Game Theory C: Nash, Dominant, and Sequential Games

in this video we're going to look at one more simultaneous game and talk about Nash equilibria and dominant strategies again and then I'm going to introduce you to the concept of a sequential game where one player has to go first and then the other player has to make their decision and so this simultaneous game here involves Walmart and Kmart having to make a choice as to whether put in a regular-sized store or to put in a super store that has everything groceries gasoline everything that you can imagine and so let's just get started here by going through the method we looked at last time looking for Nash equilibria and dominant strategies and so the method that I like to use is - let's look at Kmart's decision here it's all about making best responses to what you see the other person has done at the end of the game and so just looking at Kmart's best response over here on the left assume Walmart put in a regular store where would Kmart wish they were what would their best response be 8 million if they put in a regular store or 12 million for putting in a superstore well their best response would be the twelve and so we put a little check mark on that 12 there and so let me put a little check mark on the 12 and that way we know where Kmart would like to be and so so we have the check mark on the 12 now let's look at the case where Walmart puts in a super-sized store what does Kmart's best response is it to put in a regular store on the top row for - or a super store on the bottom 4-5 where they actually lose money well of course it's going to be the two so let's put a little check mark up there on the two now let's consider Walmart's best responses to K Mart's actions and to do that let's move these red boxes over here we're just doing this to focus on Walmart's payoffs which are on the right side here and Walmart's best response okay well suppose that Kmart put in a regular-sized store putting us in the top row what's Walmart Walmart's best response regular for nine or super store for 11 well it would be a super store for 11 and so let's put in a checkmark over here on the Elevens chick now let's move this box up and say okay what would Walmart's best response be if Kmart were to put in a superstore Walmart's best response would be to put in a regular store because one is better than -5 so let me put a little check mark on the one and then let's take a step back and see what we've discovered here all right let's move all these boxes out of the way and see where the check marks are now it looks like here we have two check marks in the Box where Kmart puts a superstore and Walmart puts a regular store what that tells us is that's a Nash equilibrium because both players want to be there given what the other player chose it's a Mutual best response it this tech mark tells us that it's Kmart's best response to Walmart's choice this check mark tells us that it's Walmart's best response to Kmart's choice and we have another set of two check marks here where Walmart puts in a superstore and Kmart puts in a regular store so our summary of this game would be that there are two Nash equilibria to places where neither person would want to change their mind two places where we have this Mutual best response and no regrets now let's ask the question does other player have a dominant strategy one best choice that they would like to make no matter what the other player does well neither player has one best choice and here's how we can see that Kmart their best choice is superstore sometimes this check mark right here and their best choice is to put in a regular store sometimes and so came does not always want to choose the bottom row for superstore and they don't always want to choose the top row for regular it depends on what Walmart does so Kmart doesn't have a dominant strategy Walmart doesn't either because sometimes their check mark for Walmart is under the column for regular sometimes their check mark is under the column for super so Walmart does not have one best choice no dominant strategies so in this case we have two Nash equilibria what's our prediction what's our best guess as to what would happen in this game I don't know it could be reasonable for either of these two to happen in a simultaneous game where one store puts in a super and the other one puts in a regular store looks like that's what the Nash equilibria are guiding us so what we want to do now is see what changes if instead of having to make these decisions at the same time simultaneously which actually doesn't make much sense in the case of building a store certainly somebody has to to make a decision first and make the commitment first let's look at what happens in a sequential game and so let's move down here where I have drawn game trees now these game trees sometimes called extensive form representations of the game extensive form and we call them extensive form games because we extend out the the matrix structure the payoff structure so now we can observe who goes first and who goes second so on the Left I have Kmart making the decision first and then what we're assuming is Walmart observes this choice and then Walmart gets to make their decision second after observing Kmart now on the right side I assume Walmart's going first and then Kmart has to follow and go second let's see what happens in this extensive form game here's how we analyze a game like this we assume that Kmart says what if they ask themselves what if questions what if Kmart says to themselves I put in a superstore what would Walmart's reaction be now keep in mind we're keeping Walmart's payoffs on the right side here what's Walmart's best response to Kmart is putting in a superstore well if one word also puts in a superstore they lose five million but if Walmart were to put in a regular store Walmart will get 1 million and so Kmart can anticipate that Walmart's response would be to put in a regular store what Kmart wants to know is what Kmart will get out of that decision if Kmart puts in a superstore they're going to get this 12 and so the way I like to do this is to say okay Kmart now knows they'll get 12 if they head down that path to putting down a super store but now Kmart needs to see what their other decision would get them if Kmart puts in a regular store what is Walmart's best response well Walmart's best response but then to be put into to be put in putting in a superstore because Walmart would get 11 instead of 9 but K Mart's interested in what Kmart will get well Kmart will it end up with this 2 right here if they decide to put in a regular store they anticipate Walmart will respond with a superstore Kmart ends up with 2 so Kmart steps back to the beginning of the game and says what's my best choice super for 12 or regular for two it's pretty obvious K Mart's best decision would be to put in a superstore so our prediction here would be that Kmart is going to put in a superstore and they're going to end up with 12 and then Walmart will respond with a regular store and Walmart is going to end up with the one to pay off of one there the procedure we just use to find that equilibrium is called backward induction and the prediction we make is called a perfect equilibrium a sub-game perfect equilibrium is the real title now what's going to happen if Walmart goes first well if Walmart puts in a superstore Kmart's best response is to put in a regular store because Kmart will then put earn two million instead of five and so we can see Kmart's response here and Walmart would end up earning eleven there so let's type that up top okay but if Walmart were to put in a regular store Kmart's best response is to put in a superstore because Walmart would get twelve Walmart will end up with one in that case so one what's Walmart's best choice what's our prediction well our prediction here is that Walmart going first we'll put in the superstore Walmart we'll end up with eleven for their payoff Kmart will respond by putting in a regular store and Kmart will end up getting two in that case so that's our prediction when Walmart goes first so two let's go back to the sequential game quickly sorry simultaneous game in a simultaneous game we had two Nash equilibria and we couldn't really tell what would happen what happens when we turn this into a sequential game is which of those two Nash equilibria happens depends on who goes first and when Kmart goes first they put in the superstore and get twelve Walmart OMA gets one but when Walmart gets to go first they get eleven and Kmart gets two you can clearly see here that who goes first is the person that does best teamwork gets twelve when they go first and only two when they go second Walmart gets 11 when they go first and only one when they go second and so a couple things we've noticed here is this game has a first mover advantage because you would like to go first in this sequential game and so we will come back and look at some other games and see what happens differently in those

Channel: BurkeyAcademy

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Keywords: How, to, set, up, and, solve, sequential, move, game., subgame, perfection.

Id: qtZIrtZmmV8

Length: 10min 57sec (657 seconds)

Published: Fri Apr 30 2010