Beating the Big Numbers on High Rollers

Trebek and Lee. They're cops!

Trebek and Lee. They’re cops!

When you think about dice and game shows, the show you probably think of first is the Heatter-Quigley show High Rollers, hosted by Alex Trebek and his awesome 70’s fro for two separate runs on NBC, and then again by Wink Martindale in 1987. Through every incarnation of the show, the bonus round remained the same: the Big Numbers.  Based on the old gambling game Shut the Box, it heavily relied on luck, but also had some strategy in how you played it.  It also seemed extremely difficult, as players rarely walked away winners.  We wondered if the difficulty of the game was due to poor strategy, or if the game was simply stacked against the contestant.  To answer that question, we first had to figure out just what the optimal strategy should be.

The rules are simple enough. The contestant is faced with the numbers 1 through 9.  In order to win, they must eliminate each number, which they do by rolling two dice.  After each die roll, the contestant chooses from the numbers they have remaining either a number or combination of numbers that equal to the number rolled, which are then eliminated from further consideration.  If the contestant manages to eliminate all nine numbers before rolling a number that cannot be matched, they win the grand prize.

During the course of the game, the contestant can also earn “insurance markers”. Every time they roll doubles, they earn an insurance marker, which essentially give the contestant an extra life – if they roll a total which they cannot match, they may instead roll again.  Contestants can use insurance immediately, even on the same roll that earned them the insurance.

The Big Numbers board in the 1987 revival, in all its glorious 80s-ness.

The Big Numbers board in the 1987 revival, in all its glorious 80s-ness.

Sometimes you can only match a roll one way.  But most of the time, especially in the first few rolls, you have several ways to match a total with the numbers remaining.  For example, there are 12 possible ways of matching a roll of 12 on the first roll!  Which of these 12 would give you the best chance of winning the game?

We decided to take a brute force approach to evaluating this game.  Since there are nine different numbers, and each number has two states (either still on the board or removed), that means that there are 512 (2 to the 9th power) possible combinations of numbers that you could be left with at some point during the game.  While that would be a large number to work out by hand, luckily we can make the computer do most of the heaving lifting for us.

First, we can trivially work out the probabilities of every situation where there are no choices to be made.  For example, consider the situation where only the number 7 is lit up and we have no insurance markers.  The only way to win in this case is to roll a 7. Seasoned gamblers would know that the chances of rolling 7 on a pair of dice is 16.66% chance, or ⅙.  However we have to remember that not all non-seven rolls will lose the game – a roll of doubles grants an insurance marker that we could immediately cash in.  That means that of the 36 possible rolls, 6 are winners, 6 let us roll again, and 24 are losers, which means the chances of winning are 6 / 30, or 20%.

Once we have these easy cases calculated, we can start working on the cases where we have a choice.  We used an iterative process for this.  We made up a list of all number combinations that we could not evaluate as above, and looked at each of them.  If we came up with a case where every option’s chances of winning had not yet been determined, we skipped it for the moment.  If we could figure out every choice’s chances of victory, we could then figure out which option would provide the greatest chance of victory, and then determine what the overall chances of winning were from that configuration of numbers.

For example, say we have the numbers 3, 4, and 7 left on the board.  We could not have figured out the chances of victory from this position earlier, since if we roll a 7, we can do one of two things: take off the 3 and 4 leaving the 7, or remove the 7 and leave the 3 and 4.  But now that we’ve determined the chances of winning from all of the simple cases, we can use those to figure out this case as well.  We’ve already figured out that the chances of victory if we have a bare 7 left on the board is 20%.  It turns out that the chances of victory with 3 and 4 on the board are slightly better: 20.83%.  That means in this instance, we should eliminate the 7 if we roll a seven with the dice, since a board with a 3 and a 4 left on it would be easier to deal with than a board with only a 7 on it.  Combine that with the chances of winning when we roll the other good numbers, none of which require making a choice (3, 4, 10, 11), we can determine that the overall chances of victory from this position are 7.64%.

After iterating through the list of unsolved configurations several times, we eventually discovered both the best strategy to use and the chances of winning from every possible combination of numbers.  From the starting position, where all nine numbers are lit and you have no insurance markers, you have a 17.1% chance of knocking all the numbers off.

Gene has a 9.32% chance of winning right now. How'd he do?

Gene has a 9.32% chance of winning right now. How’d he do?

Can we make any characterizations about the best strategy?  In general, it seems that the best strategy is to remove the largest numbers from the board that you can from your roll.  This means that on your first roll, if your roll is less than nine, you should remove just the number you rolled.  If you roll 10 or more, you should remove the 9 and either the 1, 2, or 3, depending on if you rolled 10, 11, or 12.

I do say this is the best strategy in general, but there appear to be a large number of exceptions.  Say you rolled a five to start with, and took off the 5.  If on your second roll you rolled another five, you might be inclined, following the rule of thumb above, to knock off the 4 and the 1.  Doing this leaves you with a 7.01% chance of winning.  If instead you removed the 3 and 2, you’d have a 7.47% chance of winning.  This is just one of many cases where following the general strategy is not optimal.  We tried to figure out if there was a common thread to these exceptions, but nothing jumped out at us.  Even so, We would expect that a person following the basic strategy and ignoring these exceptions would probably only cost themselves a few tenths of a percent on their overall winning percentage.

