Shapley Shubik power index part 2

I have looked previously at the Shapley Shubik power index which defines the power that each voter has depending on the number of votes that they control.

This can lead to some very interesting and unexpected results.

Imagine that there are four people owning 100 shares in a company (each share with one vote). Owner A has 44 shares, B has 29 shares, C has 20 shares and D has 7 shares.

If a simple majority is required to win a vote then the Shapley Shubik index gives A 50% of the power and B, C and D are all equal with 16.7%. Even though D has far fewer shares they are still key to the others reaching 50% of the votes.

Now C has a bright idea and decides to bring in a partner (E) and splits their 20 shares with them so C now has 10 shares and E also has 10 shares.

Recalculating the power index shows that A increases to 60% with the other four all equal on 10%. So C with 20 shares had 16.7% of the power but C and E with 10 shares each control 20% of the power between them. By splitting their shareholding C has more power than before (as long as they can trust E)

This is not always the case. A might see what C has done and decide to split their 44 shares so they have 22 shares and a new owner, F, also has 22 shares. Unfortunately for A this gives them 26.7% of the vote and F also 26.7% of the votes. Combined they now have 53.3% of the power, less than the 60% that A controlled on their own.

Could someone use these ideas to really gain more power or are these are just mathematical tricks. What do you think?

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