Popularized by movies such as "A Beautiful Mind," game theory is the mathematical modeling of strategic interaction among rational (and irrational) agents. Beyond what we call `games' in common language, such as chess, poker, soccer, etc., it includes the modeling of conflict among nations, political campaigns, competition among firms, and trading behavior in markets such as the NYSE. How could you begin to model keyword auctions, and peer to peer file-sharing networks, without accounting for the incentives of the people using them? The course will provide the basics: representing games and strategies, the extensive form (which computer scientists call game trees), Bayesian games (modeling things like auctions), repeated and stochastic games, and more. We'll include a variety of examples including classic games and a few applications. You can find a full syllabus and description of the course here: http://web.stanford.edu/~jacksonm/GTOC-Syllabus.html There is also an advanced follow-up course to this one, for people already familiar with game theory: https://www.coursera.org/learn/gametheory2/ You can find an introductory video here: http://web.stanford.edu/~jacksonm/Intro_Networks.mp4
2-5 Example: Mixed Strategy Nash
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Skills You'll Learn
Game Theory, Backward Induction, Bayesian Game, Problem Solving
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DA
Apr 14, 2019
It was such a helpful course that gave me the opportunity to learn few basic methods and terms about game theory through lots of interesting and to the point examples by three unique professors
LY
Dec 30, 2017
Easily the most challenging introductory course I've taken, but definitely worth it. I must say though that I learnt more from failing the quizzes than the lectures or practice questions.
From the lesson
Week 2: Mixed-Strategy Nash Equilibrium
pure and mixed strategy Nash equilibria
Taught By
Matthew O. Jackson
Professor
Kevin Leyton-Brown
Professor
Yoav Shoham
Professor
Hi folks. This is Matt Jackson again and so now let's take a look at Mixed
Strategy Nash Equilibrium in practice and try and understand a little bit about
what it should tell us about what we should expect to see.
So let's start with situation of soccer penalty kick.
Kicks, and this is a quite natural application of mixed strategy equilibria
because they're ubiquitous in sports and competitive games, so situations where it
actually pays to be unpredictable. So, by not knowing what the other, the
opposition is going to do. It makes it a little more difficult to
for you to pick it up the most strategy in these games where one player wins and
the other player looses. And, in particular in soccer penalty
kicks, were, looking at a situation where a kicker has to try and kick the ball
into a goal, the goalie can try to move to deflect the ball, and this happens
very quickly, so it's essentially a simultaneous move game.
The kicker is choosing to go either in, in their simplified version say to to the
right or the left. The goalie is then going to dive to the,
to one side or the other side and try to deflect the ball and if the goalie,
guesses correctly and ends up in the same.
Direction as the, the kicker, then they have a high chance of, higher chance of
saving it. If they go in the opposite direction,
they have a lower chance of saving it. Okay, so, how, how are equilibria going
to adjust to the skills of the players? So, let's suppose, for instance, as a
kicker That I might be biased. I might be able to kick the ball more
accurately in 1 direction than the other. So if you ask me to kick it towards the
left side of the goal, it might be that I hit there very accurately.
If you tell me I have to kick it towards the right side of the goal, it could be
that I'm less accurate. And I have a higher chance of just
missing the goal altogether. So is the equilibrium going to change
when we change one of the players, in terms of their skills.
So let's have a, a, a peek at this. Should a kicker who kicks penalty kicks,
worse to the right than the left, kick more often to the left than the right?
So, if I, if I'm worse in kicking towards the right, does that mean that I should
kick in the opposite direction more often? Well that's have a peak.
So, let's start with just a simple version just to get, our ideas fixed, so
imagine that the, setting is one where the kicker and the goalie, if they, so
let's have the kicker on this side. so the kicker is the role player, the
goalie is the column player and if they end up kicking, if the kicker goes left
and the goalie also happens to go to the left, then the goalie saves.
And the goalie gets a payoff of 1. The kicker gets a payoff of zero, if
instead we're in a situation where say the kicker goes left and the goalie goes
right, then the kicker scores and gets a path of 1, and, and so forth.
Okay. So this is just a simple variation on
matching pennies, and in this situation, what's the equilibrium going to be.
The equilibrium's going to be quite simple.
It's just going to be that the kicker randomizes equally between left and
right. The goal randomizes between left and
right. Each person has a probability of half to
win from kicking to the left or right, Goalie to the left or right.
So it's a very simple game, and we're have a good idea of how to solve that
one. Okay, so now what're we going to do?
Let's change things and now we've got a kicker that sometimes misses when they go
to the right. So imparticular if the goalie happens to
go to the left. And the goals wide open to the right.
