Proof of Sklansky's theorem?

[SoBeDude]

I have deciphered many proofs of different complexity.

So have I.

Show me the proof to the four color theorem.

-Scott

[SoBeDude]

yes, Fermat's last theorem is another example of a theorem without a proof...or at least it didn't have a proof for over 350 years.

I never saw it referred to as Fermat's last conjecture.

[Ralle]

I haven't seen the proof of the four color theorem. I don't
even know if it's a formal proof, or a computer search, though
I would imagine that it is a combination.

[Ralle]

Theorems are claims (usually of a mathematical kind) that can
be proven through logical (not necessarily formal logic) reasoning
within an accepted framework, i.e. based on a number of axioms
that are assumed to be true.

Sometimes the word law is used as synonymous with theorem, but
often a law would be something of empirical nature. E.g. in
physics there are a number of laws, which are not proven in the
mathematical sense, but are accepted as truths because they fit
other accepted theories and are consistent with observations.

[mostsmooth]

"Theorems are claims (usually of a mathematical kind) that can
be proven through logical (not necessarily formal logic) reasoning
within an accepted framework, i.e. based on a number of axioms
that are assumed to be true."
you say "can" be proven, are you saying they havent been proven yet ? Because if they have, then you wouldn’t use the word “can”, you would use the word “have”, no? somebody else used the word “can” as well, that’s why im wondering. If you are saying theorems “can” be proven, but don’t have to be, then I think he is ok in calling his “theorem” a theorem and proof is not necessary, no? and what the hell is this 4 color thing ive heard so much about?? [img]/forums/images/icons/cool.gif[/img]

[BenD]

Premise 1: Every play in poker has a mathematical expectation.

Premise 2: To get the most accurate calculation you must know both your hand and your opponent's hand (though you can easily "put" a player on a hand and estimate from there).

Premise 3: When choosing an action in poker (check, bet, raise, fold, etc) to play perfectly you must choose the play that has the highest mathematical expectation.

If these premises are accepted, then Sklansky's Theorem seems to follow:

"Every time you play a hand differently from the way you would have played it if you could see all your opponents' cards, they gain; and every time you play your hand the same way you would have played it if you could see all their cards, they lose. Conversely, every time opponents play their hands differently from the way they would have if they could see all your cards, you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose."

Sklansky's theorem doesn't guarantee short-term positive results. However, if you play a hand 1000 times the chance of not seeing a benefit to using the theorem are extremely small.

This is taken from the ualberta poker pages:

(http://www.cs.ualberta.ca/~darse/msc-essay/node10.html)
"The Fundamental Theorem is stated in common language, but has a precise mathematical interpretation. The expected value of each decision made during an actual game can be compared to the expectation of the correct decision, based on perfect information. Each player's long term expectation is determined precisely by the relative frequency and severity of these ``misplays''. On average, a player who makes fewer misplays than her opponents will be a winning player. The theorem may appear to state the obvious, but has many subtle implications to poker strategy, some of which are illustrated in the text."

End quote.

[SoBeDude]

As with Fermat's the four-color theorem is very old. No formal proof exists. It has been proven via computer however.

-Scott

[SoBeDude]

and what the hell is this 4 color thing ive heard so much about??

Well since this isn't a forum on advanced math, there's no need to go in to the 4-color theorem. I was using it as an example of a theorem without a proof. Fermats was another example of the same - until recently.

-Scott

[BenD]

The four color map problem essentially deals with how many colors are needed on a map so that geographic area of one color is adjacent to a geographic area of the same color. Take a map of the 48 US states, for example. You can use four colors and color each state so that the no two touching states are the same color. This idea was long thought to be true, but was generally unprovable until recently with use of computers.

[Ralle]

Yes, intuitively the theorem seems to be correct. And like you
quoted in the above post, even though it may seem trivial, the
implications are profound, and I think easily overlooked.

But, one interesting aspect, as is mentioned in The Theory of
Poker, is that the theorem doesn't necessarily hold in a multiway pot.
Now this is interesting. The question is what the fundamental
underlying reason is. Were there a proof, this would surely
demonstrate why caution is advised in multiway situations.

Also, it is interesting that even though the theorem seems to
be true intuitively, it doesn't hold in all situations. This is
why I would prefer a proof. Sometimes intuition is a reliable
guide, sometimes it is not.