# 1-4 Examples of Games

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In the celebrated example of Prisoner's Dilemma, we have this general scheme where
the two prisoners can either cooperate or not.
Defect, as it's called. If they both cooperate, they get some payoff A.
If they both defect, they get a different payoff D, where A is greater than D.
However, if they miscoordinate and one of them cooperates and the other defects,
then the cooperator gets the lowest possible payoff and the defector gets the largest possible payoff.
And that's true symmetrically here as well.
And this very well known example has a rather counterintuitive paradoxical properties.
Most games are not as conceptually confusing.
Here's an example that's conceptually very clear and these are games of pure competition.
The situation here is limited to two players, where one player's payoff is exactly the compliment of another player's payoff.
So, they always add to some constant c. Often that constant we use is zero. And we call it, for that reason,
zero sum games, as opposed to constant sum games.
And since they do sum to zero or to a constant, we only need to remember one number,
the pair to one of these and we can infer the payoff to the other player from that.
Here's the most simple version of it. This is a game of matching pennies.
So you and I each need to pick heads or tail for the coin. If we pick the same side, either heads or tails, I win.
Which means that I get a payoff of one and you have minus one.
If we miscoordinate, and so I pick heads and you tails or the other way around,
then you win. A very straight forward game of pure competition.
Here's another very well known similar game with three actions from both of us
and that's the game of rock, papers, and scissors, also known as rochambo.
So, if we pick the same action then it's a draw. And otherwise there are rules for who wins.
For example, if I pick rock and you paper then you win. I picked rock and you scissors then I win. And so on.
Again, the payoffs in both cases sum to zero.
This parenthetically, this very simple children game actually has an annual competition
that carries a nontrivial prize of $10,000 and it's actually a sobering thought that when we look at this trivial game
then, perhaps, chuckle a little bit, if we actually participated in this competition we'd actually think hard about how to play it.
Here's the other extreme, of games of pure coordination or pure cooperation.
In this case, all agents have exactly the same interest. In other words, the payoffs for every action vector that they take
is the same. And so the utility for player 'i' is always the same as the utility for player 'j'
for every action vector that they choose.
And so again we here, too, will need to write each cell of matrix only one number because
it's common to all the players. It's drives home that perhaps the unfortunate term "noncooperate" game theory
that describes this dominant strand of game theory that we are discussing for now
It's, the name was suggested these are games for, that descibe situations that are inherently conflictual
but as we see they apply also to games in which the interests of the players coincide.
So here's a game that describes the purely cooperative situation.
You and I walk towards each other on the sidewalk. We can each decide whether to go to our respective left
or respective right. And if we pick the same side then all is good. We avoid a collision.
If we don't, then we do collide and that's equally bad for both of us.
Of course in general, games will be neither purelly cooperative nor purely conflictual and here's a game
that exemplifies that. This is a game that's called "Battle of the Sexes."
So imagine a husband and a wife who want to go out to a movie.
There are two movies that they can choose from. Let's say "Battle of Armageddon" and "Flower Child."
The one, a violent war movie, and the other is a romantic comedy.
Above all they want to go together to the movie. If they go to different movies then they are equally unhappy.
So they want to go to the same movie but they have conflicting preferences.
The wife clearly would prefer to go to "Battle of Armageddon" and the husband, as romantic as he is, would like to go to "Flower Child."
So, both cooperation and competition in this game.