I wanted to have a get-together with some coworkers from a previous job. The problem was that I was unsure if the guests all got along.

I decided to do a simple test on Facebook. I personally was friends with each of the 5 people I wanted to invite.

But were each of them mutual friends with everyone too?

I went to person A’s profile and found he was in fact friends with the other guests B, C, D, and E.

The question is this: for a group of 5 people, what is the minimum number of profiles you have to check to make sure each person is a friend with everyone else? What about for n people?

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The answer

In a group of 2 people, you only have to check 1 person’s profile. If person A is friends with person B, then person B is necessarily friends with person A. In mathematical terms, the relation of mutual friends is “reflexive.”

Similarly, in a group of 3 people, you only need to check 2 people’s profiles. If person A is friends with B and C, and person B is friends with C as well, then person C is necessarily friends with A and B too.

We can generalize: in a group of n people, one needs to check n – 1 profiles. It is necessary to check the first n – 1 people’s friends’ lists to make sure each person is friends with each other and the last person in the group. But once that is verified, the last person must necessarily be friends with everyone else.

You can think about this graphically. Represent the group as a set of n points, and draw a line between points if the two people are friends. If the first n – 1 points are connected to every other point by a line, then the last point must also be connected to every other point (there are n – 1 degrees of freedom).

Thus, in a group of 5, one needs to check the friends list from 4 profiles.

The amusing part is the park is open from 5am until the unusual time of 12:01am.

This is evidently to avoid the confusion about noon and midnight:

The use of “12:00 midnight” or “midnight” is still problematic because it does not distinguish between the midnight at the start of a particular day and the midnight at its end. To avoid confusion and error, some U.S. style guides recommend either clarifying “midnight” with other context clues, or not referring to midnight at all. For an example of the latter method, “midnight” is replaced with “11:59 p.m.” for the end of a day or “12:01 a.m.” for the start of the next day. That has become common in the United States in legal contracts and for airplane, bus, or train schedules, though some schedules use other conventions

You can see an example of this at the Hertz car rental website. If you try to make a reservation for midnight, you get the following warning message:

This is all well and good, but it’s exactly the reason I tend to use the less problematic 24-hour time system.

My local grocer sells a variety of egg sizes. Can you figure out which one is cheapest?

Here are the prices:

Here is how I approached the problem.

Step 1: get the weights

To solve the problem, one needs to find the weights of the various sizes.

In my case, the weights were actually written on the egg cartons. But if they weren’t, I could have looked up the USDA regulations on egg sizes to find the following weights.

Step 2: compare unit costs

Now we can attack the problem. We will divide the cost by the weight to figure out the unit cost of each egg size.

It is clear from the table that eggs slightly decrease in unit cost the larger the size.

It’s not a big change: the jumbo sized eggs are roughly 3 percent cheaper than the medium sized ones. Still, this value adds up because I buy eggs almost every week.

Step 3: consider yolk:white ratio

There is one extra consideration. The unit cost is the appropriate measure if I were eating the entire egg.

But clearly I am going to throw away the shell. And as is the case, I typically throw the yolk away and just eat the whites.

If the extra weight from the larger eggs were due to the increased shell size or yolk size, then the larger sized eggs would not make sense.

So the question is: do the larger eggs contain more egg whites to make it worth it?

(Incidentally, buying eggs and throwing away the yolk is also a lot cheaper than buying products that only contain egg whites)

I did a bit of searching and luckily found a research paper on exactly this topic of the ratio of egg yolks and whites–it’s amazing the amount of information out there.

It turns out that the weight of the contents in an egg are proportional to its size. As a general rule of thumb, an egg’s weight is roughly 11 percent due to its shell, 31 percent from its yolk, and 58 percent from the white.

This means that egg white will increase proportionally with the egg’s size, and so the jumbo eggs are still the cheapest in unit cost.

Here are the numbers to confirm that jumbo eggs are still cheapest in terms of an egg-white only basis.

