Game Theory and Strategy End Term: 1 September 2024, Set QDB3 (May 2024 term)

A joint enterprise of three partners, 1, 2, and 3, consists of a travel agency, a car rental agency, a tour company, a souvenir shop, and a delivery service. The owner of the car rental agency, 1, forms a partnership with the owner of the travel agency, 2. Later the two purchase a souvenir shop, which they let their wives run. The owner of the tour company, 3, enters into a partnership with 2 and the two open a delivery service. Now 3 enters into the comprehensive partnership and the enterprise is the operation of all these businesses together.

We need to remember that $v(S)$ is the price that coalition $S$ can achieve in the market for all the assets that belong either to it or to its sub-coalitions. We shall describe the game in terms of the coalition function:

$\begin{array}{llll} v(1) = 20 & v(2) = 30 & v(3) = 10 & v(1,2) = 40 \\ v(1,3) = 0 & v(2,3) = 30 & v(1,2,3) = 60 & v(\phi) = 0 \end{array}$

At a certain stage the joint enterprise starts losing money and the partners decide to liquidate it.

Calculate how much each of the partners will get from the sale of the whole enterprise, if they carry out the division according to the procedure for dissolving a partnership.

Based on the above data, answer the given subquestions

Partner 1 will get ___________________

A joint enterprise of three partners, 1, 2, and 3, consists of a travel agency, a car rental agency, a tour company, a souvenir shop, and a delivery service. The owner of the car rental agency, 1, forms a partnership with the owner of the travel agency, 2. Later the two purchase a souvenir shop, which they let their wives run. The owner of the tour company, 3, enters into a partnership with 2 and the two open a delivery service. Now 3 enters into the comprehensive partnership and the enterprise is the operation of all these businesses together.

We need to remember that $v(S)$ is the price that coalition $S$ can achieve in the market for all the assets that belong either to it or to its sub-coalitions. We shall describe the game in terms of the coalition function:

$\begin{array}{llll} v(1) = 20 & v(2) = 30 & v(3) = 10 & v(1,2) = 40 \\ v(1,3) = 0 & v(2,3) = 30 & v(1,2,3) = 60 & v(\phi) = 0 \end{array}$

At a certain stage the joint enterprise starts losing money and the partners decide to liquidate it.

Calculate how much each of the partners will get from the sale of the whole enterprise, if they carry out the division according to the procedure for dissolving a partnership.

Based on the above data, answer the given subquestions

Partner 2 will get ___________________

A joint enterprise of three partners, 1, 2, and 3, consists of a travel agency, a car rental agency, a tour company, a souvenir shop, and a delivery service. The owner of the car rental agency, 1, forms a partnership with the owner of the travel agency, 2. Later the two purchase a souvenir shop, which they let their wives run. The owner of the tour company, 3, enters into a partnership with 2 and the two open a delivery service. Now 3 enters into the comprehensive partnership and the enterprise is the operation of all these businesses together.

We need to remember that $v(S)$ is the price that coalition $S$ can achieve in the market for all the assets that belong either to it or to its sub-coalitions. We shall describe the game in terms of the coalition function:

$\begin{array}{llll} v(1) = 20 & v(2) = 30 & v(3) = 10 & v(1,2) = 40 \\ v(1,3) = 0 & v(2,3) = 30 & v(1,2,3) = 60 & v(\phi) = 0 \end{array}$

At a certain stage the joint enterprise starts losing money and the partners decide to liquidate it.

Calculate how much each of the partners will get from the sale of the whole enterprise, if they carry out the division according to the procedure for dissolving a partnership.

