I usually find cheap gas based on experience. In my vicinity, I am aware of gas stations that sell for good prices. A few pennies here or there does not bother me as it is the average that matters. When I travel, I often ask a friend or I just suck it up since it’s a one-time expense.
But that is going to change. I learned of an ingenious solution from William Spaniel, who makes great instructional game theory videos.
Here is William’s story:
I ran into an interesting problem today, and I thought you might enjoy what I realized the solution was.
My friends and I were driving back from Las Vegas to San Diego. About half way through the trip, we needed gas. We weren’t worried about suddenly running out, but we definitely couldn’t make it to two towns ahead of us–we absolutely had to get gas in the town we were about to pass through. There were five exits to the freeway in this city. The goal was to get gas for as cheap as possible.The problem was that we knew virtually nothing about what the price of gas should be in this town, and we weren’t keen on going around in circles until we found the best one around.
Then, I realized that you had covered a similar problem about a year ago. Instead of cheap gas, the problem was to find true love.
The solution to maximize your probability of success was to reject the first few suitors and pick the first of the remainder who is better than those you had gone out with previously. I suggested we try a similar strategy with the gas stations, as it has the same property. We breezed by the first two (which were selling for almost $3.50 each) before stopping at one with a much more reasonable price of $2.95. As it turned out, $2.95 was the best station available.
I then explained to the other people in the car that we just solved a math problem. They may or may not think I am full of you-know-what.
I am definitely going to try this strategy! It is amazing that in statistics you can sample to find the best and save much time on your search.
Consider the picture below of bicycle tracks in the mud.
Can you figure out which way the rider was going, to the left or to the right?
This is not an easy puzzle. In fact, it was such a puzzle that once befuddled the great detective Sherlock Holmes.
But the picture provides just enough clues to be solved. Let us investigate the mystery.
Sherlock Holmes’ mistake
The background to this puzzle is the Sherlock Holmes tale The Adventure of the Priory School by Sir Arthur Conan Doyle.
At one point in the story, Holmes and Watson approach a set of bicycle tracks left in mud.
What can be deduced? Here is how Holmes reads the scene:
“This track, as you perceive, was made by a rider who was going from the direction of the school.”
“Or towards it?”
“No, no, my dear Watson. The more deeply sunk impression is, of course, the hind wheel, upon which the weight rests. You perceive several places where it has passed across and obliterated the more shallow mark of the front one. It was undoubtedly heading away from the school.”
Holmes unsurprising arrives at the right answer in the story version. But this was somewhat lucky because his logic is shaky.
The suspect part is the bit about the hind wheel obliterating the mark of the front wheel. This alone actually gives no clue about the bike’s direction.
Why? It’s because the hind wheel always follows the front wheel because it cannot turn. The hind wheel will always obliterate the front wheel marks, regardless of whether the rider was heading toward or away from the school. Nothing can be inferred on this basis alone.
Holmes was on the right track, if you will, but he did not go far enough. To solve this mystery correctly, we will take a step back for a moment.
Re-creating the scene of the crime
How can we figure out the direction of the bike rider from the track marks?
Think for a moment about how the marks are created. A rider gets on a bike, pedals to move the hind wheel, and steers by moving the front wheel. The front wheel makes the leading mark and the hind wheel follows in the same direction.
What we need to do is reverse engineer this process. We have tire marks and we need to identify which track belongs to which wheel, and more importantly, which direction the bike was moving.
And we will rely on two key facts to answer these questions:
Fact 1: the hind wheel moves in the direction of the front wheel
Fact 2: the hind wheel is a fixed distance from the front wheel
The first fact is a consequence of the hind wheel not being able to turn. The hind wheel is fixed by the bike frame in a single direction, and hence it always moves to follow the direction of the front wheel.
The second fact is a consequence of a bike design which fixes the centers of both the front and hind wheel. The hind and front wheels’ point of contact with the ground is equal to some fixed length.
So with this understanding, how can we identify the tracks and the direction of the bike?
A small and useful tangent
We will use an assist from physics (or calculus) to solve this mystery. If you’re familiar with projectile motion you can skip this. Otherwise I will give a brief refresher.
