Three people invest in a treasure dive, each investing the amount listed below.  The dive results in 34 gold coins.  Using Hamilton's method, apportion those coins to the investors based on their investment.

Investor
Investment
Allocation of 34 coins
Karissa
$8,670

Jane
$6,650

Hillary
$1,680

 
Right before the coins are distributed, the divers find one more coin they had misplaced.  Redo the allocation, now with 35 coins
Investor
Investment
Allocation of 35 coins
Karissa
$8,670

Jane
$6,650

Hillary
$1,680

 
Does this situation illustrate any apportionment issues?

Total invested = 8670+6650+1680 = $17,000.

a. Karissa: (8670/17,000) * 34 = 17.34(18 coins).
Jane: (6650/17,000) * 34 = 13.3(13 coins).
Hillary: (1680/17,000) * 34 = 3.36(3 coins).
0.34+0.30+0.36 = 1.0 coin which goes to highest investor(Karissa).

b. Repeat part a with 35 coins.

Three people invest in a treasure dive, each investing the amount listed below. The dive results in 34 gold coins. Using Hamilton's method, apportion those coins to the investors based on their investment.

Investor:

| Investor A | Investor B | Investor C |
|------------|------------|------------|
| $8,670 | $6,650 | $1,680 |

Total investment:

Total investment = $8,670 + $6,650 + $1,680 = $17,000

Fair share per coin:

Fair share per coin = Total investment / Total coins = $17,000 / 34 = $500

Investor A share:

Investor A share = (Investor A investment / Total investment) * Total coins = ($8,670 / $17,000) * 34 = 17.34 coins

Investor B share:

Investor B share = (Investor B investment / Total investment) * Total coins = ($6,650 / $17,000) * 34 = 13.3 coins

Investor C share:

Investor C share = (Investor C investment / Total investment) * Total coins = ($1,680 / $17,000) * 34 = 3.36 coins

Final allocation:

| Investor A | Investor B | Investor C |
|------------|------------|------------|
| 17 coins | 13 coins | 3 coins |

To check if the allocation is fair:

Investor A priority = Investor A share / sqrt(Investor A share + Investor B share + Investor C share) = 17 / sqrt(17 + 13 + 3) = 0.742

Investor B priority = Investor B share / sqrt(Investor A share + Investor B share + Investor C share) = 13 / sqrt(17 + 13 + 3) = 0.579

Investor C priority = Investor C share / sqrt(Investor A share + Investor B share + Investor C share) = 3 / sqrt(17 + 13 + 3) = 0.193

Investor A priority + Investor B priority + Investor C priority = 0.742 + 0.579 + 0.193 = 1.514

The sum of priorities is larger than 1, indicating an overallocation of coins. The situation illustrates an apportionment issue known as the "Alabama paradox," where adding an extra unit can change the allocation significantly.

To allocate the coins using Hamilton's method, we need to follow these steps:

1. Calculate the total investment. In this case, the total investment is the sum of the individual investments: $8,670 + $6,650 + $1,680 = $17,000.

2. Calculate each investor's share of the total investment. To do this, divide the investment of each investor by the total investment amount:

Karissa's share = ($8,670 / $17,000) * 34 coins

Jane's share = ($6,650 / $17,000) * 34 coins

Hillary's share = ($1,680 / $17,000) * 34 coins

3. Calculate the total number of coins allocated. Sum up the individual allocations:

Total coins allocated = Karissa's share + Jane's share + Hillary's share

4. If there is a disparity between the total number of coins and the actual number of coins found, adjust the allocations proportionally. In this case, since the divers find one more coin, we have 35 coins instead of 34.

The adjusted allocation for each investor would be:

Karissa's share = ($8,670 / $17,000) * 35 coins

Jane's share = ($6,650 / $17,000) * 35 coins

Hillary's share = ($1,680 / $17,000) * 35 coins

Based on this procedure, we can find the allocations for both scenarios. Now, let's calculate the allocations:

Scenario 1 (34 coins):
Karissa's share = ($8,670 / $17,000) * 34 = 17.3929 coins
Jane's share = ($6,650 / $17,000) * 34 = 13.3176 coins
Hillary's share = ($1,680 / $17,000) * 34 = 3.2894 coins

Scenario 2 (35 coins):
Karissa's share = ($8,670 / $17,000) * 35 = 17.4412 coins
Jane's share = ($6,650 / $17,000) * 35 = 13.3814 coins
Hillary's share = ($1,680 / $17,000) * 35 = 3.3365 coins

Now, to determine if there are any apportionment issues, we can compare the allocations between the two scenarios. In this case, the allocations are slightly different due to the additional coin found. While the overall proportions remain the same, the specific numbers of coins allocated to each investor change slightly. This can lead to minor discrepancies in the fair distribution of the coins, especially when the investment amounts are significantly different.

Therefore, this situation does illustrate some apportionment issues, as the additional coin found results in slightly different allocations for each investor.