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.