Solutions Problem set 7 PDF

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Solutions Problem set 7 2017_2018...

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Seminar 7 – Sequential Games with Imperfect Information Lela Mélon LL.M. Universitat Pompeu Fabra 2015/2016 Problem 1 Consider the finite game in the extensive form with perfect information in the picture.

a) How many subgames are there? Three. b) Determine all the SPNE (strategic profiles, paths and payoffs) Strategic profile (a2, (b2,b1)) Path a2 - b1 Payoffs (1,1) c) Represent the game in its normal form and determine all the Nash equilibria in pure strategies. Prove that the equilibrium found with backward induction in the extensive form of the game is actually one of the Nash equilibria.

There are two Nash equilibria  (a2, (b2,b1)) which is backward induction equilibrium  (a1, (b2,b2)) d) For all other Nash equilibria demonstrate that they do not comply with the principle of the backward induction equilibrium and that they use non-credible threats. The other strategic profile, (a1, (b2,b2)) is a Nash equilibrium but it is not perfect in the subgame. The strategic profile can be presented with It makes sense to check given what the other player is doing, that no player has an incentive to deviate. If A chooses a1, the optimal response of B is b2. On the other hand, given that B chooses (b2,b2), A does not have an incentive to deviate and choose a2; B would threaten to choose b2 and A would get -1.

So (a1, (b2,b2)) is a Nash equilibrium. But it is not subgame perfect because B does not choose optimally in the subgame that starts in the second node of the decision of B: the threat is not credible because as soon as A has chosen a2, B does not dare to be harmed with choosing b2, with utility -1, instead of b1, with payoff 1. Problem 2 Consider the finite game in the extensive form with imperfect information

a) How many subgames are there? Three. b) Determine all the subgame perfect Nash equilibria (strategic profiles, game paths and payoffs). We should start by resolving the two simultaneous games

Which have two unique equilibria in pure strategies (r,c) and (r,c) with payoffs (2,1) and (1,2) respectively. So the original game can be reduced to and it has a unique subgame prefect Nash equilibrium ((a,r,r),(c,c)) with equilibrium game path a – c- r and payoffs (2,1).

Problem 3 Consider the next finite game in extensive form with perfect information.

a) How many information sets has the player 1? And the player 2? The player 1 has three information sets and the player 2 has one information set. b) Represent the game in its normal form

c) Determine all the Nash equilibria. Which of those is subgame perfect? There are five Nash equilibria  ((U,a,a),B)  ((U,a,b), B) – subgame perfect  ((U,b,a), B)  ((U,b,b), B)  ((D,a,a),A) d) Now imagine that when 1 comes back to play the second time he must do so without knowing what the player 2 has done. a. Represent the game in the normal form

b. How many information sets does player 1 have? And player 2? The player 1 has two information sets and the player 2 has one. c. Determine all subgame perfect Nash equilibria There are two subgame perfect Nash equilibria:  ((D,a), A) with payoffs (5,2)  ((U,b), B) with payoffs (3,3) Problem 4 Consider the finite game in extensive form that we will name Game1:

a) Write the Game 1 in its normal form and find all the Nash equilibria in pure and mixed strategies. How many subgames does the Game 1 have? The game in its normal form.

There are two Nash equilibria in pure strategies (La, Lb) and (Ra, Rb). There is one Nash equilibrium in mixed strategies:

Strategic profile of this Nash equilibrium is ((8/9, 1/9), (1/3,2/3)) with corresponding payoffs (4/3, 2/3). There is one subgame. b) Consider the Game 2 that is the same as the Game 1 but now the player B can observe the actions of player A before deciding. a. Write the Game 2 in extensive form In the normal form is

b. Determine all the Nash equilibria in pure strategies. There are three Nash equilibria in pure strategies:  (La, (Lb,Lb))  (La, (Lb, Rb))  (Ra,(Rb, Rb))

c. Is there a subgame perfect Nash equilibrium? Yes, the strategic profile (La, (Lb, Rb) c) Consider Game 3 that is defined as following. The player 1 chooses between actions T and B. Selecting T implies payoff of 5 for A and 25 for B. Choosing B brings you to playing Game 1. a. Represent Game 3 in extensive form. The game in extensive form is

b. How many subgames are there? Two. c. How many information sets does player A have? What about player 2? The player A has two and the player B one. d. Determine all subgame perfect Nash equilibrium. The last subgame is the Game 1 and we saw that it has three Nash equilibria.  (La, Lb) with payoffs (4,1)  (Ra, Rb) with payoffs (3,6)  ((8/9, 1/9),(1/3, 2/3)) with payoffs (4/3, 2/3) Substitute the subgames with the payoffs of the Nash equilibria we obtain reduced games

There are three subgame perfect Nash equilibria (game path T and payoffs (5,25)

Problem 5 Consider the game in its extensive form of three players represented hereunder

a) Determine all the subgame perfect Nash equilibria. The only subgame perfect Nash equilibria is the strategic profile (b,(a,b),(a,b,b,a)) generated by the equilibrium game path b – b – a with payoffs (3, 3, 0) b) Suppose that 2 cannot observe what the player 1 does. Determine all the subgame perfect Nash equilibria. If player 2 cannot observe the actions of player 1, the game in its extensive form is

The strategic profile (3/5, ½, (a,b,b,a)) is the only subgame perfect Nash equilibrium.

c) Suppose that 3 cannot observe what the player 2, but all of them can see what the player 1 did. Determine all the subgame perfect Nash equilibria in the new game.

