so we're going to restrict the set of games here.
To all have in common, the same set of agents.
And also the same action sets so these games are only going to differ in the
utility functions. Then P is going to be an element of the
set of all possible probability distributions over games.
This is going to be our prior distribution.
So this is going to tell us how likely each of these games from the set G is.
And finally, we're going to have a set of partitions of G, one for each agent.
So this is going to be a set of equivalence classes that will say from
the point of view of an agent, certain games are indistinguishable from each
other and others are not. Let's look at an example.
So this example is a little bit contrived.
Ordinarily we're going to use Beijing games to model something that, that
kind of makes some sort of sense in the world.
And here really we're, we're looking at something kind of artificial.
But it's a small example that still lets us think about everything important about
a Beijing game. So, the first thing to notice is that
there are four possible games that might be played.
Matching Pennies, Prisoner's Dilemma, Coordination or Battle of the Sexes.
Incidentally, Prisoner's Dilemma here is being played with different payoffs then
you might have seen before, but that doesn't matter, it's, it's strategically
the same game. And we have a common prior over the
games, so there's a 30% chance that the game that the players will in fact be
playing is matching pennies. There's a ten% chance that they will in
fact be playing presidential dilemma, twenty% chance of coordination, and a 40%
chance of battle of the sexes. Now haven't marked the actions in these
games but our assumption is that they have to be the same.
So let's say player one has two choices top or bottom.
He gets to choose the top or bottom action and it's the same in every game.
And likewise, player two can choose left or right, and he gets the left or right
action in each game. So.
What is interesting here of course is that we have this information sets.
So player one gets to find out. Which of these two sets the game is in.
What that means is. In fact, nature is going to decide which
game gets played. So, randomly it's going to be decided
which of the four games being played, according to the common prior.
Let's say the most likely thing happens, and the players end up playing Battle Of
The Sexes. In that case, what player one is going to
find out is that he is in this equivalence class than this one.
So, that means, he is going to know for sure that he is not playing Imagine
Pennies or Prisoner's dilemma, but he's going to think that he might be playing
either coordination or battle of the sexes.
He's going to have no way of turning them apart.
Now player two has different equivalence classes.
So player two. Considers these two games to be
indistinguishable, and likewise considers these two games to be indistinguishable.
And, continuing our example from before. If this was really the game that was
randomly chosen by nature, then player two would find out that he was in this
equivalence class, rather than this one. Meaning, that he would think the game
being played was either Prisoner's Dilemma or Battle of the Sexes.
And, the ground, the ground truth would be in fact Battle of the Sexes was being
played. He would consider it possible that
Prisoner's Dilemma was being played. And we've already seen that player one
would consider it possible The Coordination was being played.
And what this means is that when the players are deciding what, what action to
take, they're going to have to play an action without fully knowing what game is
going to be played. And they're going to have to reason about
what their opponent is doing without fully knowing what the opponent is going
to think. They will, they do know everything about
the setup. So this whole kind of picture is
something that the players know. They know the common prior.
They know their own equivalence classes and they know their opponent's
equivalence classes. So if I'm player one and I want to reason
about what player two is going to do and I know.
That I'm in this equivalence class. That I also know that player two, that if
the game is really coordination, which I believe is possible, then player two
thinks he's in this equivalence class. And thinks that matching pennies is
possible, even though I know it's not possible.
Or, on the other hand, if Battle of the Sexes is the real game that's being
played, then I know that player two thinks he's in this equivalence class,
which means he's going to think prisoner's dilemma's possible although I
know it's not possible. And I'll, I'll leave for a future video
actually how we reason about these games. But what we've learned here is how to
define a bastion game, by writing it as a probability distribution, a common
probability distribution over multiple different normal form games.
All of which share the same number of players and the same action sets.
