Game Theory and Strategy End Term: 1 September 2024, Set QDB1 (May 2024 term)

Suppose that you receive a piece of e-mail whose subject line contains the phrase “check this out” (a popular phrase among scammers). Based just on thus and without looking at the sender or the message content, what is the chance that the message is spam? Suppose that 40% of all your emails are spam and the remaining 60% are emails you want to receive. Suppose that 1% of all spam messages contain the phrase “check this out” in their subject lines, while 0.4% of all non- spam messages contain this phrase.
Based on the above data, answer the given subquestions.

What is the total probability of having the phrase “check this out” in subject line of an email?

Suppose that you receive a piece of e-mail whose subject line contains the phrase “check this out” (a popular phrase among scammers). Based just on thus and without looking at the sender or the message content, what is the chance that the message is spam? Suppose that 40% of all your emails are spam and the remaining 60% are emails you want to receive. Suppose that 1% of all spam messages contain the phrase “check this out” in their subject lines, while 0.4% of all non- spam messages contain this phrase.
Based on the above data, answer the given subquestions.

What is the probability that the message is spam if subject contains the phrase “check this out”?

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.

If X >1, how many NEs in pure strategies does this game has

Based on the above data, answer the given subquestions.

If X <1, how many NEs in pure strategies does this game has

Based on the above data, answer the given subquestions.

If X <1, what is SPNE in this game

A market game has two buyers and one seller. Seller 1 has a watch whose worth for him is ₹2000 and he offers it for sale. Players 2 and 3 are two potential buyers who do not own a watch. The worth of watch is ₹2800 for one buyer (player 2) and ₹3000 for the other buyer (player 3). Each buyer is interested in paying the lowest price for the watch and unwilling to pay more than the watch's worth to him. Describe the game in coalition form and answer the given subquestions.

A market game has two buyers and one seller. Seller 1 has a watch whose worth for him is ₹2000 and he offers it for sale. Players 2 and 3 are two potential buyers who do not own a watch. The worth of watch is ₹2800 for one buyer (player 2) and ₹3000 for the other buyer (player 3). Each buyer is interested in paying the lowest price for the watch and unwilling to pay more than the watch's worth to him. Describe the game in coalition form and answer the given subquestions.

A market game has two buyers and one seller. Seller 1 has a watch whose worth for him is ₹2000 and he offers it for sale. Players 2 and 3 are two potential buyers who do not own a watch. The worth of watch is ₹2800 for one buyer (player 2) and ₹3000 for the other buyer (player 3). Each buyer is interested in paying the lowest price for the watch and unwilling to pay more than the watch's worth to him. Describe the game in coalition form and answer the given subquestions.

A market game has two buyers and one seller. Seller 1 has a watch whose worth for him is ₹2000 and he offers it for sale. Players 2 and 3 are two potential buyers who do not own a watch. The worth of watch is ₹2800 for one buyer (player 2) and ₹3000 for the other buyer (player 3). Each buyer is interested in paying the lowest price for the watch and unwilling to pay more than the watch's worth to him. Describe the game in coalition form and answer the given subquestions.

A market game has two buyers and one seller. Seller 1 has a watch whose worth for him is ₹2000 and he offers it for sale. Players 2 and 3 are two potential buyers who do not own a watch. The worth of watch is ₹2800 for one buyer (player 2) and ₹3000 for the other buyer (player 3). Each buyer is interested in paying the lowest price for the watch and unwilling to pay more than the watch's worth to him. Describe the game in coalition form and answer the given subquestions.

A market game has two buyers and one seller. Seller 1 has a watch whose worth for him is ₹2000 and he offers it for sale. Players 2 and 3 are two potential buyers who do not own a watch. The worth of watch is ₹2800 for one buyer (player 2) and ₹3000 for the other buyer (player 3). Each buyer is interested in paying the lowest price for the watch and unwilling to pay more than the watch's worth to him. Describe the game in coalition form and answer the given subquestions.

A market game has two buyers and one seller. Seller 1 has a watch whose worth for him is ₹2000 and he offers it for sale. Players 2 and 3 are two potential buyers who do not own a watch. The worth of watch is ₹2800 for one buyer (player 2) and ₹3000 for the other buyer (player 3). Each buyer is interested in paying the lowest price for the watch and unwilling to pay more than the watch's worth to him. Describe the game in coalition form and answer the given subquestions.

A market game has two buyers and one seller. Seller 1 has a watch whose worth for him is ₹2000 and he offers it for sale. Players 2 and 3 are two potential buyers who do not own a watch. The worth of watch is ₹2800 for one buyer (player 2) and ₹3000 for the other buyer (player 3). Each buyer is interested in paying the lowest price for the watch and unwilling to pay more than the watch's worth to him. Describe the game in coalition form and answer the given subquestions.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

Based on the above data, answer the given subquestions.

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

Consider the following game:

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

Identify the null players in this game

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 _______________________

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 0.75 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.

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 NEs, all three players enter the market

Based on the above data, answer the given subquestions.

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.

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.

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.