Chess Player's Dilemma

Imagine you are to play a game of chess, and if you lose, you die.

You are to play against the world's strongest chess player - Komodo 9 (ELO: 3324). However, you have the aid of the world chess champion - Magnus Carlsen (ELO: 2853). On each move, Magnus will suggest you a move, and it will be up to you to follow the move or make your own. The games are to continue until there is a victor.

Will you trust Magnus your game (and life)?

The dilemma arises when one considers the power difference. You, most likely, will always lose to Magnus. Magnus, most likely, will always lose to Komodo. Magnus, however, may somehow pull off a draw against Komodo - you, however, probably won't even do that with Magnus. The point is, Magnus will probably (if not certainly) never defeat Komodo. So the catch is, no matter what move you think of, it is most likely worse than Magnus's - and no matter what move Magnus thinks of, it's most likely worse than Komodo's. So, knowing that most of the time ("most" because sometime the best move is obvious - like moving the king out of check on the only possible square) there is a better move than what Magnus will suggest, will you follow Magnus - who is certain to get you killed, or make your own moves - which would get you killed if you were playing against Magnus?

HOW DO YOU WIN?!

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Comments

Let's say that it's my turn to move, and there's 100 possible "plausible" moves to make (i.e., not counting nonsensical moves no chess champion is likely to ever make, man or machine). Let's say 10 of them are on the path to victory over Komodo 9, and I need to make one of those. Magnus suggests move #59. If I reject his advice, and pick another move, the probability of making one of the 10 good moves is about 10 in 100, or about 10%. If Magnus makes his moves with this kind of probability, there is no chance he'd have the chess rating he has now. Most likely, he hits one of the 10 good moves most of the time to even be at that level--it only takes one failure to do so to lose the game to something like Komodo 9. Hence, I do not improve my odds by "not following the man that has poor odds of winning", if he happens to be far better at it than I am. I improve my odds that way only if Magnus picks a good move less than 10% of the time--a super, or champion, loser.

Michael Mendrin - 5 years, 10 months ago

That's all great, until you realize that following Magnus is guaranteed to get you killed.

Also, in Chess there's almost never more than 3 good moves - not to say anything about TEN. But I get what you're saying. But you're still going to die ;)

John M. - 5 years, 10 months ago

No, that is my point. A "super loser" is one that is "guaranteed" to get you killed. Then you don't do what he does. But Magnus is not a "super loser". He's simply just a lot worse than Komodo 9.

@Michael Mendrin – Oh but he is. He is totally a super loser. He can never win - only draw. Maybe. Very unlikely. I.e. - when he doesn't, you're dead. I only use uncertainty words because I'm just being scientifically correct and am acknowledging probabilistic miracles - like Magnus's brain neurons suddenly arranging themselves in such a combination that make him a 4000 ELO player for an hour... or some crazy thing like that.

But alright for sake of analysis let's just say A always loses to B and B always loses to C. ALWAYS. And you are A, with assistance of B, playing against C. What do you do?

@John M. – Let me think about that a bit. Whenever one says "always", it changes things.

@Michael Mendrin – Whenever someone says "always" or "never" or any other absolute without inserting"assuming" in the same sentence, I get a 404 Error in my brain. But yeah just so you know when I say "most likely" you can assume "always." Like, I won't say "You will never wall through your chair as you're sitting" - because I heard Brian Greene say if you push against a wall long enough, quantum stuff, you will fall through. Ever since then I became a paranoid for whom never is a never. But... just for reference... I will say "+c chance" instead of "most likely" when I mean something absolutely ridiculous - it's like the epsilon when finding limits... or whataever. Bottomcase c because capital C is taken ;)

But yeah personally what I'd do is drink a lot before the match - so that I can then unleash the biochemically processed nutrient-rich yellow liquid on the damn computer. Good luck calculating now ;)

@Michael Mendrin – I haven't even seen it yet, but I'm sure you'll like it.

@John M. – You're back! Yeah, I've known about the Banach-Tarski paradox for quite some time now. Now if I could only apply that to money.

@Michael Mendrin – LOL right?? That's why I chose engineering. I can do what I love while not worrying about going broke. But that is only temporary - you know, for "sustainability." Real money comes from business. But business is risky - hence one must first seek sustainability. But I have a strong feeling I must establish my OverLord Corp. before 2045... because it seems then all goes to sh!t. Besides the avatars, there are mad predictions about computing power and AI, and general technology - with quite reasonable hypotheses about the Technological Singularity. In this era, money DOMINATES like never before. But it is also great to be on the team who builds those crazy things - the team of engineers ;)

@John M. – In regards to players A, B, C, where B "always" loses to A, and C "always" loses to B, it's possible to have a game such that C will always win against A. For example, Rock, Paper, Scissors. Obviously, my choice which "always" loses against B will "always" win against A. "Always" supposes that there's never any draw.

But this is not true for "any" game.

@Michael Mendrin – Ah but it doesn't work like RPS. Here it ain't a double-edged sword; it's a ladder. That's like saying if A can lift 200 lbs, B can lift 100lbs, and C always lifts less than B, then C can lift 200lbs. Lol. I guess we can put things in better perspective. Take a 1,000 ELO player. Have him play Shredder 12 on hard - ELO 2000. Then have Shredder 12 play against Houdini 1.5a on a crappy computer - ELO 3000. The human will always lose to Shredder, and Shredder will always lose to Houdini. I'm not even joking about this - I'm personally around 1600, and I can NOT beat Shredder. Like, at all. I think once I pulled off a draw - after several takebacks, that is. So if 400 pt difference is enough to put you within the "always" range, 1000 puts you beyond it - super-always. Lol. No contest. Just like lifting.

