My attitude is that one cannot reason with someone who, for whatever cause, is being unreasonable. If that party has got it into their head that they are right, even despite blatant contrary evidence, no amount of justification will change that. Indeed, empirically, it usually makes matters worse. However, agreeing or submitting to their whims -- whilst an easy option -- is clearly not a good solution as it undermines yourself and sets a precedent for increasingly wild allegations. I therefore feel that, as one cannot "win" in this situation, the best option is to remove yourself from the situation entirely: i.e., don't "play".
However, this is not always possible. For example, if your significant other is being unreasonable, what does one do? One can listen and try to empathise with their position as much as you like, but it rarely works; meanwhile, your own position is marginalised, which causes increasing frustration. Changing the subject -- which is basically "not playing" -- can work, if whatever's being discussed is trivial enough, but for everything else, it's not going to fly. Asserting your position, as I describe above -- especially with logic -- definitely doesn't work, no matter how sympathetic you are!
Short of becoming an absolute stoic -- and I guess training yourself to not become defensive is partly a solution -- what techniques can you use to quell an unreasonable or non-evidence based argument when you're forced to engage?