How we change what others think, feel, believe and do
When people are enthused by a particular idea they may gradually acquire a more extreme viewpoint.
When looking at evidence, they will amplify confirming evidence and downplay disconfirming evidence. This contributes to their viewpoint becoming more entrenched and extreme.
This is partly due to a desire for distinctness from alternative views, leading to the person moving their position away from views that have some similarity. To agree with an opponent on a small point may seem to be a slippery slope and confuse one's clarity.
In arguing, many people like clear positions, where they are 'good' and the other person is 'bad'. This means that any agreement at all is hazardous as it makes oneself bad in any area of overlap. It may also seem as if the other person is trying to 'seduce me to the dark side' (so causing further retreat).
The easiest approach in such cases is to take an extreme position and/or push your opponent to the opposite extreme.
When the two opponents are in the same place together, the effect can be exacerbated as each seeks to avoid losing face and is constantly reminded of their differences by the presence of the other. (On the other hand, when a person is present it is difficult to objectify them).
Lord, Ross and Lepper showed how people who supported or opposed capital punishment selectively used the same body of evidence to support their own viewpoint.
A person believes in right wing politics and seeks failures in a left-wing government to prove the correctness of right-wing views. In doing so, they become even more convinced they are right.
Get a person to take an extreme position by setting someone else up in opposition to them.
One way to prevent a person polarizing away from you is simply not to be there. Use a third party to present your case or an asynchronous method such as email or promotions.
Consider how you got to the viewpoints you have, particularly if it is relatively extreme.