Optimal strategy for The Price is Right game "Punch A Bunch"

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There are $64,000 in prizes on the board. With fifty slots occupied, the average value is $1280. The expected value of a pick is a little trickier due to the second chance cards, but let's say that they add another four shots at the "average" prize, so they are each worth $1280 more than their face value. This is an additional $5120, so now we can say that a pick is worth (64,000+5120)/50, or $1382.

So a quick rule of thumb might be to continue picking, as long as picks were available, until one of the top five prizes was obtained. The EV of a pick would rise as subsequent picks were made, because after the initial low-value picks, the average of the remaining picks would rise.

The above notwithstanding, I would expect most contestants to stop and take the prize if they got $1,000, and maybe even $500.

mkl, you're mistaken in your calculation of the expected value of the first pick. This is the correct formula:

(64,000 + 4x) / 50 = x,

Where x is the expected value of the first pick (the 4x represents the 4 second chances.)

This gives EV=x=$1,391.

Since there are only a maximum of 4 picks, I know you should never settle for a prize less than $5,000.

This is illustrated by this simple example:

Suppose on your first punch you hit $1,000. Do you keep it or punch again? If you keep it you get $1,000. If you punch again, your next pick has an EV of (63,000 + 4x)/49 = x, and we get an EV = x = $1,431 which is greater than the $1,000. So you should punch again.

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Quote: cardshark

mkl, you're mistaken in your calculation of the expected value of the first pick. This is the correct formula:

(64,000 + 4x) / 50 = x,

Where x is the expected value of the first pick (the 4x represents the 4 second chances.)

This gives EV=x=$1,391.

Since there are only a maximum of 4 picks, I know you should never settle for a prize less than $5,000.

This is illustrated by this simple example:

Suppose on your first punch you hit $1,000. Do you keep it or punch again? If you keep it you get $1,000. If you punch again, your next pick has an EV of (63,000 + 4x)/49 = x, and we get an EV = x = $1,431 which is greater than the $1,000. So you should punch again.

You reached the same basic conclusion as I did--I think the difference between my calculation and yours is a slight disagreement over the effect of the second chance rule.

I come up with a slightly different answer. I assume you'll always repunch when you get another chance. Therefore, I remove these four squares and come up with a punch EV of $1372. Same basic conclusion as the above posters. Keep hitting until you have a prize of more than $1000.

Of course someone might want to keep $1000 if that money means alot to them.

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For those who are confused by "second chance" cards:
Say you only get 2 punches because you sucked at the higher/lower portion of the game. You then punch 2 compartments (#1 and #2). Drew will then pull out the #1. if it is one of the 4 second chance cards, you will then pick 1 more compartment(#3). Drew will then add #1 and #3 to get your first offer. You can then reject or accept this offer.

Quote: miplet

For those who are confused by "second chance" cards:
Say you only get 2 punches because you sucked at the higher/lower portion of the game. You then punch 2 compartments (#1 and #2). Drew will then pull out the #1. if it is one of the 4 second chance cards, you will then pick 1 more compartment(#3). Drew will then add #1 and #3 to get your first offer. You can then reject or accept this offer.

AIEEEEEEE! Drew Carey!!! You just ruined my mental image of the game, which still had Smiling Bob Barker in it!!

What happens if the player gets two second chance cards? I assumed he got to take the sum of all three monetary cards as the prize for that round. If that assumption is correct, then I show the average value of the first round is $1390.89. I had a different solution than Cardshark, which considered exactly four second chances cards, and the ranges at which they were placed. Cardshark assumed they could be anywhere, and each hole had an independent 4/50 chance of having one. However, it obviously works out the same.

I find the strategy of the game is to always trade with less than $5,000, regardless of how many rounds you have left.

I retract my agreement with cardshark's answer, if taken to the penny. The cardshark solution results in an average win per punch (plus any "second chance" punches) is $1,391.30.

I believe the correct answer is $1390.89. The flaw in cardshark's solution is that he assumes if you get a second chance, the probability of additional second chances stays the same at 4/50. The rules state there are exactly four second chances. So, if you got one, the probability of finding another in the second chance punch would be 3/49.