The idea we need has to do with modeling the motion of on a curve. A common motivating example is to consider lobbing a baseball to your friend. The ball leaves your hand with some speed and it gently rises until its maximum before falling in a symmetric curve towards your friend. The shape of the curve is a parabola and this is a canonical example of projectile motion.
While we know the path of the ball, we may wish to dig deeper. One might ask: what is the direction of the ball at any given moment? To figure this out, you might first draw the entire path of the ball, and then you can approximate. What you do is draw a line that connects a given position with a future position. This gives an approximation to the ball’s direction. The approximation can get better by decreasing the increment. And using calculus it’s possible to calculate the limit of this process, and the resulting line is known as the tangent line, the line that just kisses the curve:
The important fact is that the tangent line at a point indicates the direction of the ball at that point on the curve. More generally, on a curve, the tangent line at a point indicates the direction of the curve at that point. And this is the key to solving the bicycle mystery.
A method to solve the mystery
Let us put this all together.
We have two sets of bicycle tracks. We know the curve for the hind wheel is “following” the curve for the front wheel for the rider’s direction, but we cannot identify the curves nor can we identify the direction.
So what we will do is investigate. We take several points on both curves and we will draw tangent lines. We do not know which direction the tangent line should go (since we don’t know which way the bike was moving), so what we will do is draw the tangent line in both directions.
We know have tangent lines on two curves going in two different directions (4 possible candidates).
We will be able to identify the hind wheel’s marks going in the correct direction because it will have the following characteristics:
By fact 1, the tangent line will intersect the other curve that represents the front wheel marks
By fact 2, the tangent line in the correct direction will intersect at a fixed distance
We will consequently have our answer of which way the bike was going!
To recap, here is the process we will use:
–Choose several points on each curve
–Draw the tangent lines going in both directions for these points
–Identify which set of tangent lines best matches the characteristics of the hind wheel moving in the correct direction
This is a lot of words but it will make more sense visually, as pictured below.
Investigating the tire marks!
I have drawn out my work below.
What I have done is chosen several points, drawn tangent lines in both directions, and then color-coded them for ease.
Notice the blue tangent lines are all over the place. They do not intersect with the other curve and hence we can conclude this is the path of the front wheel.
The green tangent lines, on the other hand, are confined. They do intersect the other curve at a regular fixed length to their right. Thus we can conclude this is the path of the hind wheel going right.
(And I admit, the picture isn’t perfect because I generated these lines by sketching, but you get the idea.)
We can thus identify the path of the hind wheel, and we can conclude the bike must have been moving to the right.
Pretty cool, isn’t it?
Pat yourself on the back–you are now smarter (about this problem) than Sherlock Holmes.
Time to start thinking about taxes. Planning early is a big step in avoiding the last minute rush. It also helps to speed the receipt of an expected refund check.
But what does planning early mean? There was a time that I only had a vague idea. I went through the same unpleasantness each year. I would file away tax documents in a folder. I would schedule an appointment or get tax software. I would get psyched to do my taxes. And then I would hope I had everything I needed.
Not that I ran into trouble with missing documents at a tax preparer. Or that it mattered much when using tax software at home. But I often felt a process like taxes is about organization and in a sense being over-prepared.
What do you need to file taxes?
A while ago I learned of the idea of a “tax checklist.” A tax checklist is exactly what it sounds like. It is a checklist of all the tax documents you need to file.
Not only is the tax checklist a good way to double-check, but it is also a useful planning tool. I affix a tax checklist to the very folder where I store my tax documents. Now I can simply check off tax documents as I get them! Tax time really becomes a breeze.
A truly comprehensive tax checklist would cover every contingency, and it would be as dry and unreadable as the current tax code. So keep in mind the following list is just a starter. I trust you’ll consult an actual tax professional if in doubt.
Below is a tax checklist that contains many of the common documents. The list is based on other checklists from MSN Money, Yahoo Finance, Turbotax, and H&R Block.
For convenience, I’ve also created a printable file (and a spreadsheet) so you can affix the tax checklist to your tax folder.
And consider printing out a few more for family and friends–they’ll definitely be grateful for the tips!
