This problem is harder than it seems

nice

so guess 1/3 of the time? on average that'd result in this. problem though is if everyone passes you're guaranteed to lose but if 2 people guess theres still a chance you win, so the answer probably isn't this clean. I have no clue how to solve this shit. probability is hard.

well done

How did you model this by the way?

so it has to be assumed that the 3 people are smart enough to figure this out that this is the optimal strategy independently, and to assume that the other two are doing this also? does it even matter if one knows but 2 dont know? does it matter if one or two doesn't figure this out and how would you deal with that

1:729 I dunno,

Not that guy, but here's how you can do this: define a strategy to be a (measurable) function from the set of possible observations to the set of actions. In this case, a "strategy is simply a function s:{H,T}^2 -> {H,T,"pass"}. This is essentially the same definition of a strategy as in (economic) game theory, if you're familiar with that.

A "joint strategy" for the three players is then simply three strategies pasted side-by-side, essentially a disjoint union [math]s_1 \sqcup s_2 \sqcup s_3[/math].
You can then equip the domain with a joint probability distribution, which will generally be non-uniform because certain combinations are inadmissible (e.g. A and B disagreeing on the letter on C's forehead). Then the probability of winning, given a joint strategy [math]s_1\sqcup s_2\sqcup s_3[/math], can be found by evaluating the strategy function for each player [math]s_i[/math] for each admissible combination, and checking the proportion of them that lead to a win. Then the optimal strategy 's' is simply the win-probability maximizer over all joint strategies with [math]s_1 = s_2 = s_3 = s[/math].

In this case the strategy space is small enough that the entire process of "pushing forward" the probability distribution can be computed manually. There are tricks for speeding up the process on more complicated games, but that's not my bailiwick so I won't comment on them.

>woman stays quiet

That really does not help to find his solution.

>That really does not help to find his solution.
It does find the solution (in fact it finds ALL solutions, not just one). It just does so less efficiently since it searches the entire space of possible strategies, i.e. it is a "proof by exhaustion" instead of an assertion of a single strategy plus an information-theoretic proof of its optimality.

[soapbox]
Modelling a problem and calculating a solution are two distinct processes: the former is practical and the latter is theoretical. In general you cannot expect proficiency in either one of them to translate automatically into proficiency in the other.
[/soapbox]

Optimal strategy is to cheat

>Modelling a problem and calculating a solution are two distinct processes: the former is practical and the latter is theoretical. In general you cannot expect proficiency in either one of them to translate automatically into proficiency in the other.
OK, I see this point. I guess you are really taking my use of the word "model" in the most literal possible way, which is fine. I am interested in how one figures out that strategy which 99% of people are going to miss. The issue is that most people won't see the strategy if they put in a formulism or not. Just read this thread. Most people only saw two strategies (guess or pass), and did not further subdivide those into cases. So, in the important sense I care about, no it does not help you find a solution.

>First two people pass
>Third Person observes what is on other two peoples heads
>If both are the same, #3 should choose the opposite; there is only about 13 percent chance that all three letters are the same
>If both are different, #3 should choose the opposite of whoever flipped the coin second; only 25% chance of having the same letter.

I wanted to know how he subdivided the strategies so that he actually saw the correct one.

Just because it's "not optimal optimal strategy" does not mean it should be excluded when calculating the probability.

Nevermind, didn't read the last sentence. I think the "optimal strategy" part is purposefully ambiguous for unethical purposes...

>I guess you are really taking my use of the word "model" in the most literal possible way
The confusion isn't really over the word "model", it's in the word "this": what you wanted was a model for the solution, rather than the problem.

As I've mentioned, this really isn't my area of expertise, but the solution () mentions that strategies are subdivided by "information gain", which I understand to be essentially the same as the Kullback-Liebler divergence, or Fisher information metric. There are sketches of a combinatorial argument in there:

>each guess has a 50% chance of being wrong; this setup makes sure that all the wrong guesses are combined in a single game [...] whereas correct guesses are used maximally efficiently (only one correct guess per game)

Which reminds me of proofs in Ramsey theory, but I know even less about that than I do about probability, so I'm going to stop here.

This is so fucking easy. Each person just needs to look at the player on their right in the eye and say "I am guessing my letter is __________" and that guess is just the letter on their forehead. Everyone gets the strat, and we just guess the letter that our left hand player said, and we all win.