I also don't think the rule that the second chances are placed with the low prizes can be ignorred. Following is my solution.

First, let's calculate the expected value of a prize that is not paired with a second chance. The following table shows that average prize is $1371.40.

Win	Number	Probability	Expected
25000	1	0.021739	543.478261
10000	1	0.021739	217.391304
5000	3	0.065217	326.086957
1000	5	0.108696	108.695652
500	9	0.195652	97.826087
250	9	0.195652	48.913043
100	9	0.195652	19.565217
50	9	0.195652	9.782609
Total	46	1.000000	1371.739130

Second, calculate the average prize that does have a second chance. The following table shows that average prize is $225.

Win	Number	Probability	Expected
500	1	0.250000	125.000000
250	1	0.250000	62.500000
100	1	0.250000	25.000000
50	1	0.250000	12.500000
Total	4	1.000000	225.000000

Third, create an expected value table based on the number of second chances the player finds. This can be found using simple math. For example, the probability of 2 second chances is (4/50)*(3/49)*(46/48). The expected win given s second chances is $1371.74 + s×$225.

Second Chances	Probability	Average Win	Expected Win
4	0.000004	2271.739130	0.009864
3	0.000200	2046.739130	0.408815
2	0.004694	1821.739130	8.551020
1	0.075102	1596.739130	119.918367
0	0.920000	1371.739130	1262.000000
Total	1.000000	9108.695652	1390.888067

So the average win per punch (including additional money from second chances) is $1390.89. My previous agreement was based on cardshark rounding his answer to the nearest dollar. So, we disagree by 42¢.

Ah, I understand your argument, Wizard. I 100% agree with your calculation - thanks for enlightening me.

Quote: Wizard

What happens if the player gets two second chance cards? I assumed he got to take the sum of all three monetary cards as the prize for that round. If that assumption is correct, then I show the average value of the first round is $1390.89. I had a different solution than Cardshark, which considered exactly four second chances cards, and the ranges at which they were placed. Cardshark assumed they could be anywhere, and each hole had an independent 4/50 chance of having one. However, it obviously works out the same.

I find the strategy of the game is to always trade with less than $5,000, regardless of how many rounds you have left.

Huh? Your strategy of trading with less that $5,000 contradicts your ev of $1390.89 for 1 round. If you only get 2 rounds, I get an ev of $1388.70 for the second round if you first round was "$500 + second chance" and $1000 for a total of $1500 for round 1. I'm still working the the player's ev based on how many rounds they get, but agree with your 1 round ev.

Quote: miplet

Huh? Your strategy of trading with less that $5,000 contradicts your ev of $1390.89 for 1 round. If you only get 2 rounds, I get an ev of $1388.70 for the second round if you first round was "$500 + second chance" and $1000 for a total of $1500 for round 1. I'm still working the the player's ev based on how many rounds they get, but agree with your 1 round ev.

There are no prizes between $1000 and $5000. Perhaps I should rephrase it as trade with $1,000 or less, and keep with $5,000 or more. Still, because there are no prizes between $1390.89 and $4999.99, my advice should be correct. My multiple-punch strategies did not consider the effect of removal. With no prizes anywhere close to the breakeven point, I didn't see the benefit of fussing with that.

Quote: Wizard

There are no prizes between $1000 and $5000. Perhaps I should rephrase it as trade with $1,000 or less, and keep with $5,000 or more. Still, because there are no prizes between $1390.89 and $4999.99, my advice should be correct. My multiple-punch strategies did not consider the effect of removal. With no prizes anywhere close to the breakeven point, I didn't see the benefit of fussing with that.

While there are no cards with $1500 on them if a contestant punches the "$500 + second chance" and on the second chance, punches $1000, they will have the option of keeping the $1500 or give back the $1500.
You tube video of the game for an example.

