game theory, what do we do?

sikes

This isnt mine, obviously, but whats the answer?

---

Lucy and Pete, returning from a remote Pacific island, find that the airline has damaged the identical antiques that each had purchased. An airline manager says that he is happy to compensate them but is handicapped by being clueless about the value of these strange objects. Simply asking the travelers for the price is hopeless, he figures, for they will inflate it.

Instead he devises a more complicated scheme. He asks each of them to write down the price of the antique as any dollar integer between 2 and 100 without conferring together. If both write the same number, he will take that to be the true price, and he will pay each of them that amount. But if they write different numbers, he will assume that the lower one is the actual price and that the person writing the higher number is cheating. In that case, he will pay both of them the lower number along with a bonus and a penalty--the person who wrote the lower number will get $2 more as a reward for honesty and the one who wrote the higher number will get $2 less as a punishment. For instance, if Lucy writes 46 and Pete writes 100, Lucy will get $48 and Pete will get $44.

What numbers will Lucy and Pete write? What number would you write?

Van Dice

Does 2 seem right?! Cos Pete is semi-smart and knows Lucy picks 100. So he picks 99. But if Lucy is semi-smart she will then pick 98 etc etc. I say semi-smart cos that's what game theory is, picking an unexploitable strategy, but not the best.
This shows why some of game theory is balls, I pick 100 every time and hope the other clown has enough sense to pick 100 too. And if they didn't, I play them HU and take all $$$

HiCloy

Van Dice wrote:

This shows why some of game theory is balls, I pick 100 every time and hope the other clown has enough sense to pick 100 too.

If you're writing 100 every time I'm writing 99 every time!!!

If you think your opponent will write 100, you should write 99. Then you will get 101. However if your opponent guesses this, they will write 98 etc. So in the end they both write 2?

Brayruit

If you write 2 the most you can hope for is 4.

You cannot go for less than 98 unless your objective is to minimise what the other person is getting or unless your objective is to avoid a situation where the other person gets more than you.

So in game theory land you write 98... you will either get 96, 98 or 100 and assuming your opponent is game theory rational you will get 98.

In the real world you write 100 and tell Lucy she is a moron if she does anything else.

Brayruit

I think that this problem may not have a solution?

The argument that brings the figure down to 2 makes sense in a GT sense.

But so does the assertion that you cannot put a number lower than 98.

This means you cannot know what your opponent will write - they can rationally write 2 or 98 (or 100 using the rationale that they may decide to co-operate). Does that mean there is no solution?

gerry87

Player 1 chooses 100, player 2 chooses 99 - Player 2 ends up with 101 and player 1 ends up with 98.

So both know the other will pick 99 to max their return, so one will pick 98 so as to end up with 100.

Then they both know the other will pick 98, so the best option is to pick 97 to end up with 99.... and so on down to 0.

bohsman

99 every time, its your only chance of ending up with more than 100 and even if the other person tries to screw you over by writing 2 you still end up with 97.

bops

write 8k and your solicitors phone number

The_Daddy_H

I think the answer is 100. Assuming both players are equally infinitely intelligent, they would both reach the same conclusion. I pick 100, you pick 100, we both get 100.

The only other viable answer is 99, but both players would realise that the other guy would have thought of this strategy too and then both would get 99 by this approach and reason 100 would be the best answer.

The_Daddy_H

In reality I'd assume the guy would make a mistake or is a prick and had said 99, so I'd say 98.

gerry87

The_Daddy_H wrote:

I think the answer is 100. Assuming both players are equally infinitely intelligent, they would both reach the same conclusion. I pick 100, you pick 100, we both get 100.

The only other viable answer is 99, but both players would realise that the other guy would have thought of this strategy too and then both would get 99 by this approach and reason 100 would be the best answer.

The thing is it's assumed both players act as if the other player knows what they know.

The_Daddy_H

gerry87 wrote:

The thing is it's assumed both players act as if the other player knows what they know.

It doesn't assume anything other than that both people are smart.

nice1franko

The biggest concern would be that the other person would write down the true value of the object (assuming it is worth less than 100).

