This course will cover the mathematical theory and analysis of simple games without chance moves.

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From the course by Georgia Institute of Technology

Games without Chance: Combinatorial Game Theory

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This course will cover the mathematical theory and analysis of simple games without chance moves.

From the lesson

Week 3: Comparing Games

The topics for this third week is Comparing games. Students will determine the outcome of simple sums of games using inequalities.

- Dr. Tom MorleyProfessor

School of Mathematics

Welcome to week three of Games without Dice or Cards: Combinatorial Game Theory.

Â I'm Tom Morley. Today we want to talk about ordering

Â games. And these are our definitions.

Â They're like, games are like numbers, sort of, or at least sometimes, and ups-

Â numbers you can order. 1 is less than 2 is less than 3.

Â 2 is equal to 2, minus 1 is less than 6. And here's what this means for games.

Â To say a game is negative means that right always wins, going first or going second.

Â To say a game is zero means the first player loses, whoever moves first loses,

Â and the second player wins all the time. To say a game is positive means left wins

Â always, going first or going second. And here's a new one for you that doesn't

Â come up in numbers. A game is fuzzy with zero, that's how

Â that's pronounced. If fuzzy, F U Z, Z Y.

Â It's fuzzy with zero if the first player wins.

Â So let's look at what these mean. And how they're used.

Â First of all, what, the way they're typically used is not comparing again with

Â 0. But comparing one gain to another.

Â So, G is less than H.. Actually you just convert and, you know,

Â subtract G from both sides, and it's, this should be the same thing as saying that 0

Â is less than H minus G. That's what it is, the definition is

Â mathematically. What it means in terms of game play is

Â that H is better for left, even if it's part of a bigger gain.

Â And we'll have some examples as we, as the week progresses.

Â So, H is better for left than G, even if, they're both part of a much larger game.

Â If you're playing five games at once, and one of the pieces is G, replace it by H,

Â and the whole thing is better for left. Now you could also of course read this

Â backwards. This says G is better for right than H is.

Â Because to say that once something is less than something is the same thing as saying

Â that is bigger than the first thing. The interesting case perhaps is, is equal

Â to zero G is equal to H, and this is our definition, means H minus G is zero.

Â And what this turns out to mean is that G and H have the same outcome, the same

Â player wins, in best play, even if G and H are both part of some bigger game.

Â So you have a go game and towards the end you have this little piece in the corner

Â that's equal to G. Well replace it by that little piece in

Â the one corner by H and nothing is changed in terms of the outcome.

Â Now in real numbers you can say not only that one number is less than another

Â number. But you can also say one is less or equal

Â to two, or one is less or equal to one. And so you can combine these inequalities

Â and for instance the following way. G is less than or equal to H means just

Â like it would for real numbers, G is less than H or G is equal to H.

Â So let's see what this says. It says, to say that G is less than H

Â means that H is better for left than G is, and to say that G is equal to H means that

Â they're the same for left and right. So you put these together and less than or

Â equal to means H is at least as good, perhaps not better, but at least as good

Â for left as G. In other words, it says left with H minus

Â G going second. Okay, so that's a lot of stuff.

Â A bunch of definitions here. If you forget the definitions go back,

Â look them over. They're all right at the beginning of this

Â module. And we'll continue on in the next module

Â with some definitions with some examples.

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