If you’re like me, you have a couple of rewards credit cards. And you try to pick the best card for a given purchase.

This used to be easy. Rewards cards used to have fixed rewards categories–like 5 percent cash back on gas and groceries–and you could get in the habit of using the right card.

Now some of the big cash back rewards have moved to rotating categories. Discover, Chase, and Citi all offer cards with rotating quarterly (and sometimes monthly) rewards.

With rewards changing so quickly among so many categories, who can remember them all?

Read the rest of this entry »

Recently I considered opening an online savings account at Discover Bank.

I was intrigued for a couple of reasons. The appeal started with the rate. The savings yield was very competitive and even slightly higher than ING Direct. As of this writing, Discover Bank offers a 1.35% APY versus ING Direct’s 1.10%.

And then there was the name. The bank is associated with Discover Financial Services, and it brought to mind the good customer service and great website usability of the Discover credit card.

The high rate and the good reputation were strong reasons. I was sold on the account…almost.

Before I opened an account, I decided to look at the fees. It was there I saw what looked like a red flag.

The stupidest fee ever

Here’s a list of Discover Bank online savings account fees:

One fee stood out to me and it’s marked above–the Account Closure fee of $20 within 90 days of opening.

From a consumer perspective, it puzzled me. I’ve never heard of a bank that charges to close an account, let alone an online one that doesn’t have the overhead of a physical bank. This definitely seems like a sneaky and annoying fee.

It also struck me as a bad signal from a game theory perspective. A bank that has a high rate and a good reputation should not need a closing fee to keep me as a customer. It’s customer service should do that job. The account closure fee was like the bank saying, “Trust us, you’ll love our services. But by the way, if you don’t love our services, we’ll charge you to leave.”

This is reinforced by the 90 day lock-in period. A small period of 15 or 30 days might be understandable to discourage rate-chasers that keep moving their money around to find the highest rate. A 90 day period seems excessive. I am not going to try an untested savings account for a 90-day period, particularly in a market with so many banks that don’t charge closing fees.

If Discover Bank ever wants my business, it will eliminate this account closure fee.

And it would also help to improve its customer service–Discover Bank is getting a lot of /bad reviews.

I usually bowl a score around 130, but the other day I hit six strikes and wound up with 215.

I took a moment to revel in the high score, but then I got thinking more critically. I was curious about how good the score was in a statistical sense.

I was with some math-minded friends and our discussion brought about many questions.

How many different bowling games are possible? What’s the average bowling score? How many ways can you attain each bowling score (what is the distribution of bowling scores)?

I did a bit of research and was pleased to learn people have already done the math to answer these questions. Here is some of the interesting math.

Writing a bowling game in mathematical terms

The first step in the problem is to translate a bowling game into mathematics. The idea is to develop a short-hand notation to compactly describe the game. I’ll follow the notation developed in this article “Is the Mean Bowling Score Awful.”

Consider the very first frame of a game. What are the possible outcomes when you bowl the ball?

One possibility is you knock down all 10 pins with a strike and the first frame ends.

The other possibility is you don’t knock all 10 pins. Then you get a second chance to hit the remaining pins. If you hit the remaining pins down, it’s a spare.

Mathematically we can model the bowls as a set of two numbers: the number of pins knocked down on the first throw, and the number on the second. We can write this as an ordered set (first throw, second throw) = (x , y).

If you knock down 3 pins and then 4, the set is (3, 4). Or if you knock down 3 and then 7 to make a spare, that would be written (3,7)

In a strike you knock down all 10 pins and you don’t get a second throw. We can write this special case as (10, 0) with the understanding you never actually made a second throw.

With this notation, we can compactly describe the possible outcomes in the first frame. The outcomes can be written as an ordered pair of two numbers, where both numbers are zero or positive, and the two numbers sum to at most 10–since the most you can knock down in a frame is 10 pins.

In set notation, this is written

The first nine frames of the game operate in the same fashion.

The tenth frame of the game is slightly different. If you get a strike in the first throw, or a spare in the second throw, you get to make a bonus third throw in the tenth frame. Therefore the tenth frame has to be represented by three numbers, with special relations depending on whether a strike or a spare is made.

There are four different possibilities:

–Two throws are made, less than 10 are knocked down (no third throw)
–The second throw makes a spare
–The first throw is a strike but the second is not
–Two strikes are made

The notation gets more complicated, but essentially it’s how you would write out these four possibilities in set notation. Here is the formal description:

Therefore, we can write a bowling game as nine pairs of elements from set A and one element from set B.

In other words, a bowling game is a sequence:

And viola, we have a mathematical way to write out a bowling game.

We will address how to account for scoring of spares and strikes in a bit, as this is more complicated.

How many bowling games are possible?

This question is easier to answer since we have a notation system for a bowling game.

We know the first nine frames of a bowling game are elements from set A and the tenth frame is from set B as described above.

It remains to count the number of elements in each of these sets. Then we multiply the number of ways to get the number of total games.

The set A is the number of ways that two positive whole numbers sum to 10 or less. This is a classic combinatorics problem.

