I think a reasonable way to deal with this scenario is: each of you plays a mixed strategy with the end result that on average, both players will get 40% (meaning that 20% of the time the players will fail to reach agreement and nobody will get anything). The nice things about this strategy are (A) it gives both players at least some money and (B) it satisfies a “meta-fairness” criterion that agents with more biased notions of fairness aren’t able to exploit the system to get more money out of it.
Just wanted to note that this bit reminds me of this post: https://www.lesswrong.com/posts/z2YwmzuT7nWx62Kfh/cooperating-with-agents-with-different-ideas-of-fairness
(For those who haven’t seen it.)