The 2010 Super Bowl matchup is set. The Indianapolis Colts will face the New Orleans Saints. Already there is much speculation about who is the favorite, which quarterback will perform, which defense or offense will show up, and how these dome teams will play in an outdoor stadium.
In fact, none of these things really matter. It is possible to be ignorant and yet make a sure-fire winning gamble, both mathematically and practically.
My goal is not really to gamble. I neither gamble regularly nor do I encourage it. But I wish to illustrate an important point of gambling.
One may recall that gambling is an application of probability. In fact, it was a gambling wager in 1654 that led to the mathematical development of probability theory. The historical details are captured in letters between Blaise Pascal and Pierre de Fermat.
The idea then is the same as it it now: it’s the odds that matter. If you can play the odds, you can win. Here is how to do it in a way similar to the odds line set by Las Vegas bookies.
And yes, this works for almost any event betting on sports, politics, or otherwise.
How to make a guaranteed winning bet
Think about two people who are die-hard fans of their respective teams. Their beliefs can be written in probability terms, but keep in mind this level of precision doesn’t really matter. So let’s get started.
John believes the Colts will win with probability 3/4.
On the other hand, Michael believes the Saints will win with probability 3/5.
Both John and Michael enjoy friendly bets. And they are willing to bet as long as they think it’s a fair deal–that is, as long as the bet has a positive expectation.
Now here’s the fun part. Given these facts, it is possible to design a sure-fire bet that makes you money. It doesn’t matter which team actually wins–you will get money coming to you!
The idea is to make sure your loss for an event is always less than the gain.
Here is a specific way you can construct the bet:
–You bet with John that you will pay him $4 if the Colts win and he will pay you $5 if they lose. John will take this bet because his expected value is positive: (3/4) 4 + (1/4) * -5 = $1.75
–At the same time you bet with Michael. You bet with Michael that you will pay him $4 if the Saints win, and he will pay you $5 if the Saints lose. Michael will want to take this bet because his expected value is positive: (3/5) 4 + (2/5) * -5 = $0.40
The genius part is that both John and Michael believe they are making good bets. They will want to take these bets because their expected winnings are positive. You aren’t doing anything to pressure them or conceal the facts.
So how do you fare? The amazing part is that you net $1 regardless of who wins the game!
See this for yourself:
–If the Colts win, you win $5 from Michael but pay $4 to John. The net is you win $1.
–If the Saints win, you win $5 from John but pay $4 to Michael. Again, the net is you win $1.
You can obviously optimize the bets so you gain maximally, and you can increase the bet in multiples as long as both parties agree.
In financial terms, you have capitalized on differing beliefs to create an arbitrage opportunity. This is similar to what intelligent investors could do when mutual funds and stock options were in their infancy.
This is also something like what Las Vegas bookies do: they set a betting line to encourage both sides of a bet to have even money, and they pay out the winners a share from the pool of the losers’ money. This doesn’t always work in practice, but it is usually profitable over time.
In your case, it might be safer to hide your scheme, or John and Michael might not bet with you out of spite.
1. Above, John has a winning bet. So does Michael. In the long-run, that should mean both make money. But you too are making money for sure. How is it possible that everyone is making money? This is a betting paradox–how to resolve it?
2. You can design a winning bet as long as the beliefs are different. Construct a winning bet if Alice believes an event will happen with probability p and Bob believes it happens with q > p, and both will accept gambles that have positive expectations.
3. Research project: how do professional bookmakers make money on sports bets? Consider various types of bets like proposition or parlays.
The 2010 Superbowl matchup is set. The Indianapolis Colts will face the New Orleans Saints. Already there is much speculation about who is the favorite, which quarterback will perform, which defense or offense will show up, and how these dome teams will play in an outdoor stadium.
In fact, none of these things really matter. It is possible to make a sure-fire winning gamble, both mathematically and practically.
My goal is not really to gamble. I neither gamble nor do I encourage it. But I wish to illustrate an important point of gambling.
One may recall that gambling is an exercise in probability. In fact, you may recall it was a gambling wager in 1654 that led to the mathematical development of probability theory. The historical details are captured in letters between Blaise Pascal and Pierre de Fermat.
