Problem 13.3 (H) A team game is a cost-minimization game in which all agents have the same cost function: C₁(s) : = Ck(s) for every outcome s. In a dummy game, the cost of every agent i is =... - Ci(si, s_i) for every s_i independent of her strategy: Ci(si, s_i) and every si, s € Si. = Prove that a cost-minimization game with agent cost functions C₁,..., Ck is a potential game if and only if Ci(s) = C(s) + Cd (s)
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- Consider the following extensive form game between player 1 and player 2. T B (2, 2) L R R (3, 1) (0, 0) (5, 0) (0, 1) (a). Find the normal form representation of this game. (show the bimatrix) (b). Find all pure strategy NE. (c). Which of these equilibria are subgame perfect?Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.8) Find the mixed strategy Nash equilibrium of the following normal form game. Player 2 T1 T2 T3 2, 3 3, 5 1, 1 Player 1 S2 1, 4 4, 3 0, 5 Player 1 attaches probability (S1, S2) = () and Player 2 attaches probability (T1, T2, T3) = ( ) Player 1 attaches probability (S1, S2) = (.) and Player 2 attaches probability (T1, T2, T3) = (qi, 42, 1 – q1 – 92) where q1 , and 0 < q2 S %3D Player 1 attaches probability (S1, S2) = (G,;) and Player 2 attaches probability (T1, T2, T1) = (qı.42, 1 – q1 – 42) where 0 < qi <, and q2 = 3. Player 1 attaches probability (S1, S) = (;, -) and player 2 attaches probability (T1, T2, T3) = (1.42, 1- q1- 42) where 0 s qı s and q2 =
- Two workers are on a production line. They each have two actions: exert effort, E, or shirk, S. Effort costs a worker e > 0 and shirking costs them nothing. If two workers do action E a lot of output is produced and the workers earn £3 each. If only one worker chooses action E less output is produced and they both earn £1. The workers earn nothing if they both shirk. (i) Describe this situation as a strategic form game (assuming the workers do not observe each other's effort choice when making their own decision). (ii) For what values of e does this game have strictly dominant strategies? (iii) Describe the Nash equilibria of this game for e = 0,1, 2, 3. (iv) The workers now are re-arranged into a production line. First worker 1 moves and then worker 2 moves. Worker 2 can now see worker l's effort level before they choose their effort. Draw this extensive form game. (v) Find the subgame perfect equilibria of the production-line game for c = 1/2 and c = 3/2.2- Consider the following game. Player 2 Player 1 U 12, 2 | 3, 9 5, 8 4, 2 D (a) Find all the Nash equilibria, pure and mixed. (b) Suppose that the payoff of the column player u:(D, L) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria. (c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player l's? Give some intuition.1. Consider a sequential continuum game with two players, in which Player 1 first chooses a continuous variable £₁, and then Player 2 selects £2, with payoff functions ₁ and 7₂. Find the best response for P2, as the function î2 = BR₂(x1), for each given #₂: (a) (b) (c) T₂(x1, x₂) ) = (1400 — x1 — x2)x2 − 13x² T₂(x1, x₂) T₂(X1, X2₂) = : 100x2 + 3x1x2 − 14x1 – 30x1x² = 15x2 x² + x² +1
- Use the mixed method (Nash Equilibrium) to determine the following: What percentage of time should the maximizer play strategy H? 1. H6 17 ! 12 10 18.8% 84.6% O 15.4% 11.8%(a) Consider the following bimatrix of a normal form game. Show that the set of strategies that survive elimination of weakly dominated strategies depends on the order in which the weakly dominated strategies are eliminated. Player 2 L R 1,1 0,0 M| 1,1 0,0 | 2,1 Player 1 2,1 (b) Find an example of a game where the unique Nash equilibrium in pure strategies would not survive the eliminination of weakly dominated strategies. (c) Find an example of a game where both players' strategies are weakly dominated in the Nash equilibrium.4. Consider the following game. There are two players and two possible simultaneous move games A and B. Player 1 first chooses which game is to be played; both player observe which game is played and choose their strategy. The payoff matrices for Game A and B are given by : where k> >0. Game A U D L (10,5) (0,0) R (0,0) (5,10) Game B: U D (a) Write down the extensive form of this game. (b) Assume k = 2. Solve for all subgame perfect Nash equilibria of this game. L R (kk) (0,0) (0,0) (0,0) (c) Can you find a value of k such that there exists a SPNE in which player 1 chooses game B? Can you find a value of k such that all SPNE involve player 1 choosing game B?
- 5) Mixed strategy Nash equilibrium Consider a mixed strategy Nash equilibrium of the following coordination game: Player 2 Player 1 A B a 5.5 6.-2 b -2,6 1,1 a) In the above game, explain in words what condition player 1's probability p of playing strategy A must satisfy to induce player 2 to mix strategies between a and b in equilibrium. b) Solve for the mixed strategy Nash equilibrium where player 1 chooses A with probability P and player 2 chooses a with probability p, for 0 < p, P < 1.1. Suppose Company A is about to play a game with Company B. The following facts are known about the two players. The first Company (Company A) is a row player and uses three different strategies i.e. (Strategy X, Strategy Y and Strategy Z). Whereas the column player (Company B) has two different strategies i.e. (M and N) that can be used accordingly. The payoff matrix is given in the table below. Player B Player A M N X -5 3 Y 3 -7 Z -6 5 Answer the following questions based on the information given above a. Determine the strategies of each firm using graphical technique b. Compute the value of the game.2. Consider the following 2 → 2 normal form games: In both games Player 1 (the row player) chooses either strategy A or strategy B. Player 2 (the column player) chooses simultaneously strategy A or B. The outcomes are defined by the following matrices. The first number in each cell indicates the payoff of Player 1, the second number (after the comma) indicates the payoff of Player 2. A B Game I A B 50X, 30X 10X, 10X 10X, 10X 30X, 50X Game II A A 110X, 30X B 10X, 10X B 10X, 10X 30X, 50X where X > 0 is a payoff scale parameter. (a) For both games: find all Nash equilibria in pure strategies, if any, and find the Nash equilibrium in mixed strategies. How would you describe this strategic interaction in the two games?