Puzzles and Paradoxes

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Messages 1 - 15 of total 15 in this topic
Ben Harland

Gym climber
Kenora, ON
Topic Author's Original Post - Feb 22, 2019 - 03:27pm PT
My very first post on this site was a logic puzzle, back around 2008. It didn't generate much interest then because I didn't know what I was doing and did a poor job of it. But I'm a recidivist, so here's another. It's called the St. Petersburgh paradox. I found it in "How not to be wrong" by Jordan Ellenberg, but really it goes back to Daniel Bernoulli.

The idea is that we play a game: I flip a coin and if it comes up heads, I pay you $1. If it's tails, I flip it again. This time, if it's heads I pay you $2, but if it's tails, toss again. The game continues like this -- the game is over when heads appears for the first time and the amount I pay you doubles with every tail I throw.

As Ellenberg points out, this is a pretty good deal for you -- you make money no matter what happens -- so you must be willing to give me some money for the privilege of playing. The question is: how much?
Clint Cummins

Trad climber
SF Bay area, CA
Feb 22, 2019 - 03:49pm PT
Expected value of the game is infinite....
1/2 + 2/4 + 4/8 + ...
= .5 + .5 + .5 + ...
Ben Harland

Gym climber
Kenora, ON
Topic Author's Reply - Feb 22, 2019 - 03:53pm PT
So how high would you go, Clint? I get cold feet at around $5.
madbolter1

Big Wall climber
Denver, CO
Feb 22, 2019 - 05:31pm PT
Sloaper for the win (well, on the right track, imo).

It's a gamblers' fallacy to believe that with a fair coin there will be more than one flip. So, you have a 50% chance of making the buck, and that's really "the game." Calculating out your odds beyond that first flip is a fool's game; you might get ten in a row, but the odds are against you getting even two in a row.

https://nrich.maths.org/6954/solution

The most reasonable gamble is to risk less than you hope to gain with that first flip.

Of course, is it reasonable to risk $0.99 on the hope of netting $0.01? Thought of as one flip (avoiding the gamblers' fallacy), not so good. And each iteration you are hoping for becomes more tenuous!

Here opinions vary. Do you need to double your money for it to be a good risk? If so: $0.50. Your own risk-aversion will dominate. $0.99 for most people implies believing in the odds of having at least two successful flips, but, again, it's a gamblers' fallacy to believe that the flips are "related" in any way.

So, the "hope" of exponentially-increasing gains is a scam. This particular scam is made more interesting by the fallacious correlation between exponentially increasing gains and exponentially decreasing odds. If you risk more than $0.99 for the first flip (some would say risking even that much!), you commit to some version of the gamblers' fallacy, where you treat "the odds" as in some sense cumulative, so you are willing to accumulate "correlative" risk. But, while your risk, therefore, is real and entirely up-front, "the odds" are actually not.

What you HAVE is one flip for a buck. What's that chance worth to you?
Brandon-

climber
The Granite State.
Feb 22, 2019 - 05:50pm PT
Have any of you played ‘a fun game’ yet?

It’s literally a fun game, and I’d be into playing it.
Brandon-

climber
The Granite State.
Feb 22, 2019 - 05:55pm PT
It’s a word game. We write a word, and then everyone else joins in and writes a phrase referring to the word. Is anyone down to play the game?

The word is Alluciate
Ben Harland

Gym climber
Kenora, ON
Topic Author's Reply - Feb 22, 2019 - 06:16pm PT
Well Madbolter, whether it's a fools game is a matter of perspective. I don't see any scam, but you're right that it's a matter of how tolerant you are to risk. Supposing that you're prepared to wager $2, half the time you're out a dollar.

But the other half the time, you at least break even. As Clint pointed out, your expected winnings are tough to get your head around.

Also, I'm interested in any other games or riddles -- including yours Brandon -- I spend a couple of evenings a week with kids and some of them really dig them too.
Brandon-

climber
The Granite State.
Feb 22, 2019 - 06:24pm PT
So, let’s play. No wager, just for fun. Why not? Search my thread titled a fun game and get back to me. I like wordplay
madbolter1

Big Wall climber
Denver, CO
Feb 22, 2019 - 06:42pm PT
Well Madbolter, whether it's a fools game is a matter of perspective. I don't see any scam, but you're right that it's a matter of how tolerant you are to risk. Supposing that you're prepared to wager $2, half the time you're out a dollar.

I totally agree that it's a matter of perspective. But saying "half the time you're out a dollar" isn't really how odds work, imo. If you lose on the first flip, then you're out two bucks, not one. And that first flip is all you've really got. So, half the time you're out two bucks. On that first flip you're risking two bucks in the hope of getting one, and half the time you'll be out the two.

In this game, it takes a "sufficiently large" population to even start to talk about "the odds" in a more extended (beyond the first flip) scenario, but your particular flip(s) cannot constitute a sufficiently large population. So, if your "half the time" means some arbitrary number of iterations beyond that first flip, well, "the odds" beyond one flip are increasingly an abstraction that will not play out in your particular case (see the above article). But the money you lay out is real and up-front.

Half of a sufficiently large population will be out two bucks, as they lose the first flip; and half will be up a buck, having risked two to gain one. But any particular subset of that total population will do worse from there, and some other subset will do better. Deriving "your odds" from such population odds is the fallacy of division.

Of course, all gambling is "investing" real money in the abstraction of "odds." Many are happy to just spend a certain amount of money, figuring to just lose it, on the hope that "the odds" will go their way in particular. That is indeed a matter of perspective. People spend real money on the lottery. Hehe

https://www.oddsshark.com/super-bowl/props/coin-toss

Obviously, Superbowl coin-flips are not yet a sufficiently large population for "the odds" to play out "correctly" in reality (25 heads and 27 tails). Soooo... bet on heads (it has to hit now!). No, uh, bet on tails (it's hot right now).

LOL
originalpmac

Mountain climber
Timbers of Fennario
Feb 22, 2019 - 10:33pm PT
Sounds like trying to play a quick game of Acey-Duecy. It is never quick and the pot gets quite large, fast.
Clint Cummins

Trad climber
SF Bay area, CA
Feb 23, 2019 - 05:32am PT
https://en.wikipedia.org/wiki/St._Petersburg_paradox#Finite_St._Petersburg_lotteries
MikeL

Social climber
Southern Arizona
Feb 23, 2019 - 07:30am PT
(I must have missed something, or I've not had enough coffee this morning.)

So where is the paradox?
Ben Harland

Gym climber
Kenora, ON
Topic Author's Reply - Feb 23, 2019 - 08:08am PT
Mike, the paradox shows up when you decide that it is worthwhile to play any game where the cost to play is lower than the amount that you would expect to win, on average.

For this game, you can work out that paying, say, $100 to play is a "good deal". But nobody would ever actually pay that.

Not that I don't trust the math, but I took a "seeing is believing" approach to the game with a measly $10 cost. If you're allowed to play multiple games in a row, things don't look good after a million:

You need to go on for ~100 million to get the real payout
Ben Harland

Gym climber
Kenora, ON
Topic Author's Reply - Feb 23, 2019 - 09:53am PT
DMT -- you've reminded me of a family vacation to Florida when I was ten. We had three days at Disney, and my parents had to pry us out of the arcade where the video games were FREE!
maldaly

Trad climber
Boulder, CO
Feb 23, 2019 - 11:40am PT
The better puzzle is; Why aren’t hemorrhoids called astronauts?
Messages 1 - 15 of total 15 in this topic
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