If we were some outside civilization looking into the human creation of marriage, what would we think?  The idea of marriage is built on centuries of tradition and cultural assumptions; lending itself as a great candidate for our skeptical scrutiny.  What is the purpose of marriage and how is it different than being in a relationship?  To answer this, we need to look at the function marriage serves.

Marriage gains its value as a proof-of-work system; where each involved party must pay a cost.  A traditional proof-of-work system involves the practice of creating an artificial cost with the goal of hindering scalability.  For example, to limit email spam, an email service provider can require the sender to incur some cost (typically a CPU intensive problem) with each mail that it receives.  The sender provides a proof that it incurred the cost along with the contents of the email.  To a legitimate individual sending a dozen emails per day, the cost is paid without notice.  However, for an abusive spammer to send millions of emails per day would require additional hardware to handle the load.  This cost may outweigh the profits of sending spam and disincentivize the abuser.  For a proof-of-work system to work, it really doesn't matter what the cost is as long as the cost cannot be performed on scale.  The unscalable cost that typical marriages provide involves time, effort, and money.  Humans are not the only ones victim to this proof-of-work mating system.  A male penguin will attempt to woo his mate by collecting smooth pebbles while a praying mantis takes a more dramatic approach and allows his head to be consumed by the mating female.  As a human, it would be hypocritical to observe these acts as a waste of precious resources.  Whether it's gifting a ring worth two month's salary or conquering Mount Rainier in someone's honor, the value of the action comes from its lack of scalability, not in the action's natural worth.  Due to a shortage of resources you simply cannot marry every person you date (although some people may try).

Equally, if not more, impactful to the value of marriage is the marriage's function as a commitment device.  According to Stephen Dubner, a commitment device is "a means with which to lock yourself into a course of action that you might not otherwise choose but that produces a desired result".  By partaking in the legal contract of marriage you are adding an artificial impediment with the goal of providing mutual security in the relationship.  When deciding whether to stay in a relationship or not, the decision maker must weigh out both options and decide which choice is least painful.  The impediment of divorce (be it financial, emotional, or social) stacks the deck towards staying together.  This buffer provides stability to prevent the minor ups and downs from capsizing the relationship, similar to how insurance reduces the risk of volatility.  However, this feature of marriage also introduces two unintended consequences.

The first is that it encourages couples to stick together even if they're both worse off for it.  Delayed gratification is the practice of using self-control and forward thinking to forgo some minor pleasure in the short-term, for greater fulfillment in the long-term.  As an example, preferring to indefinitely pay a small tax every month rather than a one time large fee is a failure to practice delayed gratification.  It's easy to see how an individual could end up trapped in an harmful relationship by dreading the cost and uncertainty of separation.

The second unintended consequence is that the buffer of the cost-out can cause both parties to take advantage of the other's buffer.  Consider the following situation:
Alice and Bob just got married.  Before the marriage, the cost of separation was measured at 10 units of suffering.  This implies Bob could have incurred 10 units of suffering upon Alice before Alice should logically consider the separation option.  However, now that they're married the cost of separation increases to 30 units.  This implies Bob can now incur up to 30 units upon Alice before she will reconsider the relationship.

Neither party may be consciously aware of the increased cost's effect upon the relationship, but it would be naive to think that there are no changes due to the shift in equilibrium.  This unintended consequence leads to the stereotypical marriage jokes: weight-gain, impoverished romance, and relentless bickering.

The main takeaway is that marriage is a complex tool.  When and how to leverage the tool is up to the individual, but it's important to understand the purpose a tool serves before blindly applying it to a situation.  Undoubtedly, by stripping away the romance and reducing marriage to a utility I have offended some readers.  Offended or not, I hope that this imparts onto you the practice of questioning and re-questioning your reality by thoroughly dissecting each layer to either affirm or discard each of your preconceived, cultural habits.

East-West Confusion

How you map a problem in your head, may affect your brain's performance.

In theory, differentiating between each cardinal coordinate should be equally difficult.  To be asked “is X East of Y?” should be no more difficult than asking “is X North of Y?”.  North, South, East, West are man-made labels and are arbitrarily located.  North happens to be on the top of most maps, but there’s no reason it couldn’t equally be located on the right, left, or bottom.  Contrary to this logic, it appears that most people are worse with tasks differentiating between East and West compared to North and South.  This is likely caused by our association of North, South, East, and West with up, down, right, left respectively (see left-right confusion). 

Differentiating between East and West will require more cognitive effort than differentiating between North and South.  This implies that East/West (EW) tasks will take longer and be less accurate than the same tasks in the North/South (NS) plane.  Since this is caused by our implicit association with egocentric coordinates, we will see a similar relative difficulty in the left/right tasks compared to above/below tasks. 

Experiment (N=47) 
Participants were instructed to answer a series of randomized true/false statements accompanied by a map, using the 0/1 keys on their keyboard (1==true, 0==false).  To measure cardinality bias, the participants were shown a map with two colored locations and were given a statement “The yellow location is _____ of the red location” where the blank was filled with a cardinal coordinate (fig 1).  The same method was applied to egocentric (relative) coordinates using the positioning of a sphere and a cube (fig 2).  The egocentric questions measured the participant’s ability in left/right (LR) and above/below (AB) tasks.  The participants were given 40 questions in total; 10 from each category (NS, EW, AB, LR).

