, while the worker who attempted the hard project obtains zero.

(i)          Formalise this situation as a normal-form game, by writing it down in matrix form. (5 marks)

(ii)         Assume mixed strategies are allowed. Write down the best- response

correspondence for one of the two players. (3 marks)

(iii)        Find all pure and mixed Nash equilibria of this game. (7 marks)

(b) Consider again the game described in (a) above. Now assume the skills of player 2 are uncertain. Player 2 can be either educated (e) or unskilled (u). While player 2 knows   whether they are educated or not, Player 1 does not know it. Both players know that whether player 2 is educated or unskilled has been determined by chance, with           0≤ p≤ 1 being the probability that 2 is educated. When player 2 is educated, the

strategic situation is the same as the one described in (a) above. However, if player 2 is unskilled, the hard project will fail even if both workers attempt it, and, in that        case, they will both obtain zero.

(i)          Model this situation as a Bayesian game. (5 marks)

(ii)          Find all Bayes- Nash equilibria in pure strategies, as a function of p.

(10 marks)

(c)  Consider the following extensive form game with imperfect information:

 

(i)          Write the game in normal-form (i.e., in matrix form) and identify all pure Nash equilibria.  (5 marks)

(ii)          Identify all pure subgame- perfect equilibria. Explain your answer with

reference to subgames as appropriate.  (5 marks)

(iii)         Identify all Sequential equilibria, focusing on pure strategies. Hint:

remember that a sequential equilibrium is an assessment, and it is not just defined by strategies, but also by beliefs.  (10 marks)

  

Section B: Answer ONE question 

2.    Consider the game below between two players, for parts (a), (b) and (c).

 

 

High

Low

High

2,2

4,1

Low

1,4

3,3

Punish

-1,-1

-1,-2

(a)  First, assume the game is played once. Find the Nash equilibria of the one period game.  (10 marks)

(b) Assume the game is repeated forever with discount factor 6 = 0.75. Construct a subgame perfect equilibrium in which (Low, Low) is played along the equilibrium path.  (15 marks)

(c)  Still considering the infinitely repeated game, construct a subgame perfect                equilibrium in which (High, Low) is played along the equilibrium path (Player 1 plays High and player 2 plays Low) and also find for which values of discount factor 6 this type of equilibrium exists.  (25 marks)

 

3.    Firm U (“upstream”) makes a product (x) for a (total) cost of CU (x) = x +  (there are no fixed costs). It sells this to firm D (“downstream”). Firm D resells this to final               consumers for p per unit. The inverse demand curve is:

p(x) = 16 − 

This question concerns the firms’ bargaining over the amount x and division of profits.

(a) What value of x maximises the total profits of the two firms (几 = 几U + 几D ) if they   could practice perfect (first-degree) price discrimination and thus capture the full     welfare generated by the market w(x) =  p(x)dx − CU (x) and what is the value of this welfare (wopt)?  (6 marks)

Now consider how the firms might bargain over the division of this maximised profit.

(b) Suppose first that firm U announces a fixed unit supply price q at which it will supply the product; firm D orders the amount that maximises its profit 几D .

几D (x) = (p(x) − q)x

Compute:

(i) the amount x(w) demanded by firm D at any given supply price q;  (2 marks) (ii)  the input supply price q ∗ that maximises firm U’s profit; (2 marks)

几U (q) = qx(q) − CU (x(q))

and:

(iii) the resulting profits of the two firms.  (2 marks)

 

(c)  Conversely, suppose the downstream firm announces the input purchase price T at which it will buy the product; firm U supplies the amount that maximises its profit  几U . Compute:

(i)   the amount x(T) of the product supplied by firm U at any given price T [You should get the same amount as in part (b)(i); (2 marks)

(ii)  the input purchase price T* that maximises firm D’s profit: (2 marks)

几D (T) = (p(x(T)) − T)x(T)

and:

(iii) the resulting profits of the two firms.  (2 marks)

(d) Assume the two firms are bargaining over how to divide the maximal profit w* found in part (a); if they cannot agree, Firm U will supply the amount found in parts (b) and  (c) to firm D at the average of the prices from (b) and (c) (). Find the                     corresponding ‘threat point’ profits 几 U(t)hreat and 几D(t)hreat and use these to compute      the Nash bargaining solution 几 U(N)ash and 几D(N)ash (where 几 U(N)ash + 几D(N)ash = wopt) for     this problem. (16 marks)

(e)  Finally, suppose that firm U receives an outside offer 几 U(e)xt to supply the product to   another firm on an exclusive basis. If it uses this as its threat point payoff, while firm D sticks to the threat point profit 几D(t)hreat found in part (d) show how this affects the Nash bargaining solution? At what level of the outside offer would the bargaining     collapse, and why? (16 marks)











2022-08-19