TPIR New game: Pay the Rent
September 22nd, 2010 at 5:39:53 AM
permalink
Intro
There is a new pricing game on The Price Is Right called Pay the Rent. A player can win up to $100,000! However, it is deceptive, as the first contestant playing it found out... or did they?
Rules
The player is shown 6 grocery items, their prices unknown. They must place those 6 items on 4 levels, and each level must be higher in price than the previous level. Here is a rough illustration:The items are all placed before any prices are revealed, and are placed starting at the bottom level ($1,000). The 2nd and 3rd levels are two-item levels. The prices of the two items are combined to give the value for that level.
The bottom level price is revealed, and they are given $1,000. If they believe that the total of items on the next level is higher than that price, they may gamble for $5,000. If they are wrong at any time, they take away $0.
If they gamble, the two prices on the $5,000 level are revealed, and totaled. If they wish to go on, the total of the two items on the $10,000 level must be more than the total of the items on the $5,000. Again, going on and being incorrect loses everything.
If they believe the final item on the top level is worth more than the combined total of the two items on the $10,000, they may risk it and try to win $100,000.
The Deception
At first glance, the logical strategy seems to be to place the items in order from bottom to top. This is exactly what the first contestant did, and she did it perfectly. The cheapest item was on the bottom, the next two more expensive went above that, then 2nd and 3rd most expensive above that, and the most expensive item on top. If she were playing "Hole in One (or Two)", she would have aced it. But we're playing "Pay the Rent."
At the $100,000 level, the audience was shown on camera urging her to go for it. She looked and felt confident. She went for it. As Drew said at that point, "I hope that cat food is expensive." It wasn't. She lost everything.
Strategy
If placing items from cheapest to expensive isn't the right move... then what is?
I ran a simulation to place the items in every possible configuration. There are 6! (720) ways to place the items uniquely, but since two level require a pair of items, there are only 6!/2*2 (720/4 = 180) ways to place them. Each placement can then in turn win a maximum dollar amount. They are:
24 ways to win $1,000
89 ways to win $5,000
66 ways to win $10,000
1 way to win $100,000
(average win: $6,827.77)
Note: I suppose it is possible that the prices of items in subsequent games could change the above possibilities. These combinations are based on the actual prices given during the first live play of the game.
Here are the prices of the items from the first play (highlight to reveal):
$1.49, $2.98, $3.49, $5.49, $5.99, $7.30
There is only one correct way to stack the prices. Here is the solution to the above (highlight to reveal):
$100,000 level: $7.30
$10,000 level: $1.49 + $5.49 = $6.98
$5,000 level: $2.98 + $3.49 = $6.47
$1,000 level: $5.99
Questions
Given the prices, can you stack them up for $100,000?
If you called the products 1 2 3 4 5 and 6, from cheapest to most expensive, can you give a general solution?
If you were playing for real, would you bother going for $100,000? What level would you stop at?
There is a new pricing game on The Price Is Right called Pay the Rent. A player can win up to $100,000! However, it is deceptive, as the first contestant playing it found out... or did they?
Rules
The player is shown 6 grocery items, their prices unknown. They must place those 6 items on 4 levels, and each level must be higher in price than the previous level. Here is a rough illustration:
$100,000 .... <Item> ....
$10,000 <Item> .. <Item>
$5,000 <Item> .. <Item>
$1,000 .... <Item> ....
The bottom level price is revealed, and they are given $1,000. If they believe that the total of items on the next level is higher than that price, they may gamble for $5,000. If they are wrong at any time, they take away $0.
If they gamble, the two prices on the $5,000 level are revealed, and totaled. If they wish to go on, the total of the two items on the $10,000 level must be more than the total of the items on the $5,000. Again, going on and being incorrect loses everything.
If they believe the final item on the top level is worth more than the combined total of the two items on the $10,000, they may risk it and try to win $100,000.
The Deception
At first glance, the logical strategy seems to be to place the items in order from bottom to top. This is exactly what the first contestant did, and she did it perfectly. The cheapest item was on the bottom, the next two more expensive went above that, then 2nd and 3rd most expensive above that, and the most expensive item on top. If she were playing "Hole in One (or Two)", she would have aced it. But we're playing "Pay the Rent."