The idea then, as is now, is that it is the odds that matter. And odds are nothing more than beliefs which have interesting properties. And below I’ll teach you how you can capitalize on these beliefs.
How to win in gambling
I have two friends that are die-hard fans of the two teams in the Superbowl.
John thinks the Colts will win with probability 6/8.
How do people spend their money? Do you spend more or less than the average? Would it be cheaper to move to a new city? How would spending habits change if you got married? Or a raise?
Previously all we had were guesses. They may have been educated guesses based on customer surveys, government data, and estimates from our friends. But in the end they were general guesses. We were stuck in finding good data for our finances.
Not any more. The newly launched site Bundle is changing the game. Bundle is full of potential and its investors include Citi, Microsoft and Morningstar.
By way of background, Bundle is a site I have been working with during its private testing. At first I was impressed with the site’s comprehensive data and beautiful visualization. But what really sold me–and yes, I’ll be sappy for a moment–is the Bundle team. There are so many sites in the crowded area of personal finance and I cringe at what some of them are doing. Bundle was different and it was in line with my financial philosophy of using data to help people.
I could describe the site in words, but that would not really do it justice. I feel a visual demonstration is the best. Here is a sweet one-minute video intro to Bundle:
To reiterate a little bit from the video, Bundle is a site that lets you explore spending data in great detail. You can figure out spending data by age groups, household status, income , and geographic location. And the data is presented in a beautiful diagrams that update quickly.
There are so many exciting possibilities on Bundle. I am sure you’re going to play around with it, but here are a few features/articles to check out:
One thing I should note is that Bundle is early in its development. There are going to be many more features on the site. But even what is on there right now is impressive, and I hope you will check it out. This is a brand new tool and this is a chance to be there from the start. I am helping with content on the site so you may enjoy getting more of my thoughts on news events and economic trends.
Anyway, I hope you play around on Bundle and enjoy it as much as I do. Visit Bundle, and if you like it, you can spread the good word by becoming a fan on Facebook, following on Twitter, or using its iPhone app ViceTracker.
Nintendo’s Mario Party is a series that is a video game equivalent of board games. Gameplay involves mini-games which might include chance elements like rolling a dice or tactical elements like positioning a character on the right square. These video games consequently include many strategic elements and useful examples of game theory.
In the game Mario Party 2, for instance, there is an interesting mini-game called “Honeycomb Havoc.” I will explain the rules now, but they can also be learned by watching the youtube video below.
The rules, in brief
The game involves 4 players taking turns grabbing items from a tree. Each player hits a slow moving dice block to take either 1 or 2 items at a time. It is not possible to skip a turn and take no items, nor is it possible to take more than 2 items at a time.
The items make the game interesting. The items appear in a predetermined sequence, and they can be one of three choices. An item can be a fruit which is harmless, a coin which is good, or a honeycomb which is lethal.
The point of the game, naturally, is to avoid the honeycombs which are interspersed among the fruits and coins. The winner of the game is the player that avoids the honeycombs and is the last one standing.
The rules, in action
To see the gameplay and rules in action, check out the following video of the mini-game ”Honeycomb Havoc,” courtesy of gametheoryclips:
How can you force an opponent to pick a honeycomb? Or conversely, how can you avoid picking a honeycomb?
Think ahead, reason backwards
The game is definitely one of strategy and calculation, as the intro screen clearly indicates. As a bit of trivia, the CBS show Survivor featured a comparable game known as Thai 21, where two teams alternated turns picking 1, 2, or 3 flags. The team picking the 21st and last flag won the game.
Mathematically, this game is a specific version of a game called Nim. I will spare the gory details of solving Nim and describe how to solve this specific Mario game.
At first the gameplay is complicated with four players and three honeycombs. It is not necessarily possible to calculate what other players will do, and hence there is an interesting twist at the start.
But the game becomes more tractable as it progresses. Suppose you happen to make it to the final round. It is just you and another player, there is some number of fruits and coins ending with a honeycomb at the end. How do you play to win this game?
For illustration, let us take the scenario from the video clip above (the two player starts at about the 1:25 mark).
Yoshi gets to go first against Luigi. The line of items contains 9 items before the honeycomb. How should Yoshi play? Can Yoshi guarantee a win?