Conclusion

At these prices, it’s best for me to buy the jumbo eggs.

In general, one can compare the unit cost of egg sizes to figure out the best deal. This is true even if you care only about the egg whites or yolks, because both typically increase proportionally with egg size.

Most advisers will say to dollar cost average, but I do not think this is the right answer. Dollar cost averaging has nearly universal support among financial education teachers, banker, brokers, and money advisers. And yet, there has never really been a compelling mathematical case for dollar cost averaging.

To see why, let’s take a step back and understand the reasons people suggest to use dollar cost averaging. Then, we can explore the academic and logical arguments and suggest other alternatives. This article is outlined in 5 parts.

1. The standard explanation of DCA
2. Academic papers showing why DCA is wrong
3. Why the standard explanation is wrong
4. The people who really benefit from DCA
5. Alternatives: better ways to invest

(Standard disclaimers apply: I am not a money expert and you should always consult a professional before making your own decision.)

1. The standard explanation of DCA

Dollar cost averaging is about investing a set dollar amount of money regularly. Why would anyone do this?

Here’s an explanation that comes from our own government, in an article about investing basics at the SEC:

Consider dollar cost averaging.

Through the investment strategy known as dollar “cost averaging,” you can protect yourself from the risk of investing all of your money at the wrong time by following a consistent pattern of adding new money to your investment over a long period of time. By making regular investments with the same amount of money each time, you will buy more of an investment when its price is low and less of the investment when its price is high. Individuals that typically make a lump-sum contribution to an individual retirement account either at the end of the calendar year or in early April may want to consider “dollar cost averaging” as an investment strategy, especially in today’s volatile market.

The explanation is usually accompanied by a chart along the following lines:

table from about.com

Notice how the person ends up with more shares when the price dips and fewer shares when the price rises.

This is supposed to “reduce risk” and be good for investors. It sounds nice, but is there any merit to the argument?

2. Academic papers showing why DCA is wrong

I first read about why DCA is bunk in an excellent article by Timothy Middleton on MSN Money: the costly myth of dollar cost averaging.

Middleton cites two from a handful of papers that show DCA is not a good strategy. The first paper is from 1979 called “A Note on the Suboptimality of Dollar-Cost Averaging” by George Constantinides (pdf). This paper shows why DCA is not as good as another investment strategy in a theoretical perfect market setting.

This work was further refined by a 1993 paper “Nobody Gains from Dollar-Cost Averaging” in Financial Services Review pdf. This paper did analytical and numerical analysis to show that DCA is worse than a regular buy and hold strategy or an optimal portfolio rebalancing strategy.

These papers should end matters, but it does not explain why dollar cost averaging has universal support. Let’s explore why people think it’s a good idea.

3. Why the standard explanation is wrong

The promise of dollar cost averaging is tempting, but if you think about it, the claims are hollow.

First, there is a sleight of hand about what you are really doing. Dollar cost averaging is supposed to be an averaging mechanism because you buy more as the stock price rises, and less and the stock price falls. The problem is you are not really doing any averaging at all but instead mindlessly investing. The truth is you are putting in the same dollar amount regularly, without any regard to what the market does!

Second, the discussion of risk is very unsatisfactory. Dollar cost averaging is said to lower risk when compared to investing a lump sum all at once. But notice this: there is actually no measure of risk provided, at all! This is an obvious omission. But most people are tricked because they hear the word “average,” which is associated with the middle and avoiding extremes. This is another sleight of hand in the marketing of dollar cost averaging.

(Dollar cost averaging can avoid the risk of some extreme losses if you’re planning for retirement (paper here), but this is not the same as reducing risk. The papers cited above explain why DCA is not sensible on a risk-return basis).

4. The people who really benefit from DCA

If dollar cost averaging does not make sense, then why is it taught? I suspect there are two main reasons.

