In the last video, we discussed the prisoner's dilemma, in which both players have a dominant strategy of not cooperating, leading to an equilibrium in which both players are worse off than they would have been if they'd cooperated. Many games, though, don't have a dominant strategy. It's pretty rarely the case that your best response is the same no matter what the other person does.
Remember, that's the definition of a dominant strategy. So let's take a look at another simple simultaneous move game. Brianna and Carlos are study partners.
Here's Brianna. And here's Carlos. They work together, sorry, they work on their own during the week, and then they work together on the weekends. During the week, they can decide whether to study or slack off. These are their two strategies.
So, Carlos can study, or he can slack off, and Brianna can study, or she can slack off. Those are their two strategies. So let's draw a payoff matrix and take a look at their payoffs. If they both study, their payoff is four for each of them.
Four utility units, let's say, or four dollars worth of value. So Carlos gets four and Brianna gets four if they both study. Now if Brianna studies... and Carlos slacks off, her payoff is 3, and his payoff is 5. Why?
Because he got the benefit of goofing off all week, and then learning all the material from Brianna. Remember, payoffs include everything that goes into that payoff. And meanwhile, she had to put the work in, and Carlos can't help her out with the stuff she's having trouble with.
And it's the same story if Brianna slacks off and Carlos studies. Her payoff is 5. and his payoff is 3. Now if they both slack off, they each get a payoff of negative 2 because they're going to fail the exam. So if Carlos slacks off and Brianna slacks off, they each get negative 2. The exact numbers aren't really important, just their relation to each other. So what should Brianna do?
She's going to put herself in Carlos'shoes and consider her best response to each one of his strategies. So if Carlos studies, Carlos studies right here, her payoff is 5 for slacking off and 4 for working. If Carlos slacks off though, she's better off studying. Her payoff from studying is 3, but her payoff from slacking off is negative 2. Now notice she doesn't have a dominant strategy in this game.
Her best response depends on what Carlos does. This is more complicated than the prisoner's dilemma. where there was a dominant strategy of doing the same thing no matter what the other person did. You don't always have a dominant strategy that works no matter what the other person does.
So what's going to end up happening in this game? What's the equilibrium? The solution is to look for a combination of strategies by each player in which no one can change their choice and be better off.
Strategies that are your best response to the other person's best response. Now I know that sounds complicated, but it's actually really straightforward and in fact, you've done it thousands of times already. So let's follow the logic for Brianna. She needs to ask herself what she should do if Carlos slacks off. She should study.
That's her best response. So, if she's thinking about slacking off, she should definitely not slack off if Carlos is slacking off. I'm going to draw this arrow saying that if she's thinking about slacking off and Carlos is slacking off, She's better off moving up here to studying. Now over here, does Carlos have an incentive to change his behavior?
No, he doesn't because his payoff is 5 and 5 is greater than 4. So if Brianna has decided to study, Carlos decides to slack off, neither one of them has an incentive to change their behavior. Brianna's best response to Carlos slacking is to study and Carlos's best response to Brianna is to slack off. So if Brianna's studying and Carlos is thinking about studying or slacking off, he'll see he's better off slacking off. So he'll switch over here.
So you'll notice both of these arrows are pointing over here. Neither of them have an incentive to make to change their behavior. Now let's make this more concrete.
If Carlos is slacking and Brianna is slacking, she should decide to study instead. I've drawn the arrow to reflect that. If Brianna is studying and Carlos is studying, Carlos should decide to slack off instead.
because his payoff is higher. This is the other arrow over here. Both of the arrows are pointing to study for Brianna, slack off for Carlos, and neither one of them should change.
But what should Brianna do if Carlos is studying? What if we start off here again? And she's thinking, Carlos is studying, what should I do? She should slack off instead of studying because her payoff in that situation is higher.
So I'm going to draw the arrow pointing from here down to here. If she starts off over here, Brianna's thinking, we're both studying, I'm better off slacking off. And we go into the situation where Brianna's slacking and Carlos is studying. Now let's go back to here where they're both slacking off. Carlos is better off by switching to studying in this situation because he gets a payoff of 3 instead of negative 2. So the arrows point in this direction.
So we have, sorry, again. Neither Brianna nor Carlos should change from here. This Brianna's best response to Carlos's best response. So we can end up here or here. This is called a Nash equilibrium.
It's named after the Nobel laureate John Nash about whom the movie A Beautiful Mind was made. The concept is straightforward. A Nash Equilibrium is a set of strategies in which no player has an incentive to change.
It's the best response to the other person's best response. In the Prisoner's Dilemma example from the last video, there was one Nash Equilibrium. Both Steph and Tanya watched video games, and neither of them had an incentive to change their strategy from that point.
In this game, there are two Nash Equilibria. Now that might seem a bit odd, but it's pretty logical. Both you and your study partner... want the other person to do the work.
Now who's gonna end up doing the work and who's gonna end up slacking off? Is it gonna be Carlos studying and Brianna slacking? Or Brianna studying and Carlos slacking?
That depends more on real-world situations. Maybe Carlos is really intimidated by Brianna so he ends up studying while she slacks off. But these two right here are the Nash equilibria.
Now there are games with no Nash equilibria. in which someone always has an incentive to change their behavior in response to someone else's choice and these arrows just keep moving you around and around the payoff matrix. The best response in those kinds of games are actually only a little bit more complicated to calculate than the simple arrows that we drew for our simple games.
In more advanced game theory classes you can learn how to work them out.