Like a lot of questions, not enough info is given about the context or interaction between pete & lucy (e.g. do they both look at each other and go wink wink nod nod and rub their hands cos the're both about to get 100 dolla for their 20 dolla tat)

Is Lucy a frickin nun or Pete a priest?? If I reckon we're facing an honest joe then I'll undercut the actual price by one dollar.

I'd say the amount you gain by undercutting the top (99 or 98) will be -EV. Majority of the time I'm writing 100. If Lucy has decided to scalp me for a dollar I chuckle and kick her in the flange.

Tar.Aldarion

99 and it they write 98 so be it. I still get 96. If they write 100 then I get 101.

The_Daddy_H

The Nash equilibrium seems to be 2 for this problem, strange, I'll have to think more about it.

boogey man

I think the correct answer is 2 but depends on your read of your oppenent.
2 is the only answer that is guaranteed to get you money therefore game theory wise is correct.
However as most of us on this forum are gamblers in some sense of the word I would take a chance with 100 and hope my partner knows me well enough and is clever enough to reason this so we have a chance of maximum value.
I think this is a good example of how game theory is not the perfect strategy for poker whearas a knowledge of it would be advantageous it is certainly not the most profitable strategy.

The_Daddy_H

This is an interesting problem, there is a second game theory problem at play her e, the over arching game theory problem is to decide whether or not both players decide to reject formal game theory to maximize gains and select 100.

sikes

bohsman wrote:

99 every time, its your only chance of ending up with more than 100 and even if the other person tries to screw you over by writing 2 you still end up with 97.

you would get paid 0

The_Daddy_H

The answer must either be 100 or 2, 99 means you've already decided on the slippery slope strategy leading to 2.

sikes

We are trying to solve the game whereby each player is acting in their own best interests. So knowing the actions of the other, can they alter their strategy to gain a greater reward, and the answer is always yes, until we get to writing down 2. Once we right down 2 we are in equilibrium.

Anyway i thought that was an interesting dilema, its named the travellers dilemma and was featured in scientific america last month, but has been around for a long time.

http://www.sciam.com/article.cfm?chanID=sa006&articleID=7750A576-E7F2-99DF-3824E0B1C2540D47&pageNumber=1&catID=2

nice1franko

boogey man wrote:

I think the correct answer is 2 but depends on your read of your oppenent.
2 is the only answer that is guaranteed to get you money therefore game theory wise is correct.
However as most of us on this forum are gamblers in some sense of the word I would take a chance with 100 and hope my partner knows me well enough and is clever enough to reason this so we have a chance of maximum value.
I think this is a good example of how game theory is not the perfect strategy for poker whearas a knowledge of it would be advantageous it is certainly not the most profitable strategy.

if u write 2 then the most you can hope for is 4?

it's possible that the other person will blow their own money by writing 1 or 2just so you lose yours as well but it's not very likely.

Tar.Aldarion

No, you can stop at 99. It depends on what the other person does. If they pick 100, 99 is better than you picking 100. If they pick 98, you picking 99 is better than you picking 2. If you pick 2, and she does not, you get 4, if she does - you get 2.

Blip

Gamboool, Your always getting the odds to call!

i.e.
So I'm Pete, I write 100,
At the lowest value, Lucy will give the honest amount, say it caost 46,
So I get 44 and Lucy gets 48, so what, it only cost $2 to protentully get more back as Lucy will not under value the origonal price.

(also if Pete and Lucy are togetherrrrr then the difference is nada 92/2=46)

sikes

Tar.Aldarion wrote:

No, you can stop at 99. It depends on what the other person does. If they pick 100, 99 is better than you picking 100. If they pick 98, you picking 99 is better than you picking 2. If you pick 2, and she does not, you get 4, if she does - you get 2.

But if she picks two, the logical choice, you get nothing.

sikes

Blip wrote:

Gamboool, Your always getting the odds to call!

i.e.
So I'm Pete, I write 100,
At the lowest value, Lucy will give the honest amount, say it caost 46,
So I get 44 and Lucy gets 48, so what, it only cost $2 to protentully get more back as Lucy will not under value the origonal price.