>You two have the same/different letter

>Therefore, no one says pass.
Two pass, one guesses. 50% win chance. If two guess 25%, if three 12,5%

To be more precise/concrete, you can (assuming sufficient time+space for computation) draw up the strategy space as a directed acyclic graph, then connect an arrow from strategy S to strategy T if they differ only in a single "action" (i.e., one player decides to make a different move, everyone else's behaviour is held constant) and such that the information entropy of the distribution increases (i.e., the Kullback-Liebler divergence is positive).
Then taking transitive closures results in a kind of filtration of the strategy space where "earlier" strategies converge to "later" ones, and optimal strategies as a kind of ultrafilter. Which I suggest corresponds to the subdivision you want.

Though if you just want to give the value of the upper bound (probability of winning under optimal play) rather than proving/constructing a strategy that attains this bound, you can use the pigeonhole principle:
>each guess has a 50% chance of being wrong
>the maximum odds ratio gain of 3:1
though I expect that in the general case, the proof of existence of an optimal strategy validating tightness of this bound will be non-constructive.

Yeah, you might have an interesting perspective, but once I saw the solution I noticed it is (a+b)mod2 (where the three players are abc). And again, all this high level thinking does not help the players find their optimal strategy. I just want an easy way to write down the strategies, and I think the easiest solution is linear Cellular Automata, which I am modeling now. Regards, RR.

I think we can model the payouts this way too, and then it is just algebra. The logical rules (strategies) are trivially (exclusive or, inclusive or, an, none, always).

Most people are only seeing Always and Never when there are 5 strategies a player can take. I'll do the payouts tomorrow. I'll probably throw something up on my blog (I do econ/game theory models) later this week regardless.

Sure you can, and yes it is "just algebra": as I've alluded to in an earlier post, all this is standard probability theory as applied to economic game theory, and your suggested logical rules are essentially a reproduction of those in (finite) probability theory.

In other words, nothing I've said in this thread is new, it's merely a formal model for the computations that we intuitively know how to do, though under different names such as "basic statistics" or "basic combinatorics".

k.

cool this is what i thought too. great minds think alike user.

>whitey gets cucked by thot and the nigger rails her for good
Did I do good?

For those who are following along, and want to see how to solve these problems, I orginally did not see al the strategies which is how I got into a productive discussion with Mr. A+.

You get all the strategies just by seeing that with three members each members only sees two bits of information which can be H or T (1 or 0). This is either a basic linear relationship mod2 or a logical gate. Either way is the same. I think putting the strategies in terms of logic is easier to understand. Given information from two sources of 0 or 1 you player can:
Always Play
Never Play
OR Play
XOR Play (the winning strategy)
And Play
Negative And Play (forgot about this one)
All players play the same strategy no matter what. So you plug in and get the expected value of the payouts.
I'm tired and have been working on a hundred things tonight. Take care, friends.

actullly, logic is the only way to get all the strategies, and I haven't gone through them all so there could be overlap
later, friends
making it isn't easy
it is destiny

>not hearing about the alien hat problem

youtube.com/watch?v=N5vJSNXPEwA

it's the same thing. You can always get at most one wrong if done right.

You've left out (either by accident, or by assumption in the way you model the problem) the case where the two sources are indistinguishable, i.e. you appear not to allow for strategies like which may have s(0,1) != s(1,0).

Here I'm using my preferred lingo for the strategy function s: {H,T} * {H,T} -> {H,T "pass"}, but of course you can draw this up as a logic gate or truth-table/lookup-table if you prefer.
Though the functional form solves your subproblem by making it clear how many possible strategies (i.e. rows in your lookup table) there are in total: 3^(2*2) = 81, or 3^3 = 27 if you forbid distinguishable players which effectively reduces the problem domain to {HH, TT, HT} = {11,00,10}.

That's completely different, retard

>you appear not to allow for strategies like which may have s(0,1) != s(1,0).
The problem doesn't. It is clear on the information given. I consider players acting irrational, without strategy with given information. This does spur me to add a strategy that is unnrelated: random. A player can act under some probability function.

>Though the functional form solves your subproblem (You) by making it clear how many possible strategies (i.e. rows in your lookup table) there are in total: 3^(2*2) = 81, or 3^3 = 27 if you forbid distinguishable players which effectively reduces the problem domain to {HH, TT, HT} = {11,00,10}
This is helpful.

Let's make an algorithm, Mr. A+. If it turns out to be original we can publish it.

Question says that they all apply the optimal strategy, ie they're perfect logicians.

I work almost entirely intuitively which has mayjor drawbacks (and some bonuses). A systematic thinker might balance me. Check out my PIP model and Auction model (including the comments when I figured out a huge flaw someone like you wouldn't have made). The math isn't original (I don't think, I actually don't read academic stuff), but the applications are.

steemit.com/@roosterred

My email is defunct and I have not got a new one yet.