If you’d like to play around with these results, I’ve included a little widget at the end of this post for you to play with.  Highlight the numbers remaining on the board and the number of insurance markers you have, and it’ll outline the best strategy to follow at that point, as well as your chances of victory.  Have fun!

How to become Richer than Uncle Pennybags on Monopoly Millionaire’s Club

There have been many attempts to translate Monopoly, often considered one the worst board games ever, to television. A series of pilots in the late 80s failed to get off the ground.  ABC ran a pretty good attempt of bringing the game to the small screen as a 13 week summer series in 1990, paired with the epic Super Jeopardy! tournament. In more recent times, elements of the board game have been found on the Hub’s Family Game Night.  And now, with backing by the Multi-State Lottery Association, a new show based on Monopoly has hit syndication, Monopoly Millionaire’s Club.

logo

In an alternate universe, we’d be watching “The Landlord’s Game Millionaires’ Club”.

Monopoly Millionaire’s Club is a special subgenre of game show called a lottery show.  Depending on where you live, that term either has you nodding your head in recognition (if you live in a state with a proud history of lottery shows like Illinois or California), furrowing your brow in confusion (if you live in most of the rest of the US), or nodding your head in recognition even though you’re wrong (if you live in the UK, where “lottery show” has a different connotation).  In the US, a lottery show is a game show that is produced by a lottery commission.  It looks, sounds, and smells like a game show to the untrained eye, but there are three key differences that separate a lottery show from a standard game show.

  1. Contestants are chosen through the lottery.  Often times lottery commissions will run a series of special tickets where either the main prize or a “second chance” prize is an appearance on one of their shows as a contestant.  This is the only method of contestant recruitment, which provides a different type of contestant than the carefully controlled contestant selection process of most game shows.
  2. By law, all competition must be luck based, not skill based.  In order for the lottery commission  to get away with giving away prizes on these shows, they have to ensure that it truly remains a lottery.  You’ll never see any trivia questions or obstacle courses be played on a lottery show.
  3. The prizes on a lottery show are much, much bigger than on regular shows.  It makes sense, since the lottery commission is bankrolling the show, that they can afford to give away large amounts of money in prizes.  On most game shows nowadays, winning even a 6-figure sum is quite rare.  On lottery shows, you’ll see 6 and 7 figure prizes won with regularity.
Your contestant selection process.

Your contestant selection process.

Lottery shows are a perfect subject for us here at Game Show Theory.  To prevent the games from becoming a boring procession of “pick a number, win some money”, the show will often include an element of risk that the contestant must face, choosing to put their current winnings at jeopardy in order to win more. It’s these decisions that allow us to analyze these games and develop strategies for them.

On MMC, contestants are chosen from the audience of lottery winners to come and play a game for themselves and their section of the audience.  All money won is split half by the player, and half with their audience section.  So far, there are 8 games in the rotation, each one inspired by a different aspect of Monopoly.  Each game can see the player win a maximum of $100,000, but each game could also see a too-greedy player leave with $0.

I took a look at each game and determined the optimal strategy for playing it, usually by brute-forcing my way through all possible decision paths possible in the game.  Then, I determined the average amount of money a player would win by following that strategy, as well as their chances of winning the top prize or nothing.  It turns out that there is a large difference in what you can expect to win in each game.  The 8 games, placed in order from least profitable to most, are:

8. Community Chest

Average: $16,339.77

Chance of Winning $100,000: 4.5%

Chance of Winning $0: 31.3%

Any resemblance this game has to Deal or No Deal is purely coincidental.

cc3

Purely. Coincidental.

cc1

10 identical sealed boxes, $100,000, just one question. Have I said purely coincidental yet?

The contestant is faced with 10 boxes, each with a money amount in them, from $500 up to $5,000.  Pick a box, win that amount of money.  Simple enough.  However, there’s a temptation.  The remaining boxes all have their money amounts double.  If the contestant wants to, they can give back the money they have and select a different.  If they choose to do that, they box they select must contain a prize equal or greater in value to the one they give back.  If that’s the case, then the remaining boxes double again and the same offer is made.  This process can be repeated until the contestant wins a box worth $100,000 (if doubling a box’s value would take it past $100,000, the value instead gets set at $100,000.)  If they choose a box with a smaller value than the one they give back, they lose and win nothing.

cc2

Yeah, you’re not buying the “purely coincidental” line anymore, are you?

The strategy formula for this game is rather simple.

 

cc_equation

M is the value you currently own, n is the number of boxes left in the game, and the sum of Mp is the total money left in the remaining boxes (ignoring values that would end the game).  As long as this formula is true, you should play on.  If it’s false, you should stop with your current winnings.

This game is by far the worst game to play, partly because it’s so easy to lose before accumulating a decent sum.  For example, if you select the $5,000 box with your first selection, you’re facing a choice of either walking with that paltry sum or playing on with a 4/9 chance of being knocked out.  You have to cheat death, so to speak, many times before being able to even have a chance to earn $100,000 in this game, much less actually win it.  An average value of over $16,000 may not sound bad, but when the best game on the list has an average value three times that, you see just how bad of a game this is.


