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Hello welcome to another online lecture for rational choice theory and I'm going to do a series of online lectures where I want to go through the proof of the von Neumann Morgenstern representation theorem that we will talk a lot about this in class so I'm not going to spend too much time on the motivation so during the lecture I'll spend a lot of time motivating this theorem and explain what the theorem actually says this is more trying to make sure we understand the technicalities of the theorem okay so the assumptions is that Z is going to be a set of prizes and for the purposes of these lectures even though I don't have it written down right here it's important that Z actually is finite so Z is a finite set of prizes the theorem generalizes when Z is an infinite set of prizes but there's lots of mathematical issues that arise and in general I'm going to try to not spend too much time talking about those because those require some some some background in mathematics to really appreciate what's actually going on so we have a finite set here of prizes and what we're interested in is lotteries so I'll use the term lotteries over Z and a lottery over Z is just a probability measure over Z it's just some probability it's just a way of assigning numbers between 0 and 1 so for each prize it's each the probability of that prize is how likely you are to to get that prize if you play this lottery so we'll let capital P it's going to be the set of all functions such that P is actually a probability measure and some Z is finite we can just define it very simply as saying it's the set of all functions from Z 2 0 1 such that the sum over all of the prizes must be equal to 1 though there's lots of uh issues here if already we can see if Z is infinite we would have to give a slightly more sophisticated definition so let's not worry about Z being infinite at this stage and now we have this relation over the set of lotteries and what this means is that we're assuming that the decision-maker has opinions these are the behaviors we can observe or somehow we can elicit these opinions from the decision-maker and his opinions are not over directly necessarily over the prizes themselves but rather they're over lotteries over the prizes which are if you think about it really complex objects but so we're assuming that the decision-maker has some opinions about the different lotteries which which lotteries keep or she or she prefers over other lotteries now there's three axioms we're going to assume first of all we want to assume that this relation on P is in fact a preference relation so what does that mean well that means it's going to be a symmetric and negatively transitive and well this implies lots and lots of things about how this preference relation behaves so we've talked about this in class already you might be a good idea to remind yourself what exactly it means to be a preference relation so we're assuming that the agent has this preference relation over...