Game theory of takeovers

If you are a shareholder in a company that is the subject of a takeover bid then would you want the raider trying to acquire your company to promise to take millions out of the company once they gained control?

The natural reaction is to say ‘no’, of course you don’t want the value of your shares to be reduced after the takeover.

In fact, the promise to make a big payout might be necessary to make the deal happen in
the first place.

The corporate raider is presumably buying the company because he thinks he can add value to the business. This could be through following a different strategy or through running the existing business more efficiently. Either way he will be hoping to increase the value, and therefore the share price of the company after the takeover.

You are a small shareholder so you have effectively no influence on whether the takeover will happen or not. You see that if the takeover does happens then the price will be higher and you should wait until then to sell. Unfortunately, if all the shareholders think like that then no-one sells and no-one benefits from the higher share price.

This is a free rider problem. Everyone waits for someone else to sell, so no-one ends up selling. The only price that any shareholder would accept is the post-takeover price, but if the raider pays that then he makes no profit.

One of the ways round this is for the raider to promise to take a large payment for himself after the takeover. This will lower the post-takeover price to a point which is still higher than the current price, and so attractive to the current shareholders, but he gets his profit as well.

This problem was originally studied by Grossman & Hart in 1980.

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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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Divide and conquer

The Shapley Shubik Power Index

With Lloyd Shapley having just won the Nobel prize  it seems like a good time to look at the Shapley-Shubik power index.

This is a way to measure how much power different voters have in a vote where they control a different number of votes. It is not necessarily the case that someone controlling 20% of the votes has twice as much power as someone controlling 10% of the votes. They may have more or they may have less depending on the overall situation.

How it works

The way that the Shapley Shubik power index works is by taking every possible combination of voters and working out how many times a particular voter is pivotal. With this index the order matters so if there are three voters A, B, and C then there are six possible combinations to consider (ABC, ACB, BAC, BCA, CAB, CBA).

To work out the pivotal voter we look at how many votes the first voter has. If this is enough to reach a majority (or whatever level is needed) then they are the pivotal voter. If they are not then we move on to the second voter in the combination. If they voted the same was as the first voter and this took the total number of votes passed a majority then the second voter would be the pivotal one. It is whichever voter is the one that tips the combination passes the number of votes needed to win.

For example, assume that there are 100 votes to be cast and voter A controls 15 of them, voter B controls 40, and voter C controls 45. In the ABC combination B is the pivotal voter. This is because A has 15 votes so they do not have enough votes to get to 50 on their own. Once we add in B’s 40 votes then we get up to 55 votes so A and B together control more than 50% of the votes. This means that with this order of voters, B is the pivotal voter that takes the total over 50%.

The pivotal voter for the other five combinations is as follows:

ACB – Pivotal voter is C

BAC – Pivotal voter is A

BCA – Pivotal voter is C

CAB – Pivotal voter is A

CBA – Pivotal voter is B

So across all six combinations, A, B and C are all pivotal twice, so they are considered to have the same amount of power, even though they have a different amount of votes. This is because they each need one other player to vote with them to gain a majority. It doesn’t matter whether it is a large majority, like when B and C combine or a small majority when A combines with either B or C.

Strategy – Divide and conquer

A player with an understanding of where power really lies in a vote can use a strategy of ‘divide and conquer’ to take a disproportionate amount of power from their rivals.

If we start with the game that is described above and then imagine that B is able to change the game by introducing a new player D who takes 11 of C’s votes. This leaves A still with 15 votes; B with 40; C is reduced to 34; and D with 11.

There are now 24 possible combinations shown below with the pivotal voter in each marked in bold.





A is pivotal 4 times out of the 24 combinations and so has 16.7% of the power. B is pivotal 12 times out of 24, or 50%. C and D are both pivotal four times, the same as A.

Remember in the first game each voter had an equal amount of power, so they had 33.3% each. By managing to introduce a new player to take some of C’s votes B has greatly increased their own power from 33% to 50%. With three players they all had an equal amount of power despite their different number of votes. Now B has reduced A and C’s power while increasing their own.

In the first game any two pairs of voters could form a coalition and win the vote. By giving some of C’s votes to a new player, D, it changed the game so that B could pair up with anyone else to get a majority but none of the other players could pair up to reach a majority. This put B in a much stronger position than the first game.

Divide and conquer in action!

Image courtesy of taoty /

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