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

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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.

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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

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

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

Alright. Mean values one thing we look at a little

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

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

And in these games, people want to play it.

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

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

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

numbers. this can be generalized.

There's actually a theorem behind all this.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

game? And so this is our Japanese term sente

gote. Sente means to have Sente means you play

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

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

your, your opponent can play in some other game.

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

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

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

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

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

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

Gote. Okay, next time.

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