How is Armageddon in chess like fair cake-cutting in game theory

fair-divisiongame theoryintuition

In chess (or 9LX), Armageddon is a kind of blitz/rapid chess used as tiebreakers: Black has draw odds, but white gets more time.

Eg In the 2022 women's US Chess Championship, Jennifer Yu won the coin toss and then chose black against Irina Krush who played white. Jennifer got 4 minutes, and Irina got 5 minutes.

It says on Wikipedia:

Sometimes, players bid for Black.

Some tournaments utilise a bidding system for individual players of each match to decide how little time they would be willing to play with as black. The player with the lowest bid for each match receives the black pieces with draw odds. This system minimises the perceived unfairness of Armageddon time controls that are decided in advance before a tournament with colours randomly allocated.

Eg In the 2022 World Fischer Random Chess Championship, the finalists Ian Nepomniachtchi and Hikaru Nakamura wrote down times between 0 minutes and 15 minutes and the lower bid gets black. In their case, Nepo won the bid with 13 minutes and got black and so Hikaru got white and 15 minutes.

And then Wikipedia says:

Such an idea is reminiscent of the logical use case of fair cake-cutting.

I haven't heard of fair cake-cutting or more generally fair division before this. How is armageddon in chess like fair cake-cutting? Is bidding for lower time like bidding for a smaller quantity yet high quality slice of cake or something? Eg smaller piece but has strawberry on it. Sooo

  • smaller piece = lower time + black

  • strawberry = draw odds

Quote:

If a cake with a selection of toppings is simply cut into equal slices, different people will receive different amounts of its toppings, and some may not regard this as a fair division of the cake.

Update 1:

Alex K it says on Wikipedia 'If a cake with a selection of toppings is simply cut into equal slices, different people will receive different amounts of its toppings, and some may not regard this as a fair division of the cake.' – is this a wrong way of thinking about fair cake-cutting ? –
BCLC

No, that's correct, (…) – Alex K

Alex K what's the difference with Wikipedia's correct way of thinking and my incorrect way of thinking? I mean my naive understanding is that it's not that deep: bigger slice but without strawberry = more time + white but without draw odds. – BCLC

@BCLC I may have misunderstood what you meant at first, because I think that's right. – Alex K

Update 2:

Alex K do you think it's more like cake cutting if they did an armageddon like '1 player chooses the time for black and then the other player chooses which colour' ? I've been reading up armageddon in chess and 9LX a lot recently and then I saw this 1 suggestion. I think it's like what you said re the Alice chooses a point to cut the linear cake and then Bob chooses the slice. – BCLC

Yeah, I think that would be very much like cake cutting. – Alex K

Best Answer

I think my other answer may be wrong. I'll delete it after this is refined and discussed.

A linear cake and its values

Consider a linear cake, that can only be cut perpendicular to its axis. Alice can choose $a\in(0,1)$ as a cut. We'll call the two pieces $[0,a)$ and $(a,1]$

The perceived value of the possible cake slices may be more complex. For now, we require the following, given $a$

  • Each player has a value they assign the two slices.
  • The value is monotonic with size: a bigger piece is always more or equal to a smaller piece.
  • The value is continuous enough that equality is possible, in other words, Alice can choose an $a$ that makes the slices equal in value to her.

Normal cake cutting

Alice chooses $a$. Bob chooses the piece that has the larger value to him.

We said that the values are monotonic and allow equal values. If Bob thinks the cake slices are equal when cut at $b$, then when $a<b$, he will choose the $(a,1]$ piece; when $a>b$, he'll choose the $[0,a)$ piece.

This means that for this simple scenario, Bob doesn't have to choose a cake slice after it's cut! He could decide on $b$ beforehand.

Cake cutting with written $b$

  • Bob secretly writes down the position, $b$ at which he'd consider it a fair split.
  • Alice cuts the cake at $a$.
  • Bob gets the slice according to whether $a>b$.

If Alice sees $b$ before slicing, she can pick $a$ to get, from her perspective, more than a fair share.

If Alice sees $b$ after, she will know that she could have gotten more cake by slicing closer to $b$.

Bob, on the other hand, doesn't have the same benefit. If Alice writes $a$ prior to cutting, and Bob sees it, be can't change $b$ to his advantage.

I think this does show some similarities from Alice's perspective and differences from Bob's.

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