But I have the solution, if you're interested.

@John M. – I'm ALWAYS interested if you have a solution.

@Michael Mendrin – I see what you did there ;)

Alright so here it is:

Obtain a Time Chamber.
Place an electronic magnetic chessboard, a floor, a table, some oxygen, water, bananas, two computers - one loaded with Shredder 12, one loaded with Komodo 9 and linked to the chessboard, and one monkey.
Teach the monkey the rules of chess, and how to move pieces.
Wait.

In time, all chess games possible that can be played will be played. Thus, if $P(t)$ represents the probability of the monkey winning the game after elapsed time $t$ , then as $t \rightarrow \infty$ , $P(t)$ =1, and thus, all we have to do is let the magic happen. Now the major problem, of course, is not the simple task of building a Time Chamber - but the task of finding an infinite source of oxygen, water, bananas, and... monkeys for replenishment. OR. We could just give the monkey Avatar D, and a source of electricity, and all is good.

(Inspired by the typewriting monkey).

@John M. – First, the objective seems to be being able to beat Komodo 9 at all, ever, however many games it requires for such a victory. Given that slight change in objective, your scenario of a monkey playing at random over sufficient time depends on the fact it's possible that the monkey could play in such a way that Komodo 9 will find it unbeatable. That can't be assumed. It is LIKELY, given that Komodo 9 is just a finite computer and could play less than optimally, but it's not necessarily a theoretical fact that the monkey WILL eventually hit a combination of moves that Komodo 9, no matter how well endowed it is, will find unbeatable. Tic-Tac-Toe is analyzable enough to be able to answer this question (it will at worst be a draw for Komodo 9), but we don't know enough about the game of chess to be able to answer that. That is, we don't know if, given perfect chess players A and B, white always will win, or black always will win, or it will always end in a draw.

@Michael Mendrin – Ooh I had a big debate over the "PERFECT CHESS GAME." Numberphile made a video about estimating the total number of possible chess moves (are you subbed? they're awesome. Vsauce even awesomer). I said the moment quantum computers become consumer products I will load Komodo (or whatever the strongest engine at the time) and have it play the PCG (Perfect Chess Game). Now inevitably it is going to be a draw. I enough chess to figure it out. As long as the game starts out evenly, it will end evenly. Chess is not about making the right moves - it's about not making the wrong moves. As long as neither side obtains advantage over the other at any point throughout the game (hence, the other side will play "perfectly") - the game will draw. Make one mistake, however, and it becomes very easy for a strong player to maintain advantage and win - saying nothing about the "PERFECT" player. But the guy I was arguing with argued that given the number in the vid, such a game will be impossible to play. Then the arguer told me about EXPTIME - some weird mathematical entity about some really long time, and chess was in that exptime somehow. I argued that by the ability of the algorithm to trim TREE LINES (a tree line is a continuation of all possible moves for a specific move - say, queen takes pawn - then analyzing all possible games after that. By recognizing this as a loss, the calculation is terminated. It is, nevertheless, important to note that this termination must not occur instantly - for there are smart sacrifices in chess. And hence the dilemma in the algorithm and trimming computational needs). This is how Houdini 1.5a, on my crappy computer, can demolish Deep Blue with 15,000 moves per second analyzed vs DB's 20,000,000. But ultimately, via Bremmermann's Limit (look it up), if we take... like... 0.0001% or something of the Universe's mass and turn it into a computer, we can have googol bits per second computing speed. Maybe then we can do it? I don't even know how many bits is required per a chess flop (a flop is an analyzed position). But if that is not enough, we can get matter from other universes and turn THAT into the computer. If that won't cut it, we'll discover/invent higher dimensions and dramatically increase computing power. Ultimately humanity shall conquest existence in search of computing power capable of playing PERFECT CHESS.

As far as tic-tac-toe goes, we already know how to never lose. No brainer here. Also, I see you aren't familiar with the typewriting monkey. Basically, a monkey typing at random will, in time, type all texts that have been ever typed - infinitely many times. It will type Shakespeare's sonnets over and over - infinitely many times. You get the idea. Now with chess there's an additional complication of responding to every different position differently. This is preferable, of course, in middle games and endgames. However, in opening, it's a really bad idea to, say, never respond to e4 with c5 - the Sicilian defense. In fact, there will only be up to 400 games, max, if that is the case. Then, we need to give the monkey an opening book - so for the first about 15-20 moves, it will play the same stuff over and over. This may solve the problem. But yes, without repetition, eventually the monkey will get it right. But of course this is just theoretical hogwash - this is more impractical than falling through a solid chair at any given moment. Woah here I went again - look how much I typed. Better get outa here. Sorry for the longevity ;)

Cheers

@John M. – Intriguing choice of words, "longevity". I don't think I would have used that particular word in this context. I will get back to this later, after I've had my dinner and at least a few drinks. It's a deep subject.

@Michael Mendrin – Yeah I also thought it was awkward. But it's summer. I don't want to stress my brain fibers too much with grammatical perfection ;) Oh and something BIG coming up today or tomorrow. Hope I'll finish it by then!

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$