I stumbled over Millipede Games and obvious Dominance in the paper "A Theory of Simplicity in Games and Mechanism Design" by Pycia and Troyan.
On page 12 the authors define millipede Games.
They explain point 3 verbally in the text: "the last condition ensures that passing will be obviously dominant, since if x becomes impossible, then the agent will at least be able to return to any payoﬀ she was previously oﬀered to clinch."
Then they write that Figure 1 shows an example of a milipede Game. But in my understanding of their definition it is not a milipede Game, because Point 3 of the definition does not hold:
Player j could clinch x in her second move, then player i could not return to that payoff although it was previously offered to her.
What is my misunderstanding here? Looking for help! :)