I think I found a flaw.

I have little background in this field, so I would not be surprised to find I am wrong. Please let me know if I am and point me to where I can educate myself.

Nash equilibriums seem to apply to people who play the game with the goal that can be stated "best payoff I can get." The goal "a better payoff than anyone else" especially when you include players you are competing with only indirectly will rationally result in a different strategy.

The author is incorrectly equating those two goals, and introducing the additional factor of indirect competition. There is no nash equilibrium in such a game. If this is true, any conclusions that require its opposite are invalid.

Edit - Not changing anything above, even though it does indeed display my ignorance. The payoff wasn't to the player that was better than everyone else, it was to a random participant based on thier average payoff. That seems to me to motivate the player to seek selfishness vs superiority. If superiority isn't required, indirect competition isn't a factor. I did not find a flaw after all, and I am still bewildered.