# mod02lec16-Decisions in Game Shows: Final Jeopardy

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This topic is Decisions in Game Shows this one and the next topic we will talk about
a game show called final jeopardy. This is a popular quiz show in the US, lot of times
this is available in YouTube. I would highly recommend that you watch the final jeopardy
once at least in YouTube before you know going through this entire exercise say that one
more time. So, we will do a little case study of this
game called jeopardy. Now there are many there are many many TV shows like jeopardy where
we have to make decisions and we saw the Kaun Banega Crorepati example earlier where you
know you would say ok. Do I want to continue or do I want to stop
there are many such game shows. In fact, on TV that we see and you know this is just to
list a couple of them here, but there are many more . Now, there are some complications
here now when there is a game game means there are multiple players ok. So, unlike the Kaun
Banega Crorepati where you are the only player ok.
And so therefore, the situation is a bit different here there are multiple players. So, it adds
a little bit to the complexity we will take a look at it. I will explain the game reasonably
well, now there are three parts of the game we going to look at the last part. So, now,
essentially if you look at chess is another game; obviously, and there are many other
games. But in chess the players alternate and make
decisions the white goes first makes a decision and then back goes first by decision what
we move now turns out that they do this one after the other you can see what the other
person is done. Here what happens is we will see where all the players make a decision
in one shot one decision and they do not see what each other is doing in one shot one decision
so at the same time. So, in some sense this is like an auction
ok. There are many times you would be familiar with what is called an auction this is something
similar to that ok and this jeopardy is what we are going to be looking at and this is
a very popular TV show is been running for a long long time several decades ok. I would
highly recommend that you go to YouTube and watch one episode of jeopardy if not more.
I would recommend watching a few even if you do not have that kind of time at least watch
the last part. So, there are three parts in the jeopardy game the first part is the basic
jeopardy, then there is something called double jeopardy, then there is a final jeopardy.
The final jeopardy is all you need to see for today ok this is what we are going to
be talking about in this course ok . But I will tell you a little bit more about
what happens in final jeopardy in case you do not get a chance to see. I want to also
say that we have to just do a discussion we are not going to come up with an optimal strategy
like we saw in the Greene Cat before or you know the secretary problem all those had an
optimal strategy ok. Here we are not going to have an optimal strategy I do want to warn
you, so do not get too excited . We will just analyze I just want to give you
an appreciation for decision making under uncertainty. Now turns out and I will give
you a picture when I get a chance, but there are three players in the game and everybody
who has nonzero points. And these points are expressed in dollars actually the word dollar
there it does not make any sense. Mainly because the players do not really get that dollar
amount unlike a Kaun Banega Crorepati or who wants to be a millionaire that is the actual
amount that they go home with here the amount is just notional ok. So, so we can just think
of them as points. Now after two rounds they have certain number
of points. So, I am going to say let us say the first person has X number of points, Y
person has sorry second person has Y points and the third person has Z points ok. So,
more often than not there are three players left if somebody has 0 points they are basically
eliminated from the final round, but that happens very very rarely. So, we will just
take the more generic case of being having three players.
So, first player has X amount, second player has Y, third player has Z. Now X, Y and Z
we are going to assume is actually known not just assume is actually known to everybody
everybody can see where how much points each person has ok. Unfortunately I do not like
using upper case X, upper case Y and upper case Z. Mainly because you know these are
like random variables, but here they are not these are not at all random variables, I just
want to warn you these are not random variables ok like you seen before I do want to warn
you may be I should have used a different letter ok.
Here are the basic rules I am going to tell you the rules once it might be difficult and
I draw a picture and explain it one more time next ok. Now in a blank slide here I will
do that, so essentially this is what happens. So, when you come down to final jeopardy and
each of the player say first player has X amount of money, second player has Y amount
of money, the third player has Z amount of money; at that point the category is provided
ok So, the category of the is for the last final
jeopardy is provided, but the clue is not given. So, you do not know what question is
going to be is going to come, but you just know the category. Now once the category is
revealed ok you as a player need to wager an amount ok need to say ok I will wager this
amount of money ok and that number has to be between 0 and how much money they have.
