a
To find:
Pure Strategy Nash equilibria.
a
Explanation of Solution
In a two-player game,
Solve the following game for pure strategy Nash equilibria:
Player 1 | Player 2 | |||
D | E | F | ||
A | 7,6 | 5,8 | 0,0 | |
B | 5,8 | 7,6 | 1,1 | |
C | 0,0 | 1,1 | 4,4 |
To find the pure strategy Nash equilibria,one will use the underlining the “best response payoffs” method.
Step 1:
Underline the payoffs corresponding to player 1’s best responses. Player 1’s best response when Player 2 plays strategy D is A; one should underline the payoff corresponds to it. Player 1’s best response when Player 2 plays strategy E is B; one should underline the payoff corresponds to it. Player 1’s best response when Player 2 plays strategy F is C; one should underline the payoff corresponds to it. The matrix will be as follows:
Player 1 | Player 2 | |||
D | E | F | ||
A | 7,6 | 5,8 | 0,0 | |
B | 5,8 | 7,6 | 1,1 | |
C | 0,0 | 1,1 | 4,4 |
Step 2:
One should follow the same procedure for Player 2’s responses. One should underline the payoffs corresponding to player 2’s best responses. Player 2’s best response when Player 1 plays strategy A is E; one should underline the payoff corresponds to it. Player 2’s best response when Player 1 plays strategy B is D; one should underline the payoff corresponds to it. Player 2’s best response when Player 1 plays strategy C is F; one should underline the payoff corresponds to it. The matrix will be as follows:
Player 1 | Player 2 | |||
D | E | F | ||
A | 7,6 | 5,8 | 0,0 | |
B | 5,8 | 7,6 | 1,1 | |
C | 0,0 | 1,1 | 4,4 |
Step 3:
Now, one should look for the box where the responses of both the Players are underlined. It is the cell (C,F). This box corresponds to Nash equilibrium The given payoff is (4,4).
Introduction:
Nash equilibrium is a stable state in which different participants interact each other, in which no participant gains unilaterally, if strategy of other remains unchanged.
b)
To find:
Mixed strategy Nash equilibrium.
b)
Explanation of Solution
One should have to find the mixed strategy. Nash equilibrium for the firdt two strategies of both the players.
Player 1 | Player 2 | |||
D | E | |||
A | 7,6 | 5,8 | ||
B | 5,8 | 7,6 |
When a player doesnot have a dominant strategy, she plays a mixed strategy. Here, to get the mixed strategy, Nash equilibrium one should assume that Player 1 plays the strategy A with probability p and strategy B with probability (1-p). Player 2 plays the strategy D with probability q and strategy E with probability (1-q).
Step 1:
Here, the expected payoff of player 1 for strategy A is given by multiplying each of the payoffs corresponding to S by their respective probabilities and then summing them over. This way the expected payoff from strategy is:
The expected pay off from strategy B is:
These expected payoffs must be equal. Therefore:
Therefore, player 2 plays both of his strategy with equal probability of ½.
Step 2:
The expected payoff from strategy D is:
The expected payoff from strategy B is:
These expected payoffs must be equal.
Therefore, player 2 plays both of his strategy with equal probability of ½
Hence, the mixed strategy Nash equilibrium for the player 1 and player 2 is (0.5, 0.5)
Introduction:
Nash equilibrium is a stable state in which different participants interact each other, in which no participant gain unilaterally, if strategy of other remains unchanged.
c)
To ascertain:
Player’s expected payoffs
c)
Explanation of Solution
The expected payoff of the player for a given strategy in a mixed strategy game is given by summing over the actual probability multiplied by their respected probability.
In pure strategy equilibrium of the game described above is (4,4). That is the payoff of player A is 4 and that of B is also 4.
If player 1 choses strategy B, then the player 2 will play either of the strategy D or E with probability 0.5. Then for strategy A the expected payoff of player 1 is:
For player 2:
If player 2 choses strategy E, then the player 1 will play either of the strategy A or B with probability 0.5. Then for strategy E, the expected payoff of player 2 is:
Introduction:
Nash equilibrium is a stable state in which different participants interact each other, in which no participant gain unilaterally, if strategy of other remains unchanged.
d)
To ascertain:
Extensive form of the game.
d)
Explanation of Solution
The extensive form of a game corresponds to the game tree; where the action proceeds from left to right. The first move in this game belongs to player 1; he must chose whether to pay strategy A,B OR C. Then player 2 makes his decision. Payoffs are given at the end of the tree.
Introduction:
Nash equilibrium is a stable state in which different participants interact each other, in which no participant gain unilaterally, if strategy of other remains unchanged.
Want to see more full solutions like this?
