Game Theory 2 PDF

there's another type of domination which also makes some intuitive sense and people use in games. And that's to weaken the domination relationship and instead of, of strict domination we can consider weekly dominated strategies. What's the idea of weekly domination? it's very similar to what we had...

there's another type of domination which also makes some intuitive sense and people use in games. And that's to weaken the domination relationship and instead of, of strict domination we can consider weekly dominated strategies. What's the idea of weekly domination? it's very similar to what we had before. But instead of having the strict inequality hold everywhere, right? So instead of having this hold for all A minus I, it just has to hold some, sometimes, and you just need to weaken equality for all strategies of the others. So the idea of a, of a weekly dominated strategy. Is that it always does A prime always does at least as well as A and sometime strictly better. So you, this is still a strategy, you could say, okay A prime's really it Dominates a because it always does at least well, and sometimes strictly as well so if I'm uncertain at all, I might as well go with the one which always does as well and sometimes does strictly better. So weekly dominated strategies can be eliminated as well. You can go through, you can iterate, you know, just go through games exactly like we did before, same kind of thing. but, one thing that's true about weekly dominated strategies, is that sometimes they could be best replies, right, so, a strategy could be weekly dominated, and still turn out to be a best reply. Reply how could that happen let's suppose for instance we look at a very simple game where the role player can go up or down. If they go up they get a path of one against left and right of the column player and you know here they get two here they get three. So, this would be a situation where down weekly dominates up, right, you always get this high a payoff, and sometimes strictly higher. but none the less, it could be for instance that if left is the is, is the strategy that's actually chosen by the calm player Then upper still best reply, right. So for instance, if we put in payoffs here of, of 1 1, so the column player is exactly indifferent between these, these 2 strategies.Then this is actually a Nash Equilibrium. And so eliminating that, actually eliminates one of the Nash equilibri of the game, right? So depending on what those pay offs are, we could end up eliminating the Nash equilibrium of the game. And you know so this is, is a situation where you know, the...uh, we end up eliminating something which could be part of the equilibrium what is true is at least 1 equilibrium is always preserved. what's unfortunate is that the order of removal can matter. So which order you remove things in can begin to, to matter. there are some games which is useful to using, so for instance if you remember the Keynes Beauty contest game that we talked about earlier where people were naming injured between zero and a hundred think about trying to solve that iterative elimination of Weekly Dominated Strategies. What do you end up with so it can still be a useful logic and that logic can help you in analyzing some games but you do have to it is not as tight as [UNKNOWN] domination because there are situations where you might want to play a weekly dominated strategy. If you are sure, that the other player, was, was going to, you know, go in a certain direction. So for instance here if we put in two, one then then if we eliminate the column players dominated strat, weekly dominated strategy first, right, so they. Left weekly dominates right, we get rid of right, then what are we left with, we're left with a situation where the column then, I'm sorry the row player is indifferent between the 2 strategies right, so if we, sort of say okay look, this left dominates right so we get rid of this. Then we end up with a situation where up and left is still left. But if we removed, the row player's things first, we would remove, up first, and then, we would end up with down left. So, so, depending on, on how you go through this. You get different, different things that are left. so there are, are things that, you know, where the order matters and that's somewhat problematic. Okay, iterative strict and rationality, players maximize their payoffs. They don't play strictly dominated strategies, they don't play strictly dominated strategies given what remains. we iterate on that, nash equilibria Our subset of what remains, so it's a nice, simple solution concept that helps us, us throw things out of the game and simplify what we're looking at. We can also ask whether or not we see such behavior in reality. Do people really act in ways that are consistent with eliminating strictly dominated strategies and more over iterating on that process.