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GM
GM
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July 1st, 2021 at 4:35:44 PM permalink
Quote: Wizard

The following question I do not claim to have a correct answer for and perhaps there isn't one. It was inspired the following scene from The Dark Knight. Watching it is not required to understand the problem.

Suppose there is a game show with two perfect logicians as players. The game starts with a pot of $1,000,000.

1. Logician A will make a suggestion on how to divide the pot.

2. If Logician B accepts the suggestion, then they split it that way and the game is over.

3. If Logician B rejects the offer, the host will rake 10% out of the remaining pot.

4. Logician B will make a suggestion on how to divide the remaining pot.

5. If Logician A accepts the suggestion, then they split it that way and the game is over.

6. If Logician B rejects the offer, the host will rake 10% out of the remaining pot.

7. Go back to step 1.

The question is how should logician A initially suggest dividing the pot?


A should offer 9/19 of the money to B.
Wizard
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Wizard
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July 1st, 2021 at 5:22:12 PM permalink
Quote: unJon

Is a penny the smallest denomination? How does the host round the 10% if there’s a fraction of a penny?



Yes. He rounds to the nearest penny.
It's not whether you win or lose; it's whether or not you had a good bet.
Wizard
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Wizard
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July 1st, 2021 at 5:22:47 PM permalink
Quote: GM

A should offer 9/19 of the money to B.




I pretty much agree. I would only add he sound throw in an extra penny, so that B will take the offer. You don't want B completely indifferent. He might flip a coin and reject and offer you back $426,315.79, $100,000 less than the $526,315.78 you offered yourself initially.

I round up to $473,700, just to be nice, but I know a perfect logician would get every penny possible and offer $473,684.22.

It's not whether you win or lose; it's whether or not you had a good bet.
Dieter
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Dieter 
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July 1st, 2021 at 9:13:46 PM permalink
Quote: Wizard

He rounds to the nearest penny.



This strikes me as unusual. I would expect the host to rake the entire penny as breakage in the case of a fractional cent.

Either way is fine for the thought experiment, but after 100+ rejected compromise offers, he's earned it.
May the cards fall in your favor.
GM
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July 4th, 2021 at 2:17:04 PM permalink
Quote: Dieter

This strikes me as unusual. I would expect the host to rake the entire penny as breakage in the case of a fractional cent.

Either way is fine for the thought experiment, but after 100+ rejected compromise offers, he's earned it.


I assumed that the host would round up the fractional cents, otherwise if the total amount gets down to 4c, the game may never finish.

If we assume that the logicians only interested in maximising their own winning and are willing to accept an offer that is equal to the maximum what they could get if they refused the offer, then A should offer B $473684.21. If they are not willing to co-operate and each of them will only accept an offer that is at least 1c more than the maximum what they could get if they refused the offer, then A should offer B $473684.22.
Wizard
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July 5th, 2021 at 1:05:14 PM permalink
Quote: GM

A should offer B $473684.21. If they are not willing to co-operate and each of them will only accept an offer that is at least 1c more than the maximum what they could get if they refused the offer, then A should offer B $473684.22.



I agree.

Please put future answers in spoiler tags. That way any late-comers can enjoy the problem without seeming the answer posted.
It's not whether you win or lose; it's whether or not you had a good bet.
Dieter
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Dieter 
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July 5th, 2021 at 8:15:50 PM permalink
Quote: Dieter

This strikes me as unusual. I would expect the host to rake the entire penny as breakage in the case of a fractional cent.

Either way is fine for the thought experiment, but after 100+ rejected compromise offers, he's earned it.



I realized shortly after posting that it comes into play at far fewer rejected offers, and feel appropriately feebleminded.
May the cards fall in your favor.
Gialmere
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July 8th, 2021 at 9:26:59 AM permalink
Like the Wizard's previous problem, I don't know the answer to this one. The users at the math puzzle site where I came across it never reached a consensus for solving. It's an interesting problem involving a card game most people have played. I quote it exactly to see if consensus can be reached here...



In the game of Memory, there are many different cards, each having one matching card to make its pair. To start the game, all the cards are spread out face down. On each turn, a player chooses a card and turns it over, then chooses a second card and turns it over. If the cards match, they are removed from the play area. If the cards do not match, they are returned to their face-down positions.

Assuming you have perfect memory (you can remember where all previously seen cards are), what is the expected number of turns needed to clear the play area for N pairs?

How does this result change if two cards must first be selected, then turned over simultaneously?