There is a clever way to count the number of solutions. I found a derivation via Google Books for finding the number of ways n non-negative integers sum to an integer r:

Link to page 46 of Principles and techniques in combinatorics at Google Books

The formula is C(r + n -1, r)

In our bowling set, we want to find the way two numbers (n = 2) sum to 10 or less (r = 10, 9, 8, …, 0).

We want to calculate the formula for each value of r and then sum them all up. This is less work than it sounds.

For r = 10, we see the formula is C(11, 10) which is 11. For r = 9, the formula becomes C(10, 9) which is 10. The pattern continues for lower values of r, so in the end we want to sum up 11 + 10 + 9 + … 1.

This is readily calculated as 66. Thus there are 66 ways for each of the first nine frames.

We now want to know the number of ways for the tenth frame.

The process is the same as before. I will spare the gory details on this one and just say the answer is 241.

Now we can compute the total number of bowling games by multiplying the numbers together.

The total number of bowling games is (66 x 66 x 66 … x 66) (241) = (66 ⁹) (241), which is approximately 5.7 x 10¹⁸

This is no where near the number of possible chess games, but it is still a really large number.

To put it in perspective, it would take the entire world (6.7 x 10⁹), playing a game every day, over 2.3 million years to play that many different games.

What’s the average bowling score?

This part gets even more complicated mathematically.

The trick is to convert the sets into scores, based on the special rules for spares (bonus of next throw) and strikes (bonus of next two throws).

Then the mean can be computed by summing all possible scores by the number of games, which was derived earlier.

The mean bowling score turns out to be about 80 (or to be precise, more like 79.7).

Take that as reassurance that even a modest score like 100 is above average!

The derivation details are explained in the following excerpt:

Link to Mean bowling score at Google Books

The article is dated because it ends with leaving an open question of determining the full distribution of bowling scores. This in fact has been done.

What’s the distribution of bowling scores?

The final and really tough question is finding the bowling distribution.

That is, for each score n, what is the number of ways s(n) to achieve that score.

There are some cases where the answer is obvious. There is only 1 way to get a score of 0, as there is only 1 way to score 300, or 299, or 298, and so on until 291.

The other cases are more complicated to figure out. There are 20 ways you can score 1, and there are 11 ways to score 290.

To figure out the entire distribution requires clever computation. The results are described in this wonderful web page which contains the following nice graphic:

image source: all about bowling scores

Notice that bowling scores are heavily skewed! Scores above 120 are less likely since it requires a player getting a reasonable number of spares and strikes.

Another way to think about this is that even a modest score of 115 is in the 99th percentile.

Remember this the next time you’re out bowling. Given the range of possible bowling scores, your score is probably better than you think!

(The percentiles will change if we base the distribution on actual bowling data. Sadly I was not able to find any stats on this.)

Product placement gets tricky when a celeb’s behavior sends the wrong image.

Handbag companies were not happy to see reality star Snooki on Jersey Shore vomit in their handbags and defile their brands.

And so, they fought back in an interesting way. Via NBCPhiladelphia:

Well, it ends up that fashion powerhouses like Gucci and Coach have been allegedly sending the “Jersey Shore” train wreck [Snooki] expensive designer bags.

The kicker: Coach is not sending [Snooki] Coach bags. They’re sending her Gucci bags, and any other competing designer product they can put into her Guido-grabbing hands.

Who knew the strategies of Game Theory would come so naturally to the fashionistas who think a $5,000-price tag for a handbag is a reasonable marketing move?

It’s funny how each company is fighting by trying to destroy the competitor. The logic is something like “the enemy of my enemy is my friend.”

On closer analysis, the game is not good for the companies. The brand war is a type of Prisoner’s dilemma.

The best outcome is if no one sent Snooki a handbag. Yet each company is motivated to send a competitor’s handbag–regardless of what the other does–and so each sends a handbag as a dominant strategy. Ultimately both pay for handbags and both brands get shown negatively on TV.

In spite of the clever strategies, the losers of this “unbranding game” are the companies. The clear winner is Snooki. Ironic, isn’t it?

Why do we buy things we don’t need? What factors cause us to buy irrationally?

These are a couple of the main questions of behavioral economics. Lately I’ve been writing about this topic over at the money site Bundle.com, where I am a community editor.

A few of my friends have liked the articles and I figured readers here might appreciate them as well. I’ve tried to cover some of the latest findings on the psychology of spending. Here’s a sampling, and hope you enjoy!

How music can help you manage finances
Sounds can influence how we make decisions!

Sunny weather is ruining my finances
Sunlight makes us happy, but it raises our propensity and willingness to spend. Though admittedly I’m fine being a bit poorer and happy.

Dear rich person: that big logo doesn’t mean what you think it does
This post is closely related to the game theory of signaling. Bigger logos often mean a lower price point of a brand. A rich person doesn’t have to advertise he’s rich–he just is! I was thinking about the intuitive criterion when composing this post.

What you buy can make you feel smarter
Apparently brand images can “rub off” in your own habits. Guess that’s why so many of my friends enjoy wearing Stanford apparel, and it’s my justification for wearing all those free t-shirts I got as a student

The sounds a deal makes
This is a post about phonetic symbolism, the idea that certain sounds indicate a sense of size. Soft sounds can trick us into thinking it’s a small price and good deal–be careful!

