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Last week Slate ran a piece on how to win in the TV game show The Price is Right. Overall, the article is great and I was pleased to see that so many people shared and blogged about it (even the host of the show, Drew Carey, tweeted about it).
The main article includes tips on how to bid and win in several games, and it even includes a decision tree! But perhaps the most remarkable part of the article is a “cheat sheet” where the author describes the best strategy to win in all of the current 71 games in the show (printable version), based on a statistical analysis of hundreds of episodes. My praise goes to the author, Ben Blatt (@BenBlatt).
There were a few points in the article that merit extra discussion. I offer my reactions in this post.
Game Theory Exam 2018 Cheat Sheet Non- Cooperative Game Theory Players: N= 1;2;::;n Actions/strategies: Each player chooses s ifrom his own nite strategy set: S i for each i2Nresulting in a tuple that describes strategy combination: s= (s 1;:::;sn) 2(S i) i2N Payo outcome: u i.
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'All will be well if you use your mind for your decisions, and mind only your decisions.' Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
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Contestant’s Row Bidding Strategy
Contestant’s Row is a game that functions as a pre-test to the main games in the show. Four audience members are selected and they stand in a row. One by one each person offers a guess of the price of an item. The person who is closest without going over wins and gets to play the main game with a chance to win prizes like a vacation or a new car.
For example, let’s say the guesses were $1,000, then $1,050, then $1,100, and finally $1,200. If the actual price of the item was $1,300, then the person who guessed $1,200 would win. If, on the other hand, the actual price was $1,099, then the person who guessed $1,050 would win. Note that although the guess of $1,100 was much closer to the actual price, the person lost because the guess went over the actual price.
So what’s the strategy in Contestant’s Row? There are two key parts. One is that contestants have to guess conservatively, since guessing too high means losing for sure. The other is that the contestants name their guesses out loud and in order. This means the last contestant has a chance to bid strategically based on the other guesses.
One strategy is that the last contestant should “top” the highest bid of the others. For example, if the highest bid was $1,000, then the last contestant should guess $1,001, which allows the contestant to win if the actual price is anything more than $1,001. Ben Blatt summarizes the strategy as follows.
Game theory says that when you are last to bid, you should bid one dollar more than the highest bidder. You obviously won’t win every time, but in the last 1,500 Contestants’ Rows to have aired, had final bidders committed to this strategy, they would have won 54 percent of the time.
The statistical analysis is interesting. Why is it always so smart to bid higher than everyone else? Why not bid lower, or in between bids? My theory about this is the strategy works due to the rules of the game. Because guesses too high lose, that makes people scared to overbid, skewing the guesses downward so even the highest of the other guesses is too low.
What about bidding $1, which wins if everyone else bids too high? The article doesn’t address this point directly so I did some research. It turns out this strategy is popular but does not do well in practice. According to this site, in 1050 item bids, nearly 20 percent of the Contestants’ Row games included a guess of $1, but that guess won in less than 5 percent of rounds. This is evidence that people generally do guess too low and the last contestant should capitalize by guessing $1 more than the highest guess.
There is also another small factor to the bids. The game also offers a prize of $500 for a “perfect bid” of the exact price. While it is very hard to do–less than 3 percent of guesses–the temptation to guess perfectly is there. But contestants should probably ignore this impulse: it is much better to increase your chances of winning to play the main games, where you can win thousands, than to have a small chance of winning $500.
Now or Then game
This is a brilliant part of the article. Blatt figured out that one of the games could be won every time, even if you don’t know any of the prices!
Here is how the game works. (Alternately watch a video of the game: Video: Now or Then game)
There are 6 items displayed in a circular arrangement with prices. The contestant has to guess whether the price for each item is the current price (“Now”) or it was a price from an earlier specified date (“Then”).
The contestant wins if the contestant is correct about three consecutive items (such as items 123 or items 561, as they are adjacent in the circle). The article claims that there are always 2 prices that are “Then” and 4 prices that are “Now.”
Given the limited number of combinations, Blatt figured out a foolproof guessing strategy that wins every time! Here is a link to the decision tree.
While I like the decision tree, I felt it didn’t fully explain why this strategy is effective. So I independently derived the strategy and want to offer an intuitive explanation.
The constraints of the game mean there are only a certain number of arrangements. As 4 prices are “Now” and 2 prices are “Then,” there can only be “6 choose 4” = 15 different arrangements of the board. The next thing to realize is that once you figure out where the “Then” items are, you can be sure that all the other items are “Now.”
So here is the strategy. For items 1, 2, and 3, you should just blindly guess the prices are “Now.” Statistically you have a 4/6 chance of being right on the first guess. The chances of being right on guesses for 2 and 3 depend on whether you were right on previous guesses, but you still have the best odds by guessing the items are “Now” (math omitted, I drew out the 15 possibilities and calculated the conditional probability just to be sure).
There are a few things that could happen after your guesses of items 1, 2, and 3. Note that only 2 of the items can be “Then.” So at most you can be wrong about 2 of your guesses.
–123 are all “Now.” You have guessed 3 consecutive items so you WIN
–Your guess was wrong for 1, which means item 1 was a “Then” price. At this point you have correct guesses for items 2 and 3. Your goal is to make sure you guess correctly on item 4–so you’ll have 234 and win the game. The strategy therefore is to guess “Now” for the items 5 and 6. You don’t really care if you’re right about these; the point is that you want to know where the other “Then” price is. If it appears in 5 or 6, then you know that 4 must be a “Now.” If it does not appear in 5 or 6, then you know 4 is a “Then.” Either way, you can guarantee that you’ll get item 4 correct.
