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

99 ratings

Georgia Institute of Technology

99 ratings

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

From the lesson

Week 7: What You Can Do From Here

The topic for this seventh and final week is Where to go from here.

- Dr. Tom MorleyProfessor

School of Mathematics

Morning, afternoon, evening. Games Without Chance, and week seven, and

Â we want to talk about mean values, hot, cold, sente, gote.

Â So and if you have a Japanese dictionary, you can look up the last two terms.

Â Alright. Mean values one thing we look at a little

Â while ago at least in some of the extra problems were these games, for instance,

Â like uh,[SOUND], these, up, and these are games called check games, cashing checks.

Â And in these games, people want to play it.

Â By, when you played sums of these games, people grabbed the games.

Â If this is the game g, and you play g plus g plus g, if you played sums of

Â these, they behave just like numbers, or very much like numbers, or close to

Â numbers. this can be generalized.

Â There's actually a theorem behind all this.

Â which is called the mean value theorem. although not the same one as in calculus.

Â And so given again there's a number M of G called the mean value of G.

Â Such that oh, when a number K independent of N.

Â Such that when you add up Gn times that this minus n times the mean value if this

Â is whether a bound which doesn't depend on n over multiple of the mean value.

Â So, so large numbers of copies of G played together behave approximately.

Â Like a number and as n gets large k is, doesn't depend on n so k is small

Â relative to n, so as n gets large this approximation gets better and better at

Â least relatively, this is called the mean value term and, analysis of this together

Â with, with an actual construction of, of, of the calculation of this mean value

Â called the thermograph allows us of, of to have a number of strategies that.

Â Are not optimum but are actually computable and in some sense, within a

Â bound of being optimum, that is they're not necessarily optimum strategy, but

Â they're guaranteed to be not too bad. And these are often based on, you have a

Â sum of a large number of games, and then the real question in terms of playing a

Â game decently is which one to play in. So you have a sum of all these games.

Â Do we play G2 first, or GN, or G1 or whatever?

Â And if, our opponent plays in G2, should we respond in G2 or go to a different

Â game? And so this is our Japanese term sente

Â gote. Sente means to have Sente means you play

Â a move and you force your opponent. You have the initiative.

Â The opponent has to respond to your move, Go Tay is the opposite where you play and

Â your, your opponent can play in some other game.

Â So, in Sente, you play for instance in G2, your opponent is asked to respond

Â there immediately. in Gotay may be you play G2 no problem I

Â play in G5 over here. So there is whole class of strategies and

Â approximate strategies and interesting Constructions, having to do with the mean

Â value and various ways of finding it, which give rise to approximate strategies

Â for playing game and analyst, and analysis of whether you have Sente or

Â Gote. Okay, next time.

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