(also if Pete and Lucy are togetherrrrr then the difference is nada 92/2=46)

In this case we assume both players are game theorists. So Lucy would want to get the most for herself and so would Pete.

Tar.Aldarion

It depends on how you are playing it, if you want to make sure of making money pick 2, if you want to play for all or nothing, do it my way, it also has the opportunity that she will not pick 2, which I imagine most people I know would not.

sikes

Tar.Aldarion wrote:

It depends on how you are playing it, if you want to make sure of making money pick 2, if you want to play for all or nothing, do it my way, it also has the opportunity that she will not pick 2, which I imagine most people I know would not.

Yeah but if she isnt picking 2, we can earn more money.

Tar.Aldarion

Yeah, as I said, if she does not pick 2 we would then make 4 by picking 2. As long as she does not pick a low number like 2 or 3 we are always up on making a choice like 99. There are far more numbers she can pick that will make us more than 2/4 dollars so I don't see the point of us limiting ourselves to winning 4 dollars at most.

Hyzepher

The crux of this question is that they are not competing against ach other but against the airline. Therefore they will not bottom out at 2 as they are not trying to under value each other. They will both pick 100 to maximise profit. Of course if they are honest and the thing costs $50 they they will both get 50.

The_Daddy_H

Consider instead of $2 its $2 billion, does this change your thinking? You might get $0 or $2 billion.

nice1franko

Is the object of the game not to get as much money as possible? NOT to outdo lucy. Remember there are three players here, pete, lucy and the pilot.

Who wins if you get 4 dollars and lucy gets 2? the pilot does.

In practice, writing 100 is the only way to go. Worst (practical) scenario is you get 2 less than the actual cost. I reckon you'll both get 100 at least 40% of the time. The rest of the time you'll get 2 less than cost price.

Lucy has no vendetta against you so she's not writing 2 or anything like it. If she's a greedy cow she'll write 1 less than the cost price but so what the avg return you get on writing 100 more than compensates for it.

edit- strong case can be made for writing 99 either.... NO LESS.

sikes

Well assuming they are acting in their own self interest and rational game theorists, its quite interesting.

Tar.Aldarion

The_Daddy_H wrote:

Consider instead of $2 its $2 billion, does this change your thinking? You might get $0 or $2 billion.

The numbers should not change what you would do, be it 2c or 2 billion. That is not what this scenario is about.
As the above post says, most poeple will be out to make a maximum profit and pick 100, some might pick 2 but I think most would not. So in this way I think we should pick 99.

The_Daddy_H

It does change what people think, because responses which say 99 etc are thinking about how much better 99 is than 2, whereas from a cold rational viewpoint to maximize your earning you'd say 2 which sounds a whole lot worse but in actuality it is the correct game theoretic answer. The reason I said 2billion instead of 2c is to illustrate this point, obviously the game hasn't changed but it alters the irrational emotional impact.

sikes

yeah Daddy_H there were some surveys done amongst economics students which you might have seen.

Tar.Aldarion

The only way it could change the game is if it was some number of money that was so fantastical that 2 units of it was not different from 100 units of it, because you would not be able to spend that much money in your lifetime. Say an octillion.

The other person does not likely think that you would pick 2 when you would pick 100 and you would think that of them. It is about gain versus loss, there is much more to be gained than lost in the scenario and as the unit rises in value this becomes closer and closer, which increasingly would make me lean towards 2 eventually.

boogey man

nice1franko wrote:

if u write 2 then the most you can hope for is 4?

it's possible that the other person will blow their own money by writing 1 or 2just so you lose yours as well but it's not very likely.

2 is the minimum

CaptainNemo

Hyzepher wrote:

The crux of this question is that they are not competing against ach other but against the airline. Therefore they will not bottom out at 2 as they are not trying to under value each other. They will both pick 100 to maximise profit. Of course if they are honest and the thing costs $50 they they will both get 50.

If it was stated in the problem that Lucy and Pete hate each other and that therefore the objective is for one of them to get more than the other, then this would be an interesting game theory problem.

