Now, I will show you some interesting cases of games.
Let's look at this game, Anna versus Bill.
What do you think about this game?
So, if you do if you look at the necessary under linings,
you will see that this is a very similar game to
decide what game that I showed you before.
We have to Nash equilibrium.
However, try to take the role of one of these two players.
And think what would you play if you were Anna or Bill.
Now, if I was Bill, I would think in the following way.
I would say that OK if I play up,
I can get either zero or one dollar.
But if I played down,
I will get to either zero or one million dollars.
You know what? Forget about up.
I think I will play down and this is because I value one million dollars, it's much,
much more important for me than getting one dollar. All right.
So, this kind of equilibrium we call it a focal point.
We do have two totally legitimate equilibria.
Down and left which gives one million to each,
and then up and right which gives one dollar to each.
However, I bet you that if you play this game 100 times with
100 people that they seriously think
about the game and they just don't want to kid around,
all of them will select down if they were Bill,
and left if they were Anna.
This is a focal point.
It's one of the two equilibria that we focus on this equilibrium
because it's meaning it's much more important to us.
Some games they do not have equilibrium at all.
So, we look at these game for example,
we do the underlining process that I showed you before,
and we see that now there is no single cell that no one wants to deviate from the cell.
So, what is the outcome of this game?
What is equilibrium?
Actually there is an equilibrium but this equilibrium is not in pure strategies.
A pure strategy for player two, let's say,
would be either to play C or to play D. However, when you play,
for example, rock-scissors-paper, you do not say,
"I will always be playing rock."
You never say that. You say,
"I think I should play one-third of the times rock,
and one-third of the time scissor,
and one-third of the times paper."
This would be the best way to go since this is a random game.
But probably the same would be true here.
So, we can select an appropriate mixture of strategies and that mixture will depend on
the payoffs that will maximize our expected payoff in the end of the game.
How do we do that? By assigning a probability to each known dominated action.
If you have a dominated action,
you throw it out of the game and you don't bother with it anymore.
You take all your non-dominated actions here, C and D,
and A and B are non-dominated,
and then you assign a probability to each one of them.
How did you assign the probability?
In this game, you would assign a 50/50 percent probability of playing C or D,
or A or B according to which player you are because the payoffs are symmetric.
If the payoffs are not symmetric so, for example,
you had 0,2 1,0 1,0 0,1,
then in this case,
you would want to change the probabilities appropriately
and for how to derive this precisely you can go to
the textbook at page 225 and you will see how Church and Ware shows you
another nice way in order to calculate the exact mixed strategies equilibria.
What if you have no equilibrium at all?
No pure strategies equilibrium,
no mixed strategies equilibrium.
Well, you're in luck because John Nash proved
the theory in which says that for any game with a finite number of players,
which each player has a finite number of set of strategies,
then there exists at least one Nash Equilibrium either in pure or in mix strategies.
You can never have a zero Nash equilibria in a game.
You can always mix in a specific way and you will get
the situation a mixture from which no one wants to deviate from.
Let me show you now a slightly different case,
the case with incompleteness,
a case in which you do not know the full matrix of payoff. Consider this.
You're player one, you're the red person and you play player two.
Player two can play left and middle, you know that.
And you can play up and center and down.
And you know according to what player two plays,
you know how each of your strategies is going to reward you.
But you don't know the payoffs of your opponent.
How are you going to play this game?
Well first of all, if you have no information for player two,
you might think on the following way which I
warn you is not correct but you may think like that.
You say listen if I play down for sure no matter what I get one.
If I play up or center.
I risk. I may end up with three or I may end up with zero and
which one of the two will come up depends on
the other player which I don't know how the other player will play.
So, what do I want a random outcome between three or zero,
or a sure outcome of one?
Lots of people will say that this depends on your risk attitude,
how you see risk.
But we said from the very beginning
that risk attitude is not relevant in game by matrices.
Why? Because this attitude is already embodied in this payoffs.
So, we're looking to maximize the expected payoff.
So, how should you handle the incomplete game?
You should either pick down or
your alternative would be to pick a mix of 50/50 between up and center.
If you play down, you will get an expected payoff of one.
You will get for sure one actually and if
you play a mixture of 50/50 between up and center,
you will get three plus zero over two.
This would give you one and a half.
Since you have no risk attitude here,
you behave as if you were risk averse.
You prefer a mixture of up and center by 50/50 rather than playing down.
So, all we have here a very interesting case where
a mixed strategy dominates strictly a pure strategy.
So, a mixture of 50/50 between up and center dominates strictly the pure strategy down.
After this, I will show you the most famous example in game theory,
one that is extremely interesting,
and we're going to use in almost every lecture from now on. Stay with us.
Kosmas MarinakisAssistant Professor
