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I think the example given to show the irrationality of leximin in certain situations doesn’t do a good job of distinguishing its failings from maximin. To usefully illustrate the difference between the two I believe a another state is required with even worse outcomes for both acts (e.g. $0). This way the worst outcomes for both acts would be equal and so the second worst outcomes (a1:$1, a2:$1.01) would then be compared under the leximin strategy leading to the choice of a2 as the best act again with the acknowledgment that you miss out on the opportunity to get $10,001.01
Minimax is equivalent to maximin in a zero sum game because every resource I win you lose out on. However, in non-zero games which encourage cooperation this is no longer case. Take for example a scenario in which we share some land. Let’s assume I have one third and you have two. If I water your crops as well as my own every other day you have agreed to water mine on the other alternate days when I must look after my kids. If I don’t water your crops you won’t water mine back but you don’t have any kids so you’d actually be able to water every day without me although you’d rather hang out with your friends at the pool if you could. If I want to minimise the maximum (minimax) utility you get I wouldn’t water your crops at all and force you to water every day. Of course you’d resent me for this and wouldn’t water my crops so I’d end up with no crop. If I want to maximise the minimum (maximin) crop yield for myself, I’ll have to water your crops too even though you have twice as many crops as me and you’ll get to hang out with your friends as well!