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“I m william spaniel. Let s learn some game theory. Today s topic is the the stag hunt and pure strategy. Nash equilibrium.

And i cover this topic in lesson of game theory. 101. The complete textbook check. The video description for more information about that.

Here s a situation. We have two hunters going out to catch meat. There are two hares in the range and one stag. The hunters can only bring the equipment necessary to catch one type of animal and they have to choose this equipment without seeing what the other one is going to choose they can t really coordinated erecting.

The stag is going to be worth a lot more meat than the harris combine combined in fact we re going to get worth six units of meat while each of the individual hairs is only worth one unit of meat. But the stag is really tricky to catch and the hunters need to actually both be trying to catch the stag in order to essentially trap. It and kill it so in order to get the stag both of them have to choose the stag hunting equipment in order to be able to get it in contrast hares are really easy to catch. And so if you re a hare hunter.

You can capture all of the prey without the help of the other guy so if we condense this information into a payoff matrix here s what it looks like we have two players player 1 player. 2. Each of them can choose to hunt a stag or hunt a hare if they both on a stag. Then they coordinate and everything works pretty well they get to capture the six units of meat and split it evenly between them three apiece.

If one tries to capture a stag and the other captures a hare like in this outcome. Right here. The player who s hunting a stag in this case player. 2 fails because she needs the help of player one order to capture the stag so she gets.

0. Meanwhile player 1 is left to capture both of the hares by himself so that means he gets two units of meat lastly in this outcome. Right here both of them choose to hunt a hare and so they split the two hares evenly 1 hare. A piece now in the last two videos.

We solved games by looking for strictly dominated strategies. But here that s not going to take us anywhere let s see why suppose player 1. Knew that player. 2.

Is going to hunt a stag player. 2. In this case or player. 1.

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In this case should also hunt a stag because. 3 is greater than 2. But if player 1. Knows that player 2 will be hunting a hare.

Then player 1. Should also hunt her hair. The reason is because if he hunts a stag by himself. He s going to fail and get 0.

If he goes out and hunts a hair then he least he gets 1 so player 1 strategy. Our optimal strategy is completely dependent on player. 2 s choice and the same is true for player 2. This game is completely symmetrical respect so player.

2. Only once not a stag. If player 1. Is hunting a stag and only wants to hunt a hare.

If player. 1 is hunting a hare and so as a result. There isn t a strictly dominated strategy for either player here now given what we know about game theory for now. We can t really solve this game in any meaningful way without any further rules to implement into the game in order to figure out what is a sensible outcome here.

So we re going to introduce a new way to solve a game. And understand. What s sensible in that regard. And we call.

This nash equilibrium. So a nash equilibrium. Is a set of strategies. One for each player.

Such that no player has incentive to change his or her dad. His or her strategy now a couple of notes about this we only care about individual deviations. Not group deviations. So for checking whether some outcome is a nash equilibrium.

We don t have to see if both players can collectively change their strategies to different strategies. We only have to worry about can one player change his strategy and do better or can the other player change. Her strategy and do better. We re just worried about individual deviations.

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Not group deviations. Now the reason that nash equilibria are compelling is that they re inherently stable. What you re doing is often well given what i m doing and vice versa. So this means once we ve actually seen the outcome of the game once we ve chosen our strategies and the strategies are revealed.

We don t have any regrets about what we ve done i m happy what i have with what i ve done and you re happy with what you ve done we can t change our strategies in retrospect. Individually and do any better so let s see this in practice. How does this work well how do we find these nash equilibria. There are four different outcomes here so what we re going to do to see if any of these are nash equilibria is we re going to isolate that outcome and look to see if each or if any of the players can individually do better by changing their strategies.

So let s start out by looking at the stag stag outcome. What any player want to change his or her strategy. Given that both players are going to be playing stack well player one would not want to change his strategy. Because he s getting three for hunting.