We need to first resolve the simultaneous subgames that are played by 2 and 3. The two subgames have a unique Nash equilibrium, (b,b) and (b,a) respectively. The game can be reduced in the following way

The subgame perfect Nash equilibrium is the strategic profile (a,(b,b),(b,a)) with the equilibrium game path a - b – b and payoffs (4,0,1).

Problem 6 Consider the following game in extensive form of the players in the picture

a) How many subgames does this game have? Two. b) How many information sets does each player have? Two. c) Specify the sets of strategies of each of the players.

d) Find all subgame perfect Nash equilibria in pure strategies. We start by eliminating the subgame Nash equilibrium that starts at the third decision node of the player 2. There is a unique Nash equilibrium with strategic profile (b,d) and payoffs (1,3). The game can now be reduced in the following simultaneous game in its extensive form And its normal form being

The equilibrium strategic profile ((L, b), (b,d)) generated by the game path L – b and payoffs (4,1).

Problem 7 A company needs to send a trustworthy person to for managing for a year a new breach overseas where the living conditions are very bad. Adela, Bernard and Charles are three executive candidates to be chosen amongst. The company establishes the following procedure: it will first send Adela if she accepts the position. If she does not accept then the position is offered to Bernard. If he also passes, then Charles is offered the position. If none of them accepts it, then the company proceeds with a draw (with probabilities 1/3). The “graced” person will need to accept to be sent abroad. Adela, Bernard and Charles are making the same sum: 50.000€ annually. Let’s suppose that they are risk-neutral. Their personal situations differ vastly, which is reflected with the subjective cost of living abroad. Adela willing to resign 26.000€ to be able to still live in Barcelona the next year. Bernard is willing resign 28.000€ and Charles 37.000€.

If you go voluntarily abroad the salary increases for 40%. If you proceed to the draw the salary stays 50.000€. a) Represent the game in its extensive form.

b) Which is the subgame perfect Nash equilibrium? The strategic profile (r, a, r) with the equilibrium game path r – a and payoffs (50, 42, 50) c) What would happen if we inverted the order in which the post is proposed to the workers? The game would now be

The subgame perfect Nash equilibrium is the strategic profile (r, r, a) with the equilibrium game path r – a – a with payoffs (44, 50, 50). Problem 8 Consider the following two player game. Player 1 first has to decide between A and B. If he chooses A, then player 1 and player 2 play the game in Normal form (I) below.

If player 1 chooses B, then player 1 and player 2 play the game in Normal form (II) below

In both Normal form game, player 1 is the line player, player 2, the column player.

a) Find the pure strategy subgame perfect Nash equilibrium strategy profile of the whole game

c

Problem 9 Consider the following extensive form game

Where the first payoff corresponds to player A. Denote by G this perfect information game. Denote by G’ the imperfect information game in which A cannot see B’s previous action. a) In the imperfect information game G’, which is B’s secure pure strategy? a. Only e b. Only c c. Only d (correct) d. There is more than one secure pure strategy e. There is no secure pure strategy b) In the imperfect information game G’, mark the correct statement: a. Strategy a is not rationalizable b. Strategy b is not rationalizable c. Strategy c is not rationalizable d. Strategy e is not rationalizable e. Strategy d is not rationalizable (correct) c) In the imperfect information game G’, mark the correct statement about B’s dominated strategies a. Mixed strategy (0,1,0) dominates (0,0,1) b. Mixed strategy (1,0,0) dominates (0,0,1) c. Mixed strategy ( ½, ½, 0) dominates (0,0,1) (correct)

d. Mixed strategy (1/9, 8/9, 0) dominates (0,0,1) e. Mixed strategy (1,0,0) dominates (0,1,0) d) Mark player A’s Nash equilibrium payoffs in the imperfect information G’ a. 0 b. 1 c. 2 d. 4 (correct) e. 8 e) A change is proposed: A will observe B’s action so that instead of the imperfect information game G’, the perfect information game will be played. How much would A be willing to pay for this information? a. 1 b. 2 c. 3 d. 4 e. Nothing because he is going to be worse off. (correct) Problem 10 Consider the three player perfect information game in the figure

a) Mark the subgame perfect Nash equilibrium strategy profile a. (a, (d, d),(s,s,r)) b. (a, (c,c),(s,s,r)) c. (b,(c,c),(s,s,r)) d. (b,(c,d),(r,r,r)) e. (b,(d,c),(s,s,r)) (correct) b) One of these strategy profiles is a Nash equilibrium which is not subgame perfect. Mark it. a. (a,(d,d),(s,s,r)) (correct) b. (a,(c,c),(s,s,r)) c. (b,(c,c),(s,r,r)) d. (b,(c,d),(r,r,r)) e. (b,(d,c),(s,s,r)) c) Suppose that player 2 cannot see player 1’s actions. Mark the subgame perfect Nash equilibrium pure strategy profiles in the new imperfect information game. a. (a,(d,d),(s,s,r))

b. c. d. e.

(a,d,(s,s,r)) and (b,c,(s,s,r)) (correct) (a,c(s,s,r)) (a,(d,d),(s,r)) (b,d,(r,r,r))...