 

7. Advance to Boardwalk

Average: $28,708.46

Chance of Winning $100,000: 16.4%

Chance of Winning $0: 33.8%

Unfortunately, nothing to do with the spin-off board game from the 80’s.  The game is played on a fourteen step board.  Each step has a money amount on it starting at $1,000 and increasing by $1,000 up to $13,000 on Step 13.  Step 14, however is Boardwalk, and worth $100,000 if landed on by exact count (if a contestant rolls a number that would take them beyond Boardwalk, they do not move).  The contestant rolls a die, and moves the number of spaces rolled.  Each money amount that is landed on (not passed over) is added to their bank.  The catch is, they cannot roll the same number twice.  They are given one freebie which allows them to ignore their first duplicated roll, but any more duplicate rolls after that bankrupts them and ends the game.

atb2

The Money … Ladder? How about Money Midway?

Despite the simplicity of the game, it took a lot of effort to work out the correct strategy to this game.  We can lay out a couple of obvious ground rules.   Firstly, you should keep rolling as long as you still have your Free Roll token, since you are in no danger.  Just as obviously, you should stop whenever all of the safe rolls would take you beyond Boardwalk, as there’s no gain to be had.

What to do on the first 4 rolls is simple enough: you roll, no matter what.  There’s no combination of rolls that would give you a high enough bank to not make rolling again worthwhile.

Turn 5 is the big question.  I’ve tried to suss out some simple rules for playing this turn, but as far as I can tell there are no hard and fast rules.  I can say that you should generally play on as long as you can still land on Boardwalk, but even a rule as simple as that has exceptions.  The best that I can do is give you this (ahem) easy-to-read chart.

 

atb_chart

(Click to embiggen)

atb3

Turning the sides of the die red when they become bad rolls is a very swish touch.

That’s not pixel art or a quilt pattern, that’s the chart that tells you what to do on Turn 5. The big numbers correspond to what you rolled in the first (going up and down), and second (going left to right) turns.  Doing that will lead you to another 36-cell chart, where you can find your action by cross-referencing your third (going up and down) and fourth (going left to right) rolls.  If the box is gold, then you’ve already landed on Boardwalk, congratulations!  Similarly, if the box is black, you’ve already lost the game.  Otherwise, if the box is green, you should play on, and if it’s red you should quit.

Turn 6 is a little simpler to understand.  You can create some rules based on the space you’re currently on:

– $13,000: Always stop.

– $12,000: Roll on if 2 is still a good number and you have less than $20,000 OR exactly $22,000.

– $11,000: Roll on if 3 is still a good number and you have less than $22,000.

– $10,000: Roll on if 4 is still a good number and you have less than $23,000.

– $9,000: Roll on if 5 is still a good number.

Turn 7, if we get that far, is very easy to play.  Always stop.  It’s never worth it to play on.


 

6. Electric Company

Average: $33,074.05

Chance of Winning $100,000: 11.8%

Chance of Winning $0: 26.5%

ec1

I told you our electric bill would skyrocket if you turned on every light in the house!

The player is presented with 25 light bulbs, and 10 switches.  Each switch will light up a number of light bulbs from 1 to 10.  Each number is associated with only one switch, so if you find a switch that lights up 4 light bulbs, you know none of the other switches will also light 4 bulbs.

The goal of the game is to light as many bulbs as you can.  Each bulb lit is worth an increasing amount of money, up to $100,000 with bulb 24.  Light up bulb 25, however, and you cause a blackout, losing all your money.  You can stop at any time, leaving with the money accumulated up to that point.

We can’t give you a strict dividing line as to when you should stop and when you should continue, since that depends on the configuration of switches left in play.  What we can give you is a formula for figuring out if you should continue or not.  It’s exactly the same as the formula we used in Community Chest:

cc_equation

M is the amount of money you currently have, n is the number of switches left to pull, and Sum(Mp) is the total of the possible outcomes if you were to pull the next switch.  Again, as long as the left side of the equation is smaller than the right side, you should play on.

ec2

You want switches? We got switches!

Let’s walk through the game play in the first episode as an example.  At the first time the contestant faces a decision, she’s lit up 19 light bulbs, worth $20,000.  She has 7 switches left, worth 1, 2, 3, 5, 7, 8, and 10.  Thus, the left side of our equation is $20,000 * 7 = $140,000.  The possible outcomes if she went on are $25,000, $30,000, $40,000, $100,000, and $0 three times over.  The sum of those outcomes is $215,000.  Thus it is the right decision to move onward.

She moved onward, and flipped the switch worth 3 bulbs, increasing her total to $40,000.  Now there are 6 switches left, worth 1, 2, 5, 7, 8, and 10 bulbs.  The left side of the equation is $240,000, while the right side is only worth $150,000.  She should stop, which is what she chose to do.


 

5. Park It!

Average: $34,217.59

Chances of Winning $100,000: 32.3%

Chance of Winning $0: 56.9%

pi1

Two words that strike fear in every driving test taker.

The Monopoly Parking Garage (must have been in an edition I didn’t own) has five floors, each of which can take one car (just like the real Atlantic City, right?).  10 cars are waiting to be parked, each carrying a value between $1,000 and $10,000.  The contestant chooses a car, and once the value of the car is revealed, chooses which level of the garage to park it in.  The cars must be parked in order of value, with the highest value car on the top floor, and cannot be changed once parked.  If a car cannot be parked, it’s game over.  However, successfully parking five cars will win $100,000.  At any time, the contestant can walk away with the value of the parked cars.

This is a difficult game to win, as the majority of the time you will win nothing.  On the other hand, it’s an easy game to play, as the best strategy when it comes to placing the cars is almost always the intuitive, common sense strategy, trying to keep as many cars in play with each placement.  About the only surprise that will happen commonly comes on the first placement.  If you find the $2,000 or $9,000 car, they should be placed on the 2nd and 4th floors, respectively, to prevent a potential game over on turn 2.

pi2

The best solution for games where everybody wants to play as the race car.

