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.