game theory week1 problem set 1

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You submitted this homework on Sun 19 Oct 2014 3:01 AM PDT. You got a score of 5.25 out of 9.00.

Question 1

Dominance 

1\ 2xyza1,22,25,1b4,13,53,3c5,24,47,0d2,30,43,0

Find the strictly dominant strategy:

Your Answer ScoreExplanation 1) a;    2) b;    3) c;Correct1.00  4) d;   5) x;   6) y;   7) z   Total 1.00 / 1.00 

Question Explanation

(3) c is a strictly dominant strategy.

• Because when 2 plays x or y or z, playing c always gives 1 a strictly higher payoff than playing a, b or d.
• None of the strategies is always strictly best for player 2.

Question 2

Dominance 

1\ 2xyza1,22,25,1b4,13,53,3c5,24,47,0d2,30,43,0

Find a very weakly dominant strategy that is not strictly dominant.

Your Answer ScoreExplanation1) a;   2) b;Inorrect0.00 3) c;   4) d;   5) x;   6) y;   7) z   Total 0.00 / 1.00 

Question Explanation

(6) y is a weakly dominant strategy that is not strictly dominant.

• Because when 1 plays a, b, c or d, playing y always gives 2 a weakly higher payoff than playing x or z.
• Note that it is only weakly higher when 1 plays a, as then playing x and y gives 2 the same payoff.

Question 3

Dominance 

1\ 2xyza1,22,25,1b4,13,53,3c5,24,47,0d2,30,43,0

When player 1 plays d, what is player 2's best response:

Your Answer ScoreExplanation a) Only x    b) Only yCorrect1.00  c) Only z    d) Both y and z   Total 1.00 / 1.00 

Question Explanation

(b) only y is a best response for player 2. When player 1 plays d, player 2 earns 3 from playing x, 4 from playing y and 0 from playing z. Thus only y is a best response.

Question 4

Dominance 

1\ 2xyza1,22,25,1b4,13,53,3c5,24,47,0d2,30,43,0

Find all strategy profiles that form pure strategy Nash equilibria (there may be more than one, or none):

Your Answer ScoreExplanation1) (a, x);Correct0.08 2) (b, x);Correct0.08 3) (c, x);Correct0.08 4) (d, x);Correct0.08 5) (a, y);Correct0.08  6) (b, y);Correct0.08 7) (c, y);Correct0.08 8) (d, y);Correct0.08 9) (a, z);Correct0.08 10) (b, z);Correct0.08 11) (c, z);Correct0.08 12) (d, z).Correct0.08 Total 1.00 / 1.00 

Question Explanation

(7) (c, y) is the only pure strategy Nash equilibria.

• Check that no one wants to deviate.
• Note that c is the strictly dominant strategy and so is the only possible strategy for player 1 in a pure strategy Nash equilibrium.
• When player 1 plays c, playing y gives player 2 the highest payoff.

Question 5

Nash Equilibrium - Bargaining 

There are 2 players that have to decide how to split one dollar. The bargaining process works as follows. Players simultaneously announce the share they would like to receive s1 and s2, with 0≤s1,s2≤1. If s1+s2≤1, then the players receive the shares they named and if s1+s2>1, then both players fail to achieve an agreement and receive zero.

Which of the following is a strictly dominant strategy?

Your Answer ScoreExplanationa)1;Inorrect0.00  b) 0.5;    c) 0;    d) None of the above.   Total 0.00 / 1.00 

Question Explanation

(d) is true.

• No player has any strictly dominant strategies. Any of the options given constitutes a best response to some strategy played by the other player, and so no strategy always strictly outperforms all other strategies.
• Strategies (a) and (c) are in the set of best responses of player i when player j's strategy is sj>1.
• Strategies (b) is the best response of player i when player j's strategy is sj=0.5.

Question 6

Nash Equilibrium - Bargaining 

There are 2 players that have to decide how to split one dollar. The bargaining process works as follows. Players simultaneously announce the share they would like to receive s1 and s2, with 0≤s1,s2≤1. If s1+s2≤1, then the players receive the shares they named and if s1+s2>1, then both players fail to achieve an agreement and receive zero.

Which of the following strategy profiles is a pure strategy Nash equilibrium?

Your Answer ScoreExplanation a) (0.3, 0.7);    b) (0.5, 0.5);Inorrect0.00  c) (1.0, 1.0);    d) All of the above   Total 0.00 / 1.00 

Question Explanation

(d) is true.

• Check that no one wants to deviate.
• Note that when player i plays si<1, player j's best response is sj=1−si. This holds in a) and b). Thus, both players are best responding.
• .When player i plays si=1, player j's best response can be any number as she will get 0 no matter 1. Thus c) also forms a pure strategy NE.