How does the first result change if there are multiple identical pairs, such as in a standard deck of cards (any Queen could be paired with any other Queen)?


In all cases, assume you play to minimize the number of turns.
Have you tried 22 tonight? I said 22.
ThatDonGuy
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July 8th, 2021 at 12:32:26 PM permalink
Quote: Gialmere

In the game of Memory, there are many different cards, each having one matching card to make its pair. To start the game, all the cards are spread out face down. On each turn, a player chooses a card and turns it over, then chooses a second card and turns it over. If the cards match, they are removed from the play area. If the cards do not match, they are returned to their face-down positions.

Assuming you have perfect memory (you can remember where all previously seen cards are), what is the expected number of turns needed to clear the play area for N pairs?


I have a feeling I'm going to regret this the minute I post it, but my numbers appear to be correct for 1 and 2 pairs...


E(n) is the expected number of turns starting with n pairs

E(1) = 1
E(2) = 8 / 3
E(3) = 199 / 45
E(4) = 9,556 / 1,575
E(5) = 786,227 / 99,225
E(6) = 94,036,576 / 9,823,275
E(7) = 3,219,603,572 / 280,945,665
E(8) = 17,964,806,816,248 / 1,369,610,116,875
E(9) = 5,244,142,302,537,988 / 349,250,579,803,125
E(10) = 1,197,405,952,584,220,562 / 71,786,869,175,896,875
E(11) = 23,420,230,525,515,820,899,299 / 1,260,290,275,252,045,537,500
E(12) = 1,849,433,417,098,530,134,834,771 / 91,308,030,442,010,699,191,875
E(13) = 116,341,686,404,284,568,353,640,366,753 / 5,250,211,750,415,615,203,532,812,500
E(14) = 464,625,053,499,588,757,386,024,329,118,739 / 19,491,411,123,417,971,443,115,566,406,250
E(15) = 1,571,462,235,083,358,151,475,317,163,097,678,737 / 61,047,099,638,545,086,559,837,953,984,375,000
E(16) = 2,445,858,396,189,803,074,022,490,793,281,058,154,971 / 89,182,181,684,459,553,328,103,271,026,923,828,125
E(17) = 856,246,260,751,723,619,649,577,491,266,529,944,091,817 / 29,194,678,996,224,679,377,487,886,803,373,784,375,000
E(18) = 65,369,036,717,836,848,555,054,164,570,348,117,457,561,106,219 / 2,107,490,890,039,969,042,562,406,828,618,545,059,570,312,500
E(19) = 1,437,505,462,224,141,765,319,683,046,425,447,630,110,336,927,272,279 / 43,667,211,241,628,158,561,893,069,488,976,253,634,296,875,000,000
E(20) = 10,310,653,070,695,057,740,633,755,272,380,500,522,403,004,373,859,041 / 297,886,640,425,022,346,407,619,267,973,881,111,763,886,718,750,000
E(21) = 187,624,995,670,444,822,761,102,760,795,505,677,506,365,650,106,369,786,860,009 / 5,138,548,121,971,320,575,799,589,263,980,664,864,500,387,065,078,125,000,000
E(22) = 50,592,299,927,532,945,252,436,026,478,265,788,089,433,539,553,647,978,381,421,123 / 1,324,045,741,859,025,579,442,760,319,812,002,545,893,611,273,222,476,562,500,000
E(23) = 133,589,616,167,475,641,370,900,261,774,634,214,899,298,766,597,156,639,802,988,694,899,579 / 3,330,637,063,646,378,845,088,263,584,487,092,404,195,379,157,791,139,792,968,750,000,000
E(24) = 128,853,864,436,997,375,220,430,317,002,564,689,920,778,943,592,415,883,972,086,015,579,536,013 / 3,081,880,107,955,289,925,095,733,898,020,712,690,257,036,776,943,614,039,681,396,484,375,000
E(25) = 410,512,353,054,960,488,444,975,685,589,029,270,251,940,073,462,813,937,125,465,066,511,797,934,447 / 9,392,272,935,318,058,306,460,177,664,789,018,295,612,813,554,323,317,229,142,714,843,750,000,000
E(26) = 232,892,409,664,096,187,950,738,679,941,658,818,172,913,536,412,245,920,476,876,384,899,053,693,789,540,201 / 5,128,476,879,281,122,354,163,910,500,571,244,843,480,908,004,287,492,391,604,640,300,205,078,125,000,000

GM
GM
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July 8th, 2021 at 3:59:04 PM permalink
The denominators seem to be related to the sequence https://oeis.org/A079484.

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