Introduction to Game Theory

First Version

(5 points) Consider the following sequential game, and perform backward induction analysis.

How many positions are there in this game?
How many of these positions are start positions and how many end positions?
At the beginning, which one of moves A1, A2, or A3 will Ann choose according to backward induction?
When Beth decides between B5 and B6, which one will she choose according to backward induction?
What payoffs will Ann and Beth get in this game if both play according to backward induction?

There are 22 positions.
There is always just one start position, and there are 13 end positions, the positions that have payoffs attached.
Ann will choose A1.
Beth will chose B5.
Ann gets 3 and Beth gets 4.

(6 points) Consider the following two-player game with players Ann and Beth, where Ann has three choices, A1, A2, or A3, and Beth has the three choices, B1, B2, and B3.

	B1	B2	B3
A1	1, 3	2, 5	4, 1
A2	5, 2	4, 3	3, 3
A3	3, 1	3, 4	2, 5

What is the payoff for Beth if Ann plays A2 and Beth plays B1?
Find the Maximin moves for both Ann and for Beth.
Which moves are dominated by which moves? In each case tell whether the domination is weak or strict.
Describe all Nash equilibria.

2
A2 for Ann, B2 for Beth.
A2 strictly dominates A3, B2 strictly dominates B1.
The best responses are underlines:

B1 B2 B3

A1 1, 3 2, 5 4, 1

A2 5, 2 4, 3 3, 3

A3 3, 1 3, 4 2, 5

Therefore there is only one Nash equilibria: (A2,B2) .

(1 point) What is a zero-sum game?

In a zero-sum game, the sum of all payoffs of all players for every outcome equals 0.

(1 point) When does a sequential game have perfect information?

A sequential game has perfect information if every player, when about to move, knows all moves done by the other players (including the random player) so far.

(1 point) Assume a 3-person simultaneous game has a Nash equilibrium of Ann playing move A2, Beth playing B3, and Cindy playing C2. Assume the three players talk before playing and agree playing these moves. Why is it unlikely then that any players plays something different than agreed?

Since if only one player deviates, the payoff for that player is less or equal to the payoff in case of playing the agreed move. To get more, more than one player has to deviate.

Second Version

(5 points) Consider the following sequential game, and perform backward induction analysis.

How many positions are there in this game?
How many of these positions are start positions and how many end positions?
At the beginning, which one of moves A1 or move A2 will Ann choose according to backward induction?
When Beth decides between B3, B4, and B5, which one will she choose according to backward induction?
What payoffs will Ann and Beth get in this game if both play according to backward induction?

There are 23 positions.
There is always just one start position, and there are 14 end positions, the positions that have payoffs attached.
Ann will choose A2.
Beth will chose B3.
Ann gets 3 and Beth gets 5.

(6 points) Consider the following two-player game with players Ann and Beth, where Ann has three choices, A1, A2, or A3, and Beth has three choices, B1, B2, and B3.

	B1	B2	B3
A1	3, 3	5, 1	1, 2
A2	2, 3	3, 4	2, 4
A3	1, 2	4, 2	3, 3

What is the payoff for Beth if Ann plays A2 and Beth plays B3?
Find the Maximin moves for both Ann and for Beth.
Which moves are dominated by which moves? In each case tell whether the domination is weak or strict.
Describe all Nash equilibria.

4
A2 for Ann, B1 and B3 for Beth.
B3 weakly dominates B2.
The best responses are underlines:

B1 B2 B3

A1 3, 3 5, 1 1, 2

A2 2, 3 3, 4 2, 4

A3 1, 2 4, 2 3, 3

Therefore there are two Nash equilibria: (A1,B1) and (A3,B3).

(2 points) Assume a 3-person simultaneous game has a Nash equilibrium of Ann playing move A2, Beth playing B3, and Cindy playing C2. Assume the payoff for Ann for that outcome equals 3.

True or false: The payoff for Ann in any outcome is at most 3.
True or false: If Ann plays A3, Beth plays B3, and Cindy plays C2, then the payoff for Ann must be at most 3.