The first time you should consider stopping is after placing the third car, and that’s only if you have three of the seven remaining cars able to be placed.  As long as you can still place a majority of cars, it’s right to continue.  After placing the fourth car, compare your bank to the following formula:

pi_equation

C is the number cars that will fit in the final open slot.  If your bank is bigger than that number, then stop.


 

4. Block Party

Average: $36,434.83

Chance of Winning $100,000: 26.1%

Chance of Winning $0: 41.6%

bp1

Yes, they animate a game board in the middle of the bigger game board that makes up the set. It’s Boardception.

The contestant is presented with an array of 12 face down cards. 8 of the cards are associated with a color group on the Monopoly board, and have a cash value starting with $1,000 for Mediterranean/Baltic, and going up to $20,000 for Park Place/Boardwalk. Picking those cards lights up the respective monopoly on the board, and banks the associated cash.  3 of the cards are strikes.  Get 2 strikes, and your bank is cut in half, though you may continue to play.  Find all three strikes, and you leave with nothing.  The remaining card is the “Block Party” card.  Find that card and you can light up one whole side of the board, up to 2 monopolies.  If you light up all eight monopolies before finding the third strike, you win $100,000.  And, of course, you can leave at any time and win the value of your bank.

The Thrill of Victory.

The Thrill of Victory.

The first strategic question is how to use the Block Party card – to maximize money won or to maximize blocks won? Testing both scenarios showed that there was a clear winning strategy: light up as much as you can, regardless of their value.  It is much more profitable in the long run to light up the Dark Purples and Light Blues and only put $3,000 in your bank, than to light up only the Dark Blues for $20,000.

The second and much harder question to answer is when you you should stop.  After playing around with a bunch of different strategies, the best formula I have discovered for determining whether or not to quit is as follows:

If Block Party card is on the board:

bp_eq1

After Block Party card is found:

bp_eq2

bp2

The Agony of Defeat.

S is the number of strike cards left on the board, and P is the number of property cards left.  For example, at the start of the game, the formula would be as such:

bp_eq3

Not coincidentally, that’s pretty close to the average value of the game.  As long as that value is greater than the value of your bank, keep picking cards.


 

3. No Vacancy

Average: $38,729.46

Chance of Winning $100,000: 4.6%

Chance of Winning $0: 2.3%

nv1

From Yelp: “Very expensive for the area. Front desk forced us all to stay on the same floor. 2 Stars.”

The contestant is tasked with filling up all three floors of a Monopoly hotel.  Each floor has seven rooms.  On each turn, five cars come out, of which the contestant must select one.  Each car has a unique number of people in it, from 1 to 5.  The contestant has to choose which floor to put those people on; there must be enough room left on the chosen floor, and all people in a car must be put on the same floor.  If none of the floors have enough open rooms to fit the number of people in the chosen car, the contestant wins nothing.  Each room filled on the bottom floor is worth $1,000, the middle floor $2,000, and the top floor $3,000.  Fill up all the rooms, and you win $100,000.

This game is one of the hardest to win, but also one of the hardest to win nothing.  The best strategy is to stop playing once your top floor has 4 or more rooms filled and the other two floors have 3 or more rooms filled (thus making the car with five people in it unplayable).  There are a few exceptions to that rule where you should continue:

– 4 Rooms filled in the top and 3 Rooms filled in the middle.

– 3 Rooms filled in the middle and the bottom filled completely.

– 4 Rooms filled top, the middle filled completely, and 3 Rooms filled bottom.

– 4 Rooms filled top, and both middle and bottom filled.

– 3 Rooms filled bottom, and both the top and middle filled.

nv2

Can I ask the drivers if they drove in the the carpool lane on their way here?

Following this strategy maximizes your winnings, but will see you ending the game prematurely about 93% of the time.

What was really surprising to me was the optimal strategy to use when filling rooms. My assumption going in was that it would be very straightforward: fill the top row until you come to a number you can’t fill there, then move on to the middle row and then the bottom.  It turns out that the strategy to maximize your winnings is much, much more complex.  For example, on the first turn you should fill the rooms on the top floor, unless you pick the car which has 4 people in it, in which case they should go in the middle row!  I tried sussing out some placement rules to use, but I couldn’t generate an easy-to-follow list of rules.  I placed the optimal strategy in a spreadsheet if you’re interested in the gory results.


 

2. Bank Buster

Average: $42,592.69

Chance of Winning $100,000: 24.6%

Chance of Winning $0: 19.0%

bb1

Say we get into the cage, and through the security doors there and down the elevator we can’t move, and past the guards with the guns, and into the vault we can’t open…

The contestant is presented with a bank vault, locked with six locks.  Twelve keys are given to the contestant to select one at a time.  Each lock has two different keys that will unlock it.  Unlock a lock, and an amount of money is added to your bank depending on the lock, from $6,000 up to $20,000.  However, if you select a key that fits an already opened lock, that lock gets re-locked for good, and the money taken away from you.  You need to unlock five of the six locks to earn the top prize of $100,000.  If you manage to re-lock two locks, thus making the game unable to be won, you lose everything.

bb2

Five keys too many for Jack Narz, seven keys too few for Richard Bacon.