>Let's make an algorithm, Mr. A+. If it turns out to be original we can publish it.
Sorry to burst your bubble, but none of this is original. As mentioned earlier, this is basic stuff that can be found in any decent game theory book. I'd personally recommend the one by Maschler, Solan, and Zamir since it achieves the rare goal of being a thorough reference manual for classical game theory, while still remaining readable even for people without formal training in advanced mathematics.

>I work almost entirely intuitively which has mayjor drawbacks (and some bonuses)
>The math isn't original (I don't think, I actually don't read academic stuff)

You really should: you don't have to delve into the gory details, but understanding the gist of the results is crucial if you want to do original applied work. For example, when you claim
>but the applications are [original]
They really aren't: the idea of prediction markets is at least as old as the invention of the Arrow-Debreu security, and the PIP model is basically a restatement of the efficient market hypothesis, and the problem you mentioned in the comments has been addressed by the (in)famous Black-Scholes model which appears in literally any book on financial derivatives.
Similarly, while I'm no expert on bitcoin, I know from the theory of auction design that "honest bidding" is not something to crow about (by the revelation principle) and any auction must by mathematical necessity yield to the Myerson-Sattherthwaite theorem (see e.g. pages.wustl.edu/files/pages/imce/nachbar/my-sat.pdf), which puts a hard limit on the potential gains of a completely anonymizable trading mechanism.
These are the sort of things that intuitive thinkers ought to read about, if nothing else so they don't end up re-inventing the wheel.

You don't understand what I do, Mr. A+. Which is OK. Have a good night. When I said the math wasn't original, which is all you think of, you didn't understand.

>These are the sort of things that intuitive thinkers ought to read about, if nothing else so they don't end up re-inventing the wheel.
When Euler and others gave solutions to better shapes of sails, they didn't claim they invented calculus. Applied math is applied math.

>the PIP model is basically a restatement of the efficient market hypothesis
Read this again. It is a model to test the efficient market hypothesis in both the strong and the weak sense. The point is, with CLEAN data, we can see how the markets scale as additional options are added.

>I know from the theory of auction design that "honest bidding" is not something to crow about (by the revelation principle) and any auction must by mathematical necessity yield to the Myerson-Sattherthwaite theorem (see e.g. pages.wustl.edu/files/pages/imce/nachbar/my-sat.pdf), which puts a hard limit on the potential gains of a completely anonymizable trading mechanism.

You are missing the point of Uncapped Blind Auctions. I said the math (mostly) wasn't original. The NEW part, which seems insignificant at first glance, is that the contract algorithm chooses the greatest fundable price. That is a small change, but changes all of the math. The most important part of the new blind auction math is that, given rational bidders, the final token/marketcap is underbid slightly. That means people, under rational expectations, will always want to bid. The blind, and well known math, guarantees that speculators will risk a huge price (speculation bubbles can still happen and are literally impossible to predict). That is the point.

Now I am off to bed.

-roosterred

If the other two have identical letters guess the opposite.
If they have different letters, pass.
This way they only fail if they all have the same letter, which is 1/4.
They have a 75% chance of succeeding.

By the way, despite anyones thoughts, my ideas are mine. I give credit when I know when to. If you don't. I promise I will correct any lies.

Each player individually has 3^4 = 81 possible strategies. So together they have 81^3 = 531441 strategies. Just calculate the winning probability for each of them and you will know the optimal one.

>You don't understand what I do
No, I don't. I guess originality is relative, which is why I can look at an original application of an unoriginal algorithm, and view it as unoriginal. Fair enough, I concede this.

>It is a model to test the efficient market hypothesis in both the strong and the weak sense
Which, in my line of work (economics), is not original: building such models is a routine textbook exercise. But that non-originality is of the model rather than the application to 'clean' data (not sure what you mean by this, is current market data not 'clean' enough?), so I won't belabour the point on 'originality' any longer.

>You are missing the point of Uncapped Blind Auctions
I probably am, since I'm not involved in cryptocurrencies in anyway and all these terms are unfamiliar to me (though you've certainly managed to pique my interest), which is why I could only comment on the auction in a generic manner. (I was originally going to reply with a point about speculation bubbles, until I saw your addendum, which I had originally missed.) In my line of work I occasionally deal with auctions, but from a regulatory perspective where speculation is the major problem, but I guess as a company you will have different goals.