I will be covering more about the psychology of spending at Bundle.com so be sure to check it out. And feel free to send me fun research and tricks you’ve heard–I always enjoy reading about stuff like this!

This one has to be in the running:

Photo credit: jurvetson

The image inspired many comments on its flickr page, like whether the wires were actually lethal, whether the fine would be levied on surviving family members, and whether the sign was real.

Genuine or not, it’s a really funny sign.

There was a hand in the TV show Poker After Dark with some interesting strategy.

The hand is from Season 6, Episode 49, and here’s a clip where the hand gets interesting on the river:

Youtube video: Poker After Dark hand at 7:52

Read the rest of this entry »

Foreign investment is a tricky game. Not only must a company profit, it also has to transfer the profits home.

Companies have done creative things to convert currency, and my favorite example is Pizza Hut in the former Soviet Union:

For example, consider the situation faced by Pizza Hut when it wanted to open a chain of restaurants in the former Soviet Union. The Russian ruble was not convertible, so Pizza Hut could not take the profits from its restaurants out of the Soviet Union in the form of dollars. There was no mechanism to exchange the rubles it earned in Russia for dollars, so the investment in the Soviet Union was essentially worthless to a U.S. company. However, Pizza Hut arranged to use the ruble profit from the restaurants to buy Russian vodka, which it then shipped to the United States and sold for dollars.

Financial Management: Theory & Practice

I was always curious which vodka brand they bartered with, and whether Pizza Hut enhanced profits with the vodka trade. Either way, this example is probably the world’s most clever use of vodka.

Did you know electricity rates can change during the day, by as much as 40 percent?

You might be able to save money by planning when you do laundry or run the dishwasher. As explained on the conEdison website:

Energy-conscious customers can save money if you use electricity during off-peak periods when costs are lower.

Our Time-of-Use program is designed to encourage customers to reduce electricity use during peak hours. Under the program, you will be charged for electricity depending on when you use it. The rates are lowest on off-peak periods: weekends, holidays, and weekdays from 10 p.m. to 10 a.m., when usage and the cost of electricity are low. Rates are higher during other periods, when usage and the cost of generating electricity are much higher. It may help to view these rates in the same way as peak and off-peak telephone rates or airline fares.

Find out if your electric rates and peak/off-peak hours

One thing to be aware is not all companies charge separate peak rates. You have to do some digging to find out.

The best way to find out is to call your electric company and ask them how they charge you. See if they have different rates for peak hours, or ask if they have a time-of-use plan where you can save money by using off-peak hours.

The next way is to check your bill. There might be two separate rates for electricity which could indicate peak and off-peak rates. (My bill just has one rate as my city does not distinguish time of usage yet).

The last way is to check the website. The conEdison except above explains their off-peak hours: “the rates are lowest on off-peak periods: weekends, holidays, and weekdays from 10 p.m. to 10 a.m.” But note these off-peak hours are not standardized so be sure to check with your company.

Note: if you don’t qualify now, keep an eye out. Your city might start classifying peak use in what is often called a time-of-use plan.

How much you could save

It might not sound like a big deal to shift your energy usage, but it can add up over time.

A while ago Bankrate.com profiled this same energy saving tip, and one user explained how savings added up:

I almost saved $30 every month. It’s month after month, and it’s not that you’re doing something special. And after 12 months, you have $360. So for waiting for a couple of hours at night, you save a couple of hundred dollars a year.

This is a very large amount, I think, for simply being mindful of when to use appliances and energy.

Major appliances to shift usage

This tip isn’t about going crazy and being nocturnal just to get the lowest energy rate.

The point is to try and shift some of the major energy using devices to off-peak hours, like:

Washing machine
Clothes dryer
Dishwasher
Electric cooking

Closing trivia: pumped storage units

Wouldn’t it be great to buy electricity during off-peak hours and sell it during peak hours?

That’s essentially what electric companies are doing when they build Pumped Storage Hydroelectric plants.

Water is pumped up elevation during off-peak hours and then released down a mountain during peak hours to generate electricity.

The TVA website explains how one of its plants Raccoon Mountain works:

The plant works like a large storage battery. During periods of low demand, water is pumped from Nickajack Reservoir at the base of the mountain to the reservoir built at the top. It takes 28 hours to fill the upper reservoir. When demand is high, water is released via a tunnel drilled through the center of the mountain to drive generators in the mountain’s underground power plant.

These plants are limited because they are expensive to build and not a renewable resource, but they are still useful to manage energy supply in a weird type of arbitrage. Pretty neat!

Applying for a job is often a game of signaling. The strategy is to highlight good signals, like grades or references, while downplaying bad signals, like lack of work experience or poor teamwork skills.

The trickier part is signals you cannot control like gender, race, and height. It’s nearly impossible to conceal these signals, so you simply let them be, and you hope you get equal opportunity as the law stipulates.

But what if all your employer cared about was these genetic traits? What would you do then?

This is a question addressed in the 1997 film Gattaca. The story takes place in a not-too-distant future with advanced genetic engineering.

Read the rest of this entry »