There's also the game of SOS, which is not *quite* as bad as Brussels Sprouts, but quite close: "We begin with an n-by-1 grid of initially empty cells. Two players alternate turns, playing either an `S' or an `O' on an empty cell (each player can write either letter; this is an impartial game). You win if your move completes a string `SOS', and lose if your opponent manages this instead. If the board is completely occupied without any SOS forming, the game is declared a draw."

Nim (which of course isn't unfair in quite the same way as Brussels Sprouts, but does have the property that one who understands the game can immediately tell who can win given the initial position)

Every impartial game is as bad as Nim. #SpragueGrundy Sincerely, Adam P. Goucher

Sent: Monday, November 17, 2014 at 11:54 PM From: "Gareth McCaughan" <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Unfair Games

On 06/11/2014 14:40, James Propp wrote:

...

Does anyone have handy a copy of Martin Baxter's article "Unfair Games", Eureka, 50 (1990), 60-68? If so, and if I'm right in suspecting that it's about games like Brussels Sprouts in which the winner is predetermined, could someone list for me the games of this kind that Baxter discusses?

I have a copy. (I didn't reply earlier because I couldn't find it. Then I looked at the big pile of junk beside my desk, and all my old Eurekas were right on top of the pile!)

The discussion covers, in order:

Brussels Sprouts (including the small generalization to playing on surfaces other than the plane)

Nim (which of course isn't unfair in quite the same way as Brussels Sprouts, but does have the property that one who understands the game can immediately tell who can win given the initial position)

Lotteries and insurance (with the usual remark that they are rather alike)

Various gambling games with St-Petersburg-esque paradoxes: one where your outcome -> -oo with probability 1 even though E(outcome) = +oo, and one where at every stage E(outcome)<0 but outcome -> something positive with probability 1.

Calling these all "unfair" seems a bit of a stretch to me.

In view of the recent discussions of the late Alexander Grothendieck, I cannot resist quoting a pair of clerihews also found in this issue of Eureka.

Pierre Deligne Invented a machine, And just for a laugh He called it SGA 4 1/2.

Pierre Deligne Inventa une machine, Et pour faire rire ses amis Il l'appelait SGA 4 1/2.

-- g

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On 18/11/2014 01:54, Adam P. Goucher wrote:

...

Nim (which of course isn't unfair in quite the same way as Brussels Sprouts, but does have the property that one who understands the game can immediately tell who can win given the initial position)

Every impartial game is as bad as Nim. #SpragueGrundy

And indeed every not-impartial game of the same complete-information type. #Conway But some have more transparent strategies than others -- and Mr Baxter remarks on, e.g., the difficulty of determining what happens with best play in chess, which I think makes it less unfair in his eyes. (For my part, I agree that merely having a definite winner with perfect play does not make a game "unfair" in any useful sense. But there might be something "unfair" about challenging someone to a game of nim if you know how to compute its Grundy numbers and they're trying to analyse positions by brute force.) -- g