For questions 4-6: The restaurant chain \"Bay Crab House\" has three shareholder

Question

For questions 4-6: The restaurant chain "Bay Crab House" has three shareholders: Kristen, Dave and Garrett. The chain currently has all of its restaurants in Maryland. Kristen has proposed expanding the company by opening up a restaurant in Delaware. The shareholders must vote to either expand or not expand. Each shareholder gets one vote for each share that he or she owns. If more than half of the votes are cast for Kristen's proposal, it is approved. Otherwise, her proposal is not approved. 4. The number of shares each person owns is as follows: Shareholder Shares Kristen Dave 58 arrett 20 Find the Shapley-Shubik index for each of the three voters. Find the Banzhaf index for each the three voters. 5. Assume Kristen sells 1 share to Dave - implying that the number of shares each person owns is now: Shareholder Shares Kristen Dave arrett 20 a. Find the Shapley-Shubik index for each of the three voters. Find the Banzhaf index for each the three voters.

Explanation / Answer

Total no. of votes = 58+36+20 = 114

Minimum number of votes required for approval of proposal = 114 / 2 +1 = 58

Let, P1, P2 & P3 are Kristen, Dave & Karrett respectively.

4.

This weighted voting system is represented mathematically as {58 : 58, 36, 20}

a) Shapley-Shubik Index:

List of possible sequential coalation is as below:

{P1 , P2 , P3}
{P1 , P3 , P2}
{P2 , P1 , P3}
{P2 , P3 , P1}
{P3 , P1 , P2}
{P3 , P2 , P1}

In each of these N! sequential coalitions, the pivotal player is determined as below:

{P1 , P2 , P3} = 58
{P1 , P3 , P2} = 58
{P2 , P1 , P3} = 36 + 58 = 94
{P2 , P3 , P1} = 36 + 20 + 58 = 114
{P3 , P1 , P2} = 20 + 58 = 78
{P3 , P2 , P1} = 20 + 36 + 58 = 114

In all the 6 sequential coaltion, P1 is only pivotal.

Thus, Shapley-Shubik Index for Kristen = 6/6 =1 for Dave = 0 & for Garrett = 0

b) Banzhaf Index:

There are total 7 possible coalations. Out of which winning coalations are:

{P1} = 58

{P1, P2} = 58+36

{P1, P3} = 58 + 20

{P1, P2, P3} = 58 + 36 + 20.

In all the above mentioned winning coaltions, P1 is pivotal, as if P1 leaves the coalation losses.

Thus, total number of times all shareholders are critical = 4

No. of times P1 is critical = 4 and P2 & P3 are critical for 0 times.

Thus, Banzhaf Index for Kristen = 4/4 =1 for Dave = 0 & for Garrett = 0

5.

This weighted voting system is represented mathematically as {58 : 57, 37, 20}

a) Shapley-Shubik Index:

List of possible sequential coalation is as below:

{P1 , P2 , P3}
{P1 , P3 , P2}
{P2 , P1 , P3}
{P2 , P3 , P1}
{P3 , P1 , P2}
{P3 , P2 , P1}

In each of these N! sequential coalitions, the pivotal player is determined as below:

{P1 , P2 , P3} = 57 + 37 = 94
{P1 , P3 , P2} = 57 + 20 = 77
{P2 , P1 , P3} = 37 + 57 = 94
{P2 , P3 , P1} = 37 + 20 + 57 = 114
{P3 , P1 , P2} = 20 + 57 = 77
{P3 , P2 , P1} = 20 + 37 + 57 = 114

In the above mentioned 6 sequential coaltion, P1 is pivotal 4 times, P2 1 times and P3 1 times.

Thus, Shapley-Shubik Index for Kristen = 4/6 = 2/3 for Dave = 1/6 & for Garrett = 1/6

b) Banzhaf Index:

There are total 7 possible coalations. Out of which winning coalations are:

{P1, P2} = 57+37 = 94

{P1, P3} = 57 + 20 = 77

{P1, P2, P3} = 57 + 37 + 20 = 114

The critical players in above coaltions, i.e. if that player leaves the coalation losses, is underlined.

Thus, total number of times all shareholders are critical = 5

No. of times P1 is critical = 3, P2 = 1 and P3 = 1.

Thus, Banzhaf Index for Kristen = 3/5 for Dave = 1/5 & for Garrett = 1/5

6.

This weighted voting system is represented mathematically as {58 : 56, 37, 21}

a) Shapley-Shubik Index:

List of possible sequential coalation is as below:

{P1 , P2 , P3}
{P1 , P3 , P2}
{P2 , P1 , P3}
{P2 , P3 , P1}
{P3 , P1 , P2}
{P3 , P2 , P1}

In each of these N! sequential coalitions, the pivotal player is determined as below:

{P1 , P2 , P3} = 56 + 37 = 93
{P1 , P3 , P2} = 56 + 21 = 77
{P2 , P1 , P3} = 37 + 56 = 93
{P2 , P3 , P1} = 37 + 21 = 58
{P3 , P1 , P2} = 21 + 56 = 77
{P3 , P2 , P1} = 21 + 37 = 58

In the above mentioned 6 sequential coaltion, P1 is pivotal 2 times, P2 2 times and P3 2 times.

Thus, Shapley-Shubik Index for Kristen = 2/6 = 1/3 for Dave = 2/6 = 1/3 & for Garrett = 2/6 = 1/3

b) Banzhaf Index:

There are total 7 possible coalations. Out of which winning coalations are:

{P1, P2} = 56+37 = 93

{P1, P3} = 56 + 21 = 77

{P2, P3} = 37 + 21 = 58

{P1, P2, P3} = 56 + 37 + 21 = 114

The critical players in above coaltions, i.e. if that player leaves the coalation losses, is underlined.

Thus, total number of times all shareholders are critical = 6

No. of times P1 is critical = 2/6 = 1/3, P2 = 2/6 = 1/3 and P3 = 2/6 = 1/3.

Thus, Banzhaf Index for Kristen = 3/5 =1 for Dave = 1/5 & for Garrett = 1/5

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