Okay, so let's now discuss the phase of a relation.
So after editing has taken place, we now kind of like look at the remaining alternatives.
So like the change prospects in a sense, which we kind of like reorganized in our head.
And now we are going to evaluate this.
And the phase of evaluation and prospect theory is based on what we call a value function.
So let's have a closer look.
Oops, there we go.
What a value function is.
So the idea is that we would try to validate alternatives.
So we'll try to decide what is the best alternatives that we can actually pick.
And as I already said, in prospect theory, we talk about a value function.
We usually note it by V in order to make sure that we don't confuse it with a utility function.
And it's kind of a similar thing in a sense.
It tells us as a functional that tells us the value to us of an outcome in a specific
situation.
So it's also based on the actual outcome.
So the argument that they include into the value function is based on the outcome.
And as I was saying, we're rather thinking about gains and losses in two different dimensions.
So we'll just like very explicitly have every outcome and check whether this is actually
a gain or whether this is a loss.
So the value function is defined over gains and losses relative to a reference point.
We'll talk a little bit more about what reference points can be.
And we assume that VFX is usually concave in the gain domain and convex in the loss
domain.
Why is that?
Remember this observation from Kahneman and Tversky that individuals would rather go for
the safe option in the gain domain.
So they would rather pick the 500 for sure than a 50-50 gamble of getting a thousand.
However, we saw that individuals would actually rather go for the gamble when we're talking
about losses here.
So that's one of the things which are really like new and different in prospect theory
compared to expected utility that we assume that there is this cutoff point or there's
this reference point in a sense that everything above is a gain and we do behave differently
on gains compared to losses.
So in addition to that, we see that, so in addition to the shape of the function, we
also see that the value function is steeper in the loss domain than in the gain domain.
And that means that it's supposed to capture this effect that losses loom larger than gains
of equal size, meaning like losing 10 euro compared to a reference point hurts individuals
more than gaining 10 euros.
And then let's share a functional picture in a second, but that's really also a different
thing than the shape of the value function and the gain in the loss domain.
So the shape basically determines the risk attitude, but loss aversion is in addition
to that idea that we also do have, even though I'm risk-seeking in losses, so I'm okay with
tolerating risk when there is a loss, I still really dislike having a loss.
So therefore we do have this steeper slope of the value function.
And that's exactly the functional form that we just described.
So let's have a look at this function.
Here is the gain domain, there's a loss domain, and that's supposed to be the reference point.
And the value that we derive from like, so naturally this is where we have more to consume
and this is where we have less to consume, but the zero is not necessarily like having
nothing to consume, but it just means consuming as much as the reference point.
So make a mental note to yourself, the zero here, the origin, that is the actual reference
points.
That's not a zero, like not having any money, but it's just like if my reference point is
my previous job, I made 100 euros, then that's your reference point, that's 100 euros here.
And then if you would make more, we do see like this concave shape, it looks almost
linear, but if you have a closer look, it is a little bit, there is some concavity going
on here, but we also see, well, you have this convex shape in the loss domain if you have
a closer look, but then you also see whatever it's not continuous in the reference point.
So the value function is steeper in the loss domain.
And this is just to kind of like show or to illustrate that losses loom larger than
game is what we have actually described before, what Kahneman and Tversky have kind of found
in many actual real life decisions of individuals.
So the idea of prospect theory is really the reference dependency.
So that's a very big focus of prospect theory and that's why we present lotteries in incremental
form, how we say, so we look kind of showing them in terms of change compared to the reference
point.
So we would present a lottery or the ideas that not we would present a lottery, but people
evaluate lotteries in a sense, like looking at, okay, what is like for this outcome, what
is my gain compared to my reference point or what is my loss compared to the reference
point.
So the lottery would be kind of rewritten, not if X is the original outcomes, we would
rewrite it in terms of Y's.
And the YE for each of the state of the world is just the actual outcome minus the reference
point X0 that we have in the state of the world.
And then we can rewrite the lottery as basically this combination of like loss or gain.
So a wealth and relative wealth outcome minus reference point times the probability of being
in the state.
And those lotteries, as I already mentioned, are then called prospects in prospect theory.
So whenever I talk about prospect, this is this idea like, okay, it is a lottery, but
it's kind of like shifted in a way that we always subtract the reference point from the
actual outcome.
So it is different than expected utility because an expected utility, what we plug in into
a utility function is just the Xi, but now in prospect theory, we plug in the Y here.
So this is the relevant argument of your function.
So the lotteries are presented in incremental form, and we call them prospects.
Even though we redefine a prospect function like, or a value function here, depending
on X, it's dependent on the outcome, but it's always also dependent on the reference point.
So it depends on the outcome through the reference point.
