A diagram for a simple two-player game

(Copied from my blog)

I always have a hard time mak­ing sense of prefer­ence ma­tri­ces in two-player games. Here are some di­a­grams I drew to make it eas­ier. This is a two-player game:

1

North wants to end up on the north­ern­most point, and East on the east­most. North goes first, and chooses which of the two bars will be used; East then goes sec­ond and chooses which point on the bar will be used.

North knows that East will always choose the east­ern­most point on the bar picked, so one of these two:

2

North checks which of the two points is fur­ther north, and so chooses the left­most bar, and they both end up on this point:

3

Which is sad, be­cause there’s a point north-east of this that they’d both pre­fer. Un­for­tu­nately, North knows that if they choose the right­most bar, they’ll end up on the east­ern­most, south­ern­most point.

Un­less East can some­how pre­com­mit to not choos­ing this point:

4

Now East is go­ing to end up choos­ing one of these two points:

5

So North can choose the right­most bar, and the two play­ers end up here, a re­sult both pre­fer:

6

I won’t be sur­prised if this has been in­vented be­fore, and it may even be su­perceded – please do com­ment if so :)

Here’s a game where East has to both promise and threaten to get a bet­ter out­come:

0,1-1,3_2,2-3,0

0,1-1,3_2,2-3,0-x