Jan

20

I suggest that learning to play a game (poker, backgammon, chess, checkers or go) might teach far more than studying game theory. The big problem with drawing boards is that there's no opponent, so ideas are never subject to quite the same level of criticism, and they do not have to be quite as relevant to the very serious matter of winning.

Adi Schnytzer comments:

Game theory is not about drawing boards. People do not study game theory to help them in their game playing, believe it or not. They study it in order to understand the process of more perceived importance than board games.

Nigel Davies adds:

Please excuse my ignorance, I am a mere player. So what exactly is 'game theory' good for? And I'm talking a usable practical application that doesn't include getting a salary for teaching it to others. Please be very specific as I am very primitive.

Adi Schnytzer replies:

I recently posted the following note, which will introduce you to game theory and comment on its uses. Since it's written by the masters, it should help you out. There's nothing I can add to their wisdom.

Bob Aumann's Nobel Prize Lecture ("War and Peace") and his piece "On the State of the Art in Game Theory" are both worth reading … He also has a piece called "Consciousness," which is rather nice. These may all be downloaded here … In my view, the least (not non-mathematical) and most intuitive text available is Luce and Raiffa.

Nigel Davies adds:

There is still the problem of practical application which is what I've been going on about from the start.

In 'A Beautiful Mind,' we see that Nash figures that he and his friends should not go for the blonde because they will block each other, and somehow or other this later got him a Nobel Prize. However, it seems that Nash thought up his 'strategy' without any knowledge of the game, and from all indications, he was a virgin at the time. This sums it up - he thought he could win without any knowledge of how the pieces moved.

In a previous discussion, I brought up a similar error by a mathematician who gave a figure on the number of possible chess games. It's obvious to anyone who actually plays and knows the rules that the number has to be infinite. The guy was so arrogant and/or naive that he didn't bother to learn the rules properly before coming up with his number.

Frankly, I have the same problem with Robart Aumann's paper. It's all very well theorizing about peace, but has he actually tried to apply this? I suggest that without knowing the territory, too many assumptions will be wrong.

If it's any consolation, it seems that Lasker had a similar problem with Einstein and the theory of relativity. In Einstein's foreword to Hannak's biography of Lasker, you see that Lasker thought that there was no justification for claiming that the velocity of light in a vacuum would be infinite, unless this had been verified in practice.

This, incidentally, was one of my few moments of agreement with the Elizabethan ghost.

Ross Miller comments:

It is worth noting that the "real" John Nash never did this, just the John Nash invented by a screenwriter who got to write this movie based on his ability to write Batman movie screenplays. The example in the movie is not a Nash equilibrium. In a Nash equilibrium, you do the best you can taking everyone else's actions as given and ignoring responses to your own actions. If everyone else goes for the inferior females, you make a beeline for the superior one in a Nash equilibrium. As stated, this game has no Nash equilibrium if everyone believes that multiple hits on the same target generates no payoff from that target, but a single hit will. Nigel is correct in pointing out that solutions to this game require thinking beyond the game theoretic formalisms.

The best reason for the Nash equilibrium to get a Nobel Prize was that it facilitated the Arrow-Debreu work on a competitive equilibrium. It was because his equilibrium is an intrinsically competitive (and not collusive) concept. The screenwriter is not to be entirely faulted since the book from which the movie was based is full of technical errors and misstatements. Of course, technical correctness does not make for bestsellers and the average moviegoer is never going to understand what Nash did anyway, nor is much of anyone for that matter.

Peter Grieve offers:

My take on game theory (based on long but elementary study) is that:

1. It's not very useful in sequential games like chess, poker, etc. In chess it might help a computer make decisions based on a look ahead tree if the branches have some evaluation number. Game theory can't, of course, actually generate these evaluations, and they are quite important.

2. It's not very useful in games in which anyone has any experience. The simplifying assumptions are too great. Once in a while it could illuminate a connection that would not otherwise be obvious. But as far as selecting a detailed strategy in a real world, complex game, it would be madness to rely on game theory.

Game theory is a lot like the rest of applied mathematics. It's really strong on the simple stuff, things where its many simplifying assumptions are valid. It can act as an initial guide when there is no experience in an area. Occasionally it can suggest something new in known areas (which must then be extensively tested by experience, and often found lacking).

The problems arise when academic folks (who mostly talk to each other) get inflated ideas about the real world strength of their ideas.