Sorry for not being able to comment more on your ideas, but insights don't come as easily as I might have made it seem in this thread (the things I've suggested are "tricks" that I've accumulated from years of experience of mathematical modelling). Thanks for the discussion though, it's certainly one of the more productive ones that I've had on Veeky Forums. Good night.

Nah, I'm not going to press you to cite sources for your ideas (see above). After all, from a market-oriented perspective it doesn't really matter where the idea comes from, so long as it works, right?

The optimal strategy is to have everyone say the woman's letter out loud as their guess of their letter. The woman would then attempt to fit in by pretending to have the same letter that everyone else thinks they have, regardless of what they've got printed on their forehead.

None of you can read and you all need to get off Veeky Forums.

>no one guesses incorrectly
>simultaneous guesses
>no transfer of information possible at all
>no assumptions about who should guess and who should pass can be made at all
is the answer literally just 1/8th?

No, it's 100%.
>Everyone who sees two H's should guess T.
>Everyone who sees two T's should guess H.
>Everyone who sees one H and one T should pass.

Put duct tape over the mouths of the other contestants

> ... and nobody guesses incorrectly
HHH, first case everyone guesses T and loses.
TTT, Second case, everyone guesses H and loses.
HHT and TTH all win.
So you win 6/8=75%.

Uncapped Blind auctions work for IPOs too, not just ICOs, and they are game theoretically better than any currently used model there too.
>No, I don't. I guess originality is relative, which is why I can look at an original application of an unoriginal algorithm, and view it as unoriginal. Fair enough, I concede this.
I don't pretend to that I invented blind auctions or expected value of course. As far as I know, these are original set ups (applications).

The obvious answer is that they should randomize whether they should guess or not, that is, each person independently chooses randomly whether or not to guess. I don't feel like calculating the optimal probability (that is, the optimal probability with which to choose whether to guess or not), but a simple 50% more than doubles their odds.
Control group: everyone guesses: 1/8 chance of success
Our group: everyone independently randomizes whether to guess or pass:
1. Nobody attempts: 1/8 chance of failure, gives us 0 chance of success
2. One of them gives it a shot: 3/8 odds of that happening and 1/2 odds of him guessing correctly, gives us 3/16 chance of success
3. Two of them decide not to pass: 3/8 of that happening and 1/4 chance of success, gives us 3/32 of that happenning and winning
4. All of them try to guess, 1/8 of that happening and 1/8 of success, gives 1/64
And in total we have 1/64+6/64+12/64=19/64, while everyone guessing is 8/64 (or 1/8). So just by randomising whether to guess or not we can increase the chances of success more than twice.
Again, it's not quite optional yet but I'm a lazy fuck

I don't understand why you would default to passing instead of a random guess in the event that the other two people have different outcomes. This would keep the 75% chance of success from the 2/1 cases, but change the chance of success from 0% to whatever the odds of getting at least one heads in three coin tosses is, I think 7/8 right?

This, but unironically.

Shit, I thought that was going to be about the I.T. Crowd.

Thanks for the feedback. I think you are missing the forest for the trees. No predictions markets work like PIP markets because people likely thought they were impossible to do so profitably. I showed how a model that is basically expected value can actually be used in the real life, which is what makes it unique, and why I all it applied math. It is meant to be an actual business. This means that researchers with the data wouldn't have to make nearly as many assumptions. They would have data that said people thought there was a 25% chance an event occured for a 4x payout.

The Uncapped Blind ICO Auctions are a unique way to distribute assets, that again could work in real life, and again is never done this way.

Both models are asking what is the best workable way to do things. PIP markets are best possible prediction markets that can exist in the real life marketplace. Uncapped Blind ICO Auctions are in many ways the most game theoretically sound way to distribute stock or coins in IPOs and ICOs.

Later.

Contrast a Yes/No PIP market with Binary Options (which are usually scams). The common person is in a much better position to participate in the market because they can actually understand what is going on and how to price things. And a Yes/No PIP market is still profitable to run. It is a better way to do things.

Contrast what researchers have to assume with the data from a Yes/No PIP markets with Binary Options. As long the the PIP market was done for a short term event, there are no discounting assumptions you have to make about bidders. With Binary Options you have to basically assume a modified Black-Scholes price. This is controversal for good reason. Most people who buy Binary Options have no idea how to price them with Black-Scholes and assuming they are perfectly rational is absurd. A lot of traders, maybe even most, don't even use Black-Scholes. This is why I say PIP markets give you relatively clean data. There are a lot less assumptions about participants. You can be more confident that participants understand the market. And as an added bonus, PIP markets aren't scam factories like most Binary Options markets are.

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