# Dominion Strategy Forum

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#### Awaclus

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« Reply #450 on: May 28, 2015, 11:42:32 am »
+2

Then nobody says anything about my more defined problem...

anything about my more defined problem...

Better?
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#### mpsprs

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« Reply #451 on: May 28, 2015, 01:19:23 pm »
+2

I don't think it is so easy to assume perfectly rational. If there is a Nash equilibrium, sure you can do. But with the 100\$, n player who bids highest without tieing wins, I'm not so sure there is one.

Under the assumption that the only allowed bids are integers from 0 to 100 (or any finite set of bids), and allowing for mixed strategies (i.e. bidding x with probability p and y with probability q and so on) I'm pretty sure Nash's Existence Theorem guarantees a Nash equilibrium.  (And the only reason I'm only pretty sure is because this stuff is pretty far from my area of math).

It does not guarantee a unique nash equilibrium though.

This paper posted on the arxiv seems to address some cases of this problem (they don't deal with profit-the goal is to maximize your probability of getting that \$100, regardless of how much you spend to get it, and they also switch out highest bid for lowest bid).  For 3 players they show a computation of how to compute a Nash equilibrium where all three players use the same mixed strategy.  Modifying this for the profit version theoretically isn't hard (change some numbers here and there), but in practice as the number of players grows it will get tough.

#### ghostofmars

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« Reply #452 on: May 28, 2015, 02:29:02 pm »
0

A fun related game theory problem: You have n people bidding on a \$100 bill.  They bid secretly, can't communicate with each other, and can only bid in integer numbers of dollars.  Whoever bids the highest amount without tying with someone else wins.  How much do you bid?  (I remember hearing a problem like this but I have no idea how to solve it...)
I propose the following solution, although you would probably need to write a program to actually calculate the values. Assumption: all players play the same strategy and want to optimize the potential gain based on the probabilities p_i with which they bid the amount i.

The gain of a certain amount i is given by
g_i = ps_i * (100 - i) * (1 - sum_j>i ps_j)
where ps_i is the chance that exactly one player bid the amount i
ps_i = n * p_i * (1 - p_i)^(n-1)
Now, we want to maximize
G = sum (g_i - a p_i) + a
with respect to g_i and a. The latter is a Lagrange multiplier that ensures that the p_i sum up to 1. This will lead to a system of nonlinear equations that one should be able to solve numerically. I don't know if the solution is unique, though.
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#### Polk5440

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« Reply #453 on: May 28, 2015, 02:58:58 pm »
0

I don't think it is so easy to assume perfectly rational. If there is a Nash equilibrium, sure you can do. But with the 100\$, n player who bids highest without tieing wins, I'm not so sure there is one.

Under the assumption that the only allowed bids are integers from 0 to 100 (or any finite set of bids), and allowing for mixed strategies (i.e. bidding x with probability p and y with probability q and so on) I'm pretty sure Nash's Existence Theorem guarantees a Nash equilibrium.  (And the only reason I'm only pretty sure is because this stuff is pretty far from my area of math).

It does not guarantee a unique nash equilibrium though.

Correct on both points.
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#### GeoLib

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« Reply #454 on: May 29, 2015, 10:45:26 pm »
0

Ok I have a problem that I don't really know the answer too. Or rather, I have two contradictory answers that I'm trying to reconcile. Perhaps it's more suited to the logic puzzles thread, but I think reconciling the answers might be a math issue. Anyways:

There is a king with a kingdom of countably infinite people. He decides to play a game with his subjects. This game consists of multiple rounds. In the first round, he calls one of his subjects to play. They come to the castle and he flips two coins. If they're both heads then that person wins and the game is over. Otherwise, he calls up two people, flips two coins. If they're both heads those people both win and the game is over. Otherwise, he calls up four people, etc. The game ends as soon as someone wins. In each round twice as many people come up and their fates are still decided by two coin flips.

You are called by the King. What is your probability of winning?

So I think I have a resolution to the paradox after posing it to a friend of mine (this is really his solution, not mine).

When we do the second scenario and it's all been decided, there's an extra piece of information: that the game is over, as sitnaltax said. This eliminates the zero-probability event that the game goes on forever and there are infinitely many losers. So the issue is that we multiply zero by infinity, and this event still contributes to the probability of being a winner, lowering it from 0.67 to 0.25

« Last Edit: May 30, 2015, 01:01:08 am by GeoLib »
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#### liopoil

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« Reply #455 on: July 28, 2015, 02:17:05 pm »
+1

Still working on the real problem [What's the least prime factor of 252128 + 1?], haven't read Titandrake's solution or heron's comment on it.
Over half a year later, I went to a summer program on number theory and now am more equipped to answer this!

