The salami puzzle
December 14th, 2015 at 2:26:35 AM
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In the UK there was a TV gameshow where two contestants went head to head over a series of challenges.
One such challenge involved a salami and a set of scales. Each contestant would have to make 4 cuts behind a screen, 1 cut per round, over 5 rounds. In each round they would weigh the chunk of salami on the scales, the heaviest chunk would score one point. Best of 5 would win. You get to hear each weight as it is announced.
Assuming each salami was exactly the same, and no salami is "lost" on each cut. How would you approach this game? Part of me thinks to cut a tiny sliver on the first go, in the hope that the contestant over does it and then you end up with a larger average for each of the 4 remaining chunks. But does the other contestant know this also?
One such challenge involved a salami and a set of scales. Each contestant would have to make 4 cuts behind a screen, 1 cut per round, over 5 rounds. In each round they would weigh the chunk of salami on the scales, the heaviest chunk would score one point. Best of 5 would win. You get to hear each weight as it is announced.
Assuming each salami was exactly the same, and no salami is "lost" on each cut. How would you approach this game? Part of me thinks to cut a tiny sliver on the first go, in the hope that the contestant over does it and then you end up with a larger average for each of the 4 remaining chunks. But does the other contestant know this also?
Guess who peed in my Cheerios? Romes did...
December 14th, 2015 at 2:27:21 AM
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double post
Guess who peed in my Cheerios? Romes did...
December 14th, 2015 at 6:45:44 AM
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I have read this several times and I still don't understand the question.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
December 14th, 2015 at 7:12:50 AM
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He's asking how one would optimize this game show situation.Quote: WizardI have read this several times and I still don't understand the question.
Contestant 1 has a salami stick.
Contestant 2 has a salami stick (assumed to be the EXACT SAME WEIGHT as contestant 1).
There are 5 rounds worth 1 point each.
Each round each contestant will secretly cut off a piece of their salami sticks, then, they will be weighed against one another. Whichever contestants piece weighs more wins that round, and thus 1 point.
A contestant wins if they win 3 points out of 5. So I'd assume this is a "first to 3 points" scenario.
If I were on the show, I'd assume each contestant would do what the OP said, and play it safe to try to calculate a win. Thus, I think the first round could be stolen with a modest piece. At least, my strategy would probably be to steal the first 3 rounds and thus win; doesn't matter if you have nothing left or lose the last 2 rounds if you win the first 3 =). I would probably weight the 2nd and 3rd a bit more, but at worst I'd simply cut my salami stick in to equal thirds and use them on the first 3 rounds. Doyle Brunson says the key to no limit is aggression =P.
Playing it correctly means you've already won.
December 14th, 2015 at 9:53:21 AM
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It's a continuous variant of a game theory problem:
https://en.wikipedia.org/wiki/Blotto_games
https://en.wikipedia.org/wiki/Blotto_games
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563
December 14th, 2015 at 10:19:58 AM
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Thanks; I get it now. I'm thinking about it. My hunch is that the answer for the first cut will be to randomize between 0 and x, where x is fairly small, like 20% the length of the salami.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
December 14th, 2015 at 10:51:12 AM
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The first non-trivial case is with 3 slices (2 cuts). That seems like a better starting point.
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December 14th, 2015 at 10:52:52 AM
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After more contemplation, I update my strategy... Cut the salami in to 3 pieces.
Round 1: 1 piece (so 1/3 of salami)
Round 2: 1 piece (so 1/3 of salami)
Round 3: 0
Round 4: 0
Round 5: 1 piece (so 1/3 of salami)
Given that it's quite probable you take the first 2 rounds (assuming most people play it "safer" and bet smaller at the beginning to try to calculate how much you have left) then they would have to bet harder on rounds 3 and 4 because they can't let you win 1 more. Then in the end you should have enough of a chunk left to decimate whatever they have left. This could be a bit situational of course. If they bet enough by round 3 that you know they have less than 1/3, you win so just bet it in round 4 =p.
Round 1: 1 piece (so 1/3 of salami)
Round 2: 1 piece (so 1/3 of salami)
Round 3: 0
Round 4: 0
Round 5: 1 piece (so 1/3 of salami)
Given that it's quite probable you take the first 2 rounds (assuming most people play it "safer" and bet smaller at the beginning to try to calculate how much you have left) then they would have to bet harder on rounds 3 and 4 because they can't let you win 1 more. Then in the end you should have enough of a chunk left to decimate whatever they have left. This could be a bit situational of course. If they bet enough by round 3 that you know they have less than 1/3, you win so just bet it in round 4 =p.
Playing it correctly means you've already won.
December 14th, 2015 at 11:11:08 AM
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The interesting part about this statement of the problem is that the players can update their strategy in real time. That seems to make it far more complex than the versions where you just submit the strategy up front and see how you do.
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice."