The first is the financial incentives. If people buy into dollar cost averaging, then that means people will invest money regularly into banks and brokerages. In fact, people who truly buy into dollar cost averaging will mindlessly invest even when the markets are tumbling and there is cause for concern! It is therefore definitely in the interests of banks and brokerages to promote dollar cost averaging so that mindless investors will pour money into accounts on a regular basis.

The second reason is paternalism. The argument is as follows. The common person is afraid to invest in stocks because of perceived risk. This is to their detriment, as the market has tended to rise over longer periods of time. Dollar cost averaging is a way to reduce fears and get people to invest through small amounts. Therefore, even if dollar cost averaging is not technically correct, teaching it is a way to get more people to invest and that serves the common good.

You’d be surprised to learn how many advisors subscribe to the second view. The truth is that many advisers are smart, and they are aware that dollar cost averaging is not a good idea. But rather than trusting people to learn, they are happy to get a second best solution. After all, dollar cost averaging has such a good image, why not capitalize on it?

Or think about it another way. If an adviser tells you to put all your money in, and then the market falls 10 percent in a week, you end up an angry customer who probably fires the adviser and complains to others. So the adviser will happily say it’s fine to dollar cost average as a way to reduce THEIR risk of losing you as a client.

I can sympathize with advisers in this regard, but I still do not think this justifies the widespread agreement about dollar cost averaging.

Financial education should teach the truth and let people decide. And the numbers show that dollar cost averaging is just not a good idea.

5. Alternatives: better ways to invest

There are two major alternatives to DCA. One is to invest the money all at once, as a lump sum. This strategy does require some patience and nerves, but it will be the best if the market is expected to rise.

Another alternative is to invest the money at random days. This is a modified version of dollar cost averaging, as one is investing the same amount, but at random times.

Middleton’s article shows an example of how both methods can be better than DCA. The article was from 2005 so the example is from 2004 data:

Table data from this article

In this example, dollar cost averaging is the worst return. This example does not prove anything, but it is an illustration of what the academic papers cited above were claiming.

A third alternative is to do an enhanced dollar cost averaging method (EDCA). This is an idea I found in a 2011 paper from the University of Nebraska (pdf). In regular DCA, you mindlessly invest the same amount every month. In EDCA, you will invest extra money if the market falls, and invest less or hold off as the price rises. This strategy was tested to be better by 0.3 to 0.7 of a percent based on historical data.

The long and short: dollar cost averaging is a popular and tempting way to invest, but its track record does not back up its claims.

Alice writes a “+” or “-” sign in front of one of the numbers, and then Bob and Alice take turns.

Once the four signs are written, the arithmetic expression is then evaluated. Bob gets points equal to the absolute value of the result.

What is the best strategy and the value of the game?

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My attempt at the solution

I took a long time to solve this problem because the solution was not apparent to me. I actually wrote out a gigantic game tree in a spreadsheet and solved for each players best response manually.

(So I should mention I am fairly sure this is the answer, but I could have made a mistake in the process–I’m sure someone will point out if there’s a flaw in my argument)

Because Bob moves second, he’s at an advantage. What Bob wants to do is move the total as far away from 0 as possible.

The best strategy is to make sure that the largest terms, 3 and 4, are of the sign. So whenever Alice writes a sign for one of those terms, Bob must immediately match the sign on the other. For instance, if Alice writes +4, then Bob must write +3 right away.

This will ensure that the expression gets 7 units away from 0.

Alice, for her part, can at least throw in one term that is the opposite sign from the pair 1 and 2, to lower the absolute value to 6.

Here is an example of gameplay:

Alice writes +3
Bob writes +4
Alice writes -2
Bob writes +1

Total of game: 6

Other combinations end up at 6 as well with strategic play on both sides.

The game is quite favorable to the second player, so they should definitely take turns going first.

Extension: What would happen if the game were played with numbers up to 6, 8, or 10? What about in general 2n?

I have yet to find an answer, but I bet there’s an elegant solution.