Quote: miplet

While there are no cards with $1500 on them if a contestant punches the "$500 + second chance" and on the second chance, punches $1000, they will have the option of keeping the $1500 or give back the $1500.
You tube video of the game for an example.

You're absolutely right, as usual. Thanks for the correction. I'd certainly be interested in your two+ punch strategy. I plan to make an Ask the Wizard question out of this.

Who is that host, by the way? I don't recall ever seeing him before. Judging by the way the contestant was dressed and her hair, I would date it early eighties. Cute contestant, I was hoping she would choose one of the bottom holes and bend over to punch it, but alas, she bent at the knees.

Quote: Ibeatyouraces

I remember a bunch of years ago someone threw away the $5000 and actually hit the $10,000 and this was before the $25,000 prize.

Youtube is your friend.

Quote: Wizard

You're absolutely right, as usual. Thanks for the correction. I'd certainly be interested in your two+ punch strategy. I plan to make an Ask the Wizard question out of this.

Who is that host, by the way? I don't recall ever seeing him before. Judging by the way the contestant was dressed and her hair, I would date it early eighties. Cute contestant, I was hoping she would choose one of the bottom holes and bend over to punch it, but alas, she bent at the knees.

The host is Tom Kennedy. According to wikipedia either 1985 or 1986.

Quote: miplet

The host is Tom Kennedy. According to wikipedia either 1985 or 1986.

According to that link he hosted the "The Nighttime Price is Right." I vaguely recall that, but incorrectly thought Bob Barker hosted those too.

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Some clarification might be welcomed, given you're not wrong... depending how the point is considered.

Bob Barker did in fact host "The Nighttime Price is Right", a 30-minute syndicated version, from 1977 through its last episode in 1980. That version actually began airing in 1972, with Dennis James as host, whose contract, I understand, was not renewed. Bob Barker, already hosting the weekday-airing CBS daytime version he began emceeing since its first episode on CBS in September 1972, added that once-weekly syndicated nighttime version to his hosting duties.

The clip in question comes from an episode of the Tom Kennedy-hosted "nighttime" syndicated 30-minute version that aired in first-run in 1985 and 1986.

I hope this first post of mine is helpful and welcomed.

I think your reasoning till now has been incorrect. The solution to this problem requires dynamic programming. The description of the game, found here:


has one additional parameter you must take into account: Namely, the number of punches one receives at the beginning of the game is not fixed. It itself is a random variable from 1 to 5 (depending on how many of the "small" prizes the player can guess are high or lower than the "listed" price). I would argue that the strategy should be completely different depending on how many punches one initially gets (e.g. imagine you got 50 punches, then the strategy would not be to stop at 1,391, just keep punching, you will eventually get the 25,000!!!), and at what stage of the game the player is at (e.g. a player after punching 2 holes with 3 more punches remaining and receiving a prize of 1000 in the second hole might have a different strategy then a a player with only 1 punch remaining).

This game is very interesting and seems to have many properties similar to UTH or MS (who knew, the "building blocks for many casino games seem to be "hidden" in the price is right! Now I know why I loved watching this game so much while growing up!). To that end I will write a program and let you know how you should play this game and what "return" to expect if you ever end up playing this game...

jet

So I can't speak for its accuracy, but I came across this article today and immediately thought of the WoV The Price is Right Threads:
http://www.slate.com/articles/arts/culturebox/2013/11/winning_the_price_is_right_strategies_for_contestants_row_plinko_and_the.html

This was discussed already

Quote: jet

I think your reasoning till now has been incorrect. The solution to this problem requires dynamic programming. The description of the game, found here:

http://priceisright.wikia.com/wiki/Punch-A-Bunch

has one additional parameter you must take into account: Namely, the number of punches one receives at the beginning of the game is not fixed. It itself is a random variable from 1 to 5 (depending on how many of the "small" prizes the player can guess are high or lower than the "listed" price).

Actually, it's 0 to 4, not 1 to 5. (1 to 5 is only in Plinko.)

Quote: Wizard

This was discussed already

Note that the discussion includes the four "Second Chances", which are no longer part of the game.