mathematics - Can political debates really work?

In the far-off country of Politica, there are three main parties: the Left, the Right, and the Centre. In the last election, there were 19 million Left voters, 21 million Right voters, and 23 million Centre voters.

Every time two people who support different parties get into a debate, they each end up convincing the other that their party is terrible and not worth supporting, and both decide to switch their vote to the third party. For instance, a debate between a Left voter and a Right voter will turn both of them into Centre voters. This is the only way anyone's chosen party can change: if more than two people get into a debate, they'll just end up fighting instead.

Is it possible that by the next election, everyone will be agreed on the same party?

_{For the sake of simplicity, assume a fixed population: the set of voters is exactly the same in the next election as the last one.}

_{This problem is inspired by one in the 1984 Tournament of Towns.}

Answer

I am assuming that

these numbers are all exact. It will quickly become apparent why I need this.

So, consider

the difference between the number of Left and Right voters. When two L,R voters argue this doesn't change. When an L and a C argue, we turn +1 into -2. When an R and a C argue, we turn -1 into +2. In any case, the total mod 3 remains fixed.

Now,

this difference starts out at 4 million. Since the total number of voters is a multiple of 3, we want it to end at zero. But 4 million is not a multiple of 3. Therefore: no, it is not possible.