Exposition
Suppose that there are 100 prisoners (arbitrary amount) who have to follow a strict set of rules.
One of these rules is that they must be in bed by 10pm every single night. However, the majority of prisoners want this changed.
The prison guards allow the prisoners to vote to change their bed time, but require a super majority to agree that the bed time should be changed: if >=75% of votes agree that yes, the bed time should be changed, then the prison guards will change it.
The Dilemma
The prisoners are given the following option: they may give up a fixed amount of food ration in order to gain an extra vote. In fact, they may obtain an extra vote for every food ration that is relinquished.
Assume that the prisoners have a vested enough interest in the vote to give up their food rations. Also assume that there is no demographic correlation between one's overall willingness to give up food and their vote.
If you assume that a yes voter and no voter are just as likely to purchase extra votes, these votes should essentially cancel out.
The Question
Can you expect that no voters have a higher incentive to purchase extra votes, given that the threshold of voting is 75%?
If the amount of votes bought per capita is the same on either side, the votes cancel, but should you naturally expect there to be higher votes bought per capita for no voters?
A no vote carries more power than a yes vote, so I can see this being the case, but I am not sure.
This depends on how many times each side out thinks each other:
1st move: YES voters are confident and don't spend rations.
2nd move: If NO voters out-think them they spend more rations to vote NO
3rd move: If YES voters figure NO voters would try to spend more rations to change the vote they do the same
4th move: If NO voters think this far ahead they quit altogether as there is no point in wasting rations
5th move: Like move one YES voters don't bother spending rations because they foresee move 4
i.e moves 1-4 iterate and in only move 2. That gives quantifies 'incentive' as a quarter which is not very appealing
Alternatively:
If we say that infinite rations are allowed instead of move 4 you'd reach:
4th move: NO voters further sacrifice rations to boost their NO votes
5th move: YES voters further sacrifice rations to boost their YES votes
This pattern re-iterated instead causing half the total moves to be in favor of NO voters and half to be in favor of YES voters causing the incentive to be a half which means no equal incentive.
However:
The fact that if there was no extra votes the general result would be half likely (75-100 people vote yes) doesn't make a difference. due to how the chances are fair. So technically, this is experimentally an even vote. The fact that 75% is the required majority for YES people and over 50% of people are YES people cancel out
We can demonstrate this in this way:
Now that we have our 'quantified incentivity' we can now approach what would happen in a change of threshold. Our current incentivity is:
0.75 for Yes or 0.5 for Yes
0.25 for No or 0.5 for No
Since we have seen that the 75% threshold splits the boundary of YES voters in half the incentives are multiplied as such:
0.0375 for Yes or 0.25 for Yes
0.0125 for No or 0.25 for No
...essentially the ratio between the two stays the same. However if we bring the threshold up (such as 99%) the probability for YES to lose generally is greater which multiplies to the YES incentive to get more votes. The probability for NO to lose goes down so that is multiplied to the NO incentive. Hence:
0.75
X((100-99)/(100-50))for Yes or 0.25X((100-99)/(100-50))for Yes0.0125
X(1-((100-99)/(100-50))) for No or 0.25X(1-((100-99)/(100-50))) for No