Based on the above data, answer the given subquestions

Partner 3 will get ___________________

Consider the following game:

$\begin{array}{lll} N = \{1, 2, 3, 4\} & v(1) = v(2) = 0 & v(3) = v(4) = 1 \\ v(\phi) = 0 & & \\ v(1, 2) = 0 & v(2, 3) = 1 & \\ v(1, 3) = 1 & v(2, 4) = 1 & \\ v(1, 4) = 1 & v(3, 4) = 2 & \\ v(1, 2, 3) = 1 & v(2, 3, 4) = 2 & \\ v(1, 2, 4) = 1 & v(1, 3, 4) = 2 & v(1, 2, 3, 4) = 2 \end{array}$

Based on the above data, answer the given subquestions

Identify the null players in this game

Consider the following game:

$\begin{array}{lll} N = \{1, 2, 3, 4\} & v(1) = v(2) = 0 & v(3) = v(4) = 1 \\ v(\phi) = 0 & & \\ v(1, 2) = 0 & v(2, 3) = 1 & \\ v(1, 3) = 1 & v(2, 4) = 1 & \\ v(1, 4) = 1 & v(3, 4) = 2 & \\ v(1, 2, 3) = 1 & v(2, 3, 4) = 2 & \\ v(1, 2, 4) = 1 & v(1, 3, 4) = 2 & v(1, 2, 3, 4) = 2 \end{array}$

Based on the above data, answer the given subquestions

Choose the correct alternative.

A machine can be in two possible states viz. Good (G) and Bad (B). It is in the Good state 90% of the time. An item produced by the machine is Defective (D), 1% of the time if it is Good and 10% of the time if it is Bad.
Based on the above data, answer the given subquestions.

What is the probability of the item being defective

A machine can be in two possible states viz. Good (G) and Bad (B). It is in the Good state 90% of the time. An item produced by the machine is Defective (D), 1% of the time if it is Good and 10% of the time if it is Bad.
Based on the above data, answer the given subquestions.

What is the probability that the machine is good if the item is defective

Five bidders with private valuations uniformly distributed between 0 and 1 participate in a sealed bid first price auction. Assuming that everyone is playing the symmetric equilibrium bidding strategy answer the given subquestions:

Optimal bid for a bidder with valuation 0.6 will be

Five bidders with private valuations uniformly distributed between 0 and 1 participate in a sealed bid first price auction. Assuming that everyone is playing the symmetric equilibrium bidding strategy answer the given subquestions:

The expected revenue of the seller in this auction scenario will be

Based on the above data, answer the given subquestions

Consider following two statements about this game and choose correct alternative
P: D is weekly dominated by U
Q: L is weekly dominated by R

Based on the above data, answer the given subquestions

Consider following two statements about this game and choose the correct alternative P: (D, R) is a Nash equilibrium.
Q: Elimination of weakly dominated strategies also eliminates (D, R), a Nash equilibrium.

Given the following preference structure, find a stable matching system using the Gale–Shapley algorithm, when the women propose:

Women: A, B, C.

Men: a, b, c, d.

Based on the above data, answer the given subquestions

Mr. a will be paired with

Given the following preference structure, find a stable matching system using the Gale–Shapley algorithm, when the women propose:

Women: A, B, C.

Men: a, b, c, d.

Based on the above data, answer the given subquestions

Mr. b will be paired with

Given the following preference structure, find a stable matching system using the Gale–Shapley algorithm, when the women propose:

Women: A, B, C.

Men: a, b, c, d.

Based on the above data, answer the given subquestions

Mr. c will be paired with

Given the following preference structure, find a stable matching system using the Gale–Shapley algorithm, when the women propose:

Women: A, B, C.

Men: a, b, c, d.

Based on the above data, answer the given subquestions

Mr. d will be paired with

Based on the above data, answer the given subquestions

How many pure strategy equilibria are there in this game?

Based on the above data, answer the given subquestions

In one of these pure strategy NEs, all three players enter the market

Based on the above data, answer the given subquestions

Based on the above data, answer the given subquestions

The cost to be paid by player 1 will be ___________________

Based on the above data, answer the given subquestions

The cost to be paid by player 2 will be ___________________

Based on the above data, answer the given subquestions

The cost to be paid by player 3 will be ___________________

Based on the above data, answer the given subquestions

a= ___________________

Based on the above data, answer the given subquestions

b=___________________

Based on the above data, answer the given subquestions

c= ___________________

Based on the above data, answer the given subquestions

d= ___________________

Based on the above data, answer the given subquestions

e=___________________

Based on the above data, answer the given subquestions

f=___________________

Based on the above data, answer the given subquestions

g= ___________________

Each one of two political parties can choose to buy time on commercial radio shows to broadcast negative ad campaigns against their rival. These choices are made simultaneously. Due to government regulation, it is forbidden to buy more than 2 hours of negative campaign time so that each party cannot choose an amount of negative campaigning above 2 hours. Given a pair of choices $(a_1, a_2)$, the payoff of party $i$ is given by the following function: $v_i(a_1, a_2) = a_i - 2a_j + a_i a_j - (a_i)^2$. Write the normal form representation of this game and answer the given subquestions.

