What you are actually trying to do
Not win. Winning a political argument with someone you know is close to impossible and, on the rare occasions it works, tends to cost more than it returns.
The goal is narrower and more achievable: leave one question with them that they will meet again on their own, every time a political issue comes up. Force, or persuasion?
A conclusion someone reaches themselves holds. One handed to them does not, and usually produces the opposite of what was intended, because being argued into a position feels a lot like being pushed — which is the very thing you are there to talk about.
Move one: affirm the value, genuinely
Start where they are. "I think compassion for people who are struggling is genuinely important." "I agree that personal responsibility matters, and that a lot of policy pretends it doesn't."
This only works if you mean it, and you will almost always be able to, because the underlying value is usually one you share. The disagreement is nearly never about whether children should be fed or whether people should be free to build things.
If you say this as a tactic, they will hear that it is a tactic. People are very good at detecting the maneuver where someone agrees with them in order to set something up.
Move two: ask the method question, and stop talking
"We agree on the goal. I'm curious — do you think the best way to get there is through government force, or through voluntary cooperation?"
Then let it sit. Do not answer it for them, and do not fill the pause. The question is doing the work, and it only works if they are the one who answers it.
Most people have never separated the two questions before. They have spent years arguing about goals with people who share their method, and the distinction can be genuinely new to them.
Move three: the personal test
"Would you do it yourself? Would you go to your neighbor's house and take the money for this?"
Almost everyone says no. Then the only question left is the useful one: what is it about the government doing it that makes it different?
That question has real answers, and some of them are decent — scale, coordination, the free-rider problem, democratic authorization. Let them make those arguments. You are not there to defeat them; you are there to make sure the question gets asked out loud, which for most people it never has been.
Move four: leave it open
Do not push for agreement. Do not summarize. Do not wrap it up.
"I don't have all the answers here either. But I think that question — force or persuasion — is worth thinking about." And then change the subject.
The move that ruins this conversation is the one where you win it. A question someone is still turning over is worth more than a concession they made to end the discussion.
Most people will not agree immediately, and that is the expected outcome rather than a failure. You planted something. It grows on its own schedule, and usually not while you are watching.
The better version of all of this
If the conversation is going badly, or you can tell it is about to, there is an option that outperforms every move above: do not make the argument at all. Send them something and let them find it themselves.
A question someone works through alone, with nobody watching to see whether they concede, is not the same object as a question put to them by a friend who clearly has a preferred answer. The first one they can change their mind about at no cost. The second one has their pride in it.