It turns out the answer is no. Yoshi should lose if Luigi plays properly, and here is the reason why.
The strategy with two players
A player’s strategy is to force the other to have to take the honeycomb. What circumstances will make this possible?
One obvious answer is when a player faces 1 or 2 items before the honeycomb. In this situation, the player can swoop up all of the fruits or coins and leave just the honeycomb, which the other has to take because skipping is not allowed.
We can label the above situation as a winning position. It is possible, with proper play, to win when facing 1 or 2 items before the honeycomb.
Knowing this, we can reason one step further. A player facing 3 items before the honeycomb, therefore, must be in a losing position. This player can only take 1 or 2 items, which sets up the other player in a winning position.
The logic can again be extended. If 3 items before the honeycomb is losing, then the previous player who forced that position, must have been in a winning position. Hence, a player facing 4 or 5 items before the honeycomb is in a winning position.
A pattern soon emerges that a player facing 3, 6, 9, etc. (multiples of 3) items before the honeycomb is in a losing position. No matter what this player does, the other player can, with proper strategy, force him into a losing position. Or to say it positively, a player facing 1, 2, 4, 5, 7, 8, etc. (not multiples of 3) items before the honeycomb is in a winning position and has the ability to force the other player into a losing position.
Returning to the video, Yoshi faces 9 items before the honeycomb. It is be possible for Luigi in theory to win the game.
Unfortunately that is not how things turn out. The game starts out well. Yoshi takes an item and Luigi responds by taking two, leaving 6 before the honeycomb, putting Yoshi in a losing position.
It is the next round where things fall apart. Yoshi takes two items, leaving 4 items before the honeycomb. At this point, it is evident that Luigi should take only one item to leave Yoshi at 3. But Luigi actually takes two items, leaving Yoshi at 2 items before the honeycomb. Yoshi smartly grabs both of them up, forcing Luigi to grab the honeycomb and lose. But let us not be so critical of Luigi–game theory takes time and practice to master.
The strategy with more than two
The game is not as easy to analyze with three or four players. I don’t think it is possible to calculate a surefire strategy, like in the two person case. But if you have one, do let us know
The reason I think this so is the following setup that has two different outcomes. Suppose there are three total players, and suppose you end your turn with 4 items before a honeycomb.
Situation 1: If the following player picks a single item, then the player after that will have to pick one or two. Either way, you start your turn with one or two items before the honeycomb and you can survive.
Situation 2: Suppose instead you end with 4 items and the following player picks two items. In that case, the next player could pick up the remaining pair of items before the honeycomb and you will be forced to lose. Or he might be nice and just take 1 allowing you to live, but the player following you to lose.
What happens depends on how strategic and cutthroat the other players are, and whether you have a pact with anyone before the final round.
Discussion questions
1. It was shown above that having 3, 6, 9, etc (multiples of three) items before the honeycomb was a losing position. What are the analogous losing positions if a player could take 1, 2, or 3 items at a time?
2. What are the losing positions when a player can take 1, 2, 3, …, k items at a time?
3. In Honeycomb Havoc, there is just one line of items, or “heap” that all players are choosing from. How does the game change when there are more heaps? See the following discussion for a mathematical answer.
I didn’t learn how to draw, or paint, or mold clay. I slightly improved artistically, if at all. But I still feel I learned a lot from my eighth grade art class.
It all started with the first assignment. We were to create an art portfolio case from construction paper. The portfolio would house all of our assignments for the year. And to complete the project, we had to write the teacher’s classroom rules on the cover of the portfolio.
The portfolio was a clever idea. I have preserved it as a memento, and consequently I still have all of my eighth grade art projects. What’s more is that I have my transcription of the classroom rules.
Back then I felt the rules were sensible. And now more than 10 years later, I feel they still capture many important values. These simple lessons have helped me in life and in managing money.
Here is a sampling of some of the rules and lessons from art class:
1. 90 percent of my success is effort. Honest, daily effort
I practice this lesson by keeping a daily record of my spending habits in my budget and expense tracking spreadsheet. By recording what I do, I stay conscious of my habits and keep my effort level high.