–Your guess was wrong for 3. This is analogous to being wrong about 1. You have correctly guessed 1 and 2, so you want to make sure you guess item 6 correctly, creating a 612 winning board. You should guess “Now” for items 4 and 5. Again, you don’t care if you’re right: you just want to know where the final “Then” piece is. If it’s in 4 or 5, then you guess “Now” for item 6; otherwise you guess “Then.”
–Your guess was wrong for 2. This is a bit more complicated since you have guessed 1 and 3, but they are not connected. Again the trick is to figure out where the final “Then” price is. You should guess “Now” for items 4 and 6. As there is only 1 more “Then” price, you will be right on at least one of them. Now you know the prices of all but one item, which means you can easily deduce if item 5 is a “Now” or a “Then.” You’ll end up winning with consecutive items 561 or 345, or perhaps both.
–Your guess was wrong for two of the items. That means 12, 13, or 23 were “Then” prices. Now you’re golden: every remaining item will be a “Now” and you can definitely win by guessing correctly on 456.
Game Theory Cheat Sheet Template
This strategy works 100 percent of the time, so long as the rules stay the same. If I were in charge of the show, I’d probably introduce some penalty to incorrect guesses (say losing $100 of the eventual prize) or mix it up by having some boards with 1 or 3 “Then” prices.
Analysis of Squeeze Play
This part of the article includes suggested strategy for a couple of games. In the game Squeeze play, a contestant has to choose one of three digits, call them digit 2, 3, and 4. Apparently the producers have a tendency towards a pattern.
The reason that it’s better to always pick digit No. 3 in Squeeze Play is that the show’s producers are clearly not placing the numbers randomly. The wrong digit was placed in the third slot 49 percent of the time as compared with 22 percent and 28 percent for the second and fourth, respectively.
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What is clear is that guessing the middle digit is the safest bet in Squeeze Play.
This is an interesting statistical result. It’s probably not a game theory equilibrium, however, as the show’s producers can easily change the distribution at any time. It’s like saying you should pick “Rock” because it would wins in historical data for “Rock-Paper-Scissors”–an opponent can change and the history does not matter.
Strategy for spinning the wheel
The “Wheel” game is also analyzed. The game works as follows: three contestants spin a wheel, in turn. The wheel includes all multiples of 5 from 5 to 100, inclusive. The goal is to get as close to 100, without going over. The contestant with the highest total proceeds to the Showcase round, with a chance to win prizes worth $20,000 or more.
As with Contestants’ Row, the final contestant has an advantage in knowing how the others have done. What’s the optimal strategy?
Coincidentally, I covered this exact topic in a post a couple of years ago, and there was some very lively discussion: Price is Right Wheel game best strategy. Be sure to read my comment where I provide the answers.
A puzzling point: why can’t you learn prices?
Blatt’s article emphasizes that the strategies work without knowing anything about the prices of the items. For some odd reason, the author then only considers a “game theory strategy” to mean a strategy that does not use any knowledge of the prices.
For example, the cheat sheet on several games includes the sentence, “There is no game theory-based strategy that is more advantageous than guessing randomly.” But in the “Balance Game,” a random strategy only wins 33.3% of the time. But contestants have historically won 40.1% of the time. Clearly they are doing much, much better than chance, probably by knowing the prices!
In fact, it is a very, very good idea to research the prices. One contestant claims he used detailed history of prices on the show to come up with a perfect bid on a Showcase. As written in Esquire, “Terry Kniess studied prices. He saw that virtually every prize on The Price Is Right, from a pack of gum to the flashiest car, repeated. He and Linda memorized their values the way Terry had learned to count cards.”
In short, a game theory strategy is a mapping from available information into an action. Price history is available information: use it.
The show’s strategy
I will end with something that stunned me. The Slate article has one glaring omission: it is an article about game theory that is only told from the perspective of the contestant. Articles that tell people smart strategies are popular, but they are the equivalent of learning fighting skills by practicing how to break wooden boards. As Bruce Lee famously points out, boards don’t hit back.
If the Slate article actually increases the winning percentage of a few contestants, that will come at the expense of other contestants. This is a result of the economics of the show–they have a fixed budget of prizes.
As explained in this wonderful article from Esquire
The Price Is Right pays out of pocket for most of the prizes that it gives away, and the prize budget is fixed. If it’s been giving away too many cars especially, it’ll pull out some of the harder pricing games, Range Game or That’s Too Much, to balance the books. They’re not rigged, but they rely on the natural tendency of most contestants to guess somewhere in the middle. In the first instance, contestants almost always stop the game too early; in the second, they almost always stop it too late. The further the producers push the prices toward the extremes of possibility, the less likely someone will win.
This little paragraph explains exactly why the “cheat sheet” is not a foolproof guide: if contestants exploit tendencies of the show, the producers do the same for tendencies of the contestants.
6 Rules Of Game Theory
Running a game show is a balancing act: contestants have to win prizes so that it’s interesting to watch and people want to participate, but they cannot win too many prizes or the show will lose money. And that’s the real game theory–the real bottom line–about The Price is Right.
Game Theory Cheat Sheet Cheat
Slate: Winning The Price Is Right