Okay, so we do have a couple of assumptions on the value function in prospect theory.
So it's the idea, the value function is a mapping of Y, that's the outcomes in relation
to the reference points, or these are the outcomes of the prospects, and then we'll
just assign a number to this.
And then again, the numbers here don't have like as much of a cardinal meaning in a sense,
but it just says like more is better, so more gives you more value in the actual number.
If your value function is three, that's not really including a lot of information what
it means that your value function is three, it just tells you, well, if the value function
is three for this prospect, and it's two for this prospect, well, then three is better than
two, that's basically what I'm saying.
So the assumptions that we have that V is continued in the loss and in the gain domain,
we do have like a little bit like where we kind of like at the reference point level
will do see it's not smooth in a sense, but other than that, it is actually continuous.
And that's usually assumed for making things more tractable.
And we also assume V is strictly increasing.
And that seems to make sense, right?
We have that in expected utility, and if V depends on X, yeah, or V depends on Y via
X, so like my gain also increases in the outcome if I fix the reference point, right?
So it seems to make sense, like if you get more money eventually out that we should get
a higher value.
In addition to that V of zero is equal to zero.
So and then again, remember this zero is not consumption of zero or wealth level of zero
necessarily, it is the reference point here, that the value of the reference point is equal
to zero.
Let's just like normalizing it.
Yeah.
Okay, so we also see that V is concave of Y is greater or equal to zero.
So we have some kind of declining sensitivity for gains in a sense, like if I already gained
a lot, then the next unit of gain is not going to give me like the same extra utility here.
So the idea is like kind of getting into the gain domain is really important, but like
well, like how the value increases of the prospects kind of decreases if I already looking
at larger gains.
At the same time, these also convex if Y is less or equal to zero.
So we do have declining sensitivity for losses.
So it's really painful to lose, but then eventually if the losses become so large, then individuals
do not lose as much utility anymore.
And then in addition to that, so there are these conditions four and five is basically
the four is a condition that individuals are risk averse when it comes to gains, but
they're risk-loving when it comes to losses.
And then the last condition that we're looking at here is that we're saying like minus V
of minus X is greater than V of now, so minus V of minus Y is greater than V of Y. And that's
basically this idea that the value function is steeper when we're in the last domain.
So this is what we call a loss aversion and there's quite some large empirical support
that individuals are really loss averse.
So people really dislike losing here.
Then this is a function that Kahneman and Tversky proposed two years, that was quite
a couple of years later when they originally included this.
So it is like, I mean, how do these things develop?
It's descriptive decision-making theory.
So people are proposing, like Kahneman and Tversky were proposing all those thoughts
about how we value prospects.
And then a lot of people were basically like looking at observations and trying to fit
the model and find a good way to describe those functions.
And then what Kahneman and Tversky proposed is a value function, which is in the gain
domain here, it's X or the actual outcome minus reference point to the power of alpha.
And in the last domain, we have this lambda here.
So that's what's ensuring a loss aversion, it's a loss aversion parameter and other than
that it's X, the outcome minus X, not reference point, minus the outcome to the power of beta.
And they kind of like empirically adjusted this, I would say like alpha and beta is probably
should be something around 0.9.
So that's comparable in a sense, but then the lambda would be the parameter that determines
loss aversion.
But that's just like what they propose.
And there's been a lot of discussion how to best describe alternatives, because of course
it's difficult as well, right?
Do we find a functional representation for all individuals?
Well, that's probably hard to tell because we may also differ on our preferences.
We may differ, for example, on the degree of loss aversion, but those are values have
been like empirically found to like at least describe decisions of many individuals also
at the same time acknowledging that not everybody is the same, but it's like this idea of describing
decisions in the aggregate.
Okay, and then the big question for prospect theory is like, it was a really like new way
of thinking about things like saying like, well, it's not the outcome that matters, it's
kind of the outcome compared to what my reference point is.
So what I want to have that really determines behavior and if the outcome is greater, I
completely behave differently.
And if the outcome is smaller than my reference point, right?
So that's something that expected utility really doesn't capture.
But then the big question did arise like, what's the reference point, right?
In what kind of prospect, what kind of reference point do we rely on?
And Kahneman and Tversky are relatively silent on this.
So they propose different things, but they also say it's context dependent.
So it depends on the context, what your reference point is.
So for example, my reference point when I look for a new job could be my old salary.
So oftentimes the status quo is used and there's also like some empirical evidence.
I just recently read a paper published last year.
So that is where it seems to be a very common reference point utilize that some of this
could also not be my income, it could be also other people's income, right?
So for example, if you're thinking about finding a job after you graduate and you may have
a student job right now, right?
