Here are 44 questions to help prepare you for your Game Theory midterm. You can see the solutions in the solutions.pdf file. You can see the resources I used in the resources section.
- Represent the following game in normal form:
Alice, Bob and Celine are childhood friends that would like to communicate online. They have a choice between 3 social networks: facebook, twitter and G+.
Clearly state the players, strategy sets and interpretations of the utilities.
- Represent the following game in normal form and then analyze it.
Assume two neighbouring countries have at their disposal very destructive armies. If both countries attack each other the countries’ civilian population will suffer 10 thousand casualties. If one country attacks whilst the other remains peaceful, the peaceful country will lose 15 thousand casualties but would also retaliate causing the offensive country 13 thousand casualties. If both countries remain peaceful then there are no casualties.
- Attempt to predict rational behaviour using iterated elimination of dominated strategies for the following:
- Attempt to predict rational behaviour using iterated elimination of dominated strategies for the following:
| 2, 11 |
1, 9 |
3, 10 |
17, 22 |
| 27, 0 |
3, 1 |
1, 1 |
1, 0 |
| 4, 2 |
6, 10 |
7, 12 |
18, 0 |
- Attempt to predict rational behaviour using iterated elimination of dominated strategies for the following:
- Attempt to predict rational behaviour using any method you know.
| 3, 2 |
3, 1 |
2, 3 |
| 2, 2 |
1, 3 |
3, 2 |
- Analyze the following game.
| 7, 3 |
0, 2 |
| 2, 1 |
6, 1 |
| 4, 0 |
4, 2 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
| 5, 3 |
70, -1 |
4, 2 |
| 6, 7 |
71, 2 |
2, 1 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
| 6, 7 |
2, 1 |
4, 6 |
| 0, 4 |
3, 8 |
2, 3 |
| 1, 2 |
1, 5 |
1, 1 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
- For what values of α does a Nash equilibrium exist in pure strategies for the following game
- Attempt to predict rational behaviour using iterated elimination of dominated strategies for the following:
- Analyze the following game.
-
Define Strictly dominated strategy.
-
Analyze the following game.
-
Define Weakly dominated strategy.
-
Define Normal form game.
-
Analyze the following game.
| -1, -1 |
-4, 0 |
| 0, -4 |
-3, -3 |
- Analyze the following game.
| 5, -1 |
11, 3 |
0, 0 |
| 6, 4 |
0, 2 |
2, 0 |
- Analyze the following game.
| -1, -1 |
-4, 0 |
| 0, -4 |
-3, -3 |
| -2, -3 |
-5, 1 |
- Analyze the following game.
- Analyze the following game.
- Attempt to predict rational behaviour using iterated elimination of dominated strategies for the following:
| 4, 3 |
5, 1 |
6, 2 |
| 2, 1 |
3, 4 |
3, 6 |
| 3, 0 |
9, 6 |
2, 8 |
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Suppose everyone in your town selects a real number between 0 and 100, inclusive (i.e. 0 and 100 are both possible choices, as is any other number between). The winner is the individual (or individuals) who selects the number closest to 2/3 of the average of numbers chosen. What number do you choose? Why?
-
Define Best Response.
-
Analyze the following game.
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
| 5, 4 |
3, 3 |
5, 2 |
3, 2 |
| 3, 2 |
9, 3 |
5, 2 |
7, 1 |
| 4, 5 |
2, 1 |
5, 4 |
5, 6 |
| 2, 2 |
4, 0 |
5, 0 |
8, 1 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
| 1, 1 |
2, 2 |
4, 1 |
| 2, 2 |
5, 5 |
3, 6 |
| 1, 4 |
6, 3 |
0, 0 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
| 9, 9 |
2, 2 |
4, 1 |
| 2, 2 |
5, 5 |
3, 6 |
| 1, 4 |
6, 3 |
0, 0 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
| 1, -1 |
-1, -1 |
-1, 1 |
| -1, -1 |
-1, 1 |
1, -1 |
| 0, 0 |
0, 0 |
0, 0 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
| 1, -1 |
-1, -1 |
-1, 1 |
| -1, -1 |
-1, 1 |
1, -1 |
| -1, -1 |
1, -1 |
-1, 1 |
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There are two candidates running for president. Each candidate must position themselves along the ideological left-right axis in order to appeal to voters. Suppose there are ten possible ideological locations along this axis. Suppose further that voters are evenly distributed along this axis, i.e. approx. 10% of the pool of voters are located at each of the ten ideological positions. Which position should they pick?
-
Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
- Plot the utilities for player 1.
| 5, 1 |
0, 2 |
| 1, 3 |
4, 1 |
| 4, 2 |
2, 3 |
- Plot the utilities for player 1.
| 4, -4 |
9, -9 |
| 6, -6 |
6, -6 |
| 9, -9 |
4, -4 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
-
Investment game: Suppose everyone in your town is given a chance to invest 10$ or not. If more than 90% of the town invests, each person will get 15$ in return but if less than 90% invest, they lose their money. What are the pure nash equilibriums of this game?
-
Find the mized nash equilibrium of this game.
| 50, 50 |
80, 20 |
| 90, 10 |
20, 80 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
- Attempt to predict rational behaviour in this game.
player 3 - D
| 5, 5, 5 |
3, 6, 3 |
| 6, 3, 3 |
4, 4, 1 |
player 3 - C
| 3, 3, 6 |
1, 4, 4 |
| 4, 1, 4 |
2, 2, 2 |
- Compute the Nash equilibrium (if they exist) in pure strategies for the following game.
no rain with probability p
rains with probability 1 - p
| -2, -2 |
-2, 0 |
| 0, -2 |
-10, -10 |
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Cournot duopoly: There are two firms operating in a limited market. Market production is: P(Q)=a-bQ, where Q=q1+q2 for two firms. Both companies will receive profits derived from a simultaneous decision made by both on how much to produce, and also based on their cost functions: TCi=C-qi. What is each players best response and how is it different from the monopoly best response?
-
Two players bargain over how to split $10. Each player i ∈ {1, 2} chooses a number si ∈ [0, 10] (which does not need to be an integer). Each player’s payoff is the money he receives. We consider two allocation rules. In each case, if s1 + s2 ≤ 10, each player gets his chosen amount si and the rest is destroyed.
- In the first case, if s1 + s2 > 10, both players get zero. What are the (pure strategy) Nash equilibria?
- In the second case, if s1 + s2 > 10 and s1 != s2 , the player who chose the smallest amount receives this amount and the other gets the rest. If s1 + s2 > 10 and s1 = s2, they both get $5. What are the (pure strategy) Nash equilibria?
- Now suppose that s1 and s2 must be integers. Does this change the (pure strategy) Nash equilibria in either case?
