Here is a rehash of some of my thoughts from the recent past. I had posted this on my blog some time back and thought it might provide some food for thought.
What is an interesting problem? A subjective question at best. What is interesting to some could be dis-interesting to others and vice versa. Understanding how people react in different situations is an interesting but complex problem. Abe Lincoln once said "When I'm getting ready to reason with a man, I spend one-third of my time thinking about myself and what I am going to say and two-thirds thinking about him and what he is going to say". This quote is inspiring in the sense it made me think a little bit harder on something that is infinitely simple yet with no solution for all practical purposes.
Let us think of a game with two players. Each player is asked to write either "a" or "b" on a piece of paper. If either states "b" the game ends there and the hypothetical pay off is 0. But if both write "a" the game goes moves on to the second stage. If the game proceeds then the setting is as follows. Each player is asked to write down the largest possible integer that they can think of. The winner gets a hypothetical $1000 and the loser gets $ 500.
A simple game one should think! Well if we think about it, it is obvious that both players will choose "a" in the first stage since $ 500 is strictly greater than zero. The problem really is in the second stage. Here a unique solution does not exist. Both players cannot play the best response strategy because the best responses are not defined. Example: say if player 1 is thinking of writing 999, he knows that player two will play 1000, which makes player 1 think of playing 1001 and then the cylce goes on and on. The problem is a technical difficulty in optimization theory. It is a mathematical impossibility to arrive at a result because we are maximizing over an open set.
Can we find a game as trivial as this? Just choose a number and the person choosing a bigger number wins. A seemingly innocous problem but in reality unsolvable at this level. Note that it is quite possible someone might say that they dont care about winning $ 500 dollars as compared to $ 1000. Let me assume that one always wants to win which is fair enough (dont u think?).
Monday, October 30, 2006
Food for thought
Posted by Mad Max at 5:16 PM
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7 comments:
If I were to compete in the number game, I would follow the following strategy.
I think it is in a way determined by how much time is allotted to write down the number. Let us say, it is 60 seconds. I initially resorted to saying infinity, but then, it is not a quantifiable number. So, I thought of writing 99999999.... until time permits. But then, I thought exponents work good. So, one could write 9999^9999 (there is an optimization problem on where to use the exponent symbol which we shall ignore for now..may be space numbers and put the exponent symbol at the end) and so on. One could get to hexadecimal arithmetic and use F (16) instead of 9 to get the maximum bang for the buck. What is the probability that the other person employs the same technique? Then it comes to how fast a person can write... If you want to be creative, you could use a pen ink which doesnt dry for atleast 60 seconds in which case you can fold the paper multiple times to create more numbers out of the written numbers....Yet another random rambling of mine..:)
I do agree that some of the simplest of the problems are the most difficult to solve.
The term "infinitely simple" was interesting. How can something be infinitely simple, if it is not finite..;)
just figured that the exponent sign after the first digit gives the maximum value...So, w^xyz is always greater than wx^yz or wxy^z or wxyz (w,x,y,z > 1). So, F^FFFFFF... will be bigger. Considering that 9 can be written faster than F, there could be a trade-off...but, what the heck...:)...With the halloween celebration within the company, I am getting to relax after a loooooong time..
@ Mindframes: Okie so your proposed solution is set a time limit on the game. The mechanism which can unravel the problem would be the time limit. I agree that makes a lot of sense. But the only thing which intrigues me is that here we seek to differentiate the winner not based on his ability to pick the larger number but rather on his ability to write faster. Am I correct? If yes then I would say that it defeats the entire purpose of the experiment because test of writing speed can be done by more direct experiments.
But I definitely think that the problem can be set up in such a way that a unique answer can pop up using the time limit mechanism.
BTW: Infinitely simple...hmm..really dont know what i was thinking when i came up with that. Anywayz did a google search and landed on this one... "Thomas Aquinas, in Summa Theologiae, wrote that because God is infinitely simple, he can only appear to the finite mind as though he were infinitely complex" ... whatever that means...heheheh
I have to say.. I don't understand the crux of this blog.
Does it have anything to do with NP or NPN complete problems?
@ Brainwaves: Not really. It is more of a game theoretic problem and a simple property in optimization theory which makes it next to impossible to choose thelargest possible integer.
A Problem is only as complicated as we want it to be. Complexity boggles the human mind. Is this not why we rely on assumptions and abstractions? For example, "...with all other things equal..." or "ceteris paribus" when we know for a fact that all other things are not equal.
@ Bumblebee: I surely agree that we need to impose restrictions...in a way I guess ur point is more connected to "bounded" rationality...
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