Game theory

No, he is almost certainly has an 1 or else he wouldn't offer to trade.

Even if there were 100 cards, labelled 1 to 100 and you got a 2 it would still be a stupid idea to accept the trade.

>>7909560

In that case it wouldn't tho

Monty hall problem anyone?

He's offering a trade so either he's stupid or he has a low numbered card (3,2 or 1). Can't be 2 because that's your card, so it's either 3 or 1.
But he knows that you probably won't accept the deal unless you have a low numbered card. So he wouldn't want to make that deal if he already had the card 3.
That means his card is 1, you should'nt accept the deal

so the players each only draw 1 card, they possibly trade, and then the game ends?

>>7909579

yep

HE offers the trade, do not accept.

The trade is a mandatory offer, accept it because there's a 2/3 chance to make your position better off than current.

>>7909696
>2/3
4/6?

I'm sorry?
Don't we mean 4/5?

>>7909780
Yes, you're right. I forgot no replacement so I'm retarded.

IF WE ARE ASSUMING YOUR OPPONENT IS RATIONAL AF AND SO ARE YOU, then yes, you should accept, see reasoning below, interesting problem OP.

If you're opponent has a 6, he definitely will not offer a trade.

If your opponent has a 5, he can only get better by getting a 6, he knows that if you a have a 6 you won't be down for a trade, so he won't offer a trade if he has the number 5.

If your opponent has a 4, things just got a little complicated for him, he might think about it and realize that he is better off only if you have a 5 or a 6. If you have have a 6, then you will decline the trade (obvious reasons), if you have a 5, you will be better off only if your opponent has a 6 and if your opponent has a 6 then he would never think of giving you an offer and you would know this, so you would also decline the trade if you had a 5. Therefore, the opponent can never be better off by offering to trade if he has a 4.

(cont.)

do I have to assume he might bluff?
fuck

>>7909572

This is a good solution.

>>7909793

Now we go from the bottom:

If your opponent has a 3, then he can only be better off post trade if you have a 4, 5, 6.
If you have a 6, straight no.
If you have a 5, no because only better with 6 and opponent will not trade if he has 6.
If you have a 4, better off with 5 or 6, opponent will not offer trade if 6 (obvious) opponent will not offer trade with 5 (see point two in above post) so you know he has to be lower than 4, you will say no.

If your opponent has a 1, he will offer.

So you say no, opponent will only offer if he has a 1.

>>7909793
>>7909806

>then yes, you should accept, see reasoning below, interesting problem OP.
>So you say no, opponent will only offer if he has a 1.

Silly me, the things I definitely don't do to get more people to comment on my answers.

>>7909799
>bluff?
You mean, he might assume you are stupid?

If he thinks you aren't rational, then he might say, hey, if that guy was rational he would definitely not accept my offer if I have a 3 and he has a 4 (>>7909806) but if he is stupid, he just might, maybe I can try.

So he offers you a trade even if he has a 3.

TL;DR: If your opponent thinks your dumb enough to accept a trade with a 4, you can win by accepting the trade with a 2 (or rather, have a 50-50 chance of winning since he would offer a trade if he had a 1 too).

>>7909571
that's not monty hall, idiot

>>7909540
No
If your opponent had 4,5,6 he would hurt his chances by trading so would not have offered.

If your opponent had 3, trading would help him on average, allowing him to win if you had a 4,5,6. If you had a 4,5,6 you would decline the trade, guaranteeing a loss.

So your opponent wouldn't offer to trade with 3,4,5,6, he must have a 1. Don't accept.

>>7909572
>>7909793
does this work for n cards as well?

https://en.wikipedia.org/wiki/Unexpected_hanging_paradox

>>7909846

It applies just as well

>>7909937
not really

>>7909846

Suck my unicorn tip, dickhead

So will this work with some arbitrarily large number of cards like 1000 where you draw something arbitrarily small but larger than 2, like 14?

Can you do the reverse induction to get to 1?

>>7909540
what does a trade mean? you give him your card and he takes yours?

>>7909540
how does game theory interact with the subjective values that your opponent attached to his card that motivated his offer. yes, you take it. this has litte to nothing to do with monty hall

>>7910051
No, you give him a cow for his card.

>>7909806
By that logic, a rational player would never offer a trade at all.

>>7909540
This isn't a math problem, it's a psychology problem. If you throw out the psychology of *why* he wants to make the trade, then his card is either 1, 3, 4, 5, or 6, meaning that there's an 80% chance that he has a higher card, so you should trade. If you only allow simplistic analysis of his motive, then he must have drawn 1 or 3, meaning 50% chance it's a good trade. If this is a real world analysis, as in something like "making a read" in a poker game, then the options are:

a) he drew a 1 (20% chance)
b) he drew higher than a 2 and thinks you outdrew him -- meaning he cannot have a 6
c) he drew a 6 and is retarded
d) he drew a fairly good card, possibly even 6, and thinks this is the dumbest game he's ever played (rightfully so) and he just wants to make it interesting
e) he won't actually make a trade and is only making the offer to see if he can get a read on what you have by your reaction

In any of the cases above, except (a), you should agree to make the trade. If you're wrong, then he drew a 1 and you drew a 2, which just makes you very unlucky.

>>7910147
A rational player might offer a trade if they have a 1. At least, there's no reason NOT to offer, even if a rational opponent would never accept.

>>7909812
Upboated

The amount of people who don't instantly recognize this problem makes me realize how many of you on /sci/ didn't attend extracurricular math activities in elementary/middle school.

>>7910178
>it's a psychology problem

Game theory is basically mathematical psych applied to benefit and utility analysis. That's why behavioral economists study it, it gives interesting insights into how a rational game player acts.

One interesting thing about game theory is that one of the guys who started studying it in the beginning believed we needed to nuke the Soviets because no matter if they nuked us or not, our nuking them gave us a benefit in both cases, making mutual nuclear warfare the nash equilibrium.

Of course you fucking take it, you have a 1/6 chance of winning with your card and he either has a 3 or 4 card.

>>7910286
It's okay anon we were all in high school once

>>7910287
Fuck off troll

>>7909572
He could have a 4,3 or 1. With a 4 card, he still has a 1/3 chance of losing.

Usually, these problems include something about both players having perfect logic, and thats the crux of this one. If they each have perfect logic, and know the other has perfect logic, then these posts are all correct
>>7909793
>>7909806
>>7910147

So the whole premise of the problem is illogical. Without the assumption of perfect logic, the problem becomes unsolvable by pure logic.
Theres a problem with the problem. Kind of like..... so you have this square circle.....
only more subtle

>>7910327
without the assumption of pure logic the problem becomes extremely simple

>>7910297
He can't both have a 4 and make a deal, unless he's a retard.
That would mean he excepts you to accept the deal even if you have a 5 or 6 in your hand. But it's obvious you wouldn't want to change your card if you had a 5 or 6