这个作业是完成博弈论相关理论问题

Math 486 Midterm – Fall 2020

1. Consider the extensive form game shown below having the three players P1, P2, and P3. Notice

that not all game paths result in a turn order of P1, then P2, then P3 (be careful about the player

assigned to each node).

(a) Determine the number of strategies for each player. You do not need to write out the strategies

for each player.

(b) Analyze the game using backward induction. Indicate the path through the game tree that will

be taken (according to your analysis). You can do this by making a sketch of the game tree and

highlighting the path through the game tree that will be followed. You do not need to provide

explanations for the steps in the backward induction.

2. Consider the two player game shown below where Player 1 has the strategies a and b and Player 2

has the strategies c and d, where x is a positive parameter, x ∈ (0, ∞).

Player 2

c d

Player 1 a x − 4, 5 8, 0

b x, 7 x, x

(a) Find the range of values for x where player 1 has a strictly dominated strategy. State which

strategy is strictly dominated.

(b) Find the range of values for x where player 2 has a strictly dominated strategy. State which

strategy is strictly dominated.

(c) Find the range of values for x where neither player has a strictly dominated strategy.

(d) Choose a value of x where one of the players has a weakly, but not strictly, dominated strategy.

What is the rational outcome of the game if that player does not play their weakly dominated

strategy?

Math 486 Midterm – Fall 2020

3. Consider the following Cournot Duopoly game, in which two firms simultaneously choose quantities

q1, q2 ∈ [0, ∞) and the market price is given by

p =

(

132 − 2q1 − 2q2, if 0 ≤ q1 + q2 ≤ 66

0, if q1 + q2 > 66

and each firm has the same cost of 12 per unit:

c1(q1) = 12q1, c2(q2) = 12q2

The payoff function for each firm is the total profit.

(a) Write the payoff function πi for each firm.

(b) Find a Nash equilibrium strategy profile.

(c) Are there any other strategy profiles (q1, q2) where both firms could get a higher payoff than

the payoff they get in the Nash equilibrium? Explain or demonstrate with an example.

Math 486 Midterm – Fall 2020

4. Consider the following coordination game, where the payoffs are in dollars.

Player 2

L R

Player 1 U 4, 1 0, 0

D 0, 0 1, 4

(a) Find any Nash equilibria for this coordination game.

(b) Suppose that before the game is played, there is an initial stage where player 1 chooses whether

or not to burn $2. Player 2 knows that Player 1 has this option and Player 2 observes Player 1’s

action. Then the players play the coordination game shown above. Since player 2 knows that

player 1 has the option to burn the money, this “money burning” could be viewed as a way that

Player 1 communicates their intention to Player 2.

Draw an extensive form game that includes both stages of this new game: stage 1, where Player

1 chooses whether or not to burn $2, and stage 2, when the players play the coordination game.

Note that whenever Player 1 burns $2, this decreases their own payoff by $2 (this should be

reflected in the payoffs of the extensive form game).

(c) Write out the normal form game for the 2-stage extensive form game and identify any Nash

equilibria. I usually do not ask you to draw out very large normal form games – this is a bit of

an exception – you should arrive at a normal form game where 1 player has 8 strategies and the

other has 4.

(d) Use iterative elimination of weakly dominated strategies to reduce the game. There is a way to

reduce this (over multiple iterations) to a game where player 1 has 2 remaining strategies and

player 2 has 1 remaining strategy.

(e) Based on your result in the previous part, summarize the way Player 1’s option to burn money

affects the outcome of this two stage game (compared with the coordination game without

money burning).

Math 486 Midterm – Fall 2020

5. Suppose there are six people who work together (players 1, 2, 3, 4, 5, and 6). There is an infectious

disease that spreads easily between people who spend time in proximity to one another.

• People can practice social distancing measures to lower their risk of infection. In this example, we’ll assume that each person experiences a different “cost” of enacting social distancing

measures: Person i experiences a cost of i if they choose to socially distance (at work and

elsewhere).

• The benefit of working together for someone who does not get sick is 6.

• If a person does get sick, they have to isolate and experience a benefit of zero.

• Assume that a person taking social distancing measures will not get sick, but for a person who

does not take social distancing measures, the probability of getting sick is n

6

·, where n is the

total number of people who do not practice social distancing.

Thus, if one person is not social distancing, the probability of that person getting sick is 1

6

.

If there are 3 of the 6 people not practicing social distancing, then they each have a 1

2

probability

of getting sick.

We are assuming that the risk of infection scales with the number of people who exhibit risky

behavior.

Each player seeks to maximize their expected payoff. Each player decide simultaneously

whether to socially distance (S) or not to socially distance (N) (we could assume that the

players follow the same protocals at work, but it is the choices they make outside the workplace

that introduce risk of infection). Thus, for each player i, the strategy set is Si {S, N}

(a) Show carefully that when players 1, 2, 3, and 4 socially distance, and players 5 and 6 do not,

this not a Nash equilibrium.

(b) Show carefully that when players 1,2, and 3 socially distance, and players 4, 5 and 6 do not,

this is a Nash equilibrium.

(c) There are players in this game that have dominated strategies. Which are they?

Math 486 Midterm – Fall 2020

6. Consider a game with incomplete information where there are two players, P1 and P2. There are

two possible interactions that could occur between the players, type A and type B.

• The interaction will by of type A with probability 1

4

and of type B with probability 3

4

.

• Player 1 knows the type, but player 2 does not.

• Player 1 observes whether the interaction will be type A or type B and chooses to move either

“Up” (U) or “Down” (D). If Player 1 chooses “Down”, the game ends.

• If Player 1 moves “Up”, then Player 2 may move “Left” (L), or “Right” (R). Player 2 observes

player 1’s move, but does not observe the interaction type.

• Refer to the extensive form game shown below, which includes the payoffs.

(a) Create a Bayesian Normal form game.

(b) Identify any Bayesian Nash equilibria. No explanation is needed. It is enough to identify best

responses in the Bayesian normal form game in the usual way and then use the information

about best responses to identify any Bayesian Nash equilibria.

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