Question

Pure strategy Nash equilibrium

3. In the following games, use the underline method to find all pure strategy Nash equilibrium.

(B ) [0, 4, 4 0, 5, 3]

[4, 0 0 4, 5, 3]

[3, 5, 3, 5 6, 6]

(C) [2, -1 0,0]

[0,0 1,2]

(D) [4,8 2,0]

[6,2 0,8]

(E) [3,3 2,4]

[4,2 1,1]

4. In the following 3-player game, use the underline method to find all pure strategy Nash equilibria. Player 1 picks the row, Player 2 picks the column, and Player 3 picks the page. For each outcome of the game, the three players’ payoffs are listed in order: 1’s payoff, 2’s payoff, 3’s payoff. For example, when the outcome of the game is (row 1, column 1, page 2), the payoffs are (3, 3, 6) for Players 1, 2, and 3, respectively.

page 1

[5, 5, 5 3, 6, 3]

[6, 3, 3 4, 4, 1]

page 2

[3, 3, 6 1, 4, 4]

[4, 1, 4 2, 2, 2]

I will get you started, so you can see the idea. The best response to (row 1, column 1) is page 2, so I underlined the 6. The best response to (column 2, page 1) is row 2, so I underlined the 4. Each player will have 4 situations to best respond to, so there should be 12 things underlined in total.

5. Is the above 3-player game dominance solvable? If not, explain why not. If yes, write down an order of deletion that solves the game.

Answer #1

In this game we have following Pure Nash Equilibrias

Row is Player 1 & Column is Player 2

(0,4) | (4,0) | (5,3) |

(4,0) | (0,4) | (5,3) |

(3,5) | (3,5) | (6,6) |

Hence Best Responses for plater 1 for Player 2's repsonses from column 1 to column 3 are Row 2, Row 1 and Row 3 respectively (Row2,Col1);(Row1,Col2);(Row3,Col3)

SImilarly for player 2 we have Col1,Col2 & Col3 respectiely for Row1, Row2& Row3 Hence (Row1,Col1);(Row2,Col2);(Row3,Col3)

Intersection between these two players is (Row3,Col3)=(6,6) Pure Nash Equilibriuim

Game C)

(2,-1) | (0,0) |

(0,0) | (1,2) |

In the above game we have Nash Equilibria equals to (1,2)

Game D)

(4,8) | (2,0) |

(6,2) | (0,8) |

Hence Nash Equilibria for this game is (2,0)

Game E)

(3,3) | (2,4) |

(4,2) | (1,1) |

Above Game has NE at (2,4)

QUESTION 3
Below is a game between player A and player B. Each player has
two possible strategies: 1 or 2. The payoffs for each combination
of strategies between A and B are in the bracket. For example, if A
plays 1 and B plays 1, the payoff for A is -3 and the payoff for B
is -2.
Player B
Strategy 1
Strategy 2
Player A
Strategy 1
(-3,-2)
(10,0)
Strategy 2
(0,8)
(0,0)
How many pure strategy Nash...

For each of the following games:
1) Identify the Nash equilibrium/equilibria if they
exist, 2) identify all strictly dominant
strategies if there are any, and 3) identify the
Pareto-optimal outcomes and comment whether they coincide with the
Nash Equilibrium(s) you found. Also,
4) would you classify the game as an invisible hand
problem, an assurance game, a prisoners dilemma or none of
these?
Row Player
(R1)
(R2)
Column
Player
(C1)
(C2)
(-1,-1)
(-5,0)
(0,-5)
(-4,-4)

19. How can a pure strategy Nash equilibrium be accurately
described?
a.) One exists in all games
b.) It is always the overall best outcome
c.) It can only be reached by collusion
d.) It’s an outcome in which players implement each dominant
strategies

Consider the following two-person zero-sum game. Assume the two
players have the same three strategy options. The payoff table
below shows the gains for Player A.
Player B
Player A
Strategy b1
Strategy b2
Strategy b3
Strategy a1
3
2
?4
Strategy a2
?1
0
2
Strategy a3
4
5
?3
Is there an optimal pure strategy for this game? If so, what is it?
If not, can the mixed-strategy probabilities be found
algebraically? What is the value of the game?

Consider the following simultaneous-move game:
Column
L
M
N
P
Row
U
(1,1)
(2,2)
(3,4)
(9,3)
D
(2,5)
(3,3)
(1,2)
(7,1)
(a) Find all pure-strategy Nash equilibria.
(b) Suppose Row mixes between strategies U and D in the
proportions p and (1 − p). Graph the payoffs of Column’s four
strategies as functions of p. What is Column’s best response to
Row’s p-mix?
(c) Find the mixed-strategy Nash equilibrium. What are the
players’ expected payoffs?

1. For the following payoff matrix, find these:
a) Nash equilibrium or Nash equilibria (if any)
b) Maximin equilibrium
c) Collusive equilibrium (also known as the "cooperative
equilibrium")
d) Maximax equilibrium
e) Dominant strategy of each firm (if any)
Firm A
Strategy X
Strategy Y
Firm B
Strategy X
200
23
250
20
Strategy Y
30
50
1
500
2. For the following payoff matrix, find these:
a) Nash equilibrium or Nash equilibria...

Two equally sized companies, Auburn Motorcar Company and Cord
Automobile Corporation, dominate the domestic automobile market.
Each company can produce 500 or 750 midsized automobiles a month.
This static game is depicted in the following figure. All payoffs
are in millions of dollars.
a. Does either firm have a dominant strategy?
b. What is the Nash equilibrium strategy profile if this game is
played just once?
c. What strategy profile results in the best payoff for both
players?
d. Is...

This problem refers to the following game:
A
B
A
(2, 4)
(0, 2)
B
(1, 1)
(3, 4)
a. What are the pure-strategy
Nash equilibria?
b. Is there a mixed strategy
Nash equilibrium where both players mix A and B? If so, ﬁnd the
equilibrium. If not, explain why not.

(4) In this game, each of two players can volunteer some of
their spare time planting and cleaning up the community garden.
They both like a nicer garden and the garden is nicer if they
volunteer more time to work on it. However, each would rather that
the other person do the volunteering. Suppose that each player can
volunteer 0, 1, 2, 3, or4 hours. If player 1 volunteers x hours and
2 volunteers y hours, then the resultant garden...

Mike and Jake are basketball players who sometimes flop (that
is, intentionally fall or feign an injury in order to cause a foul
on another player). It is better for each of them if both play
fairly and don't flop then when both flop. But each player is
better off flopping than not flopping. Their payoffs are given
below.
Jake
Mike
Flop
Not Flop
Flop
(3, 3)
(4, 1)
Not Flop
(1, 4)
(1, 1)
For instance, if Mike flops...

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