I don't think you can change people's minds by just telling them they're wrong, or by comparing the costs and benefits of two alternatives, without taking into account their values. After all, you're picking the benefits--you may have excluded something important to them, because it's not important to you. Or maybe it's even something you see as an evil. There's a lot of unexpressed assumptions in most arguments I've read.
You might be able to do it (change their minds) by showing how your proposal is in accord with their values. I'm sure you can do it by changing their values, if you can manage that.
One person told me I changed their mind by presenting information they already had in a different way, which I suppose can be called perspective. (I wouldn't call it framing. I think framing has a deserved bad reputation--it's like push advertising. "If only we choose the right vocabulary we can get those dimwits to make the right decision!")
I learned a long time ago that most people learn best (most easily and most thoroughly) when they learn something themselves. Facts you can just tell, I mean drawing conclusions and figuring out problems. They keep the knowledge, they incorporate it into their worldview, when they work it out on their own rather than being told the solution. That was a hard truth for me: I like to tell people things. I like being the one that knows! And I like being helpful. But when I discovered the best help I could be would be to step back and let them figure things out, I started doing that as a first step in helping.
So, this started out being about the United States moving toward some kind of government-provided health care for all (citizens? persons within our borders?) using tax dollars. I haven't paid much attention-I'm not willing to pay the price of trying to affect this process, so the details are useless to me until it's in place in whatever form it ultimately takes. But what keeps catching my attention is the underlying struggle between two approaches to the problems of life: one, where we (as individuals) decide that this problem is best solved as a community, and agree to impose the costs of a solution on all of us (some voluntarily, some through coercion) by using government as a tool to provide that solution; and two, where we (as individuals) decide that this problem is best left in individual hands and to individual judgement about individual solutions, which also imposes some costs on everyone, just different costs.
People in all kinds of political associations (or lack of them) in all kinds of positions on all kinds of problems have this decision somewhere in the basis of their approach. And they decide it differently depending on the problem. What I rarely see is argument addressing this decision, this core choice about what solutions to try and how to make those solutions happen. Instead in the health care debate I've read mostly comparisons of the costs: if we stay with our current system X people pay Y cost, and if we move to government-mediated health care A people will pay B cost. Then the other side picks apart your cost analysis (which can always be done) and you've moved nobody from one position to another.
I think it's possible to move this decision one way or another by looking at values instead of costs. I know, I know--my friends in favor of single payer or a national health service will say that the other side doesn't care about poor people because the other side discounts societal effects on choice, and my other friends in favor of private health care will say the first side doesn't care about encouraging self-reliance and growth (of all kinds: intellectual, economic, research) because the other side thinks it's worthwhile to take away the fruits of personal labor to pay for other people's bad judgement.
I think it would be more useful to admit that one set of people thinks giving the government control of my health care is worth it because only government is big enough to provide care for everyone, and medical care is important enough to do it this way, and our current system is failing a lot of people. And that another set of people thinks giving the government control of my health care means imposing the worst of government bureaucracy on decisions that are very personal, that emotions matter to health care decisions and don't matter to government rules, and that every time I cede power over my life to the government I become more vulnerable to human mistake, human corruption, and clerical error. I think both sets of people would benefit from examining what governments in the United States do well and what they really get wrong; our culture is important to this discussion, more important than how other countries' health care systems work.
I don't think that conversation is taking place.