The stag and only two for hunting a hare so he s satisfied maintaining his strategy and player. Two is also in the same boat. He crate xie. She hits three for hunting.

The stag and only two for hunting the hare so she s happy maintaining her stag strategy and so we know that this outcome. Right here. The stag stag outcome is a nash equilibrium every player is happy with this outcome. Neither player can change his or her strategy and expect to do better now.

This should be intuitive. Because the stag outcome represents. The best possible outcome for both players right they both get three in this situation. And these three points is better than any other outcome for both of the players.

So we should expect this to be a stable strategy given that it s the best thing that both players can do. But we need to try to see if there are more nash equilibria in this game. Games. Don t always have one.

Nash equilibrium. There can always be more than one nash equilibria and so we should really check to see if there are any more here. So let s take another outcome. Let s look at this outcome.

Right here. When player 1 hunts. A hare and player. 2 hunts.

.

A stag is this a nash equilibrium. Is this inherently stable. The answer is no why is that look at player ones choice. If player 1.

Knows that player. 2 is hunting a stag. We ve seen this before he s going to want to change from hunting a hare to hunting a stag biga. So you can go from 2 and move up to 3.

That s a profitable deviation to change from hare to stag. So that means this can t be a nash equilibrium. Because there exists an individual deviation that leaves that player better off. Now.

We could just be done with that and not look to find any more deviations. But we could also note that player. 2 has a profitable deviation in this case right because she s playing a stag here and she s earning a zero. But she could switch over to here and she could do better that way so.

This is also a profitable deviation. This is superfluous information. If we only care about finding nash equilibria because we already know based off a player ones deviation that this can t be a nash equilibrium. But we could have also shown it by looking at player twos deviation in this case alright.

So we know that this is an inertia calabrian. What about this case up here. Where player one s hunting. A stag and player two hunting.

A hare well that s essentially identical to the situation. Down here. And we can see that this isn t a nash equilibrium. Because player.

One would want to change his strategy. He could go from hunting a stag and getting zero to hunting a hare and getting one. So that s a profitable deviation for him that means this can t be a nash equilibrium. And again.

We don t have to track player twos deviations. Because as soon as we find a single deviation in this case. Player ones. We know that this outcome can t be a nash equilibrium.

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That leaves us with one outcome to check. Which is when they re both hunting hares. So do any of the players have a profitable deviation given this outcome is going to have an expectation well. The answer is yes look at player ones choice player one could switch from hunting a hair to but rather neither player has a profitable deviation player.

One can t switch over from hunting a hair tonic a stag because he goes from earning one turning zero that s not good for him. He would be sad. He would be upset. If he switched his strategy from hare to stag given that player twos won t be hunting a hare.

Same is true for player two. So if player. Two expects player one to be hunting a hare she earns one for maintaining her strategy and zero for changing her strategies. That s not a profitable deviation.

She would be unhappy if she did that so that means this outcome is also collectively stable. This is a nash equilibrium. So we found two nash equilibria. Here.

There s a nash equilibrium. Where they both hunt. A stag. And there s a nash equilibrium.

Where they both hunt. A hare now this one was obvious. This was much more counterintuitive because this is worse for both players than if they both collectively chose stag. However when you re unable to coordinate like this if your expectation is that you re going to be hunting a hare today for example perhaps outside the hunting range.

It said today is hare hunting day that sets this expectation in mind where both players are going to be hunting this hare and so this guarantees that they re going to be doing. Something a little bit better when they re hunting hares and as a result this can sort of sneak in there and trap. Them into this inefficient situation. Where they re both hunting hares instead of getting the stacks.

So nash equilibria aren t always efficient always good. But they are inherently stable and no one has any regrets hunting hares in this particular case given that the other one is going to be doing the same thing. So. That s why we search for nash equilibria that s one way of finding nash equilibria and in the next video.

We will learn about what nash equilibrium is intuitively join me then ” ..

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