So, for example, player 1 can wager between 0 and x of little x is the amount that the
player wagers that puts down then then that that person that x little x must be between
0 and big X ok. Likewise if player 2 wagers y which should
be a number between 0 and upper case Y and wager, the third player wagers z ok. Now the
wagers are such that if you win if you get the answer correct then that x y or z gets
added to your score. If the wager is wrong it gets subtracted from your score ok. So,
now, everybody writes down their wagers at the same time nobody knows what each other
other people have wagered. So in fact, you see a big screen on your left
and your right and when you see it on TV. You will not see what the other person are
writing down. In fact, you will never know how much they wagered until the very end ok.
So, now what happens is the so now so here are the steps step one is the category ok.
Step two is to is the wager is written, but it is hidden wager is written down, but it
is hidden. The next step is the clue is given or the question is given the quiz question
is given and the players write down the answers on their screen ok much like the screen that
I am writing down right now they write down the answer.
Then the quiz master are most like Alex Bellbrook will show will actually show what each person
has answered. So, it goes the first player and looks at his or her answer and if the
response is correct then the person would now get x added to this course. This score
will be little x plus big X, now if y also gets it right there was going to be y plus
big Y and z plus big Z ok. . Now, the answer question is answered incorrectly
then the x amount or y amount or z amount gets subtracted. So, this is when the the
wagers are revealed wagers are revealed and you know how much they all better. So, you
could as a player either have 0 or twice how much exaggerate. So, if you wager all your
money then basically if you got the answer correct you would double and if you got the
answer wrong you would have 0. So, you would be anywhere between 0 and 2
x if you are player 1, 0 and 2 y if you are player 2, 0 and 2 z if you are player 3. Now
the winner of the jeopardy is the one with the highest final total this is after adding
or subtracting ok. We will not worry about ties, we will ye for the purposes of this
course we will ignore that . So, the question is what should the players wager? How much
should each player wager? So, what should little x be? What should little
y be and what should little z be? So, this is the question that is what we are trying
to decide. This the decision you need to make under uncertainty. What is uncertain? While
in the well making the wager what is uncertain is what question is going to come.
However, what is known a little bit about it some information is known which is what
category it is. So, if it is an easy category you might think oh maybe I will get the answer
correct. It is a hard category may be oh may be not ok and something like that, so so that
is what is is is going to happen ok. So, so so let me give you an example ok. So,
let me give you an example just to illustrate this whole process ok . So, when you look
at the screen you will see player 1 and then they will have a screen here this is player
2 will have a screen here, and player three have a screen here. And when they give out
the category let us say the category is mathematics ok . So, they tell you the category is mathematics
alright. So, that is that is what Alex Bellbrook will come out and say ok the category is mathematics.
Now, this person has initial amount like the beginning amount beginning amount. Let us
say this person has 10000 and say this person has 7000, and this person has 4000 alright.
Now the category is mathematics alright. Now what happens is this person needs to decide
ok mathematics is something that I am strong at. So, maybe I will wager all my money ok.
Now the others cannot see this this is secret nobody can see this other than the player
ok this is kept secret. Now this person said oh I do not know I am not very strong as mathematics
I am going to be wager 0. This is also a secret ok and the third person says ok mathematics
I am ok with it maybe I will wager 7000 ok and this is also a secret.
So, you do not know how much they wager. So, right now the question has not it come. So,
they put down this amount of money they cannot change this till the end of the game now the
question is revealed ok. So, now, these people have made their decisions based on how strong
they feel about the subject of mathematics ok the first person feels a bit strong they
put down all their money the second person is not very strong. So, puts down zero thinking
whatever they ask I am not going to be able to answer .
So, I will just put down zero the third person is kind of ambivalent and said ok I will put
downs a half way ok 7000. We will talk about what is a good number to put down these are
not the best numbers this is just to illustrate what could happen, but you need to be thinking
about how each person should be wagering at this point of time.
Now let us say the question comes up saying for example, what is the smallest number or
actually the only number, 3 digit number that equals the cube, or the the sum of the
cube the sum of the cube of the digits ok. So, that is a question alright.
