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

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

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

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Alright. 
Mean values one thing we look at a little 

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

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

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And in these games, people want to play 
it. 

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

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

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

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numbers. 
this can be generalized. 

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There's actually a theorem behind all 
this. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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game? 
And so this is our Japanese term sente 

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gote. 
Sente means to have Sente means you play 

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

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

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your, your opponent can play in some 
other game. 

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

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

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

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

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

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

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Gote. 
Okay, next time.