Yale University OCW-Game theory (Note of lesson two)

The followings are about the Yale University OCW(耶鲁大学开放课程)- Game theory, lesson two, mainly about the necessary of putting yourselves into other people’s shoes when you play a game.

Firstly, what are the ingredients of a game?

It includes players, strategies set, and payoffs.

If we develop some notation, so the standard notation for players we can use things like little i and little j, use little “si” to be a particular strategy of player i, and use capital “Si” to be the set of alternatives. Besides, we also could use “U” for utile, to be player i”s payoff. So “Ui” will depend on player i’s choice. So do the “Uj” and all the way up to player N’s choices. And the “Ui(s)” depends on the

profile.

 

Take the last game the doctor left us to play for example; we call such a particular play of the game as “a

strategy profile/vector/list”. And we could know that as follows.

Just get “0”.

But before we analyze a game, we reasonably assume that everybody knows the possible strategies and everyone else could choose and everyone knows everyone else’s payoffs.

By the way, “S-I” means a strategy choice for everybody except person “I”.

And this is a definition of dominated strategy:

Play I’s strategy “s’i” is strictly dominated by player i’s strategy “si”, if “Ui” from choosing “si”, when other people choose “s-i” is strictly bigger than UI(s’i) when other people choose “s-i”. And the key part of the definition is for all “s-i”.

Secondly, this is a game to show somethings.

there is a game, in which an invader is thinking about invading a country and there are two ways through which he can lead his army, and you are the defender of this country and you have to decide which of these passes into the country you are going to choose to defend. And the catch is, you can only defend one of these two routes.

The key here is going to be that there are two passes. One of these passes is a hard pass. And the other one is an easy pass. If the invader chooses the hard pass, he will lose one battalion of his army. If he meets your army, which ever pass he chooses, then he’ll lose another battalion.

So from that above, if we take the battalions the invader loses as payoffs for each other, we could make such a payoffs form as follows.

	Easy pass	Hard pass
Easy pass	1,1	1,1
Hard pass	0,2	2.0

From the case we know there’s a weak notion of domination here for the invader.

And this is a definition for it: player i’s strategy “s’i” is weaking dominated by his strategy “si” if player i”s payfoo from choosing “si” against “s-i” is always as big as or equal to her payoff from choosing “s’i” against “s-i”. And this has to be true for all things that anyone else could do.

So in order to mix our payoffs, we should never choose a weakly dominated strategy.

Besides, we should put ourselves in other people’s shoes and figure our that they’re not going to play strongly or strictly or weakly dominated strategies.

Thirdly, let’s look back at the game the doctor left us last lesson.

From the Yale students’ strategy profile, we know most of them are not random machine. So the average number will not be 50, so the answer will not be 33 or another numbers around.

And as it’s almost impossibly that all of students choose the umber 100, so the answer will be below 67. And then those number larger than 67 will be certainly dominated. And if we go on thinking in this way, once we delete 68 through 100, we know that the strategies that are less than 67 but bigger than 45 are not weakly dominated in the original game, but they are dominated they’re weakly dominated.

…….

And keep our minds in shoes argument, which is an in “shoes in shoes” argument, at which point you might want to invent the sock. And then we will get the number 1, which seems the most likely right answer. But it is also almost impossible. Because the answer asks everyone who play need to be rational himself and he need to know that others are rational too. That makes sense and sounds like philosophy.

The technical expression of that in philosophy is common knowledge.

So it asks you to put yourself into other people’s shoes and think about how sophisticated are they at playing games and you need to think about how sophisticated do they think you are at playing games and you need to think about how sophisticated do they think that you think that…….

That is an infinite sequence.

At last, the doctor pointed out the differences between the common knowledge and mutual knowledge.

Common knowledge is a subtle thing, a statement about not just what I know the other person knows that I know that the other person know my thoughts like that.

And mutual knowledge doesn’t imply common knowledge.

It sounds complicated, right? But it is interesting.

by cloudTed(观尔腾)

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