Obvious Dominance and Milipede Games (Question to Pycia and Troyan (2019)

Question

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! :)

Answer 1

It is a convoluted definition because condition 3:

"At all $h$, if there exists a previously unclinchable payoff x that becomes impossible for agent $i_h$ at $h$, then $C_i^\subset (h) \subseteq C_i^h (h)$."

means that at every history if there is a payoff that player $i_h$ could not have secured (or clinched), but was feasible at every previous history where some other player was taking decisions and now is not feasible, then the set of clinchable payoffs cannot decrease.

That is, the set of clinchable payoffs is not allowed to decrease, except when some special condition is met.

Figure 1 is an example of a millipede game because the set of clinchable payoffs never decreases for players $j$ and $k$. It only decreases for player $i$ when (it goes from $\{x,y,z\}$ to $\{x,y\}$ (from the second to the third move), but this is fine because in player $i$'s third move the only payoff that was previously unclinchable is $w$, but it has not become impossible at that history (since the only impossible payoff is $z$ that was taken by player $j$).

Hope this helps.