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

There have been a few posts about a few The Price is Right games and the strategies involved like "Pay the Rent" and "Showcase Showdown" where they spin the wheel. I was watching TPIR the other day when the contestant got to play Punch A Bunch and it got me thinking as to the optimal strategy as to when to keep the amount you win or throw it away to try and win a bigger dollar amount. Here are the rules from the TPIR website:

This is a game in which the contestant can win up to $25,000.

The onstage contestant will see and have described four minor prizes. In order, a false price will be shown for each and the contestant will judge whether the actual retail price is higher or lower. For each correct judgment, the contestant eventually gets a chance at the Punchboard. After the contestant has determined the number of chances at the Punchboard, the contestant then takes his chances one at a time.

There are fifty compartments on the Punchboard. In each compartment, there is a card with a specific amount in dollars. At one side of the board in full view is a card with the breakdown in dollars and the quantity of each denomination as follows:

(1) $25,000 (1) $10,000
(3) $ 5,000 (5) $ 1,000
(10) $ 500 (10) $ 250
(10) $ 100 (10) $ 50
(4) Second Chance cards

(One of each $50, $100, $250 and $500 will also have "Second Chance" printed on the card). These fifty cards will be placed at random in the fifty slots under the supervision of a representative of Program Practices.

The contestant selects one of the fifty slots and punches through the paper at his selection. The host will remove, read and show the card confirming what the contestant has won on that chance. If that is the contestant's only chance, the game is over and the amount shown is what he has won.

If, however, the contestant has another chance (or chances), the contestant must decide whether to keep the amount shown and conclude play or give up the amount shown and take another one of the chances. This decision is made with any subsequent chances the contestant has earned until the contestant either decides to stop or runs of out of chances.

If the contestant punches the hole in which the card reads "second chance", the contestant gets another free chance at the board. That amount is added to the original amount before either the play is ended or the decision to continue with the contestant's chances or stop is made.


Now my question is, depending on how many punches you get and/or have left, what would be the optimal prize to keep and stay with or to toss it and try for a larger amount? Obvioulsy, the number of punches left would have to way in your decision also. I guess an easier way to say this is, what would I stay with with either 3,2, or 1 punch left?

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.

Not that I am planning on being on TPIR but I thought this would be a good math question here.

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.

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.

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.