Not only is this game very likely to end well for the contestant, but the optimal strategy is also very simple.  The first time you want to consider stopping is after your 4th key.  If you have re-locked a lock and are in danger of losing, stop if your bank is $26,000 or more (except in the very specific case that you’ve unlocked the $6K and $20K locks, and the re-locked lock is the $7K lock), otherwise go on.  After you’ve picked 5 keys, stop the game with $21,000 or more in your bank (unless you have exactly $21,000 and the $20K lock is still available to open).  Finally, stop with 6 keys if your bank has reached $34,000.  The game is guaranteed to end one way or another after 7 keys have been chosen.


 

1. Ride the Rails

Average: $49,942.86

Chance of Winning $100,000: 39.5%

Chance of Winning $0: 7.1%

rtr1

I’ve sold monorails to Chesapeake, Gulf Coast, and Washington, and, by gum, it put them on the map!

This game is pretty similar to ITV’s 2009 show The Colour of Money.  That should have sent a shiver down your spine if you’re unlucky enough to remember The Colour of Money.

The contestant is presented with a list of ten railroad lines.  Each line has a railroad engine carrying between one and ten boxcars, called “cash cars”, and a caboose.  Each train has a unique number of cash cars between 1 and 10, like the switches in Electric Company.  Once a contestant chooses a line, that line’s train will slowly make it’s way across the floor, revealing its cash cars.  Each cash car revealed is worth money: $1,000 for the first line chosen, $2,000 for the second, $3,000 for the third, and $5,000 for the fourth.  At any time, the contestant can stop the train and bank the value of the cash cars revealed.  This is important, because if the caboose comes out, all of the money earned on that train is lost (money won on previous trains is kept, though).  If the contestant manages to earn $50,000 after four trains, then their total is doubled to $100,000.

Thanks to combinatorics, we know there are only 5,040 different combinations of trains that could be selected.  We can also create an exhaustive list of strategies, where a strategy is the number of cars we will let pass before hitting the brakes in each round.  Since the possible number of cars in a train changes in later rounds based on what was picked in earlier rounds, we’ll define strategies in terms of the length of possible trains in each round.  For example, if a strategy told us to stop after the 3rd train in the second round, that would mean stopping after 3 cars if we picked a train from 4-10 in the first round, or stopping after 4 cars if we pick a train from 1-3 instead.  To reflect the incentive of trying to reach $50,000 to win the maximum prize, we’ll also add a special strategy that can be chosen in the fourth round: “End”, where we will let the train continue until we’ve hit $50,000 or we see the caboose.

rtr3

Stop on a Whammy!

After testing every possible strategy against every possible configuration of trains, the winning strategy is fairly simple: 6, 5, 4, End.  In other words, we let 6 cars pass in the first round before stopping.  In the second round, let the number of cars pass equal to the 5th longest train in the second round.  In the third, we want to see a number of cars equal to the 4th longest train left on the board.  Finally, we go for $50,000 or bust in the last round.  This strategy will win us, on average, almost $50,000, making this game the most profitable game to play.


 

BONUS: Go for a Million

Average: $176,140+

Chance of Winning $1,000,000: 7.8%

Chance of Winning $0: 15.2%

gftm1

The big board in action, with the (sigh) “Monopoly Rock ‘n Roller” die roller in the middle.

The end game is almost identical to the end game to the 1990 version of Monopoly, for those of us who remember it.  The contestant starts on GO, and has 5 rolls to traverse the 40 squares of the board and return back to GO.  Along the way, the contestant earns money for each square he lands on, which he can stop with at any time.  If the contestant rolls doubles, he gets another roll, but like in the board game, three straight doubles and you’re off to jail with no money.  Likewise, landing on Space #30, “Go To Jail”, is also an instant lose.  You can also lose by landing on Community Chest and Chance spaces; contestants have to draw a card when they land on that space, which like the game could be a “Go To Jail” card.  If you make it around the board and pass GO, you win $200,000.  However, if you land right on GO, you win $1,000,000 for yourself, and your audience section gets a rolling Audience jackpot.

gftm5

“Can’t I just pay $50 and try again?”

In order to play this, you first have to give back whatever you’ve won in the first game.  Since the expected value of the game is over $176,000, you should always give back anything you won previously, even a maximum prize of $100,000.  I’m not able to calculate the exact value of the game, since I don’t have a list of how much money you win by landing on every property.  The figure listed above is only counting the $200,000 for passing GO, and $1,300,000 for landing on GO, assuming the value of the Audience jackpot to be $300,000.

gftm_win

What happens when you win $1,000,000. Sliding into GO optional.

Also, I don’t really have a strategy for whether you should stop or not.  I doubt that you should ever stop, since the chances of failure are so small compared to the potential payout.  I suppose it might be possible if, as a worst case scenario, you are on Indiana Avenue, seven squares away from “Go To Jail”, after having rolled two doubles with only one roll left.  Since you have a 33% chance of failure in that scenario (⅙ of the time you’ll roll a 7 to land in jail, ⅙ of the time you’ll roll your 3rd double) with no chance of hitting the jackpot, you’re probably better off quitting then.  But situations like that are so rare that it’s not worth the effort to try to quantify them.  Just keep rolling, and hopefully you’ll be rolling in the dough before long.

1000 Heartbeats: When to Cashout?

Logo

You know what they say about the Deputy Undersecretary of the Interior, they’re only 1,000 heartbeats away from the Presidency.

