Those of you who got the reference might be able to guess what I *hope* this thread can be dedicated to. I had the idea of thinking of questions with no apparent single correct answer. While its fun and all to debate this with myself, its truly gratifying to get a discussion going with several people.
So here's the deal: I'm going to post a question, logic puzzle, whatever every so often (Probably every week or so, depending on my personal schedule). Some of these might be things I've thought of; others may be obscure questions I find around the web* or in books. Post on here with what you think the "correct" answer is. Debate the merits of each solution posted. At the end of the time period, I'll look at all the solutions and pick out which one I feel is the best, which is not necessarily the one I most agree with (hopefully I can be objective enough to keep to this). I'll consider the effectiveness of your argument, as well as the uniqueness of your answer. The winner will get a cookie and, time permitting, I'll provide comments on the top solutions.
This thread may sputter and die, but for now lets see where it goes. The first question is below, and it falls into the category of a logic problem, but I'm not sure if it necessarily has an clear answer**
Imagine, for a second, there are two dwarves with respective piles of gold coins. Each dwarf is perfectly logical and realizes that the other is perfectly logical. In addition, each is infinitely greedy and wants to maximize the amount of gold coins he has. Dwarf A has some large, arbitrary amount of coins, and dwarf B has 2 fewer coins than that amount.
They are placed in this situation. The dwarf with the smaller amount of coins may remove 3 coins from the other's pile and then add one to his own. The other 2 are vaporized (just go with it). So if A = 102 coins, and B = 100, B could remove and add so it ended at A=99 and B=101. A could, if he chooses, then do the same thing ending with A=100 and B=98 This process continues until the dwarf with the lesser amount declines to remove coins. Then they leave with whatever they have left
Given that each dwarf wants to maximize their personal stash, and is unable to deviate from the confines of the rules of the situation, what happens? How many "processes" are gone through? Are any gone through? Explain your answer.
Have at it.
*Web questions are hard, cause then people realize they have a close personal friend named Google.
**Please also tell me if my questions are lacking, or if I missed something that actually lends it to be more formulaic and less philosophical.
Given that A and B are rationale and B has the sole choice of initiating the process, B will do so and take coins from A. Dwarf A will then declare the process at an end.
Look at it this way -- is it rationale to alternate taking coins to reduce the piles to nothing? No. Given that, it would be rationale to do it once, if you move first, knowing your opponent will not behave irrationally and continue the process.
z
I can't spare a moment for the dog faced boy I won't lend another hand to the worm girl of Hanoi Don't deplete my oxygen for the guy who's turning blue But ask me, and I'll do anything for you
Assuming this isn't laid out on parchment for them, they'd probably go though 2-4 repetitions before totally stopping. It's a losing proposition for both of them to keep going. Being logical beings should prevent them for continuing when they see no one is going to ever win.
If it was laid out (such as this), I don't think either would ever take any coins from the other.
Given that A and B are rationale and B has the sole choice of initiating the process, B will do so and take coins from A. Dwarf A will then declare the process at an end.
Look at it this way -- is it rationale to alternate taking coins to reduce the piles to nothing? No. Given that, it would be rationale to do it once, if you move first, knowing your opponent will not behave irrationally and continue the process.
z
I think this question has an issue with the definition of "perfectly rational".
"Perfectly rational", in this context as I've seen it used, has meant "they will always make the decision that has the best value for them, where 'best' takes into account only the valued objects specified in this situation". (So none of them will take into account things like, agreeing that "we won't take coins from each other now and will never take any coins from each other in future occurrences, and we'll invest the 202 coins in a startup where we each own exactly half". This, by the way, is my gripe with situations like this, as such an agreement would be a lot more rational and a lot more profitable than this.) However, in this case, there is no best value - each choice, "take" or "quit", has the advantage that you end up with a coin, and the disadvantage that you give the other guy the chance to make you lose coins. So the debate here is whether the advantage or the disadvantage has the higher value.
That said, I would say that neither dwarf would take any coins. Dwarf B, who chooses first, has the ability to gain one coin and take three from A, but in doing so, gives A the same choice, thus if B chooses to take one coin, he takes on the risk of losing three.
Now when it's said that "both dwarves are perfectly rational, and both know that the other is perfectly rational", I take that to mean that since one of these choices is best to a "perfectly rational" person, whichever choice dwarf B considers correct, dwarf A would also consider correct, and both know this. Also, I don't believe that the number of coins they have matters - what matters is the potential gains and losses. That is, that each dwarf would make the same decision when they have 102 and 100 coins, as 1002 and 1000, as 12 and 10.
So if my inferences in the above paragraph are right, then dwarf B, who chooses first, would choose to keep his coins and leave. If, as I said above, both dwarves would make the same choice in the same situation, then if dwarf B chooses to take 3 and keep 1, then it must be the best choice, which means that dwarf A would do the same, and both know this. That means that dwarf B choosing to take 3 and keep 1 causes dwarf B to lose 2 coins, and he knows this. Since dwarf B is perfectly rational and infinitely greedy, he will choose not to lose 2 coins, thus he will choose to quit immediately.
