The Price is Right Lucky Seven game – a good example of dominated strategies

The game show The Price is Right has so many fun games to analyze.

One of my favorites is called “Lucky Seven” or “Lucky $even.” This is a guessing game and the prize is a brand new car.

Here is how the game works. You start out with seven one-dollar bills. You must then guess four digit’s of the car’s price in succession. After each guess, the actual digit of the price is shown.

If your guess was correct, you hold on to your cash. If your guess was wrong, then you have to pay the difference between the actual number and your guess. So if you guessed 5 and the actual was 7, you lose 2 dollars.

You lose the game if you run out of money. You win the game if you have at least one dollar remaining, and you can use one of those dollars to “buy” the car. Pretty cool way to win a car if you ask me!

An example of game play

It’s definitely easier to see how this game works than to read its rules.

So here’s a clip of the game, courtesy of CBS:

The Price is Right Lucky Seven

The strategy on the first three guesses

If you watched the above clip, you can see that there’s no easy way to guess the thousands digit, or the hundreds, or the tens. These are dictated more by common sense like whether a car is worth around $12,000 or instead $17,000. It would make sense to guess near the middle if you have no clue, like a 4 or 5 or 6 to minimize losses.

Something that may help is an unwritten rule that the show has not used zeros as digits since the early 1980s (undocumented but written on Wikipedia).

Other than that, it’s something of crapshoot. If you’re skillful or lucky enough to make it to the final digit, then you can use a bit of strategy.

The final digit: think dominated strategies

Suppose you’ve guessed the first three digits right on the nose and you’ve made it to the final guess with all seven dollars. What should you do?

The problem can be solved by thinking about which guesses are sensible. Take the guess of 1. Does it ever make sense to guess 1?

The answer is clearly no. With seven dollars, you can be off by six and still win the game. For a guess of 1, that means you will win if the actual digit turns out to be between 1 and 7. You will still lose the game if an 8 or a 9 show up!

Notice that instead you could have guessed 2 which wins for digits between 1 and 8. This is definitely better, but you still lose if a 9 shows up.

You can instead guarantee a win by guessing 3. Now the entire range of 1 to 9 is covered by a difference of six from your guess.

This argument shows that 1 and 2 are dominated strategies. Regardless of your beliefs, there is simply no way to justify them. It is always better to guess 3 if you have seven dollars remaining.

By symmetric reasoning, the guesses 8 and 9 are dominated by the guess of 7.

The right guesses–the ones that cover the entire range and win for sure–are therefore 3, 4, 5, 6, and 7.

The final digit: a cheat sheet

I will not belabor the calculations, but it can easily be shown that you can win for sure if you have 6 dollars (and guess 4, 5, or 6) or even if you have 5 dollars (and guess 5). The other choices in these situations are dominated strategies and should never be guessed–though I have seen people guess wrong and lose!

The game is not as simple when you have 4 dollars or less. Now it is not possible to guarantee a win. But it is possible to eliminate bad guesses by considering dominated strategies.

Suppose you have 4 dollars on the last guess. Would you ever want to guess 3? The answer is no. With 3, your range is 1 to 6, and you’d be better off guessing 4, which has a larger range of 1 to 7.

The logic can be extended to consider the other dollar amounts of 3 and 2. You can usually eliminate at least one number and increase your odds of winning.

I’ve compiled a small cheat sheet table below.

*can win for sure

Here’s an example just to reiterate. If you have 3 dollars at the end, you should definitely guess between the digits 3 and 7. You cannot guarantee you’ll win but you will give yourself the widest possible ranges by avoiding the guesses 1, 2, 8, or 9. Any of the remaining guesses are reasonable depending on your instinct.

(And my wildest dream is that this table actually helps someone. Please memorize this table if you’re going to be on the show–and good luck!)