Question 1:

• Competition among firms will drive the price down to 11.50, because at this price profits are zero.
• Sellers will not produce if the price is below 11.50. However, if only the bad quality product is produced, then buyers will not buy at a price above 8 Dollars. Hence, there is no production in equilibrium.
• If equal quantities of both good and bad quality goods are produced, then buyers are willing to pay at most 0.5(14)+0.5(8)=11. However, at this price firms are not willing to produce. Thus, there is no such equilibrium.
• Again there is no market, as firms would have the incentive of only producing the low quality product.
• If there is no ban on producing low quality goods, then there is no production. On the other hand, if there is a ban on producing low quality goods, then production will occur and consumers are strictly better off.

• The probability of getting a good car is 1/2. Thus, buyers are willing to pay at most (0.5)2500+(0.5)300=1,400 Dollars. Owners of good cars would not be willing to sell. Thus, only lemons will be traded. The price will be between 200 and 300.
• Now the probabilities are 0.6 and 0.4, respectively. Thus, they are willing to pay at most (0.6)2500+(0.4)300=1,620 Dollars. Now owners of good cars will sell. Thus, there exists an equilibrium in which all cars are sold at a price between 1,500 and 1,600 Dollars.
  Again, there is also a lemons equilibrium, where the prices is between 200 and 300 and lemons are sold.

Question 3:

• The dominant strategy is "Wait" for the little pig. There is no dominant strategy for the big pig.
• The only Nash equilibrium is "Wait", "Press".

Question 4:

• If all other males are doves then the payoff are 10 and 4, respectively.
• Thus, if one knows that the other player is a Dove then it is optimal to be a Hawk.
• If all other males are hawks then the payoff are -5 and 0, respectively.
• Thus, if one knows that the other player is a hawk then it is optimal to be a dove.
• The average payoff for a dove is 0p+4(1-p)=4-4p.
• The average payoff for a hawk is -5p+10(1-p)=10-15p. Thus, the payoffs are the same if and only if 4-4p=10-15p. Thus, p=6/11
• If there are more hawks then it is better to be a dove. Similarly, if there are more doves then it is better to be a hawk. Thus, the percentage will always move towards the equilibrium.

Question 5:

• You should always call heads.
• You should again always call heads.
• In this case you can randomize, i.e., it does not matter whether you call heads or tails.
• You should choose a fair coin.
• The mixed strategy equilibrium is for the first player to choose a fair coin and for the second player to randomize with 50% heads and 50% tails.

Question 6:

• If the goalie jumps left with probability p_G then the probability of scoring when kicking right is 0(1-p_G)+p_G=p_G.
• If kicker chooses left then the probability of scoring is 0p_G+p(1-p_G)=p(1-p_G).
• For the kicker to be indifferent we need p(1-p_G)=p_G. Thus, p_G=p/(1+p).
• If the goalie jumps left with probability p_K, and the goalie jumps right then the probability of not scoring is. p_K
• If the goalie jumps left with probability p_K, and the goalie jumps left then the probability of not scoring is. (1-p)p_K+(1-p_K).
• p_K=1/(1+p).
• p_K decreases as p is increased. As the kickers weak side gets stronger, the goalie will jump more often to the kicker's weak side. This, however, means that the goalie jumps less often to the kicker's good side. As a consequence, in order the kicker will slightly increase the probability of kicking to his good side.