As it is stated, it's trivial. Both want to get maximum value for themselves and are smart people therefore both write 100.

sikes

no CaptainNemo, they are not getting maximum value, 101 is the maximum value. They are not a team, there are two players with payoffs for each decision who are bound by the logic and reason governed by game theory, no emotional attachment to money, or loyalty or anything like that.

Blip

The_Daddy_H wrote:

It does change what people think, because responses which say 99 etc are thinking about how much better 99 is than 2, whereas from a cold rational viewpoint to maximize your earning you'd say 2 which sounds a whole lot worse but in actuality it is the correct game theoretic answer. The reason I said 2billion instead of 2c is to illustrate this point, obviously the game hasn't changed but it alters the irrational emotional impact..

No it does'nt, because all values will have to change in equal proportion, choice is now between 1B and 100Billion, but furthermore the purchased items are also increased so too is the buyng power of Pete and Lucy, so there is no greater emotional impact.

The_Daddy_H

Blip wrote:

No it does'nt, because all values will have to change in equal proportion, choice is now between 1B and 100Billion, but furthermore the purchased items are also increased so too is the buyng power of Pete and Lucy, so there is no greater emotional impact.

Obviously all values dont change in equal proportion when taking emotional considerations into account. Would you risk a guaranteed 2 billion for the chance of 100 billion, I somehow doubt it.

CaptainNemo

sikes wrote:

no CaptainNemo, they are not getting maximum value, 101 is the maximum value. They are not a team, there are two players with payoffs for each decision who are bound by the logic and reason governed by game theory, no emotional attachment to money, or loyalty or anything like that.

I thought there was an assumption that the normal rules governing money and people would apply - i.e. any normal human being would rather have as much of it as possible. I don't think it was stated that the two players would try to seek the game theoretical equilibrium situation?

sikes

Im not sure you are following the point of this at all. Read the article I linked too. Yes 100 is a better answer, but its illogical.

Blip

The_Daddy_H wrote:

Obviously all values dont change in equal proportion when taking emotional considerations into account. Would you risk a guaranteed 2 billion for the chance of 100 billion, I somehow doubt it.

The values have to change for the given example to work

Anyway, You are not risking a guaranteed value.

Edit, Also If I had a cazillion then I would risk 100 for 2, stay within the context of the tread!

yahoo_moe

CaptainNemo wrote:

Both want to get maximum value for themselves and are smart people therefore both write 100.

But this strategy is exploitable - if they play "as a team", they maximise their combined yield by each choosing 100 (200 is the max they can get between them) but writing 100 NEVER gets "maximum value for themselves" individually

Choosing 100 to maximise joint value means that they each sacrifice (and must rely on the other person sacrificing) the extra 2 you would get as a reward for choosing 99 (101 total) if you know the other person is going to choose 100... that's why it's interesting

CaptainNemo

Hi Sikes, thanks for the link, I understand what this whole problem is getting at now. I think that's what I was trying to say myself, albeit in my fumbling and non-mathematical way: that if 2 is the logical choice then there's something wrong with the logic.

The_Daddy_H

Blip wrote:

The values have to change for the given example to work

Anyway, You are not risking a guaranteed value.

Edit, Also If I had a cazillion then I would risk 100 for 2, stay within the context of the tread!

This is within the context of the thread, I'm trying to show the common fallacy of reasoning.

The values dont change according to a simple linear proportionality.

You do have a guaranteed 2 billlion. By simply saying the value is 2 billion you automatically can make 2 billion.

gerry87

Why does it stop at 2? If one person choose 1, then they would end up with three and the other person -1. Or can there be no neg amounts?

edit: forgot the question said between 2 and 100, my bad

Tar.Aldarion

It says in the question that it stops at two, that is why.

Blip

Sikes,

One thing he does'nt mention in TD senario is the cost of the items. I think the value is extremly important.

if the items cost $50 why price more than $2 below this?

me too.


I get to thinking this is more of a hidden moral question.........

gerry87

Blip wrote:

Sikes,

One thing he does'nt mention in TD senario is the cost of the items. I think the value is extremly important.

if the items cost $50 why price more than $2 below this?

me too.


I get to thinking this is more of a hidden moral question.........

Because if you know the other person is going to price at 50, pricing 49 it will allow you to ge 51.