-- Girolamo Cardano, 1563
December 14th, 2015 at 1:03:54 PM
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Reminds me of tournament Blackjack...
You gotta win now, quickly....
You gotta win now, quickly....
Youuuuuu MIGHT be a 'rascal' if.......(nevermind ;-)...2F
December 14th, 2015 at 4:12:43 PM
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Quote: teliotThe first non-trivial case is with 3 slices (2 cuts). That seems like a better starting point.
I was thinking that too. I've started to tinker with that case. It would seem to be a pretty difficult problem to solve, considering the simplicity of it.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
December 14th, 2015 at 4:27:00 PM
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Or even simpler. A stack of 6 poker chips. Three streets.Quote: WizardI was thinking that too. I've started to tinker with that case. It would seem to be a pretty difficult problem to solve, considering the simplicity of it.
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December 14th, 2015 at 5:28:10 PM
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Quote: teliotOr even simpler. A stack of 6 poker chips. Three streets.
Or even simpler than that, suppose it is down to two streets, A has 5 chips, B has 3. What is the optimal strategy?
On the first turn, A should play as follows:
1 chip with prob. 1/3
2 chips with prob. 1/6
3 chips with prob. 1/6
4 chips with prob. 1/3
On the first turn, B should play as follows:
0 chips with prob. 1/3
1 chip with prob. 1/6
2 chips with prob. 1/6
3 chips with prob. 1/3
On average, A will have a net win of 5/6 rounds. I hope that is right.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
December 14th, 2015 at 5:35:44 PM
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I really don't understand how you guys would think there is a "strategy" for it. You think he is gonna cut thin, he thinks you think he is gonna cut thin so he cuts thick, you think he thinks you think ........
it's just like rock paper scissor, would you have a "strategy" for rock paper scissor?
it's just like rock paper scissor, would you have a "strategy" for rock paper scissor?
December 14th, 2015 at 7:16:44 PM
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Quote: AyecarumbaActually, yes.
I have read the post and honestly don't see any valid argument.
and if you are referring to this
http://www.datagenetics.com/blog/march52013/index.html
it's just an article referring to a simulation in which the initial population of RPS are not equal. Really had nothing to do with people playing with each other.
I know there are actual RPS competitions but I think the skill involved is reading tells and recognizing pattern LIVE during actual game play, rather than making up "strategy" beforehand.
December 14th, 2015 at 8:58:52 PM
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I show it comes down to this matrix showing the expected net wins by A according to the first play by A (across) and B (down).
I have a general idea how to solve for the strategy but it would be extremely tedious and I would probably make a mistake somewhere.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 0 | 0 | -0.666667 | -0.5 | 0 | 1 | 1 | 1 |
| 1 | 0.666667 | 0 | -0.5 | -0.166667 | 0 | 1 | 1 |
| 2 | 0.5 | 0.5 | 0 | -0.5 | 0 | 1 | 1 |
| 3 | 0 | 0.166667 | 0.5 | 0 | -0.5 | 0.5 | 1 |
| 4 | -1 | 0 | 0 | 0.5 | 0 | 0.5 | 1 |
| 5 | -1 | -1 | -1 | -0.5 | -0.5 | 0 | -0.5 |
| 6 | -1 | -1 | -1 | -1 | -1 | 0.5 | 0 |
I have a general idea how to solve for the strategy but it would be extremely tedious and I would probably make a mistake somewhere.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
December 14th, 2015 at 10:45:19 PM
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What if there is a tie? (e.g., 0-0). Must each round have a winner?
Simplicity is the ultimate sophistication - Leonardo da Vinci
December 15th, 2015 at 10:18:21 AM
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Quote: andysifI really don't understand how you guys would think there is a "strategy" for it. You think he is gonna cut thin, he thinks you think he is gonna cut thin so he cuts thick, you think he thinks you think ........
it's just like rock paper scissor, would you have a "strategy" for rock paper scissor?
You could not be more wrong.
For a number of reasons;
1) game is over 5 rounds
2) the volume of salami is finite
3) we learn after each round what the opponent cut (and therefore has remaining)
4) the first round is pretty much a punt, granted. But after the first round, we have some information that affects later decisions points. Think more blackjack shoe and less roulette wheel.
Guess who peed in my Cheerios? Romes did...
December 15th, 2015 at 10:21:59 AM
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Quote: AyecarumbaWhat if there is a tie? (e.g., 0-0). Must each round have a winner?
I am sure 0 was not an option, so lets assume some volume of salami was cut, but the scales are super precision. So some number that is not zero, but tiny and that ties are not possible
Guess who peed in my Cheerios? Romes did...
December 15th, 2015 at 11:41:00 AM
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As interesting reading someone once held a competition using 10 games using non-negative integers adding up to 100 ( http://www1.maths.leeds.ac.uk/~pmt6jrp/personal/blotto.html ). It seems intuitive to try and win 6, but it seems trying to win 5 was a better strategy.