ITV debuted a new game show on February 23rd, 1000 Heartbeats.  The main gimmick of the show is that the contestant’s own heartbeat determines how long they have to play.  It’s a stylish show, complete with a live string quartet providing the music at the same tempo as the contestant’s heart rate, and has been getting favorable reviews.

The contestant is given a “clock” of 1000 of their own heartbeats (measured by a well-hidden heart monitor) to play a series of minigames testing the contestant’s skills in anagrams, mental arithmetic, and general knowledge. Each successfully completed minigame increases the potential winnings of the contestant, up to a maximum of £25,000.  However, if they run out of heartbeats, their game is over and they leave empty-handed.  After each game is completed, the contestant is given a preview of the next game and, taking into consideration the number of heartbeats they have remaining, may either play on for more money or stop.  However, before the contestant can walk away with their money, they must play one final minigame with the remainder of their heartbeats, named Cashout.

gameplay

My cardiologist’s new stress test has yet to be endorsed by the American Heart Association.

Cashout is a fairly simple game.  As their heartbeats tick down, the contestant is given a series of True or False statements, and must correctly answer 5 in a row in order to win.  Giving an incorrect answer not only forces them to start their chain of 5 answers over again, but also deducts 25 heartbeats from their clock.  It’s an effective denouement, and has provided us with tense finishes already in the show’s short history.  But watching it got me thinking – how many heartbeats would you want to bring into Cashout to maximize your chance of success? And when during the course of the game is it ideal to play Cashout instead of pressing onward for a potentially higher payday?

strings

People were shocked to see the bold new direction being taken by Brian Eno.

Before we can measure how successful a contestant will be in Cashout, we first need to determine what metrics we can measure that will allow us to estimate a contestant’s success.  I’ve identified three metrics: what their heart rate is, how often they give the correct answer, and how long each question takes to read and answer.  Using those three variables, we can figure out the odds of completing a sequence of 5 correct answers, and how many heartbeats would elapse during that time.

These metrics will be different for each player, but for the purposes of this article, we will create an average contestant, using the data from the contestants who played during the first five episodes.  In Cashout, the average contestant would answer 70.14% of the True/False questions correctly, while taking 6.45 seconds per question with a heart rate of 128 BPM.  Using these values, we ran a Monte Carlo simulation to determine the chance that the average contestant will successfully complete Cashout given the number of heartbeats they started with.  The results are displayed in this graph.

Cashout Chart

It’s a curve that starts declining fairly gently, but increases in slope before crashing to 0% around 70 heartbeats   That’s the minimum number of heartbeats needed to see and answer 5 questions correctly with no wrong answers  It may not have have happened during the first week of shows, but it should happen 17.5% of the time.  Starting Cashout with anything above 250 heartbeats leads to a better than 50% chance of succeeding.

We can use these values to help decide whether or not to proceed to the next round during gameplay.  At any given point in a player’s game, we can calculate the Expected Value (EV) of the game, which is the amount of money the player would win on average if they played Cashout.  That value is determined by the amount of money banked so far multiplied by their chances of completing Cashout with their remaining heartbeats.  For example, if a contestant has £500 banked, and has a 90% chance of winning Cashout with their remaining heartbeats, the EV of their game at that point would be £450.

If it’s a good idea for the contestant to play onward, the EV of their game after the next round must be higher. We can represent this in the following formula:

Equation1

M is the amount of money currently banked, H is the number of Heartbeats remaining, and C is the function that tells us the chances of winning Cashout given a number of heartbeats, as defined by the chart above.  M’ and H’ are the money and heartbeats left after the next round is played.

This function will be easier to work with if we divide both sides by M’, as so:

Equation2

Money LadderWe now see that the contestant should want to move on if their future chances are greater than their current chances multiplied by the ratio at which the banked money will increase.  Looking at the money ladder, we can see that in rounds 2, 3, 4, and 6 the money doubles, and M divided by M’ would be .5.  Thus, the contestant should move on if their chances of winning Cashout are greater than half of their current chances.  In rounds 3 and 7, the jump is greater, as the money is increased 150%.  M divided by M’ in this case would be .4, so the contestant’s future chances can drop by as much as 60% of their current level before moving on becomes a bad idea.

So, we know the ratio at which our money rises, and we can determine our current chances of winning Cashout.  But how can you determine your future chances?  After all, you don’t know exactly how many heartbeats you’ll have left.  This is the position in the game where an element of estimation comes into play.  Taking a look at your past performances as well as the difficulty level of the next game, you’ll have to estimate how many heartbeats you’ll think you’ll need to successfully complete the next game.

Here’s that data broken down in graph form.  The two lines represent the break-even points of how many heartbeats you can spend given your current number of heartbeats, depending on what round you are playing.

HB Chart

Click to see the full-size chart.