- Consider the two period Repeated Prisoner's Dilemma Game where each player is interested in the SUM of the payoffs she gets in each period. Players see the outcome after the play in each period. (The period payoffs are 10,5,1,0.) (i) Write out this game in its strategic form. (ii) Find all Nash equilibria and all Subgame Perfect Nash Equilibria.arrow_forwardi. ii. QUESTION ONE A. A Nash equilibrium is a strategy profile such that every player's strategy is the best response to all the other players. It requires that each player makes a best response and that expectations regarding the play of other players are correct. Below is the table showing strategies and payoff for Player 1 and Player 2. PLAYER 1 R1 R2 R3 R4 C1 0,7 5,2 7,0 6,6 C2 2,5 3,3 2,5 2,2 PLAYER 2 C3 7,0 5,2 0,7 4,4 CA 6,6 2,2 4,4 10,4 REQUIRED; Transform the normal form game above into an imperfect extensive game form Find the Nash equilibrium for the game above using iterative deletion of strictly dominated strategies. Find the Nash equilibrium using brute force or cell by cell inspection.arrow_forwardConsider the following game in normal form. Not cooperate Cooperate Not cooperate 20,20 50,0 Cooperate 0,50 40,40 What is Nash equilibrium? Is it efficient? Why? What needs to be complied with so that the players would like to cooperate? What happens when one of the players does not cooperate? Why? Define trigger strategy. Calculate the discount factor (δ) that would make both players decide to cooperate.arrow_forward
- Consider the following game : Stag Rabbit Stag 9, 9 0, 8 Rabbit 8, 0 7, 7 The first payoff is that of player 1 and the second that of player 2. a. ) Draw the extensive form of the simultaneous game. Find all the Nash equilibrium. p. Suppose player 1 moves first or we are in a sequential game now. Draw the extensive form in the sequential version. c. What is the subgame perfect Nash Equilibrium (SPNE) in the sequential version? d. ) Explain why it is an SPNE.arrow_forwardUse a matrix to model a two-player game of rock-paper-scissors with payoff of 1 if you win, -1 if you lose, and 0 if you tie. In this game, how many pure-strategies Nash equilibria exist?arrow_forwardConsider the game with the payoffs below. Which of the possible outcomes are MORE efficient than the Nash Equilibrium (NE)? Note, they do NOT need to be Nash equilibria themselves, they just need to be more efficient than the NE. Multiple answers are possible, but not necessary. You need to check ALL correct answers for full credit. JILL High Medium LowMAGGIE Left 3,4 2,3 2,2Center 4,8 9,7 8,7Right 7,6 8,5 9,4Group of answer choices (Left, Low) There is no strategy combination that is more efficient than the Nash equilibrium for this game. (Right, Medium) (Left, High) (Center, Medium) (Center, High) (Center, Low) (Left, Medium) (Right, Low) (Right, High)arrow_forward
- A and B are competitors in the mobile phone industry. Both A and B have to decide whether to participate or not to participate in a Phone for the Future Trade Fair next month. The matrix payoff below shows the profits (USD million) corresponding to their actions. a) What is the Nash equilibrium of the above game? b) Is the Nash equilibrium Pareto Optima? Explain. c) Suppose B is pessimistic of A's rationality, what is B's strategy? Compare and comment on B's strategy in (a) and (c). A Participate Do not participate B Participate Do not participate 400,1000 200,200 500,500 1000,400arrow_forwardSwitch the payoffs in cells (A, A) and (D, D). What is the pure strategy Nash equilibrium outcome if there is one?arrow_forwardHere is a payoff matrix that is interesting. I am not in an imaginative mood, so I won't try to tell a story about why the payoffs come out this way. Find all the pure-strategy Nash equilibria. Is this a game where you expect that players would end up in a Nash equilibrium? What would you expect to happen and why? Row Top Row Bottom Column Left 4, -3000 12,8 Column Right 10,6 5, 4arrow_forward
- Use the following payoff matrix for a one-shot game to answer the accompanying questions. Player 2 Strategy X Y Player 1 A 50, 50 -110, 6 B 6, -110 26, 26 a. Determine the Nash equilibrium outcomes that arise if the players make decisions independently, simultaneously, and without any communication.check all that apply (26,26) (50,50) (-110,6) (6,-110) Which of these outcomes would you consider most likely?multiple choice 1 (-110,6) (50,50) (6,-110) (26,26) b. Suppose player 1 is permitted to “communicate” by uttering one syllable before the players simultaneously and independently make their decisions. What should player 1 utter?multiple choice 2 B A What outcome do you think would occur as a result?multiple choice 3 (26,26) (50,50) (-110,6) (6,-110) c. Suppose player 2 can choose its strategy before player 1, that player 1 observes player 2’s choice before making her decision, and that this move…arrow_forwardUse the following payoff matrix for a one-shot game to answer the accompanying questions. Player 2 Strategy X Y Player 1 A 30, 30 16, -50 B -50, 16 50, 50 A. Determine the Nash equilibrium outcomes that arise if the players make decisions independently, simultaneously, and without any communication. check all that apply (16, −50) (−50, 16) (30, 30) (50, 50) Which of these outcomes would you consider most likely? multiple choice (16, −50) (50, 50) (−50, 16) (30, 30) B. Suppose player 1 is permitted to “communicate” by uttering one syllable before the players simultaneously and independently make their decisions. What should player 1 utter? multiple choice A or B What outcome do you think would occur as a result? multiple choice (−50, 16) (16, −50) (30, 30) (50, 50) c. Suppose player 2 can choose its strategy before player 1, that player 1 observes player 2’s choice before making her decision, and that this move structure is…arrow_forward• (1,0) (1,1) Player 2 G D H (2,0) Action Player 1 Player 1 Action B Player 1 (3,-1) E Player 2 (0,1) (-1,-1) (a) Find all the Subgame Perfect Nash equilibria in this game. (b) Find all the Nash equilibria in this game. (Hint: write the game in strategic form.)arrow_forward
- Managerial Economics: A Problem Solving ApproachEconomicsISBN:9781337106665Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike ShorPublisher:Cengage LearningManagerial Economics: Applications, Strategies an...EconomicsISBN:9781305506381Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. HarrisPublisher:Cengage Learning
- Microeconomics: Principles & PolicyEconomicsISBN:9781337794992Author:William J. Baumol, Alan S. Blinder, John L. SolowPublisher:Cengage LearningExploring EconomicsEconomicsISBN:9781544336329Author:Robert L. SextonPublisher:SAGE Publications, Inc