## How to

When solving these problems, learning by doing is much better than learning by reading. I encourage to you read only as far in the solution as you need, then trying to solve the problem. If you get stuck, try reading a little further. And of course, let me know if you find a better solution!

## Friday, March 2, 2012

### How to win on, "The Price is Right"

Problem:  In the popular game show, "The Price is Right," four contestants try to get nearest to the actual retail price, without going over, for a presented product.  The contestant who gets closest wins the round gets to compete in a pricing game to win additional cash and prizes.

What is the optimal bid strategy for each of the contestants?

Is the optimal strategy commonly pursued and how can you tell?

Solution:
We'll get to the solution in a sec, but if you watch the show, this is one of the most exciting parts.  After the first round, a new contestant joins three existing ones with the famous phrase, "Come on down, you're the next contestant on The Price is Right."  An enthusiastic college student, marine, etc. comes running down, excited to play and win prizes.

Then, a prize is brought out and they each try to bid and get closest without going over.  The winner gets to play a pricing game, while the losers must wait and stew and try to win the next game.

Let's start by doing an example where there are four bidders, in order:  Alice, Bob, Candice and David.

A prize is brought out -- let's say a portable jacuzzi.  How much should each person bid?

Pic 1:  A portable jacuzzi

The first thing that everyone does, is assign a price to this.  Let's say that:
Alice thinks that the price is \$1000,
Bob thinks it is \$800,
Candice thinks it i, \$1200 and
David thinks it is \$1000.

How much should each person bid?  (and then we have to go back and account for the complete set of bidders and how bid order affects the choice)

Let's accept a few assumptions to start:
1.  The error range on price is quite high, probably +/- 30% and equal on both the high and low side.
2.  The likelihood of anyone correctly guessing the correct amount is low (approximately 1 in several hundred if we accept assumption 1)
3.  Everyone acts rationally to maximize his/her choices (which admittedly doesn't always happen)
4.  All contestants are equally good at guessing prices (also, not always true)

Well, Let's start with Alice:  One strategy is for Alice to bid exactly \$1000.  This has the benefit that if she is right, she immediately wins (and even gets a bonus).

However, if she is over by even \$1, she immediately loses.  So, there is little reason bid \$1000, as she wants to be under.

This logic would suggest to bid \$1, however there are three more bids coming, and she must assume that they will not be wildly off on the price and they will act rationally.

So, Alice must be under, but in a range that she thinks: 1) will not go over, and 2) will be close enough so that future bidders will either be under, to which she is indifferent or future bidders will go over the actual price.

Alice should keep a tight range, but go lower than her actual price.  A good suggestion for Alice would be to bid \$900-\$950.  Let's say \$950.

Now comes bidder #2, Bob.  Bob is faced with a few options.  He may think his price is over Alice's, or it may be under Alice's.  If his desired price is over Alice, his optimal strategy would be to bid \$1 over Alice.  If he bids anything more than \$1 over Alice, he is essentially giving up range in which he can win.  And, he is no worse off that someone in the future may bid \$1 over him, because that can happen at whatever price he chooses.

If he believes that the price is under Alice's he then wants to bid enough under Alice to maximize his range, but not so low that he is likely to be pipped (or have someone bid \$1 more than he).

So, let's say \$750 for Bob.

Bob now owns the range \$750-950 and Alice \$950+.

Now, Candice must choose knowing that only one bidder, David, is coming after her.  She actually has a very simple choice to make.  As the third bidder, she must choose if she believe the product's price is either: 1) below the lowest bid, 2) between the two bids, or 3) higher than the higher bid.

In each scenario, she would seek to maximize her chance within the range.  As she knows that David will choose his range optimally, she knows that he will bid \$1 over the lowest number that gets him into his range.  So, she simply wants to guess the correct range and hope that David disagrees.

So, in her choices to maximize her range:
Scenario 1 -- Candice thinks the price is below lowest bid:   bid \$1.
Scenario 2 -- Candice thinks the price is between the two bids (e.g., \$750-\$950):  Bid \$1 more than the lower bid, \$751
Scenario 3 -- Candice thinks the price is higher than the highest bid (e.g., \$950):  Bid \$1 more than the highest bid:  \$951.

In this case, Candice should bid \$951.

David's strategy is almost identical to Candice's.  He looks for the range of bids that he thinks the bid is in, and should bid \$1 more than that bid.  In this case, David believes the price is \$1000 and the outstanding bids are:  \$750, \$950 and \$951.

In this case, David should bid \$952.

So, the ideal bid strategy depends upon you location.  If you are fist, you should go under, but close enough that your price is not so low you're likely to have a range so big that someone bids \$1 more than you.  If you are second, it is a similar concern, but you need to account for the first bid so that you are in your desired range.  Third and fourth bid are automatic about maximizing your range by bidding \$1 more than the floor of your range.

Do people do this?
In fact, the results show that people very rarely do this.  If they did, you would see many more bids which were \$1 higher than someone else's and many more bids of \$1.  While you do see these, you do not see them to the extent you would expect:  at least two such bids per game and always by the third and fourth bidders.

Why not?  There are a two reasons:
- People do not know how to optimize their chances of winning
- Behavioral economics come into play and people do not seek to maximize their chances.  It is considered rude and unsporting to bid \$1 more than someone else and try to game ranges and not bid on what you think the price is.  So, people just do as they're encouraged and try to guess the price

Some people just have challenges too with this game.

Oh, and the actual retail price of the hot tub in  in pic 1 is:  \$899.  So, Bob wins in this case.

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