Game Theory and Strategy · End Term · Sep 2025 · FN

Three machines, A, B, and C, produce respectively 50%, 30% and 20% of the total no. of items of a factory. The percentages of defective output of these machines are 3%, 4% and 5%. Based on the above data, answer the given subquestions.

If an item is selected at random, the probability that the item is defective is ____________

Three machines, A, B, and C, produce respectively 50%, 30% and 20% of the total no. of items of a factory. The percentages of defective output of these machines are 3%, 4% and 5%. Based on the above data, answer the given subquestions.

Suppose an item is selected at random and found to be defective. The probability that the item was produced by machine A will be ___________

Consider the market for a single network good as discussed in the lecture, and write a different consumer’s utility function for joining the network as

capturing the idea that it may be more plausible that a user who has a higher value for the standalone component of a technology also assigns more importance to the size of its network (i.e., consumers differ in their valuation of both the standalone and the network benefits). Here, a is the standalone benefit,

measures the network effect,

is the expected number of users joining the network and

is uniformly distributed on the unit interval. Based on the above data, answer the given subquestions.

The value

corresponding to an indifferent consumer for a given price

and a given expected network size

will be

Consider the market for a single network good as discussed in the lecture, and write a different consumer’s utility function for joining the network as

capturing the idea that it may be more plausible that a user who has a higher value for the standalone component of a technology also assigns more importance to the size of its network (i.e., consumers differ in their valuation of both the standalone and the network benefits). Here, a is the standalone benefit,

measures the network effect,

is the expected number of users joining the network and

is uniformly distributed on the unit interval. Based on the above data, answer the given subquestions.

The maximum value of p to have buyers will be

Consider the market for a single network good as discussed in the lecture, and write a different consumer’s utility function for joining the network as

capturing the idea that it may be more plausible that a user who has a higher value for the standalone component of a technology also assigns more importance to the size of its network (i.e., consumers differ in their valuation of both the standalone and the network benefits). Here, a is the standalone benefit,

measures the network effect,

is the expected number of users joining the network and

is uniformly distributed on the unit interval. Based on the above data, answer the given subquestions.

The mass of buyers (n) will be

Consider the market for a single network good as discussed in the lecture, and write a different consumer’s utility function for joining the network as

capturing the idea that it may be more plausible that a user who has a higher value for the standalone component of a technology also assigns more importance to the size of its network (i.e., consumers differ in their valuation of both the standalone and the network benefits). Here, a is the standalone benefit,

measures the network effect,

is the expected number of users joining the network and

is uniformly distributed on the unit interval. Based on the above data, answer the given subquestions.

The willingness to pay

for the

unit of the good when

units are expected to be sold will be

Consider the market for a single network good as discussed in the lecture, and write a different consumer’s utility function for joining the network as

capturing the idea that it may be more plausible that a user who has a higher value for the standalone component of a technology also assigns more importance to the size of its network (i.e., consumers differ in their valuation of both the standalone and the network benefits). Here, a is the standalone benefit,

measures the network effect,

is the expected number of users joining the network and

is uniformly distributed on the unit interval. Based on the above data, answer the given subquestions.

is __________ (more than one options are correct)

Consider the market for a single network good as discussed in the lecture, and write a different consumer’s utility function for joining the network as

capturing the idea that it may be more plausible that a user who has a higher value for the standalone component of a technology also assigns more importance to the size of its network (i.e., consumers differ in their valuation of both the standalone and the network benefits). Here, a is the standalone benefit,

measures the network effect,

is the expected number of users joining the network and

is uniformly distributed on the unit interval. Based on the above data, answer the given subquestions.

The fulfilled-expectations demand curve in this setting will be **(consider (1-n)=m)**

Two roommates need to each choose to clean their apartment, and each can choose an amount of time

to clean. If their choices are

and

, then the player i’s payoff is given by

. (This payoff function implies that the more one roommate cleans, the less valuable is cleaning for the other roommate.) Based on the above data, answer the given subquestions.

What is the best response of roommate 2 if roommate 1 chooses

?

___________

Two roommates need to each choose to clean their apartment, and each can choose an amount of time

to clean. If their choices are

and

, then the player i’s payoff is given by

. (This payoff function implies that the more one roommate cleans, the less valuable is cleaning for the other roommate.) Based on the above data, answer the given subquestions.

Any

is dominated by

.