The reference point for you to some extent, maybe your current income, but it may also
be, I don't know what your peers are making.
So a reference point could also be like dependent on peers because you're thinking like, well,
I knew kind of entering like the job market in a different way.
Now I got my master's degree and I'm comparing my income to the income of other people who
are in the same situation as well.
So there's like a lot of different things that could potentially be reference points.
Sometimes we also think about expected values.
So what can I expect from this prospect?
And then the expected value doesn't have to be like 100% exact, but it's just like, oh,
this is an investment.
And my reference point is like what I'm expecting, what the return will be for me in order to
determine if I make a loss or gain on this investment.
It can also be past values.
So like it doesn't have to be the status quo right now, but it could also be like, I don't
know, you had a very high paying job five years ago and you had to kind of change to
a lower job.
But then when you ever made new opportunity, you just still compare it to the income level
that you had in your previous job because you kind of like not have really included
or accepted that this is the income that you're making right now when you just think this
is what I should compare my alternatives to.
It could be something like initial or purchase price or the reference point could also be
just zero.
Well, the reference point is not getting anything, but I mean, most of the time it would probably
not think this is something we should compare ourselves with, but it's like sometimes it's
like, well, if you're in this situation, if you don't have anything to lose, then the
reference point could actually be zero.
And in addition to that, there's also the question, our reference points are constant
over time.
And the answer is probably not right.
It really depends on the current situation and how also like lotteries or how prospects
are presented to individuals.
And we'll have the second part of this module 4.2, we'll talk a lot more about like how
we perceive alternatives and what happens and how we kind of like work with different
interpretation.
In addition to that, there's also sometimes this idea like maybe it's absolute reference
points or just based on the value relative as it was saying, compared to peers.
So I'm comparing my salary to what others are making.
And yeah, and I mean, if you think about, yeah, once you get like a job after you graduate
and at the beginning, and this is supposedly illustrating that reference points, I'm not
necessarily saying it was at the beginning, it feels like tons of money.
So I remember that when I graduated from the university, and then I got a job and was
making, I don't know, more than 40,000 euros a year, and that totally seemed to me like
I was being rich.
So I was like, oh, wow, this is pretty cool.
And that was a large, again, compared to like living on a student, student budget.
So I was very excited about this.
And then a couple of weeks into my job, I figured out some of the others are making
more than I am and we're doing the same kind of job.
And then it didn't feel like a gain anymore, it just rather felt like a loss.
So just to kind of like illustrate this, reference points do change.
So they really change depending on the situation.
And that's also something that expected utility doesn't really capture, right?
Expected utility will just say, oh, this is the amount of money that you consume and that
should give you a certain happiness level.
But then it really also sometimes matter if you put this into context.
OK, so this was the value function.
And the second thing, which is really like was a new way of thinking, it was this idea
that people do something to probabilities.
So in expected utilities, we have kind of this assumption that individuals work with
objective probabilities in the sense that for a risky gamble, they know the probabilities
and this is what they use, or they may still work with subjective probabilities, but they
just assume them to be given.
However, perfect theory allows more flexibility with this regard.
So the idea is that people do something with probabilities that somehow digest them.
And the idea is that individuals attach decision rates to probability.
So they have a function based on probability if the probability is p and the function is
pi that assigns those weights to probabilities.
And that could either be because I may or may not like be able to exactly evaluate this,
or it could also be just something that's very rational in the sense that I know, OK,
I have a 5% chance, I don't know, I have a 1% chance of developing this specific cancer,
but because the outcome is so bad, I overweight this chance because I just really want
to not really don't want to get cancer.
So the idea is of decision rates, they are very flexible in the sense that they could
either be like some non-rational underlying process of changing probabilities, which
individuals do, or in addition to that, they can also be like an active decision saying
like, I want to I know the probability is small, but I want to attach more weight to
this specific probability for some reasons of this state.
OK, so this is a couple of assumptions that Panama and Spain made about this probability
weighting.
And I also want to put this in context for you a little bit.
We will later also discuss cumulative prospect theory, because this is the part where the
probability weighting, which they were bringing in, it was a fantastic new way of thinking
about things, saying like, well, probabilities, and it's also a complicated concept to individuals.
So we'll don't always rationally work with a probability that we're given, but we'll
do something to it.
But at the same time, there was quite some backlash, or quite some, not backlash, but
a lot of research done showing it doesn't always work the way that Panama and Spain
were thinking when they first thought about probability weighting.
But let's discuss this here, how we do it in prospect theory.
And then the second part of the module, and also in rank-dependent utility and accumulative
prospect theory, we're going to discuss a different way of assessing probabilities and
attaching decision weights to probabilities.