I can quickly see if my spending is sustainable, and if not, I compensate quickly. I have found that saving is not about making obscene money or spending too thriftily. It is largely about honest, daily effort and maintaining that effort.
2. Rushing your work is not high effort—slow down
I wish we had this rule when I worked in consulting! Seriously, people too often confuse frantic work with productive work.
Perhaps the worst money habit is apathy. Feeling “I can’t save,” or “I can’t invest,” or “I can’t get out of debt” can be very discouraging.
Things often get so emotionally bad that one feels like it’s not even worth trying. But success can only come from trying and from effort. I am reminded of Wayne Gretzky’s quote, “You miss 100 percent of the shots you never take.”
4. I will be prepared with both supplies and ideas
What a novel idea. This applies both for art class and for money management. You need the supplies to enact plans and the ideas to get you moving.
This applies for most loans—credit cards, student debt, and mortgage payments. Many financial problems could be helped simply by preparation. Avoiding late fees can help you improve your finances forever.
6. I can always find more to do
In art class this lesson was about not sitting idle and asking the teacher what to do. I apply this lesson to money management by remembering perfection is a concept, not a goal.
I will do a lot and keep busy when possible. But I remind myself that it will never be sensible to optimize all of my spending, from clipping coupons for groceries to learning how to do an oil change. I will take care of the big picture and remember I could always do more, though I will be content to do just a bit less.
And to further this point, perhaps this is a fitting lesson on which to end
Strangely, in many auctions it is a huge mistake to win. Winning can either be unprofitable or less profitable than expected, kind of like a Pyrrhic victory. The tendency of the winner to overpay is more commonly known as the winner’s curse.
I have earlier described how the winner’s curse affects salary negotiations, oil drilling, and broadcast rights in my article understanding the winner’s curse. The essential feature of such auctions is the winner is bidding too high based on his information. As a consequence, the bidder wins in unprofitable or less profitable situations.
How can the winner’s curse be remedied? While there is no hard and fast rule, there is a principle that can help.
The idea is to bid like a winner or a consequentialist. This idea is described in the fine book The Art of Strategy and I have recast a problem from the text to illustrate.
Buying a blog
Suppose you are in the process of buying a small blog. You wish to buy it and improve it a bit. How much should you offer to pay?
The question naturally depends on how much you think the blog is worth. You happen to know a thing or two about small blogs. After some research you conclude the small blog is worth somewhere between 4 thousand and 12 thousand. Besides this range you cannot estimate any more specifics. You think it is equally likely for the blog to be worth any of the values in the range.
You also have an advantage in the process. You happen to know a lot about websites and blogs, and you figure you can improve the website. You can apply your web savvy to increase the blog’s worth by 20 percent. You are certain of this as you have built up other blogs before.
The one thing you are not able to do is find an exact worth of the blog. The best you know is that the blog is equally likely to be worth any value from 4 thousand to 12 thousand, with an average value of 8 thousand.
It’s important to understand how the uncertainty affects the offer. Suppose you offer 5 thousand. There are various ways this could turn out.
If the blog turns out to be actually worth 4.5 thousand, then you would be able to raise its worth to 5.4 thousand. Since you have paid 5 thousand, you will make a decent profit.
If the blog turns out to be actually worth only 4 thousand, then you can only raise its worth to 4.8 thousand. Since you have paid 5 thousand, you will lose a bit of money.
Your first priority in this process is to make sure you make a sensible offer. You want to be sure you are not systematically overpaying and suffering from the winner’s curse.
So you ask yourself: what is the most you should offer for the blog? In other words, at what offer will you exactly break even on average? Once you know this, you will end up shaving a little bit more so you can make an offer that will actually profit.
The wrong reasoning = winner’s curse
I’ll admit it: when I first heard this question I came to the wrong answer.
Here was my logic. I figured the company was worth 8 thousand on average. Since the buyer could raise the value of the company by 20 percent, I calculated the average buyer value as 8 x 1.2 = 9.6 thousand. Therefore, a buyer could offer 9.6 thousand and just break even on average.
Do you see what is wrong with this logic?
The issue is the answer does not take the seller into account. The seller often knows more than the buyer. The fact that the seller is willing to accept an offer is bad news. It means the seller thinks the blog is worth at most that offer.