So, the first person thinks about it and is not sure and puts a question mark. The second
person who remarkably things about it and then just writes down 847, and the third person
thinks about it and says ok I know this this is 153 which by the way is the right answer.
Because 1 cubed is 1, 5 cubed is 125, 3 cube is 27, so 1 plus 125 plus just 27 is 153 ok.
So, that is the only number that can be written in that form.
So, now what when they reveal the answer this person had does not write anything, so they
would go home with 0, this person has 7000 this is the final the endpoint goes with 0.
Second person goes at 7000 and the third person gets 21000 and the winner. So, this person
is the winner of that day this is the winner because they have the maximum amount of money
. So, now, the question is what is a good number? So, what should this number be? What
should this number be? And what should this number be will be our study next.
So, let us look at this so so let us say the category is world geography ok. So, now this
is another different category. Now I am giving you different numbers for what X, Y and Z
are? So, let us say the first person has 14000, the second person has 11000, and the third
person has 9000. The question is what should each person do? What should each of the players
do? Now when a category something like world geography.
You will see that most of these guys are pretty good then world geography. So, you know they
would feel somewhat confident they unlikely that anybody is going to put down a 0 for
a topic like world geography . So, especially you would expect ok question that they could
potentially answer ok. So, I do not want to say that now the topic
is somewhat important although some of the websites do not say too much about it. And
so what should each player do ok. So, now let us check of the first let us go player
by player. In fact, let us take the first player.
So, now if the first player has the maximum number of points 14000 ok now what could happen
the worst thing that could happen is the second player wagers all their money and gets the
answer right and thereby going to 22000. So, Y could be Y’s amount could be 22000 . So,
X must wager just enough. So, that if they get it right so which is 14000 plus 8001 to
get 22001 and thereby win ok we do not want a tie for now.
Let us think let us pretend that that is that is the case. So so so what we will do is we
will say ok. let us go ead and wager 80001, so little x is 80001, this is a reasonably
good strategy. Now this is a very standard tactic almost all the top player generally
do that especially when the topic is not terribly weird. Let us say for example, the topic is
Shakespeare and you are not strong in that yes it does not make sense for you to wager
somewhat lesser. But otherwise it is a pretty good choice.
Now the second player their best hope is they do know that this being a very popular strategy
this is what the first player is going to do. So, the second player is the second highest
scorer is going to say well I can only hope that they get the answer wrong and if they
get the answer wrong they be down to 5999 ok. However, though what could happen is the
third player who is at 9000 could double and go to 18000 we do not want that. So, we want
to top that one because five this guy goes to 5999, then you know then that is ok you
know you are still at 11000. So, that is not a problem.
However, if the person gets it wrong in this situation that will be an issue how? So, given
that this topic is something that they feel confident about they would wager 70001. So,
that 70001 plus 11000 gives me 18001. So, 70001 plus 11000 if they get the answer correct
is 18001 and they can win provided the first person gets it wrong ok. Now the lowest player
are their best bet would could be several things one good bet is for them to wager 3000.
So, if everybody gets the answer wrong then the lowest player is going to win 6000 because
they know that they probably guessing that this is a strategy that the others are going
to pick. Now we could really talk about you know is this the best strategy probably not
ok, but this is a strategy and you also want to pick strategies that you know that that
are dependent on the topic and a whole bunch of other things. The other thing is if a category
is very tough you could do something different you could think where everybody is going to
get it wrong. So, I will be conservative and put down 0
and topics like world geography maybe you are one person who feels a little bit weak
about the topic, but you are opponents may not. So, then you say ok let us go ead and
and do that ok. So, that is that is enough you know I have given you two websites I am
not going to click this, but this website is called the jeopardy fan dot com go ead
and click it and look at the analysis that they provide it is really interesting. And
also go ead and click this one I am not going to do it.
Here it is called the final wager dot com click that and give you some information about
various ways that people who have played this game a lot or at least have been in the show.
I tell you how best and what are the good strategies to adopt and I would be interested
in hearing about what you think should be a good strategy for just this one. So, feel
free to email me your personal options. Thank you very much.

1. Debu says:

boring