The case in strategy games is not intransitive. Given any distribution, there is an optimal play against it. For example, if my opponent played 40% rock, 30% paper, and 30% scissors, I would prefer paper, then rock, then scissors. If your opponent plays all three equally, there are no preferences, not circular preferences. Randomization is used to prevent the opponent from gaining information about you. If you could use a pseudorandom method to exploit failures in their cognition and win in the long run, that is a preferable strategy.
In none of these cases would I decide to throw rock, then realize paper is better and change my choice, then realize scissors it better ad infinitum. Paper is not necessarily a better choice than rock, it would just beat rock in a game. Equating these two concepts is a level confusion.
Transitivity may not hold in situations where all the choices are not available at once. For example given activity choices A(fishing), B(dancing), C(reading) I may pick A>B>C when made aware of all choices, but in isolate may pick A>B, B>C, C>A.
Either way, if axiom 2 were interpreted as referring to choices made when all options were known, for example if you knew you could fish, dance, or read and were asked to rank among all of them, the VNM theorem would still work. In this case, you would never say C is better than A because you would always be aware of B.
The case in strategy games is not intransitive. Given any distribution, there is an optimal play against it. For example, if my opponent played 40% rock, 30% paper, and 30% scissors, I would prefer paper, then rock, then scissors. If your opponent plays all three equally, there are no preferences, not circular preferences. Randomization is used to prevent the opponent from gaining information about you. If you could use a pseudorandom method to exploit failures in their cognition and win in the long run, that is a preferable strategy.
In none of these cases would I decide to throw rock, then realize paper is better and change my choice, then realize scissors it better ad infinitum. Paper is not necessarily a better choice than rock, it would just beat rock in a game. Equating these two concepts is a level confusion.
Would you really call that rational? If my brain behaved this way, I would attempt to correct it.
Either way, if axiom 2 were interpreted as referring to choices made when all options were known, for example if you knew you could fish, dance, or read and were asked to rank among all of them, the VNM theorem would still work. In this case, you would never say C is better than A because you would always be aware of B.