Question 7

Bertrand Duopoly 

• Two firms produce identical goods, with a production cost of c>0 per unit.
• Each firm sets a nonnegative price (p1 and p2).
• All consumers buy from the firm with the lower price, if pi≠pj. Half of the consumers buy from each firm if pi=pj.
• D is the total demand.
• Profit of firm i is:
  • 0 if pi>pj (no one buys from firm i);
  • D(pi−c)/2 if pi=pj(Half of customers buy from firm i);
  • D(pi−c) if pi<pj (All customers buy from firm i);

Find the pure strategy Nash equilibrium:

Your Answer ScoreExplanation a) Both firms set p=0.    b) Firm 1 sets p=0, and firm 2 sets p=c.    c) Both firms set p=c.Correct1.00  d) No pure strategy Nash equilibrium exists.   Total 1.00 / 1.00 

Question Explanation

(c) is true.

• Notice than in a) and b) at least one firm i is making negative profits since pi<c and it sells a positive quantity. Thus, firm i would prefer to deviate to pi>pj and earn a profit of 0.
• It is easy to verify that p1=p2=c is an equilibrium by checking that no firm wants to deviate:
  • When p1=p2=c, both firms are earning null profits.
  • If firm 1 increases its price above c (p1>c), it will still earn null profits.
  • If firm 2 decreases its price below c (p1<c), it will earn strictly negative profits.
  • In both cases, either the firm is indifferent or strictly worse off. Then, it does not have incentives to deviate given the other firm's strategy.

Question 8

Voting 

• Three voters vote over two candidates (A and B), and each voter has two pure strategies: vote for A and vote for B.
• When A wins, voter 1 gets a payoff of 1, and 2 and 3 get payoffs of 0; when B wins, 1 gets 0 and 2 and 3 get 1. Thus, 1 prefers A, and 2 and 3 prefer B.
• The candidate getting 2 or more votes is the winner (majority rule).

Find all very weakly dominant strategies (there may be more than one, or none).

Your Answer ScoreExplanation a) Voter 1 voting for A.Correct0.25  b) Voter 1 voting for B.Correct0.25  c) Voter 2 (or 3) voting for A.Correct0.25  d) Voter 2 (or 3) voting for B.Inorrect0.00 Total 0.75 / 1.00 

Question Explanation

(a) and (d) are (very weakly) dominant strategies.

• Check (b): for voter 1, voting for candidate A always results in at least as high a payoff as voting for candidate B and indeed is sometimes strictly better (when the other players vote for different candidates).
  • When voters 2 and 3 vote for B, voter 1 is indifferent between A or B (since B will win anyways).
  • When either 2 or 3 (or both) vote for A, voter 1 strictly prefers to vote for A than for B.
• Check (c): for voter 2, voting for candidate B is a very weakly dominant strategy.
  • When voters 1 and 3 vote for A, voter 2 is indifferent between A or B (since A will win anyways).
  • When either 1 or 3 (or both) vote for B, voter 2 strictly prefers to vote for B than for A.
• (b) and (c) can't be very weakly dominant strategies, since they sometimes do worse than the other strategy.

Question 9

Voting 

• Three voters vote over two candidates (A and B), and each voter has two pure strategies: vote for A and vote for B.
• When A wins, voter 1 gets a payoff of 1, and 2 and 3 get payoffs of 0; when B wins, 1 gets 0 and 2 and 3 get 1. Thus, 1 prefers A, and 2 and 3 prefer B.
• The candidate getting 2 or more votes is the winner (majority rule).

Find all pure strategy Nash equilibria (there may be more than one, or none)?

Your Answer ScoreExplanation a) All voting for A.Inorrect0.00  b) All voting for B.Inorrect0.00  c) 1 voting for A, and 2 and 3 voting for B.Correct0.25  d) 1 and 2 voting for A, and 3 voting for B.Correct0.25 Total 0.50 / 1.00 

Question Explanation

(a), (b) and (c) are pure strategy Nash equilibria.

• It is easy to verify that (a), (b) and (c) are equilibria by checking that no voter wants to deviate:
  • When all voters vote for the same candidate, no single voter has any incentives to deviate because his/her individual vote can't modify the outcome of the election.
  • In (c), voter 1 is indifferent between candidates A and B, and voters 2 and 3 are best responding to the strategies played by the remaining voters (if voter 2 votes for A, candidate A wins; if voter 2 votes for B, candidate B wins).
• (d) is not an equilibrium, since voter 2 has incentives to deviate and vote for candidate B.