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“This video. I introduced the concept of an extensive form game. And then i demonstrate demonstrate the method of how to solve extensive form games. Called backward induction.
An extensive game is a structured way to analyze how two or more individuals interact strategically with one another. Suppose that we have two players in this game. One s name is draco. Other one s name is harry.
And these players can take actions. Such as casting spells for example. Draco. Can cast the spell named a or a spell named c.
And harry can cast a spell. Named x. Or a spell named s. Harry.
Spell. X. After a may have a different effect than x. Prime.
After a which may be the exact same spell. But it s a different effect and it shows that the timing of this of the actions. Actually has an important consequence here when a game is in equilibrium. None of the players have an incentive to unilaterally deviate from that equilibrium play in a two player game.
Means that each player s strategy is a best response to the other strategy. Now what do i mean by strategy well strategy is a contingent plan of what the agent wants to do given the actions of the other player. We ll be looking for pairs of strategies. Harry and draco.
Which are a best response to the other strategy to take a concrete example in this game draco. If draco casts spell. A and then subsequently harry cast spell f. Both both players would receive a payoff of five.
We have written this drinka receives the first five and harry receives the second five. Now if it was c. And then x prime draco would receive four and harry would receive five this is what s going to motivate our solution to the extensive form game notice that if break o started off right here. He has no idea what his payoffs will be later on down down the tree without reasoning through what harry would do and then what he would subsequently do so a natural place to start is here at the bottom of the tree because at the bottom of the tree.
The individual knows exactly what the stakes are then reasoning backwards harry can put himself and draco shoes deduce. What he would do and deduce. What his payoffs will be from that so on and so forth up to the up to draco s first known and what we would get is we would get a solution to this game. This process is known as backwards induction and so let s go ahead and plot a backwards induction solution to this game.
And see how all this works. So let s start at the lower level of this tree or draco is faced with two distinct choices on the left branch of the tree. Where you see the draco has a choice between a and c. Now draco could choose c and get three or you can choose a and get two so he s going to choose c and get three.
If we go over to the right branch. A draco would choose a because five is better than two if we go up to the second level of this game. Will see is that draco s actions will just translate it these payoffs and so what we can do we can rewrite the tree to represent this fat okay. So what i did is i promoted the payoffs up to the next level.
The paths of three and four given that drako chose c..
In the past of five and three given the draco chose a prime. But we can keep track of what precisely draco s moves are throughout this game. So harry gets to this node. It s going to be weighing whether he wants five or four.
So first more or less. It s going to choose x over here on this side is going to be choosing between five and three those are harry s choices and we get to promote these payoffs. So now we get to the final step of the problem we get to break o s first node and what we ll see is that now draco is choosing between big. A and big c.
Draco perverse five two four and so he ll choose a now rewriting the game tree back to its original full form we can see where the equilibrium lives and where these out of equilibrium strategies fall as well i can see this blue path denotes the equilibrium outcome. This is the outcome that we actually can we get a and x. So draco plays spell. Hey harry plays spell x.
And they realize a payoffs of five and five. We can also see that if we actually got to this note break a workplace spell. See we got four this note. Harry would play spell x.
We got to this node draco would place about a prime now these are all specified in these strategies. Which are the equilibrium strategies that are the best response to one we have the equilibrium strategies been delivering outcome. We have the equilibrium past reach the edge. And so this gives us a nice sense for how these two individuals interact strategically let s consider a slight tweak on this game.
Let s see these games are not always as simple as this one to solve now. This is an identical game to the one that we just solved except for i changed this payoff that s now four. I changed it from five this will give us a real sense for why we have established these concepts of strategy. Payoff and outcome as distinct objects in this extensive game theory.
It will give you a sense for how important. It is to have these concepts clear in your head. We re going to use backward induction and drink up on the third stage is going to solve back up. He s going to choose c on the left and choose a prime on the right side because those were his optimal choices.
Too before now again just as before we can look at this right note that harry has it looks just the same as before and what we ll see is the harry will choose x prime just as he did before but here s where we run into trouble at this node harry is indifferent between x. And s note that he gets a payoff or four now regardless of which option he picks. So he might as well pick x or he might as well pick s. But it doesn t matter what he does in fact he could put a probability weight that he plays deck.
So he might actually act randomly that s what s called a randomized strategy or a mixed strategy. Let s do know the probability that he places on action x. As p. What is the payoff for for a draco of choosing a what we can go ahead and compute that just by computing.
The probability weighted average of the payoffs. So draco s expected payoffs 3 5 times p plus. 3. That s one minus t.
When is an optimal for drinker to choose a versus choosing c. It s going to be optimal to choose a when this paths going down. The a bridge is bigger than forth its optimal that you see and then it s the other way around p. Where p is bigger than 1 2.
Drakkar will find it up to choose a in his strategy. But if it is less than 1 2. And drago will find it out that you see so let s consider the two cases if p is bigger than 1 2. This strategy pair says that draco s best response is to play a we will see him little a from before which got us these payoffs in the first place.
And harry s best response to that or one of mary s best responses..
Because he s indifferent above the value p. This is to pick p bigger than 1 2. And pick x prime on the right hand side with the tree. This will realize payoffs of 5p plus 3.
1. Minus p for drako and 4 for harry. Now let s consider case. 2.
Where p is less than 1 2. And that implies that draco s optimal choice is to choose see when drink ll chooses c. We get the path of 4 5. They have strategies of draco choosing c.
Little c. And. A prime and harry chooses p. Less.
Than 1 2. And extra. And the payoff in equilibrium is just for fun this game has an order of magnitude more complication than the previous game. And it goes to show how complicated game there you can actually count just with one little tweak of another when we did that the whole complexion of the game changed we have to consider a whole variety of actions and those actions could induce multiple equilibria to the game.
But fortunately our our strategy of using backward induction. Didn t fail us in fact it allowed us to do the under uncover some valuable insights into the strategic nature of these two players in ” ..
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