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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
as it is in perfect information games.
https://karpathy.ai/lexicap/0012-large.html#00:16:26.240
Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And I don't wanna say it's simple there either.
https://karpathy.ai/lexicap/0012-large.html#00:16:29.320
Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
It's of course very complicated computationally there too,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
but at least conceptually, it's very straightforward.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
There's a state, there's an evaluation function.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
You can try to learn it.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Here, you have to do something more.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And what we do is in one of these papers,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
we're looking at where we allow the opponent
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
to actually take different strategies
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
at the leaf of the search tree, if you will.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And that is a different way of doing it.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And it doesn't assume therefore a particular way
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
that the opponent plays,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
but it allows the opponent to choose
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
from a set of different continuation strategies.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And that forces us to not be too optimistic
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
in a look ahead search.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And that's one way you can do sound look ahead search
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
in imperfect information games,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
which is very difficult.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And you were asking about DeepStack.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
What they did, it was very different than what we do,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
either in Libratus or in this new work.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
They were randomly generating various situations
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
in the game.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Then they were doing the look ahead
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
from there to the end of the game,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
as if that was the start of a different game.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And then they were using deep learning
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
to learn those values of those states,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
but the states were not just the physical states.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
They include belief distributions.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
When you talk about look ahead for DeepStack
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
or with Libratus, does it mean,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
considering every possibility that the game can evolve,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
are we talking about extremely,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
sort of this exponentially growth of a tree?
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Yes, so we're talking about exactly that.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Much like you do in alpha beta search
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
or Monte Carlo tree search, but with different techniques.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So there's a different search algorithm.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And then we have to deal with the leaves differently.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So if you think about what Libratus did,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
we didn't have to worry about this
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
because we only did it at the end of the game.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So we would always terminate into a real situation
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
and we would know what the payout is.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
It didn't do these depth limited lookaheads,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
but now in this new paper, which is called depth limited,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
I think it's called depth limited search
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
for imperfect information games,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
we can actually do sound depth limited lookahead.
https://karpathy.ai/lexicap/0012-large.html#00:18:43.880
Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So we can actually start to do the look ahead
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
from the beginning of the game on,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
because that's too complicated to do
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
for this whole long game.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So in Libratus, we were just doing it for the end.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So, and then the other side, this belief distribution,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
so is it explicitly modeled what kind of beliefs
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
that the opponent might have?
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Yeah, it is explicitly modeled, but it's not assumed.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
The beliefs are actually output, not input.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Of course, the starting beliefs are input,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
but they just fall from the rules of the game
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
because we know that the dealer deals uniformly
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
from the deck, so I know that every pair of cards
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
that you might have is equally likely.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
I know that for a fact, that just follows
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
from the rules of the game.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Of course, except the two cards that I have,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
I know you don't have those.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Yeah.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
You have to take that into account.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
That's called card removal and that's very important.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Is the dealing always coming from a single deck
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
in Heads Up, so you can assume.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Single deck, so you know that if I have the ace of spades,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
I know you don't have an ace of spades.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Great, so in the beginning, your belief is basically
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
the fact that it's a fair dealing of hands,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
but how do you start to adjust that belief?
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Well, that's where this beauty of game theory comes.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So Nash equilibrium, which John Nash introduced in 1950,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
introduces what rational play is
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
when you have more than one player.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
And these are pairs of strategies
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
where strategies are contingency plans,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
one for each player.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So that neither player wants to deviate
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
to a different strategy,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
given that the other doesn't deviate.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
But as a side effect, you get the beliefs from base roll.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So Nash equilibrium really isn't just deriving
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
in these imperfect information games,
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
Nash equilibrium, it doesn't just define strategies.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
It also defines beliefs for both of us
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
and defines beliefs for each state.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
So at each state, it's called information sets.
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Tuomas Sandholm: Poker and Game Theory | Lex Fridman Podcast #12
At each information set in the game,
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