<p>It is a convoluted definition because condition 3: </p>

<p>"At all <span class="math-container">$h$</span>, if there exists a previously unclinchable payoff x that becomes impossible for agent <span class="math-container">$i_h$</span> at <span class="math-container">$h$</span>, then <span class="math-container">$C_i^\subset (h) \subseteq C_i^h (h)$</span>." </p>

<p>means that at every history if there is a payoff that player <span class="math-container">$i_h$</span> 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. </p>

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

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

<p>Hope this helps. </p>

CC BY-SA 4.0
2020-01-18T11:01:15.937Z
Thanks a lot!

You are right I think. I did not correctly take the "previously unclinchable" into account. Only if something that was previously unclinchable becomes impossible, the citated statement holds:

Thanks :D
Tobias Pastoors
CC BY-SA 4.0
2020-01-19T00:23:43.813Z
@TobiasPastoors Can you please mark the answer as accepted?
Regio
CC BY-SA 4.0
2020-01-19T11:11:32.337Z
of course! I was just not aware of that possibility
Tobias Pastoors
CC BY-SA 4.0
2020-01-16T18:19:27.277Z
2020-01-16T18:19:27.277Z
Regio
<p>I stumbled over Millipede Games and obvious Dominance in the paper <a href="http://people.virginia.edu/~pgt8y/Pycia-Troyan-Simplicity.pdf" rel="nofollow noreferrer">"A Theory of Simplicity in Games and Mechanism Design" by Pycia and Troyan</a>.</p>

<p>On page 12 the authors define millipede Games.</p>

<p>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."</p>

<p>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:</p>

<p>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.</p>

<p>What is my misunderstanding here? Looking for help! :)</p>

<game-theory><extensive-games>
Obvious Dominance and Milipede Games (Question to Pycia and Troyan (2019)
Tobias Pastoors
# Obvious Dominance and Milipede Games (Question to Pycia and Troyan (2019)

(Asked by: username_0 on 2020-01-09)

I stumbled over Millipede Games and obvious Dominance in the paper ["A Theory of Simplicity in Games and Mechanism Design" by Pycia and Troyan](http://people.virginia.edu/~pgt8y/Pycia-Troyan-Simplicity.pdf).

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! :)

**Tags:** game-theory, extensive-games

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## 1 answers

### Answer by username_1 on 2020-01-16. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: Thanks a lot!  You are right I think. I did not correctly take the "previously unclinchable" into account. Only if something that was previously unclinchable becomes impossible, the citated statement holds:  Thanks :D  
> **username_1**: @TobiasPastoors Can you please mark the answer as accepted?  
> **username_0**: of course! I was just not aware of that possibility  

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