40.316-Homework 3 Solved

Question 1: Using the principle of backward induction find the subgame perfect equilibrium (SPE) for the following three player extensive form game. How about the outcome path? State all the assumptions you make.

 

   Question 2: Consider the following entry deterrence game:

 

  Question 3: Assume that two firms, A and B, dominate a specific market. The marketing departments of both companies have discovered that increased advertising expenditure, all other things remaining the same, has a positive effect on that firms sales. However total sales also depends negatively on how much the other company spends on advertising. This is because it adversely affects the market share of the first company. If we assume there are only two levels of advertising (high or low expenditure) which each firm can carry out, then the payoff matrix in terms of profits (£millions per year) for the two companies is given in the following figure.

 

 

 From this normal form game both firms are better off if they each maintain low advertising expenditure compared to both incurring high advertising costs. This is because if both firms increase their advertising expenditure simultaneously then market shares are unaffected, but due to increased advertising costs profits fall. However each firm has an incentive to try and increase their level of advertising above their competitors level as this increases market share and overall profits.

If the interaction between the firms is infinitely repeated then it is possible for the two 
firms to coordinate their actions on the efficient outcome. Design a trigger strategy with 
punishment to enforce the cooperation and estimate the requirement on the discount factor.

   Question 4: Consider a scaled down model on workings of OPEC. Countries aim to collude on production, drive up price and profits and return to equilibrium if someone deviates. Let P = 300 - 5Q world demand for oil (where Q is total production and qi be the production of country i). Let marginal cost for production be “c” for all countries. Assign a suitable payoff function that gives the profit to each country: (P -c)qi. Assume there are 4 countries and c = 20.

(a)    Using the payoff function of previous question find static Nash equilibrium(where each country tries to maximize its profit).

(b)    Now suppose countries in an infinitely repeated game try to enforce a “Grim Trigger” strategy by keeping qi = 7 each unless someone deviates. If deviation, go to q = 11.2 each forever. Is it enforceable and feasible given which requirement of discount factor?