An example of a situation where game theory would be valuable is the following. Suppose you where playing a game of Rock-Scissors-Paper with a really smart, vastly superior opponent who knew a lot about your mind. How can you at least break even in this game? Game theory tells us the answer. Roll a die, if it comes up 1-2, choose Rock, if 3-4, Paper, if 5-6, Scissors (roll the die in secret, of course). You can even tell the opponent that you will use this selection method, and it doesn't help him beat you (unless he can guess the way the dice will come up). He can use this same strategy on you, making sure he breaks even, and the game is at equilibrium. This seems intuitively obvious, but what if Rock breaks Scissors wins double? What sort of die should one roll then? Game theory will tell us.

Of course if Nigel reads this, he will immediately think of several possible strategies to bamboozle game theoretically inclined, mammoth brained opponents in Rock-Scissors-Paper. But if he is to win anything, he will have to bluff the opponent out of using the above strategy (perhaps by artfully convincing the opponent that his (Nigel's) mind is "primitive").

During the Cold War, everyone wanted to hire Air Force generals with lots of nuclear war experience, but there were none (General Ripper was long gone). The think tanks used some game theory. Thank goodness we never found out how valuable it was.

Adi Schnytzer comments:

Three points only:
1. Game Theory was used successfully to win a battle in the Pacific during WW2, though I don't have the details on hand.
2. Without game theory, a simple dumb computer would never have beaten the World Chess Champion!
3. Aumann's insights on war are useful, but make sense only to those living somewhere nuts like the Middle East. Those in cocoons who believe that the problem rests in a failure to love their fellow man (read: "Liberal Europe At Large") will never understand.

Nigel Davies adds:

Without game theory, a simple dumb computer would never have beaten the World Chess Champion!

How do you come to the conclusion that 'game theory' should take the credit? Why not Faraday, Edison or Graham Bell? As far as I know, none of the programmers studied game theory, but there were a few chess players on the Deep Blue team. If game theorists are claiming this, then by the same token shouldn't one be able to claim that the big bang was only possible thanks to physics professors? Now that would really be a feather in their cap - they might get two Nobel prizes!

Aumann's insights on war are useful, but make sense only to those living somewhere nuts like the Middle East. Those in cocoons who believe that the problem rests in a failure to love their fellow man (read: "Liberal Europe At Large") will never understand.

Ghengis Khan would probably have sorted the Middle East out in no time - old Ghengis was a good player in his day. OK, I guess you're going to claim that the Mongolian hordes had their own 'game theory' which enabled them to win their battles etc. So the academics can take the credit after all …

Game Theory was used successfully to win a battle in the Pacific during WW2, though I don't have the details on hand.

As should be clear from the above, I think specifics are needed in order to see why game theorists are taking credit for this one and why it's good shooting with one's howitzers, or even luck. And how many battles were lost by the way? Or weren't these retrospectively scored?

Stefan Jovanovich adds: 

There are only two reasons why the Americans had any chance in the Battle of Midway:

(1) Admiral Nimitz trusted his Navy code breakers and their analysis of the limited decryptions they had under Commander Rochefort. By translating messages and studying operational patterns, the code breakers predicted future Japanese operations. Relying on those predictions, Nimitz sent to sea the only three American carriers he had at Pearl Harbor and positioned them on the flank of the predicted Japanese line of attack.

(2) When an American scout plane sighted the Japanese fleet, Admiral Spruance put all of the American planes in the air for an all-out attack. In terms of conventional doctrine at the time, this was a highly suspect move, and its initial results were terrible. The Japanese fleet's air cover fighters and anti-aircraft gunnery annihilated the attacks by the Marine Corps scout bombers, Navy torpedo bombers, and U.S. Army Air Force torpedo-carrying "Marauder" bombers. The Army Air Force "Flying Fortress" high altitude bombers also failed but did not suffer any losses. The next attack by Navy torpedo bombers was literally wiped out; there were no planes and only one pilot survived. Only the last attack - by Navy dive bombers - succeeded.

If "game theory" includes cryptographic analysis, then its contribution to the Pacific War effort was, indeed, invaluable; but it required the willingness of Admiral Spruance to go "all in."

Adi Schnytzer replies: 

Thanks Stefan. No, it wasn't the cryptography I had in mind. According to Careers in Mathematics,

Game theory, a part of operations research, was used to select a strategy for the Battle of Midway, a turning point in the Pacific arena during World War II. The U.S. Navy was on one side of Midway Island, and the Japanese Navy on the other. We calculated our probability of winning in the four cases of our going north of the island or south of it, and the same for the Japanese. Game theory was then used to select the winning strategy.

As I recall, breaking the codes told the U.S. where the Japanese fleet was going, and game theory told them how to place their limited resources optimally. But since this isn't nearly as important as winning a chess game, why are we bothering?


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