Let’s use this data to take a closer look at the two contestants who completed games on the show that aired on March 2nd.

contestant1

Luanne had a relatively poor first round of Contrast, and continued the theme with an even worse attempt at Unravel in round 2.  By the time Round 3 came along, she only had 366 heartbeats left to play Assemble. According to the chart above, Luanne should have continued to play if she thought that she could complete Assemble in fewer than 200 heartbeats.  Since she hadn’t done that yet in her game, it was probably a wise move that she opted to play an early Cashout.   Taking 366 heartbeats into Cashout should be enough for an average contestant to win about two-thirds of the time.  In this particular example, she managed to play Cashout successfully, ending with a scant 6 heartbeats remaining and taking home a hard-earned £500.

contestant2

Andy blew threw Contrast and Unravel before hitting a stumbling block in round 3 with Assemble.  He righted the ship in Round 4 with Link, and faced a decision on whether to play Keep Up, a mathematical game, in round 5.  It seems like most players are loathe to play these games requiring mathematical computation, but Andy opted to continue playing.  This could be seen as an aggressive move, but is a move that should lead to more money won as long as he spends fewer than 231 heartbeats on the game.  He only spent 190, which increased the expected value of his game by £414.25.  With only 204 heartbeats remaining, he had an easy choice to opt for Cashout at this point.  Unfortunately, he quickly exhausted his heartbeats with a series of wrong answers, and wound up not converting his banked £5,000.  Andy made a risky, yet mathematically sound choice to play Keep Up, but was not rewarded in the end.

Estimating how many heartbeats each minigame would take to complete would be very useful, We could could look at the past playings of each minigame and determine the time taken to complete it.  However, after only five episodes, the data we would get would not be very reliable due to the small sample size.  Perhaps if ITV gives this show the run it deserves, we will revisit this topic in a future article.

Making the Most out of the Money Cards

The game of Card Sharks is one of the best remembered shows to come out the Goodson-Todman stable. It ran for three years from 1978 – 1981, had a successful revival run from 1986 to 1989, and spawned several international versions, most famous of which is the British version (titled “Play Your Cards Right”), which ran for a total of 13 years.

The game itself was based on the old gambling card game of Acey-Deucey. As such, it is ripe for strategic and mathematical evaluation.  While we will no doubt one day look at the proper strategy in the front game, today we’re going to take a closer look at the “Money Cards” bonus round.  This was one of the most exciting and lucrative bonus rounds around in the 1970’s, where a contestant, if armed with enough skill and backbone, could parlay $200 into $28,800.  But how should one play the Money Cards to maximize one’s winnings?  Should a player even play to maximize their winnings?

The Rules

Seven cards are laid out on three rows, as seen in the image to the right. An eighth card is turned face up, and the contestant wagers any or all of their bank on whether the next card will be higher or lower (Aces are high). The contestant is staked $200 at the beginning of the round, and must bet in $50 increments, with a $50 minimum bet. Guessing correctly wins the amount bet, guessing incorrectly (even tying) results in the bet being lost.

After the fourth card is turned over, play proceeds to the second row and the contestant is given another $200. Contestants keep wagering, following the same rules up until the last higher/lower decision on the top row, where the minimum bet becomes half of the player’s total. In addition, the player may choose to swap the face-up card before the first, fourth, and seventh decision (the first card on each level) with a new card from the deck.

While some of the rules changed during the run of the show (the ramifications of which we will address later), this set of rules will be the basis of our discussions.

Strategy

Some of the simpler aspects of the game are easy to analyze.  We know that in a deck of cards, the “8” card is the midpoint – there are 24 cards lower and 24 cards higher than it.  We should thusly wager on the next card being higher when our base card is lower than 8, and lower when the base card is higher than an 8.  When the card is an 8, we can count the cards we’ve seen (not a difficult task in the end game, as almost all cards remain visible when after they are played), and determine whether there are more higher and lower cards in the deck, and predict using this knowledge.

When it comes to choosing whether or not to switch our base card when given the opportunity, we can analyze our chance of success with every base card, as seen in the following table:

Using this data, we can determine that we have on average, a 72.4% chance of calling the next card correctly given a random base card.  Therefore, it makes sense to switch our base card when our chances of winning are below that number. Using the table above, we see that we should switch on any card between a 5 and Jack.

Betting Strategy – The Naive Approach

It’s clear that we should bet the minimum when we are faced with an 8 that we can’t change – barring a large number of lower or higher cards already used, it’s going to be a net loser, so we should minimize our losses.  But what about the other 12 cards?

For our purposes, let’s define a betting strategy as a series of 6 percentages, representing the percentage of your bank that you should bet depending upon your face up card. Why only 6 instead of 12?  Because the chances of winning with a 2 are the same as winning with an Ace, the chances of winning with a 3 are the same as a King, and so forth.  We only need 6 percentages to cover all twelve possibilities.

For example, the following would be a betting strategy: 30%|50%|70%|80%|90%|100%.  These represent the percentage of your bank that you will bet, depending on the face-up card.  If you are facing a 7 or a 9, you bet 30% of your bank, the left-most percentage. If you are blessed with an Ace or a 2, you bet the right-most percentage.  If a betting strategy’s wager falls below the minimum bet allowed, then you would instead bet the minimum.

Let’s envision a theoretical contestant, All-In Adrian. He’s a smart gambler, so he knows that when you’re offered even money on a better than even money proposition, you should bet heavily.  In this case, he’s going to bet everything on every card, as long as it’s not an 8.  His betting strategy would be 100%|100%|100%|100%|100%|100%.

Adrian is correct, you should bet heavily when the odds are in your favor.  He’s found the strategy that will earn him the most money on average.  But is that the necessarily the best strategy?

Betting Strategy – The Nuanced Approach

Let’s look at another theoretical contestant, Nervous Nick.  Nick hates gambling and hates to lose. He’s going to do nothing but wager the minimum bet of $50 every time except for the last, where he’ll bet the new minimum of half of whatever he has left.  His strategy would be written as 0%|0%|0%|0%|0%|0%.