Consider any positive odd (since 252 is even) prime p which divides 252128 + 1. Then 252128 is congruent to -1 (mod p). Since the group of nonzero integers modulo p (Up) is cyclic, there exists a g with period length (order) p - 1, and gp - 1 = 1. Then -1 = g(p - 1)/2, and we can replace 252 with gn. Then we have that 128n is congruent to (p - 1)/2 (mod p - 1). Then 27n = (2k + 1)(p - 1)/2 for some k. 2k + 1 is odd, so p - 1 must be divisible by 2^8. 257 is the lowest such prime p, so let's test it!

252 = 22327, so 252128 = 225632567128. Then by Fermat's little theorem, which was implicitly used above as well, this is the same as 7128. Now we need to determine whether this is 1 or -1, and if it is -1, then it works since adding 1 gives 0, so then 257 would be a divisor. It must be 1 or -1 since it squares to 1, and 257 is prime.

(-1)(257 - 1)(7 - 1)/4 = 1, so by quadratic reciprocity, which is hard to state on this forum, 7 is a square modulo 257 iff 257 is a square modulo 7. 257 is 5 (mod 7), which is not a square mod 7, so 7 is not a square modulo 257. Therefore 7 = gn, where g is a generator of Up and n is odd, since if it were even 7 would be a square. Then 7128 = g128n where n is odd, so since g has order p - 1 and 256 does not divide 128n for odd n, 7128 is not 1 (mod 257). Therefore it is -1, so 257 is the least prime factor of 252128 + 1.

I spent the last 6 weeks rigorously proving all of these results and lemmas, from the ring axioms. Pretty cool to see them defeat a problem that was way over my head in January.

EDIT: Hey, this matches up quite nicely with what Heron and Titandrake did way back when!
« Last Edit: July 28, 2015, 02:24:17 pm by liopoil »
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#### Kirian

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« Reply #456 on: July 28, 2015, 02:47:35 pm »
+1

Good work liopoil.  And here I had assumed the answer was Moat.
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#### Kuildeous

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« Reply #457 on: July 28, 2015, 03:12:43 pm »
0

Good work liopoil.  And here I had assumed the answer was Moat.

Were you working in Base 15? Because that is totally the answer in Base 15!
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#### heron

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« Reply #458 on: July 28, 2015, 07:58:28 pm »
0

Nice liopoil. Which summer program did you go to?
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#### liopoil

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« Reply #459 on: July 28, 2015, 08:01:37 pm »
+1

Nice liopoil. Which summer program did you go to?
Ross
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#### heron

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« Reply #460 on: July 28, 2015, 09:55:57 pm »
+2

Nice liopoil. Which summer program did you go to?
Ross

Cool! I'm at Canada/USA mathcamp right now.
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#### Witherweaver

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« Reply #461 on: July 28, 2015, 09:57:39 pm »
+3

Nice liopoil. Which summer program did you go to?
Ross

Cool! I'm at Canada/USA mathcamp right now.

Canadian math is confusing, because you have to avoid using "a" as a variable name.
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#### Ichimaru Gin

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« Reply #462 on: July 28, 2015, 10:08:50 pm »
0

eh?

#### Titandrake

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« Reply #463 on: July 28, 2015, 11:10:43 pm »
0

Nice liopoil. Which summer program did you go to?
Ross

Cool! I'm at Canada/USA mathcamp right now.

Oh, cool! I went there around 5 years ago (2010 + 2011)
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#### scott_pilgrim

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« Reply #464 on: August 06, 2015, 10:58:46 pm »
0

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
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#### skip wooznum

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« Reply #465 on: August 06, 2015, 11:35:20 pm »
+1

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
my high school math would tell me 12.  Could you explain why it wouldn't be?
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#### ConMan

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« Reply #466 on: August 06, 2015, 11:37:07 pm »
+7

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
Poorly written, and it appears to be another one of these "write the expression in an ambiguous manner so people can argue about order of operations and feel superior to each other" questions.

That said, typically you would read 3x as being a single "unit" that is equal to 6, so that the entire expression is equal to 3. If it had been written 18/3x it would probably be worse, and that's why mathematicians tend to take up lots of space to write fractions so you can clearly see what's in the numerator and what's in the denominator and didn't we argue about one of these in this thread about ten pages ago oh my god my head hurts.
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#### Watno

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« Reply #467 on: August 06, 2015, 11:51:05 pm »
+2

I would definitely interpret it as 18/(3x). If something after the division sign wasn't part of the denumerator, why not put it in front of it?

