Game Theory 101 (#13): Weak Dominance

hi i'm william spaniel let's learn some game theory today's topic is weak dominance I cover it in lesson 1.4 of game theory 101 the complete textbook check the video description for more information about that now in this entire chapter we've only been looking at simultaneous Move games and your goal whenever you get a simultaneous move game is always going to be to find all of that games Nash equilibria now we started this unit by talking about iterated elimination of strictly dominated strategies and I told you that you should always just eliminate a strictly dominated strategy whenever you see it and the reason for that is that a strictly dominated strategy can't be a part of a Nash equilibrium remember a set of strategies and a Nash equilibrium must be unable to be improved upon so that means that if I were to play a strictly dominated strategy in a Nash equilibrium well that's impossible I must not be able to improve on my payoff by playing this strategy in a Nash equilibrium but if I'm playing a strictly dominated strategy then by definition the strictly dominant strategy must pay more than the strictly dominated strategy and so that means I can profitably deviate from the dominated strategy to the dominant strategy so that's why I strictly dominated strategy can't be a part of a Nash equilibrium which is why we're safe eliminating it as soon as we see it but I want to look at something a little bit weaker in this video and in fact the name of this thing is weak dominance so to illustrate weak dominance of look at this game right here player one simple two moves up and down player two also simple two moves left and right now compare player two's left option with her right option imagine she knows player one's going to go up then left is better than right because one is greater than zero now imagine player one is going to go down you can see that it doesn't really matter what she picks here she's stuck getting two so if you combine these two pieces of information together that means left is always at least as good as right and sometimes better so in this case it's better and in this case it's equally as good and so we refer to this as weak dominance we're left weakly dominates right if left were to strictly dominate right it would require this indifference here to go away would require that left always be better than right and here it's not always better it's just equally as good in this case up here it's actually always better but here it's just equally as good and so this equally is good part means that this is not a strictly dominant strategy left does not strictly dominate right it only weakly dominates it so to put that into words here left weakly dominates right for player two and that's because the left is as least as good as right and sometimes better okay so imagine we could just take weak dominance and treat it just like strict dominance and work our way through a sequence of iterated elimination of weakly dominated strategies well you can do that and what you'll get after that is going to be a Nash equilibrium of the original game so if we were do that here if we just eliminated right because right is weakly dominated then we're left with left and from here it's just a decision of player once to make sure he's getting the most out of his choice 3 is greater than 2 so that means he's going to want to go up and so that leaves you with up left as being a Nash equilibrium so after using iterated elimination of weakly dominated strategies any remaining Nash equilibrium must be a Nash equilibrium of the original game so if we're looking at the original game here we now know that this is a Nash equilibrium but there's a huge black coming there may be more Nash equilibria and you won't know until you go back to the original game and check so as soon as you eliminate a weakly dominated strategy for the game from the game you might also be eliminating Nash equilibria from the game and you don't really know sometimes you won't be but sometimes you will be the only way to know for sure is to go back and check and in fact by eliminating the right strategy of player twos you are actually eliminating ash equilibria so to see that I'm going to show you that this is in fact a Nash equilibrium and all you get to do to check on that is to see if the players can actually deviate profitably and the answers that they can't so to illustrate that player to can't switch to left and get a better payoff right she's perfectly happy playing right if player 1 is playing down because 2 is equal to 2 and player 1 doesn't want to switch from down to up because he's currently getting 2 for the Nash equilibrium if he switches he's going to get 0 that's not a profitable deviation and so that means that this is a Nash equilibrium so I know that a lot of people when they first think of we dominus they think well that's really silly no one would ever want to play a week a weekly dominated strategy why can't we eliminate it and part of the answer is in formal terms is that it eliminates Nash equilibria but then they counter well you know that doesn't make any sense still I don't want to play these weakly dominated strategies because I'll always get something better or at least as good by playing the strictly dominant strategy but I'd like to point out here that in this game the two different pure strategy Nash equilibria actually have distributional consequences for player two player two actually prefers the equilibrium outcome where she plays the strictly dominated strategy in the Nash equilibrium she prefers this Nash equilibrium where she's getting a two to this Nash equilibrium where she's getting one so she actually has incentive to credibly commit to playing right and forcing player one to play down and go along with this Nash equilibrium and it's actually in her best interest to credibly commit to playing this weakly dominated got to be careful and I say of that weakly dominated strategy and so there is in fact reason why you might want to place a weakly dominated strategy which is why we can't just throw those things away really quick and easily as we can with a strictly dominated strategy all right that wraps up this video in the next video we will deal with games that have infinitely many Nash equilibria join me then

Channel: William Spaniel

Views: 75,587

Rating: 4.951807 out of 5

Keywords: game, theory, 101, introduction, economics, political, science, mathematics, calculate, payoffs, pure, mixed, strategy, Nash, equilibrium, strict, weak, dominance, strictly, weakly, dominated, iterated, elimination

Id: OlV190noZOw

Length: 5min 49sec (349 seconds)

Published: Wed Sep 05 2012