So if a seller accepts that offer, then the correct range is the blog is worth somewhere between 4 thousand to 9.6 thousand, or an average of 6.8 thousand. The buyer will be able to raise the blog up to a value of 6.8 x 1.2 = 8.16 thousand on average. But this is not good enough–the buyer has offered 9.6 thousand and will lose on average. It’s the dreaded winner’s curse!
The right answer
The correct offer is one where in which the offer exactly equals the expected worth of the blog. You need to shave your offer to account for the information advantage of the seller.
Which offer works?
The correct answer is 6 thousand. When a seller accepts this offer, the blog is actually worth somewhere between 4 and 6 thousand, with an average of 5 thousand. You can expect to raise the blog’s value to 5 x 1.2 = 6 thousand, which is precisely the amount you have offered.
The math behind the answer
The way to solve for the offer is to set up an equation between your offer and the blog’s expected value to the buyer.
The two quantities are your offer, let’s say x, and the blog’s expected worth.
What is the blog’s expected worth? As argued above, it’s important to consider the seller. A seller accepting an offer means he thinks the blog is worth at most that amount.
So a seller accepting an offer of x means the blog is worth somewhere between 4 thousand and x, for an average of (4 + x) / 2. This is the blog’s average worth to the seller.
On top of that, you can raise the value by 20 percent. Hence, the expected blog’s worth is 1.2 multiplied by (4 + x) / 2.
The equation is therefore:
x = 1.2 (4 + x) / 2 x = 0.6 (4 +x) x = 2.4 + 0.6 x
0.4 x = 2.4 x = 6
Now you know 6 thousand is the highest you should pay.
Getting to a profitable bid is another exercise in negotiations and game theory. I will not discuss the issue here but I will mention that in such situations patience can be a negotiating asset.
Discussion questions
1. The above discussion was about buying a blog. How could you apply the same principle in other situations where the winner’s curse applies–like signing a star athlete or bidding on broadcast rights for the Olympics?
2. Write an equation for the break-even offer if the bid is x, the valuation is equally likely to be between a and b, and the buyer’s skill leads to improving the company worth by y percent.
3. (more advanced) Rewrite the equation in 2 if the valuation is between a and b, but with a probability density function f(t).
4. Would the bidding strategy change if there were two or more buyers?
5. How would the bidding change if instead the winner paid the second highest offer?
6. Would additional bidders help the seller or the buyers?
7. How would the game change if the seller was less informed than the buyer about the worth of the seller’s company?
One day I was planning to buy some personal care items. On a whim, I checked if they were available at Amazon.com. To my surprise, many of the items were listed. And to my delight, they were cheaper at Amazon.
The good news turned out to be even better. Some of the items qualified for Amazon’s Subscribe and Save option. The Subscribe and Save option means an extra 15 percent or so discount and such items always ship for free.
The offer sounded great, but I wanted to try it out before sharing with all of you. I have tried Subscribe and Save for a few months and I am very pleased with it.
I want to share my experience so you can decide whether it will be useful for you too.
How Amazon’s Subscribe and Save works
One thing I have really enjoyed is the Amazon Subscribe and Save option is very easy to choose and manage.
You choose a quantity and delivery schedule (every 1 month, 2 months, 3 months, or 6 months)
You buy the item using a credit card and entering the shipping address
You click “sign up” for the program
You can manage your deliveries and orders from your account
Now here is the good part. With the Subscribe and Save program, you will often get 15 percent off your item. You will also get free shipping. This is true even if your order is below the normal $25 minimum.
Naturally I loved the “save” part of the program. But I was originally worried about the “subscribe” part. It turned out my fear was unfounded.
The “subscribe” part is really a misnomer. I’ve discovered a few useful strategies to make the most of this offer.
You can cancel without penalty after your first order.
You may be able to customize your delivery schedule for total flexibility like I did
Let me explain a bit more about these.
You can cancel any time
At first, I was worried about the delivery schedule. When you order, you have to specify shipments every 1 month, 2 months, 3 months, or 6 months. That felt like a commitment I was not willing to make.