In fact I have a paper here that uses such a notation (1/16n³), but that same paper also caused me lots of confusion by writing log n M and meaning log(nM), so it's probably not an example of good notation.
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#### scott_pilgrim

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« Reply #468 on: August 06, 2015, 11:57:36 pm »
+1

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
my high school math would tell me 12.  Could you explain why it wouldn't be?

No, I can't explain why it wouldn't be, but almost everyone I asked said 3 (you resolve the 3x first for some reason).  They couldn't give me a good reason for it though.

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
Poorly written, and it appears to be another one of these "write the expression in an ambiguous manner so people can argue about order of operations and feel superior to each other" questions.

That said, typically you would read 3x as being a single "unit" that is equal to 6, so that the entire expression is equal to 3. If it had been written 18/3x it would probably be worse, and that's why mathematicians tend to take up lots of space to write fractions so you can clearly see what's in the numerator and what's in the denominator and didn't we argue about one of these in this thread about ten pages ago oh my god my head hurts.

I'm not trying to start an argument, I legitimately want to make sure that there's not some rule I'm somehow unaware of that says you do "smash up multiplication" before everything else.  And I want to be very sure that I'm correct, because the book I was working from consistently resolved the 3x first, and I had to try to explain to a student why they were doing that, even though it seems very wrong to me.  I really think the book is wrong, but I want to be 100% sure of that before I tell my boss that the book has a major error in it that needs to be fixed.

Of course I agree that you should always write it in a less ambiguous way, but I have to be able to tell students what to do if they see it written like that.

The argument for 12 is that multiplication and division have equal priority, so you go from left to right.  18 divided by 3 is 6, 6x is 12.  The only possible argument for 3 that I could think of is that 3x is somehow fundamentally different from 3*x (with a symbol between them), since I think we could all agree 18÷3*x is 12.  Then I'd have to ask about 18÷3(x), and 18÷3(2).  But I would think 3x is exactly the same as 3*x, and I wasn't aware of any special rule that distinguishes them.
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#### skip wooznum

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« Reply #469 on: August 06, 2015, 11:58:16 pm »
+1

typically you would read 3x as being a single "unit"
never mind me, if you ever were. This sounds familiar. I concede.
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#### skip wooznum

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« Reply #470 on: August 06, 2015, 11:59:52 pm »
+1

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
my high school math would tell me 12.  Could you explain why it wouldn't be?

No, I can't explain why it wouldn't be, but almost everyone I asked said 3 (you resolve the 3x first for some reason).  They couldn't give me a good reason for it though.

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
Poorly written, and it appears to be another one of these "write the expression in an ambiguous manner so people can argue about order of operations and feel superior to each other" questions.

That said, typically you would read 3x as being a single "unit" that is equal to 6, so that the entire expression is equal to 3. If it had been written 18/3x it would probably be worse, and that's why mathematicians tend to take up lots of space to write fractions so you can clearly see what's in the numerator and what's in the denominator and didn't we argue about one of these in this thread about ten pages ago oh my god my head hurts.

I'm not trying to start an argument, I legitimately want to make sure that there's not some rule I'm somehow unaware of that says you do "smash up multiplication" before everything else.  And I want to be very sure that I'm correct, because the book I was working from consistently resolved the 3x first, and I had to try to explain to a student why they were doing that, even though it seems very wrong to me.  I really think the book is wrong, but I want to be 100% sure of that before I tell my boss that the book has a major error in it that needs to be fixed.

Of course I agree that you should always write it in a less ambiguous way, but I have to be able to tell students what to do if they see it written like that.

The argument for 12 is that multiplication and division have equal priority, so you go from left to right.  18 divided by 3 is 6, 6x is 12.  The only possible argument for 3 that I could think of is that 3x is somehow fundamentally different from 3*x (with a symbol between them), since I think we could all agree 18÷3*x is 12.  Then I'd have to ask about 18÷3(x), and 18÷3(2).  But I would think 3x is exactly the same as 3*x, and I wasn't aware of any special rule that distinguishes them.
I actually think I remember such a rule, yes.
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#### scott_pilgrim

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« Reply #471 on: August 07, 2015, 12:03:09 am »
0

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
my high school math would tell me 12.  Could you explain why it wouldn't be?

No, I can't explain why it wouldn't be, but almost everyone I asked said 3 (you resolve the 3x first for some reason).  They couldn't give me a good reason for it though.

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
Poorly written, and it appears to be another one of these "write the expression in an ambiguous manner so people can argue about order of operations and feel superior to each other" questions.