So the first thing is that, or the first assumption that Kahneman and Tversky made about
probabilities is that they said it's kind of a standardized function.
So it's an increasing function in terms of P. So the more likely, the more decision weight
we attach to this.
And we have that pi of 0 is equal to 0.
So if it's not happening, then we just don't give more weight to this state.
And if it's happening for sure, we also let the state be equal to 1.
But not necessarily if you had like, if you work with a state of probability, it would
be between 0 and 1.
And then we would just like draw a straight line.
But that's not what Kahneman and Tversky were thinking.
They're thinking like there's things going on between 0 and 1.
But we know that the probabilities in the probability, the decision rate increases in
probability.
In addition to that, they assume something which they call sub-additivity and what the
sub-additivity means.
So if you have a scaling factor R, the probability of R time or the decision rate of R times
the probability is greater than R times the probability of P if the P is small.
So that basically means like if you would assume this is 0.5 and then this is a 1% chance
that this decision weight of 0.5% is greater than 0.5 times the decision weight of 1% that
we're attached to.
So it's kind of like this idea that like if it gets smaller, we kind of like attach more
decision rate to this.
In addition to that, Kahneman and Tversky also said that we overweight small probabilities
so that individuals do put more weight on probabilities when they're very small.
And that's this idea like, well, I only have a 1% chance, a 0.1% chance of an adverse event.
But I still pay a lot of attention to this.
In addition to that, what Kahneman and Tversky said is that when we evaluate risky prospects,
so it's a probability P, then we have 1 minus P, which is basically the probability of the
other risky outcome that if you attach the weight to this, this is less than 1, meaning
we because both of those states of the world are risky, we just never feel that we are
absolutely certain, even if we combine those two states.
And that's what we call the sub certainty.
In addition to that, there's also the sub proportionality where the intuitive explanation
is more and like it's rather this idea that we have like two probabilities.
And then if you scale them with a factor which is less or equal to 1, and then looking at
this fraction, at this fraction of pi of P2 and pi of P1 is less or equal to pi of
R times P2 and pi of R times P1.
And that's like one of the assumptions that's more technical ones that did not come with
like a lot of intuitive explanation here.
And then in addition to that, we assume that probabilities are not clearly defined for
very small P, or for very large P, but are still like not equal to 1 or 0 yet.
So it's like we're getting very close to certainty, there is some fuzziness going on in terms
of how we value those probabilities.
So eventually, they thought about probability rating being a function looking like this,
right?
So you see that the over-weighting of very small probabilities, you see it's not really
defined for very, very small probabilities and very large probabilities.
It is increasing, but it's not equal to the 45 degree line.
Well, as I was saying, the assumption of probability did not really, of probability rating did
not actually hold up to some empirical findings.
And that's where Kalman and Tversky also kind of like improved their work later with cumulative
prospect theory.
So in order to sum up this chapter or this module on prospect theory, let's make it quick.
It says assessment, we found that research specifically or later on empirical validation
of this theory, specifically about, found that some of the assumed characteristics
of the probability rating function was kind of unsatisfying in a sense that they wouldn't
have a lot of empirical support.
So for example, that probability rates do not have to sum up to one, seem to be something
that we don't often find or don't always find.
And in addition to that, because of the probability rating, then sometimes dominated alternatives
would be chosen.
And that seems to be also in contradiction of this idea that individuals kind of like
rearrange prospects in the editing phase and be checked, or turn into a dominated alternatives.
However, what we find is that we find a lot of support for specifically loss aversion.
So it's kind of like, seems to me that individuals are oftentimes we can, despite behavior, pretty
well with prospect theory, and at the same time, expected utility is not useless.
So we kind of like when we're looking into new evidence here, that both expected utility
and prospect theory seem to have their justification as descriptive theories.
Even though EUG is a normative theory, we can still apply this as a descriptive theory
and oftentimes explain empirical findings.
And it's rather this idea that maybe some of us are equated utility people and others
rather make decision based on prospect theory.
And then the big question in prospect theory is always like, what is the actual reference
point?
And it's really hard to empirically validate this, because prospect theory allows for different
reference points.
So in a different individuals in the same decision, this situation could have very different
reference point, but that makes it so hard to fit those functions.
Because if I don't know the reference point, and two people have different reference point,
and we'll just observe the outcome, and we'll assume they have the same reference point,
you obviously make a mistake while fitting those functions.
Okay, so this sums up the chapter on prospect theory.
This is the first part of module two.
The next part will look into two more descriptive theories, rank-dependent utility theory and
cumulative prospect theory, which you also sometimes you know it by CPT.
And then, yeah, we'll talk more about some biases and other interesting findings and
decision making.
So I hope you enjoyed this lecture and take care.