 Hi, folks. Let's take a peek at a game now where we can begin to see whether iterative elimination of strictly dominated strategies has any bite in in application and In order to do this we're going to look at an experiment It was done by Baldwin and Mies in the late 1970s and they were actually looking at social behavior in pigs So so this the players in our game here are going to be pigs and There's actually a sort of an interesting discussion of this this comes from Joe Harrington's book game strategies and decision-making and It is Is sort of interesting? You've got two pigs in a cage. Okay, so they're in a cage several meter cage One of the pigs is larger than the other. We'll call that the dominant pig or I mean, we'll sorry for the terminology so we'll actually use the word larger pig and In what they need to do is is food arrives, but they need to press a lever to get the food to arrive Okay, and the critical thing is that the lever is on one side of the cage You go over you press the lever and then the food arrives on the opposite side of the cage Okay, so the pigs are put in a cage the cage They learn eventually that if they hit this lever food appears on the other side So what they would have to do is if they want to eat They would have to move over run out to one side of the cage hit the lever run back to the other side Get the food that comes out eat the food a few pallets of food Then they go back hit the lever again run back to the other side of the cage get some food Okay, now the difficult part is that there are two pigs in the cage so if we put two pigs in the same cage and One is larger Then when they're both trying to get the food which comes out and and these are pigs The larger one will end up getting the food and the smaller one will get less food Okay, so that's the basic idea here and then we can analyze this as a game So in particular, let's take a look in more detail at how the payoffs work here so when the food comes out there's 10 units of food 10 say pellets of food that come out and Let's look at what the typical Split ends up being if there are two pigs in the cage one is larger one smaller and If the larger one gets to the food first Then there's basically a one nine split So that means that the small pig ends up with just with one pellet and the larger pig would end up with nine units of the food So if the large one gets there first It's very hard for the small one to get anything they tend to get one on average, but no close to nothing If the small pig gets to the food first then they end up with a four six split Okay, so then it's 40 60 So is the small pig still gets less the bigger pig still gets more but the small pig at least has a Literally a fighting chance here now if they get to the food at the same time then the It's a three seven split. So the bigger pig gets the little more and One of the thing is that you know running over and pressing the lever actually consumes some calories So let's take that they take it to say two units of food in terms of energy Okay, so so we've got these different splits and so forth. So what we can do is is write out a simple Normal form game for that. So given all these numbers Here's the small pig over here large pig over here and now they have two choices They can either run over and press the lever Or they could sit there and sit by the food side and wait for the other pig to press the lever Now if you both go to press the lever, then they get back to the food at the same time It's going to be a three seven split, but it's a three seven split And then you subtract off two for the cost of running back and forth So they each lose two units of food. So three minus two we get the one Seven minus two we get the five. So we end up with one five if they both do it at the same time in a situation where Say the small pig presses the lever and the big pig just waits there Then it's a one nine split, but the small pig ends up losing two units of energy They actually end up with a net negative in that situation and so forth So you can go through and put this into a normal form game And what we end up with is a simple matrix form two by two game, which looks like this Okay, so we can analyze this game quite simply Let's take it and let's analyze it via the iterative elimination of strictly dominated strategies So what's true in this game? Well, where does anybody have a strictly dominated strategy? The big pig the large pig doesn't have a strictly dominated strategy They would like to wait if the other one presses they would prefer to to press if the other one waits So no domination here, but notice that the small pig Always gets a higher payoff four versus one Zero versus minus one. They would always prefer to wait So in this particular situation The small pig has a strictly dominated strategy of waiting So we should get rid of press as a strategy for the small pig And once we've done that now in what's left The big pig should press and so what we end up with is When we iteratively eliminate strictly dominated strategies We end up with a prediction that the small pig should wait and the big pig should be the one that presses the lever Okay So let's look at how they actually behaved the pigs in there in the experiment And so what they did here is they gave them 15 minutes of doing this They did 10 tests where the pigs were alone in the cage So the first you 10 tests where the pig just sits there for 15 minutes and learns how to To press the lever and get food And then they put the pigs together and do another 10 tests each for 15 minutes as well Okay, so what they're doing here is this is the frequency of pushing the lever per 15 minutes And we can look at what happens when the large pigs When the pigs are separated they're both alone There we see roughly the the large pigs going about 75 Times per 15 minutes and pressing the lever. I mean these pigs are really moving back and forth to get the food The small pigs say 70 times running back and forth to get the food So they're both if they're the left alone they go they press the lever they run back and forth If they're together Then what happens? well, the prediction was that the larger pigs should do the pressing right they should do the pressing and The smaller pigs should do the waiting and indeed How frequently do they push the lever if the small pigs very seldomly only five times now Whereas the large pigs are doing it about a hundred and five times So indeed we are see them seeing them pressing and waiting in conjunct as predicted by the theory And in fact, the large pigs are doing more pressing and partly possibly because they're getting fewer pellets out Since the small pig is sitting there by and and eating some of the pellets that the large pig is Producing by pressing the levers Okay, so what do we learn from this our pigs rational. Do they know game theory? Well, they probably didn't sit down and solve the the normal form game and iteratively eliminate strictly dominated strategies But one thing is true is they do learn and they did seem to respond to incentives And and and I think the iterative elimination of strictly dominated strategies is something which nicely Captures learning so you learn not to play a strictly dominated strategy, right? Because it's always giving you a lower payoff eventually discard such strategies, so Players won't be sitting there playing a strictly dominated strategy if they ever have some experience with other strategies Once they stop playing nose then you learn not to play strictly dominated strategies out of what remains So the small pig can learn that it just doesn't pay to run over and press the lever because the big pig gets everything So they stop pressing the lever. They just sit there and wait and eventually the big pig does the pressing so you learn not to play strictly dominated strategies out of what remains and and so the idea here is that learning evolution survival of the fittest these are powerful game theoretic tools and iterative elimination of strictly strictly dominated strategies in games where where there is some power to these things ends up You know making some predictions which can be quite powerful