That said, typically you would read 3x as being a single "unit" that is equal to 6, so that the entire expression is equal to 3. If it had been written 18/3x it would probably be worse, and that's why mathematicians tend to take up lots of space to write fractions so you can clearly see what's in the numerator and what's in the denominator and didn't we argue about one of these in this thread about ten pages ago oh my god my head hurts.

I'm not trying to start an argument, I legitimately want to make sure that there's not some rule I'm somehow unaware of that says you do "smash up multiplication" before everything else.  And I want to be very sure that I'm correct, because the book I was working from consistently resolved the 3x first, and I had to try to explain to a student why they were doing that, even though it seems very wrong to me.  I really think the book is wrong, but I want to be 100% sure of that before I tell my boss that the book has a major error in it that needs to be fixed.

Of course I agree that you should always write it in a less ambiguous way, but I have to be able to tell students what to do if they see it written like that.

The argument for 12 is that multiplication and division have equal priority, so you go from left to right.  18 divided by 3 is 6, 6x is 12.  The only possible argument for 3 that I could think of is that 3x is somehow fundamentally different from 3*x (with a symbol between them), since I think we could all agree 18÷3*x is 12.  Then I'd have to ask about 18÷3(x), and 18÷3(2).  But I would think 3x is exactly the same as 3*x, and I wasn't aware of any special rule that distinguishes them.
I actually think I remember such a rule, yes.

Huh...I'm going to have to look into it more.  I'm not sure what to search for though...

So what happens with 18÷3(x)?  That would still be 3, right, since the parentheses get resolved first (doing nothing)?  And 18÷3(2) is also 3, because it becomes 18÷32 (where 32 is a "smash up" 3*2, not the number thirty-two)?
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#### skip wooznum

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« Reply #472 on: August 07, 2015, 12:07:58 am »
+1

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
my high school math would tell me 12.  Could you explain why it wouldn't be?

No, I can't explain why it wouldn't be, but almost everyone I asked said 3 (you resolve the 3x first for some reason).  They couldn't give me a good reason for it though.

Okay, I really want a definitive answer from someone with a solid math background on this (input from anyone is welcome though).  I have a Bachelor's in math, but so does one of my co-workers, and we couldn't agree on this.

If x=2, what is 18÷3x?
Poorly written, and it appears to be another one of these "write the expression in an ambiguous manner so people can argue about order of operations and feel superior to each other" questions.

That said, typically you would read 3x as being a single "unit" that is equal to 6, so that the entire expression is equal to 3. If it had been written 18/3x it would probably be worse, and that's why mathematicians tend to take up lots of space to write fractions so you can clearly see what's in the numerator and what's in the denominator and didn't we argue about one of these in this thread about ten pages ago oh my god my head hurts.

I'm not trying to start an argument, I legitimately want to make sure that there's not some rule I'm somehow unaware of that says you do "smash up multiplication" before everything else.  And I want to be very sure that I'm correct, because the book I was working from consistently resolved the 3x first, and I had to try to explain to a student why they were doing that, even though it seems very wrong to me.  I really think the book is wrong, but I want to be 100% sure of that before I tell my boss that the book has a major error in it that needs to be fixed.

Of course I agree that you should always write it in a less ambiguous way, but I have to be able to tell students what to do if they see it written like that.

The argument for 12 is that multiplication and division have equal priority, so you go from left to right.  18 divided by 3 is 6, 6x is 12.  The only possible argument for 3 that I could think of is that 3x is somehow fundamentally different from 3*x (with a symbol between them), since I think we could all agree 18÷3*x is 12.  Then I'd have to ask about 18÷3(x), and 18÷3(2).  But I would think 3x is exactly the same as 3*x, and I wasn't aware of any special rule that distinguishes them.
I actually think I remember such a rule, yes.

Huh...I'm going to have to look into it more.  I'm not sure what to search for though...

So what happens with 18÷3(x)?  That would still be 3, right, since the parentheses get resolved first (doing nothing)?  And 18÷3(2) is also 3, because it becomes 18÷32 (where 32 is a "smash up" 3*2, not the number thirty-two)?
I doubt it, but that's just speculation. I assume the rule is specifically for varibles with a coefficient. Is 3(x) considered a coefficient? Idk.
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#### pacovf

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« Reply #473 on: August 07, 2015, 12:09:50 am »
+4

So what happens with 18÷3(x)?  That would still be 3, right, since the parentheses get resolved first (doing nothing)?  And 18÷3(2) is also 3, because it becomes 18÷32 (where 32 is a "smash up" 3*2, not the number thirty-two)?

If you ever see written 18÷3(x) or, may the Lord have mercy on our souls, 18÷3(2), burn whatever book you are reading to ashes and never look back.
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