Two roommates need to each choose to clean their apartment, and each can choose an amount of time

to clean. If their choices are

and

, then the player i’s payoff is given by

. (This payoff function implies that the more one roommate cleans, the less valuable is cleaning for the other roommate.) Based on the above data, answer the given subquestions.

If the unique, symmetric Nash equilibrium in pure strategy for this game is represented by

then the value of

will be __________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(1)= _________________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(2)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(3) =____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(4) = ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(1,2) =____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(1,3)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(1,4)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(2,3)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(2,4)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(3,4)=____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(1,2,3)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(1,2,4)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(2,3,4)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(1,3,4)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

v(1,2,3,4)= ____________

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

Consider following two statements and choose the correct alternative (I) This game has symmetrical players. (II) Players 1 and 2 are only symmetric players.

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

Consider following two statements and choose the correct alternative (I) This game has a null player. (II) Player 4 is null player.

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

Given the game [61; 40, 40, 30, 10]. Write the game as a coalition function and find out symmetric and null players. Also, calculate Shapley-Shubik power index for the game. Based on the above data, answer the given subquestions.

A cable TV station wants to connect six new customers to its network. The connections to the network are described in the following tree graph:

Calculate the distribution of the costs among the six players if all six decide to connect to the network. Based on the above data, answer the given subquestions.

The cost to be paid by Player 1 will be ___________

A cable TV station wants to connect six new customers to its network. The connections to the network are described in the following tree graph:

Calculate the distribution of the costs among the six players if all six decide to connect to the network. Based on the above data, answer the given subquestions.

The cost to be paid by Player 2 will be __________

A cable TV station wants to connect six new customers to its network. The connections to the network are described in the following tree graph:

Calculate the distribution of the costs among the six players if all six decide to connect to the network. Based on the above data, answer the given subquestions.

The cost to be paid by Player 3 will be ___________

A cable TV station wants to connect six new customers to its network. The connections to the network are described in the following tree graph:

Calculate the distribution of the costs among the six players if all six decide to connect to the network. Based on the above data, answer the given subquestions.

The cost to be paid by Player 4 will be ___________

A cable TV station wants to connect six new customers to its network. The connections to the network are described in the following tree graph:

Calculate the distribution of the costs among the six players if all six decide to connect to the network. Based on the above data, answer the given subquestions.

The cost to be paid by Player 5 will be ____________

A cable TV station wants to connect six new customers to its network. The connections to the network are described in the following tree graph:

Calculate the distribution of the costs among the six players if all six decide to connect to the network. Based on the above data, answer the given subquestions.

The cost to be paid by Player 6 will be _____________

Six 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. Based on the above data, answer the given subquestions.

Optimal bid for a bidder with valuation 0.9 will be

Six 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. Based on the above data, answer the given subquestions.

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

Consider two competing firms in a declining industry that cannot support both firms profitably. Each firm has three possible choices as it must decide whether or not to exit the industry immediately, at the end of this quarter, or at the end of the next quarter. If a firm chooses to exit, then its payoff is 0 from that point onward. Every quarter that both firms operate yields each a loss equal to −1, and each quarter that a firm operates alone yields a payoff of 2. For example, if firm 1 plans to exit at the end of this quarter while firm 2 plans to exit at the end of the next quarter then the payoffs are (−1, 1) because both firms lose −1 in the first quarter and firm 2 gains 2 in the second. The payoff for each firm is the sum of its quarterly payoffs. Formulate this as a normal form game considering that E denotes immediate exit, T denotes exit this quarter, and N denote exit next quarter. Based on the above data, answer the given subquestions.

There are no strictly dominated strategies in this game.

Consider two competing firms in a declining industry that cannot support both firms profitably. Each firm has three possible choices as it must decide whether or not to exit the industry immediately, at the end of this quarter, or at the end of the next quarter. If a firm chooses to exit, then its payoff is 0 from that point onward. Every quarter that both firms operate yields each a loss equal to −1, and each quarter that a firm operates alone yields a payoff of 2. For example, if firm 1 plans to exit at the end of this quarter while firm 2 plans to exit at the end of the next quarter then the payoffs are (−1, 1) because both firms lose −1 in the first quarter and firm 2 gains 2 in the second. The payoff for each firm is the sum of its quarterly payoffs. Formulate this as a normal form game considering that E denotes immediate exit, T denotes exit this quarter, and N denote exit next quarter. Based on the above data, answer the given subquestions.