If player 1 chooses 1.5 hour as his action, the best response of player 2 will be

Each one of two political parties can choose to buy time on commercial radio shows to broadcast negative ad campaigns against their rival. These choices are made simultaneously. Due to government regulation, it is forbidden to buy more than 2 hours of negative campaign time so that each party cannot choose an amount of negative campaigning above 2 hours. Given a pair of choices $(a_1, a_2)$, the payoff of party $i$ is given by the following function: $v_i(a_1, a_2) = a_i - 2a_j + a_i a_j - (a_i)^2$. Write the normal form representation of this game and answer the given subquestions.

How much negative campaign time each player will choose under a symmetric and unique Nash equilibrium?

Each one of two political parties can choose to buy time on commercial radio shows to broadcast negative ad campaigns against their rival. These choices are made simultaneously. Due to government regulation, it is forbidden to buy more than 2 hours of negative campaign time so that each party cannot choose an amount of negative campaigning above 2 hours. Given a pair of choices $(a_1, a_2)$, the payoff of party $i$ is given by the following function: $v_i(a_1, a_2) = a_i - 2a_j + a_i a_j - (a_i)^2$. Write the normal form representation of this game and answer the given subquestions.

Equilibrium payoff of each player will be

Each one of two political parties can choose to buy time on commercial radio shows to broadcast negative ad campaigns against their rival. These choices are made simultaneously. Due to government regulation, it is forbidden to buy more than 2 hours of negative campaign time so that each party cannot choose an amount of negative campaigning above 2 hours. Given a pair of choices $(a_1, a_2)$, the payoff of party $i$ is given by the following function: $v_i(a_1, a_2) = a_i - 2a_j + a_i a_j - (a_i)^2$. Write the normal form representation of this game and answer the given subquestions.

Now assume that this game is repeated infinitely. If the parties cooperate and sign a binding agreement on how much to campaign, they choose

$a_1 = a_2 = 0.$

What will be the payoff of each player in each stage game if they stick to this agreement

Each one of two political parties can choose to buy time on commercial radio shows to broadcast negative ad campaigns against their rival. These choices are made simultaneously. Due to government regulation, it is forbidden to buy more than 2 hours of negative campaign time so that each party cannot choose an amount of negative campaigning above 2 hours. Given a pair of choices $(a_1, a_2)$, the payoff of party $i$ is given by the following function: $v_i(a_1, a_2) = a_i - 2a_j + a_i a_j - (a_i)^2$. Write the normal form representation of this game and answer the given subquestions.

Now assume that this game is repeated infinitely. If the parties cooperate and sign a binding agreement on how much to campaign, they choose

$a_1 = a_2 = 0.$

What should be the level of campaign a player should choose if he wants to deviate from this agreement to get the payoff of $\frac{1}{4}$

Each one of two political parties can choose to buy time on commercial radio shows to broadcast negative ad campaigns against their rival. These choices are made simultaneously. Due to government regulation, it is forbidden to buy more than 2 hours of negative campaign time so that each party cannot choose an amount of negative campaigning above 2 hours. Given a pair of choices $(a_1, a_2)$, the payoff of party $i$ is given by the following function: $v_i(a_1, a_2) = a_i - 2a_j + a_i a_j - (a_i)^2$. Write the normal form representation of this game and answer the given subquestions.

Now assume that this game is repeated infinitely. If the parties cooperate and sign a binding agreement on how much to campaign, they choose

$a_1 = a_2 = 0.$

For which discount factors $\delta \in (0, 1)$ can this agreement cannot be supported as a subgame perfect equilibrium of the infinitely repeated game?