What if I was trying a new product and I hated it? What if I got sick of something? What if I just changed my mind later? Was I still obligated to a future delivery?
Luckily, there is no issue here. There is no commitment with the Subscribe and Save program. It is possible to cancel a subscription at any time penalty-free.
I have done this myself. And it is also mentioned on the site:
How do I cancel my subscription?
There are no commitments. Cancel your subscription any time, online, 24 hours a day using Manage Your Subscribe & Save Items through Your Account. You will only be charged for orders that have shipped.
*Source: Amazon FAQ
How could you practically use this information?
Well, let’s say you are trying a new product. You may want to select a 6-month delivery schedule so you have ample time to test it out. If you don’t like the product, it will be easy to cancel before your next shipment.
This is a virtually error-proof strategy, as I can attest to. Once I completely forgot to cancel an item I disliked. But it turned out just fine. Amazon sent me an email reminder about an upcoming shipment, and consequently I could log in and cancel from my account. It was really easy and made me feel a lot happier with the program.
You may be able to customize delivery
The last strategy was all about ordering less. But what if you really like an item? What if you originally picked 6 months but now you need more at month 4.5?
I learned something cool: I was able to order more or less depending on my needs. Essentially I could customize my delivery schedule.
In my experience, I was able to do two things with my items:
–I could order more if I was running low, even before the next shipment
–I could skip a shipment if I had enough on hand
These options appeared under the “manage your subscribe & save items” section in my account.
These features really took the cake for me. I was able to get deliveries when I wanted them, and for free, at an incredible price.
*I will hedge this strategy because it is not mentioned in the official terms of the program. But I suspect it is possible with many items.
How to get started:
There are two ways you can get started with Subscribe and Save.
One way is to browse the list of eligible items in the Subscribe and Save store. This is great if you want to see what is available and browse popular items. The most popular items seem to be packaged groceries like coffee or health bars, baby and child care like diapers, and health care like antacids.
The other way is simply to search for a product you want on Amazon as you regularly would. If the product is eligible for the Subscribe and Save program, you will see a blue textbox indicating so under the product description and next to the area you buy the item. It will be obvious enough.
Let me know your thoughts
Before writing this article, I searched around and amazingly very few websites and forums discussed Amazon’s Subscribe and Save program. Many were filled with more questions than answers. And for some of them the authors had never even used the program!
That is why I felt it necessary to share my opinion and clear up some of my confusions. I would love to hear other experiences as well. If you have used this program, please share your thoughts so others can make an informed decision. I am especially curious about not so great experiences since these are the ones we can learn the most from. Thanks!
(To download, go to the “free budget spreadsheet and expense tracker” file in Financial Tools page)
Last year a lot of people took a fresh look at their finances. And that meant starting with the basics of comparing income to expenses.
I have long been recommending such expense tracking. For four years I have made a free spreadsheet available. And it has helped over 5,000 people with the finances.
In this year’s version, I elected to add a small budgeting component. You can now write in monthly budget amounts for different categories, and these are compared against the year-to-date running average. This provides an idea of how your budget estimate compares to your actual spending. This gets more interesting the more time you track.
(Personally I don’t budget and I leave these entries blank. Instead I analyze actual spending to revise my financial habits).
By and large, the spreadsheet is the same as the simple expense tracker from past years. You put in your categories, expenses, and income, and the spreadsheet tallies your savings and yearly totals. You can get an exact idea of where your money is going.
The spreadsheet can be downloaded in the Financial Tools section. (And if you’re wondering, I maintain single page with all the site’s spreadsheets to avoid broken and outdated links in individual posts).
A preview of the spreadsheet
Here is a sample of the expense tracking spreadsheet (click image for larger preview):
Using the spreadsheet in Open Office, Google Spreadsheets, or Microsoft Excel
You can use the budgeting spreadsheet into most common spreadsheet programs (if you see something odd, use common sense or email me please).
Time commitment
Some people worry it will take a long time to budget and track their money. It takes me about two minutes every day. When I get busy, I just collect all of my receipts and write them down as soon as I get free.
(To download the file, see the “free budget and expense tracker” file in Financial Tools page)
option. The Subscribe and Save option means an extra 15 percent or so discount and such items always ship for free.