weakly dominated strategy in this game is

Consider two competing firms in a declining industry that cannot support both firms profitably. Each firm has three possible choices as it must decide whether or not to exit the industry immediately, at the end of this quarter, or at the end of the next quarter. If a firm chooses to exit, then its payoff is 0 from that point onward. Every quarter that both firms operate yields each a loss equal to −1, and each quarter that a firm operates alone yields a payoff of 2. For example, if firm 1 plans to exit at the end of this quarter while firm 2 plans to exit at the end of the next quarter then the payoffs are (−1, 1) because both firms lose −1 in the first quarter and firm 2 gains 2 in the second. The payoff for each firm is the sum of its quarterly payoffs. Formulate this as a normal form game considering that E denotes immediate exit, T denotes exit this quarter, and N denote exit next quarter. Based on the above data, answer the given subquestions.

Pure strategy Nash equilibria in this game are (More than one options correct)

Consider two competing firms in a declining industry that cannot support both firms profitably. Each firm has three possible choices as it must decide whether or not to exit the industry immediately, at the end of this quarter, or at the end of the next quarter. If a firm chooses to exit, then its payoff is 0 from that point onward. Every quarter that both firms operate yields each a loss equal to −1, and each quarter that a firm operates alone yields a payoff of 2. For example, if firm 1 plans to exit at the end of this quarter while firm 2 plans to exit at the end of the next quarter then the payoffs are (−1, 1) because both firms lose −1 in the first quarter and firm 2 gains 2 in the second. The payoff for each firm is the sum of its quarterly payoffs. Formulate this as a normal form game considering that E denotes immediate exit, T denotes exit this quarter, and N denote exit next quarter. Based on the above data, answer the given subquestions.

This game has unique mixed strategy equilibrium

Consider two competing firms in a declining industry that cannot support both firms profitably. Each firm has three possible choices as it must decide whether or not to exit the industry immediately, at the end of this quarter, or at the end of the next quarter. If a firm chooses to exit, then its payoff is 0 from that point onward. Every quarter that both firms operate yields each a loss equal to −1, and each quarter that a firm operates alone yields a payoff of 2. For example, if firm 1 plans to exit at the end of this quarter while firm 2 plans to exit at the end of the next quarter then the payoffs are (−1, 1) because both firms lose −1 in the first quarter and firm 2 gains 2 in the second. The payoff for each firm is the sum of its quarterly payoffs. Formulate this as a normal form game considering that E denotes immediate exit, T denotes exit this quarter, and N denote exit next quarter. Based on the above data, answer the given subquestions.

A man dies, leaving an estate worth 500 units. The deceased has two creditors; one’s claim is 450 units, and the other’s claim, 250 units. The division of the estate between them is carried out according to the contested garment principle. Based on the above data, answer the given subquestions.

Creditor 1 with claim 450 units gets

A man dies, leaving an estate worth 500 units. The deceased has two creditors; one’s claim is 450 units, and the other’s claim, 250 units. The division of the estate between them is carried out according to the contested garment principle. Based on the above data, answer the given subquestions.

Creditor 2 with claim 250 units gets

Suppose a search engine has two ad slots that it can sell. Slot **a** has a clickthrough rate of 10 and slot **b** has a clickthrough rate of 5. Three advertisers are interested in these slots. Advertiser **x** values click at 3 per click, advertiser **y** values click at 2 per click, and advertiser **z** values click at 1 per click. Based on the above data, answer the given subquestions.

Total valuation of socially optimal allocation will be ___________

Suppose a search engine has two ad slots that it can sell. Slot **a** has a clickthrough rate of 10 and slot **b** has a clickthrough rate of 5. Three advertisers are interested in these slots. Advertiser **x** values click at 3 per click, advertiser **y** values click at 2 per click, and advertiser **z** values click at 1 per click. Based on the above data, answer the given subquestions.

VCG price for slot **a** will be ___________

Suppose a search engine has two ad slots that it can sell. Slot **a** has a clickthrough rate of 10 and slot **b** has a clickthrough rate of 5. Three advertisers are interested in these slots. Advertiser **x** values click at 3 per click, advertiser **y** values click at 2 per click, and advertiser **z** values click at 1 per click. Based on the above data, answer the given subquestions.

VCG price for slot **b** will be ____________

Suppose 6 numbers are drawn independently from the uniform distribution on the interval [0, 1] and then sorted from smallest to largest. The expected value of the number in third position on the list is

Choose the correct alternative