Something funny happens when you compare Nick and Adrian’s strategies.  On average, Adrian is going to win a heck of a lot more money than Nick.  But if you were to look at who would be more likely to have more money after each contestant played one round, Nick would be ahead most of the time.

This speaks to the risk involved in Adrian’s strategy.  Sure, in the long run, he would come out ahead.  But in the meantime, he’s going to go bust about 70% of the time, while Nick is guaranteed to walk away from the game with some money.  And, unfortunately for Adrian, there is no long run.  Most contestants would only get the privilege of playing the Money Cards once or twice before losing the front game and having to leave the show.  Adrian’s strategy would be great if he could play the game unlimited times, but that’s just not the case.

Risk

The story of Nick and Adrian shows that we can’t just look at the raw value of a strategy in our analysis; we have to consider risk as well.  And that’s going to complicate matters.  We can’t find a strategy that creates a mathematically proven equilibrium between value and risk – such a strategy doesn’t exist.  As any economist or sociologist would tell you, different people quantify the risk/reward ratio differently.  But, just because we can’t provide the solution, doesn’t mean we can’t provide a solution.

What we want to do is analyze every possible betting strategy.  We’ll look at the expected value of the strategy, and to represent the risk we’ll use the standard deviation that the strategy creates.  The higher the standard deviation, the more volatile the possible range of values, and the higher the risk. However, analyzing every possible combination of six percentages would be extremely overwhelming.  We’ll limit the amount of possible strategies by doing two things:

  • Every percentage must be less than or equal to the next percentage in line.  This makes sense – there’s no reason why you would ever, say, wager 40% on a 6 but only 30% on a 5.
  • We’ll limit our percentages to increments of 10%.  We’re already losing precision in our potential betting strategies by being forced to wager in $50 increments, so the precision we lose by limiting ourselves to 10 percent increments is not that much.

Those two rules took our potential number of strategies down to a manageable 8,008.  We then ran a Monte Carlo simulator to determine the expected value and standard deviation of each strategy. When looking at the results, we found that we could eliminate a large number of potential strategies from consideration.  To see why, we plotted each strategy’s expected value and standard deviation onto a scatter chart:

The general shape of the graph forms a curve sloping upwards.   In modern portfolio theory, the points on the curve form what is known as the efficient frontier.  The points that form this curve are the strategies that offer the highest value for a given level of risk.  Strategies that do not form part of this curve are not optimal, as they offer less return for their risk than another strategy would.

Let’s take a look at an individual example involving two potential strategies: Strategy A (40%|40%|40%|50%|50%|90%) and Strategy B (0%|0%|30%|50%|60%|100%)  Strategy A was determined to have a value of $1,768.28 with a deviation of $1,562.92.  Strategy B has a value of $1.807.79 and a deviation of $1,496.47.  Since Strategy B has both a higher value and lower risk than the Strategy A, there’s never a reason to use Strategy A.  Thus, we can discard Strategy A from further analysis.

Removing these inefficient strategies leaves us with a group of 647 viable strategies. Updating our scatter point chart shows us the efficient frontier in clearer detail.

 

Unfortunately, this is where our search for an “optimal strategy” has to end.  Each of these 647 strategies is as viable as the next from a mathematical viewpoint.  It only depends on how much risk you wish to undertake in the pursuit of profit.

Personal Strategies

Now, we could leave it here, and allow you to select a strategy based on your personal level of risk, but standard deviation doesn’t do a very good job of measuring risk aversion. If a person says “I’m willing to accept a standard deviation of $3,000”, what does that really mean?  It’s clear we have to reframe the question if these results are going to be worth anything.

Luckily, we can use the standard deviation of a distribution to determine how likely a given event is to occur.  Given a dollar value, we can find how many standard deviations away from the mean each strategy is, and find the strategy that will give you the highest chance of achieving that dollar value by finding the strategy the lowest number of standard deviations away from the mean.

For example, say you wanted to find the strategy that gave the you highest chance of winning at least the modest sum of $500.  Then you’d want to use the strategy 0%|0%|0%|20%|30%|50%, where $500 is -1.012 standard deviations away from the mean, corresponding to an estimated chance of 84.4% of making at least $500.  If you wanted to shoot higher and maximize your chances of making $2,500, then the riskier strategy of 50%|90%|100%|100%|100%|100% would give you the best chance.

Rule Changes

The Money Cards underwent several rule changes over the years, each in the favor of the contestants.  Two years into the show’s run, the rule concerning ties changed. Where before ties were considered losses (meaning it would be possible to lose on an Ace or 2), ties were now considered pushes – no money was gained or lost if the same card came up twice.  For one thing, this now meant that all strategies should end in 100% for Aces and 2’s, since there was no possible way to lose money.  It also would allow contestants to play more conservatively to find the best strategy for minimum values.

When the show was revived in 1986, two major rule changes happened to the Money Cards.  Firstly, the amount of money given to the contestant upon reaching the second row was increased to $400, making the top prize a round $32,000. The second big change had to do with when the contestant could change cards.  Now, instead of just the first card on each level being changed, you could change any card at any time, but only once per level.  This complicated the strategy involving choosing when to change cards, the analysis of which we will leave for another time.

Our Final Answer

In case you want to try these strategies out yourself, we’ve created a series of charts which will give you the ideal strategy given the minimum amount you want to win for any of the three rule sets.  While we can’t point to a single strategy and say it’s the best strategy to follow, looking at these charts and gauging how risky you would want to be would take you a long way to maximizing your potential earnings.