Comments on: Let’s Play a Game, Part 2: Game Trees and Totally Finite Games

By: Melvin

Melvin — Sun, 11 Oct 2009 11:28:55 +0000

Hmm.. I never knew about this before^^ Very interesting.. So there’s no such thing as infinite games. Nice writing!

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By: Michael Norrish

Michael Norrish — Tue, 11 Mar 2008 00:08:17 +0000

#2 also violates the finite-ness requirement: each player could keep specifying supergame.

By: musesusan

musesusan — Sun, 09 Mar 2008 04:28:28 +0000

*grin* Knew you guys would get it.

By: John Armstrong

John Armstrong — Sat, 08 Mar 2008 22:09:46 +0000

Yes, of course, miller’s right. I’m partly jumping ahead conceptually and partly forgetting that different parameter settings count as different games. That is, G(6,3) is not the same game as G(5,3).

Another way to put it would be to point out that each move changes the game entirely. Let’s say we’re playing G(6,3) still. Then if player 1 takes one stone, we’ve now ended that game. Instead we’re playing G(5,3) and the other player is player 1. Of course, if player 1 in G(6,3) had taken two stones, then we’d be playing G(4,3).

By: miller

miller — Sat, 08 Mar 2008 19:27:17 +0000

I’ve always called it a cat’s game too, and I never get strange looks. Of course, I don’t play tic tac toe much on account of it being too easy to solve.

I’m thinking the supergame violates (5), but isn’t there a way to fix this without destroying the paradoxical part?

The Modified Supergame:
Play exactly like the Supergame, only the first player must choose a totally finite game from the following set: {G(5,3), The Modified Supergame}

By: musesusan

musesusan — Sat, 08 Mar 2008 17:28:25 +0000

Ack! Thanks for catching that mistake! (Guess I got so excited by all the pretty colors I just went crazy.)

And there’s actually a completely non-self-referential proof for #2, if you think carefully about the five axioms.

By: John Armstrong

John Armstrong — Sat, 08 Mar 2008 17:13:25 +0000

Shouldn’t the rightmost circle in the middle row be red? It’s red’s turn and he has a red option below. Then the rightmost in the next row up should be red because it’s blue’s turn and she has no blue options. And then the rightmost circle in the next line is red, but the top stays blue.

There’s an interesting bit of self-reference in the proof of number 2, rather reminiscent of Gödel’s famous proof. Basically, I don’t see which axiom of totally finite games it violates, but I can show that it’s inconsistent for it to be a totally finite game.