The kicker scores 75% of the time, but actually misses completely 25% of the
time. Okay, so this is the, a, a kicker who's
still . does well if, if they go left and the
goalie goes to the opposite direction. But now they have a lower probability of
winning when their kicking right and they have a wide open goal.
Okay, so how should this adjust, whats, what Should the new equilibrium look
like? So let's suppose let's first of all try and keep the kicker indifferent.
So let's think of the goalie going left with probability P, right with
probability 1 - P. For the kicker to be indifferent what has
to be true? Well what's their payoff if they go left? Their payoff to going left.
Left is just 1 x 1 - P. There kick-off, there payoff to going
right is .75 x P. These 2 things have to be equal, in order
to have this, thee kicker being different.
So what do we end up with? We end up with .75P is equal to 1 - P.
so we end up with, 1 = 1.75p. or p is = to 1 over 1.75.
Which is = to, 4 over 7. Okay? So that tells us that the.
Goalie should be going left with probability 4/7, and right with
probability 3/7. Okay? So we know what the goalie's
supposed to be doing, so, so now the fact that we changed, the goalie's payoffs
haven't changed, but the fact that we changed the kicker's payoffs, meant that
the goalie actually had to adjust. Right? So even though the goalie's
payoffs haven't changed at all in this game, the new equilibrium has a different
set of, of probabilities for the goalie in order to keep the kicker at different
now. Okay? So now let's, let's see what the
kicker's going to do. So how are we see what's so, so let's
suppose that the kicker goes left with probability q, right with probability 1 -
q, and let's solve for q. Well, for the goalie to be indifferent,
what is their payoff if they go left? If they're going left, they're getting and
they're getting a q probability that they match.
So they get q + 0.25(1 - q) if they go left.
If they go right instead what are they getting? They're just getting 1 minus q.
Right? So these 2 things have to be equal so we end up with, q = .75 * (1 - q).
So q / (1 - q) = .75. What does this tell us about q? It tells
us that q = 3/7. Okay, so what's going to happen, when we
work out this? We get 3/7 for the probability that the kicker's going to go
left. And 4/7 for the probability that the
kicker is going to go right. So overall what do we have now? We have
the strategies looking like this as we made this adjustment.
And we notice two sort of interesting things about this.
one is that the goalie's pay-offs didn't change, but they still had to adjust
their strategies. And the second is that the kicker is
actually kicking more often to the weaker side, right? So the, the, the right foot
got worse than it was before and they are actually going in that direction more
often. and why is that? It's because the kicker,
the, the goalie has also made an adjustment in this game.
And so, the, the comparative statics in mixed strategy Nash equilibria are
actually quite subtle, and somewhat counter intuitive in terms of what you
might expect you're, you're, you get a bias so that this becomes a weaker
direction, and the equilibrium adjusts. So that the player goes in that direction
more often. So, let's have a, a look just through the
intuition here. Again the goalie strategy must have the
goalie indifferent and so when we went through those payoffs, the kicker, the
goalie goes left more often than right, and the kicker, actually, so sorry
there's a type here. The kicker, actually goes right More
frequently, right, goes right with probability now 4/7, so they've increased
their probability on that. And when we end up, what we see is the
goalie's strategy is adjusting, but we also see that the kicker adjusts to
kicking more toward their weak side. so the, the goalie now actually has a
slight advantage. So if you go through and calculate the
probability that the goalie's going to win, they're going to win 4/7, so the
time in this, in this match. And, and we can think what would happen
if the goalie Actually just stayed with our old strategy of still going 50 50.
Then the kicker could always go left, and win 1/2 the time instead of 3/7.
So the, the fact that the goalie has to make an adjustment is because they have
to de, defend more to the left side to defer because now the, the, the kickers
has a, a better Chance of winning on that side.
So the goalie goes more in that direction.
That pushes the kicker towards their weaker side.
In order to make sure that the goalie is willing to go to the left side with
higher frequency. So, by adjusting to strategy to keep the
kicker indifferent the goalie takes advantage of the kicker's weak right kick
and wins more often. Often.
Okay, so just in terms of summary and, and mixed strategy in soccer penalty
kicks, in general. Players must be indifferent between the
things that they're randomizing over. that produces very interesting and subtle
Comparative statics. and you know there, there's a question
that might come up in your mind, do people really do this? I mean, this is
fairly complicated, right? So the, you know, 50-50 we can figure out one we get
these, to these games where a player has an advantage The advantage 1 way or
another, then the actual mixture becomes fairly complicated.
And it's not so obvious that players will actually do that.
so we'll, we'll take a look at that and see if, if this actually bears out in, in
practice.
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