At the $100,000 level, the audience was shown on camera urging her to go for it. She looked and felt confident. She went for it. As Drew said at that point, "I hope that cat food is expensive." It wasn't. She lost everything.
Strategy
If placing items from cheapest to expensive isn't the right move... then what is?
I ran a simulation to place the items in every possible configuration. There are 6! (720) ways to place the items uniquely, but since two level require a pair of items, there are only 6!/2*2 (720/4 = 180) ways to place them. Each placement can then in turn win a maximum dollar amount. They are:
24 ways to win $1,000
89 ways to win $5,000
66 ways to win $10,000
1 way to win $100,000
(average win: $6,827.77)
Note: I suppose it is possible that the prices of items in subsequent games could change the above possibilities. These combinations are based on the actual prices given during the first live play of the game.
Here are the prices of the items from the first play (highlight to reveal):
$1.49, $2.98, $3.49, $5.49, $5.99, $7.30
There is only one correct way to stack the prices. Here is the solution to the above (highlight to reveal):
$100,000 level: $7.30
$10,000 level: $1.49 + $5.49 = $6.98
$5,000 level: $2.98 + $3.49 = $6.47
$1,000 level: $5.99
Questions
Given the prices, can you stack them up for $100,000?
If you called the products 1 2 3 4 5 and 6, from cheapest to most expensive, can you give a general solution?
If you were playing for real, would you bother going for $100,000? What level would you stop at?
-Dween!
September 22nd, 2010 at 6:43:57 AM
permalink
Here is a link to a video of the game on YouTube. You can forward to about the 2:45 point to skip the Contestants Row portion.
As was said, the strategy is deceptive. Here is my basic strategy. Highlight to read it:
6
1,5
2,3
4
Are there any possible sets of prizes than can win, where this strategy isn't correct?
In my opinion proper strategy would be to stop at the $10,000 level. It should be fairly easy to get there.
As was said, the strategy is deceptive. Here is my basic strategy. Highlight to read it:
6
1,5
2,3
4
Are there any possible sets of prizes than can win, where this strategy isn't correct?
In my opinion proper strategy would be to stop at the $10,000 level. It should be fairly easy to get there.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
September 22nd, 2010 at 6:46:17 AM
permalink
Cool... this is going to take the poor contestants a while to figure out. If you know there is a solution (probably fair assumption for TPIR), then I would guess
6
14
23
5
would work most of the time. Although, it doesn't have to: the order of the 2nd and 3rd rows could be reversed. This is a tricky game if you are only guessing prices. Even if you could order them perfectly, you might get tripped up on the 2nd/3rd row order.
Now that I know the strategy, if I got all the way to level 4 and had played properly, and still felt that the top prize was the most expensive, I would go for the big one. 10 to 1 is a good deal if you get that far and have thought about the game this way.
6
14
23
5
would work most of the time. Although, it doesn't have to: the order of the 2nd and 3rd rows could be reversed. This is a tricky game if you are only guessing prices. Even if you could order them perfectly, you might get tripped up on the 2nd/3rd row order.
Now that I know the strategy, if I got all the way to level 4 and had played properly, and still felt that the top prize was the most expensive, I would go for the big one. 10 to 1 is a good deal if you get that far and have thought about the game this way.
Wisdom is the quality that keeps you out of situations where you would otherwise need it
September 22nd, 2010 at 7:45:55 AM
permalink
Quote: Wizard
Are there any possible sets of prizes than can win, where this strategy isn't correct?
Sure :)
1,2,4,8,16,32
Also, in Dween's example last step isn't working with your strategy: $7.48 > $7.30
"When two people always agree one of them is unnecessary"
September 22nd, 2010 at 9:13:15 AM
permalink
Jeez. I immediately thought the best strategy would be to take $5000 or $10000, depending on how confident you were on the prices at those levels, and the heck with the $100000. Frankly, it looks like an easy $10000 to me, a lot easier than Plinko. Plus it's cash, no screwing around with disposing of garbage prizes or paying sales tax and getting 1099'd on a car.
A falling knife has no handle.