Ah, three responses and 3 different answers. This is starting off marvelously.
Ambitious](So none of them will take into account things like, agreeing that "we won't take coins from each other now and will never take any coins from each other in future occurrences, and we'll invest the 202 coins in a startup where we each own exactly half". This, by the way, is my gripe with situations like this, as such an agreement would be a lot more rational and a lot more profitable than this.)
Well, yeah, but logic puzzles aren't supposed to make sense wrote:
(So none of them will take into account things like, agreeing that "we won't take coins from each other now and will never take any coins from each other in future occurrences, and we'll invest the 202 coins in a startup where we each own exactly half". This, by the way, is my gripe with situations like this, as such an agreement would be a lot more rational and a lot more profitable than this.)[/quote] Well, yeah, but logic puzzles aren't supposed to make sense
So if my inferences in the above paragraph are right, then dwarf B, who chooses first, would choose to keep his coins and leave. If, as I said above, both dwarves would make the same choice in the same situation, then if dwarf B chooses to take 3 and keep 1, then it must be the best choice, which means that dwarf A would do the same, and both know this. That means that dwarf B choosing to take 3 and keep 1 causes dwarf B to lose 2 coins, and he knows this. Since dwarf B is perfectly rational and infinitely greedy, he will choose not to lose 2 coins, thus he will choose to quit immediately.
Both dwarves do not have the option of making the same choice in the same situation. Dwarf B moves first with Dwarf A having taken no action against him. If Dwarf A chooses to play, it will be with the knowledge that Dwarf B took his coins. Completely different.
I think it's somewhat of a moral question. How much do you risk to retaliate against someone that has wronged you? There are people that would give their lives and/or livelihood to strike back at someone. Others will walk away. If I was Dwarf A, I would have no problem walking away if B took my coins.
z
I can't spare a moment for the dog faced boy I won't lend another hand to the worm girl of Hanoi Don't deplete my oxygen for the guy who's turning blue But ask me, and I'll do anything for you
Both dwarves do not have the option of making the same choice in the same situation. Dwarf B moves first with Dwarf A having taken no action against him. If Dwarf A chooses to play, it will be with the knowledge that Dwarf B took his coins. Completely different.
I think it's somewhat of a moral question. How much do you risk to retaliate against someone that has wronged you? There are people that would give their lives and/or livelihood to strike back at someone. Others will walk away. If I was Dwarf A, I would have no problem walking away if B took my coins.
z
"Perfectly logical" means that if there is a best choice, they will take it.
It also means that they both will come to the same "best choice" in the same situation.
If Dwarf B stole (I'll use the words "stole" and "quit" to avoid ambiguity, without intending any of the moral connotations normally associated with those words), then it must have been the most profitable choice for him. The only thing that could make it a different situation for Dwarf A is that in Dwarf A's case, Dwarf B has already chosen to steal. So the question is, does that make a difference?
But, both of them know that the other is perfectly logical. Dwarf B, using his perfect logic, would envision a situation where the roles were reversed - where A had fewer coins, and thus got to choose first whether to quit or steal. Dwarf B, knowing that A was perfectly logical, and knowing that if he (B), a perfectly logical dwarf, chose to steal when picking first, then so would A. Thus, just as Dwarf A going second knew that Dwarf B was willing to steal, Dwarf B going first knew that Dwarf A was willing, thus there is no difference. So Dwarf B would quit.
"Perfectly logical" means that if there is a best choice, they will take it.
It also means that they both will come to the same "best choice" in the same situation.
If Dwarf B stole (I'll use the words "stole" and "quit" to avoid ambiguity, without intending any of the moral connotations normally associated with those words), then it must have been the most profitable choice for him. The only thing that could make it a different situation for Dwarf A is that in Dwarf A's case, Dwarf B has already chosen to steal. So the question is, does that make a difference?
But, both of them know that the other is perfectly logical. Dwarf B, using his perfect logic, would envision a situation where the roles were reversed - where A had fewer coins, and thus got to choose first whether to quit or steal. Dwarf B, knowing that A was perfectly logical, and knowing that if he (B), a perfectly logical dwarf, chose to steal when picking first, then so would A. Thus, just as Dwarf A going second knew that Dwarf B was willing to steal, Dwarf B going first knew that Dwarf A was willing, thus there is no difference. So Dwarf B would quit.
Ambitious, I know you're logical. If you were Dwarf A, would you continue and steal back from Dwarf B knowing that you could lose even more? If so, given that this is the logical thing you would do, the correct move for Dwarf B is to steal from your pile knowing you won't retaliate.
z
I can't spare a moment for the dog faced boy I won't lend another hand to the worm girl of Hanoi Don't deplete my oxygen for the guy who's turning blue But ask me, and I'll do anything for you
