0:00

Yeah, hi folks. Let's take a peek at a game now where we

Â can begin to see whether iterative elimination of a strictly dominated

Â strategies has any bite in, in application.

Â And in order to do this we're going to look at an experiment that was done by

Â Baldwin and Meese in the late 1970s and they were actually looking at social

Â behavior in pigs. So, so this, the players in our game here

Â are going to be pigs. and there's actually sort of an

Â interesting discussion of this. This comes from Joe Harrington's book,

Â Games, Strategies and Decision Making and it is, it, it is sort of interesting.

Â you've got two pigs in a cage. Okay.

Â So they're in a cage, a several meter cage.

Â one of the pigs is larger then the other. We'll call that the dominant pig or

Â [LAUGH] I mean, well, sorry for the terminology.

Â So, we'll actually use the word let's say, the larger pig.

Â and in, in, what they need to do is, is food arrives, but they need to press a

Â lever to get the food to arrive. Okay? And the critical thing is that the lever

Â is on one side of the cage, you go over, you press the lever.

Â And then, the food arrives on the opposite side of the cage, okay? So the

Â pigs are put in a cage. The cage, they, they learn eventually

Â that if they hit this lever, food appears on the other side.

Â So what they would have to do is if they want to eat they would have to move over,

Â run over to one side of the cage, hit the lever,

Â run back to the other side, get the food that comes out, eat the food a few

Â pellets of food, then they go back hit the lever again,

Â run back to the other side of the cage, get some food.

Â Okay. Now, the difficult part is that there are

Â two pigs in the cage so if we put two pigs in the same cage and one is large,

Â then when they're both trying to the food which comes out and, and these are pigs

Â the larger one will end up getting the food and the smaller one will get less

Â food. Okay? So that's the basic idea here and

Â we can analyze this as a game. so in particular, let's take a look in

Â more detail at how the payoffs work here. So when the food comes out, there's ten

Â units of food, ten say pellets of food that come out.

Â And, let's look at what the typical split ends up being if there are two pigs in

Â the cage, one is larger, one is smaller, and, if the larger one gets to the food

Â first, then there's basically a 1,9 split.

Â So that means that the small pig ends up with just, with one pellet, and the

Â larger pig would end up with nine units of, of the food.

Â so if that large one gets there first, it's very hard for the small one to get

Â anything. They, they tend to get more on average,

Â but no, close to nothing. if the small pig gets to the food first,

Â then they end up with a 4,6 split. Okay? Then it's 40,60, so the, the small

Â pig still gets less, the, the bigger pig still gets more,

Â but the small pig at least has literally a fighting chance here.

Â now if they get to the food at the same time, then the it's a 3,7 split,

Â so the bigger pig gets a little more. And one other thing is that, that, you

Â know, running over and pressing the lever actually consumes some calories, so let's

Â take that they take it to say two units of food in terms of energy.

Â Okay? So, so we've got these different splits and so forth.

Â So what we can do is, is write it out a simple normal form game for that.

Â So, given all these numbers here is the, the small pig over here, large pig over

Â here, and now they have two choices. They can either run over and press the

Â lever or they could sit there and sit by the food side and wait for the other pig

Â to press the lever. Now, if you both go to press the lever,

Â then they're, they get back to food at the same, it's going to be a 3,7 split,

Â but it's a 3,7 split, and then you subtract off two for the cost of running

Â back and forth, so they each loose two units of foods.

Â So 3-2, we get the 1, 7-2 we get the 5, so we end up with 1,5 if they both do it

Â at the same time. in a situation where say the small pig

Â presses the lever and the big pig just waits there, then it's a 1,9 split, but

Â the small pig ends up losing two units of energy,

Â they actually end up with a negative, negative in that situation and so forth.

Â 4:41

So you can go through and put this into a normal form game,

Â what we end up with is, is a simple matrix form 2x2 game,

Â which looks like this. Okay. So we can analyze this game quite simply.

Â Let's take it and let's analyze, analyze it via the iterative elimination of

Â strictly dominated strategies. So what's true in this game?

Â Well where does anyone have a strictly dominated strategy? the big pig, the

Â large pig doesn't have a strictly dominated strategy.

Â They would like to wait if the other one presses,

Â they would prefer to, to press if the other one waits, so no domination here,

Â but notice that the small pig always gets a higher payoff, four versus one, zero

Â versus negative one, they would always prefer to wait.

Â So, in this particular situation the small pig has a strictly dominated

Â strategy of waiting. So we should get rid of press as a

Â strategy for the small pig, and once, we've done that now and what's left the

Â big pig should press. And so, what we end up with is when we iteratively eliminate

Â strictly dominated strategies, we end up with a prediction that the

Â small pig should wait and the big pig should be the one that presses the lever.

Â Okay. So let's look how they actually behaved,

Â the pigs in their, in the experiment. and so what they, they did here is they

Â gave them 15 minutes of, of doing this they did ten tests where the pigs were

Â alone in the cage. So the, you first, you, ten tests where

Â the pig just sits there for 15 minutes and learns how to, to press the lever and

Â get food. and then, they put the pigs together and

Â do another ten tests, each for 15 minutes as well.

Â Okay, so what they're doing here is this is the frequency of pushing the lever per

Â 15 minutes. and we can look at what happens when the

Â large pigs are, when, when the pigs are separated they're both alone, there we

Â see roughly the large pigs going about 75 times per 15 minutes and pressing the

Â lever. I mean these pigs are really moving back

Â and forth to get the food. The small pig say 70 times running back

Â and forth to get the, the, this, the food.

Â so they're, they're both, if they're left alone, they go, they press the lever,

Â they run back and forth. if they're together, then what happens?

Â well, the prediction was the larger pig should do the pressing, right? They

Â should do the pressing and the smaller pig should do the waiting and indeed how

Â frequently do they push the lever? And the small pig is very seldomly, only five

Â times now whereas the large pigs are doing it about 105 times.

Â So indeed, we are see the, seen them pressing and waiting in conjuction and,

Â as predicted by the theory. And in fact the, the large pigs are, are

Â doing more pressing, and partly possibly because, they're getting fewer pellets

Â out since the small pig is sitting there by and, and eating some of the pellets

Â that the large pig is producing by pressing the levers.

Â Okay. So what did we learn from this? Are pigs rational? do they, do they know game

Â theory? well, they, they probably didn't sit down and solve the, the normal form

Â game and iteratively eliminated strictly dominated strategies.

Â and, and, and I think that iterative elimination of, of strictly dominated

Â strategies is something which nice, nicely captures learning.

Â 8:27

So you learn not to play a strictly dominated strategy, right? Because it's

Â always giving you a lower payoff, eventually, discard such strategies,

Â so players won't be sitting there playing a strictly dominated strategy if they

Â ever have some experience with other strategies.

Â once, once they stop playing those, then you learn not to play strictly dominated

Â strategies out of what remains. So the small pig can learn that it just

Â doesn't pay to run over and press the lever because the big pig gets

Â everything. So they stop pressing the lever, they

Â just sit there and wait, and eventually, the big pig does the pressing.

Â So, you learn not to play strictly dominated strategies out of what remains.

Â And, and so the idea here is that learning evolution survival of the

Â fittest, these are powerful game theoretic tools.

Â And iterative elimination of strictly, strictly dominated strategies in games

Â where, where there is some power to these things ends up you know, making some

Â predictions, which can be quite powerful.

Â