Fig 1: Cardinality Task Example Fig 2: Egocentric Task Example

Results and Discussion 
To measure the difference in response times and prevent data skew, I measured the relative percentage of time spent in each of the four categories and took an average of the percentages.
Percentage of Time Taken by Category(ms) Accuracy by Category (%)

Participants took 8% longer on EW tasks compared to NS tasks and 15% longer on LR tasks compared to AB tasks.  In terms of accuracy, NS tasks outperformed its counterparts by 9% while AB and LR tasks showed no difference 

Experiment 2 (N=248) 
In the first experiment the participants were asked to respond to spatial true/false statements with the 0/1 keys.  Because a participant will typically use their left and right hands to press these keys, it could interfere with quality of the response with questions regarding left/right and East/West (Stroop Effect).  To take this into account, a second test was released which instead required the participants to use the up/down arrow keys to respond true/false. 

Results and Discussion
Percentage of Time Taken by Category (ms) Accuracy by Category (%)

In this version of the experiment, the relative response time difference increased compared to the previous results.   EW tasks took 24% longer and LR tasks took 20% longer.  Participants were 4% more accurate with NS and 3.5% more accurate with AB tasks than their counterparts. 

General Discussion 
There exists a bias in both cardinality and egocentric identification ability in most humans.  It is likely that we associate the two spatial coordinate systems with each other and have a difficult time in the cardinality system purely due to this association. 

In both experiments, the IP and geolocation were recorded from each participant.  A vast majority of participants were located in North America and Europe. 

Task Specialization 
Since the questions were given at random to each participant, I could measure the speed of the testers as they progressed through the experiment, independent from how they performed on the categorical tasks.  On average, the participants increased in response speed over time and failed to suffer from cognitive fatigue.

Time Taken Per Question (ms)


I have to give a huge thanks to the Reddit community for providing all the data points during the experiment.

Take the test yourself to test your own bias.
Often when arguing a position in a blog post, I'll find an interesting study someone has already performed that helps strengthen or exemplify a point with data.  These studies, when properly interpreted, are great checks to ensure that you're bringing something concrete to the table rather than just spewing airy opinions.  But what happens when you have a question that you can't find the answer to with real-world data?  You make the data yourself, of course!  Help me test a theory and be a part of this new study by taking this 2-minute test.  I'll follow up here in two weeks time with the full results. 

Spatial Bias Test

The Paradox of Akrasia 


How to Trick Yourself Into Making Smarter Choices

Imagine what you'll be like one year from now.  Of course you'll be smarter, healthier, and more successful in every possible way.  This is our projected self.  The greater the disparity between our projected self and actual self, the more unhappy we are.  Being unhappy sucks, so it's a no-brainer that as the rational, dopamine-seeking beings that we are, we'll do whatever it takes to align to our projected self... right?   Fast forward a year and odds are you'll be in the same place that you started; no bad habits broken nor good habits kept.  One study found that only 8% of people who set New Year's resolutions accomplish their goals and 44% quit before the first month[0].  Well what happened?  The short story is that we got lazy.  Despite our best intentions, we slowly fall into our old ways.  Quitting after gaining new information is not always a bad thing.  Maybe you learn that running a marathon is just not your thing; that's okay.  However, if you truly want to accomplish something, you shouldn't let laziness become your hurdle.

Why are we lazy?

Like all successful species we're built to reproduce.  The intelligence of our human brain took us leaps and bounds past other species in optimizing for the ability to survive and therein become efficient reproducers.  This generic tool did a great service for the reproductive capability of the species, but there was no way to restrict it to only focus on reproduction.  Of course our genes didn't build a brain without strings attached.  The genes have engineered a brain that could be manipulated in real time with the help of neurotransmitters; dopamine, which heightens the attraction between individuals, oxytocin, the "cuddle hormone", and norepinephrine to get us out of bed in the morning, just to name a few.  Sometimes you can actually feel the different "strings" being pulled.  "Directly after copulation, the devil's laughter is heard".  The "devil's laughter" that Schopenhauer is referring to is the realization that despite our higher intelligence, we're still slaves to our genes.  Because our genes are optimizing for reproduction we often act in a ways that does not rationally align with our goals (which I presume is more than just making babies).

Laziness is another example of a genes vs intelligence conflict.  To conserve precious calories (wild cheese burgers are hard to come by) we're genetically wired to be lazy; however we're ironically miserable as a result of our laziness.  In a study from the University of Chicago[1], when given the option to wait 15 minutes at the current location or make a 15 minute round-trip walk to another location to submit a paper, only 32% of participants chose to make the walk.  They performed a separate experiment where they did not give the participants an option but instead randomly chose some participants to stay idle and some to make the walk to the more distant location.  The participants that were forced to walk reported higher satisfaction in the experience than the people who stayed.  Is it that we are bad at predicting the type of things that makes us happy?  According to the study, that's not the case.  A third group of participants were asked to predict which task would lead to greater happiness.  A majority of the participants accurately predicted that the busy task would produce greater happiness than the idle task.  The paradox is that we will willingly choose the path of least happiness.  Laziness is that disagreeable friend of our business partner (our genes) whom we're forced to hang out with simply because we share the same investment.

It might be a bit depressing realizing that when it comes to laziness, the cards are stacked against us.  It's true that if we're optimizing for the intelligent self, too often weaknesses are introduced by the selfish drive of the genes.  However, occasionally there's a silver lining and we can make these weaknesses our allies.  Here are three examples of how you can make human fallacies work in your favor, specifically to help with self control.

Nudging to a Better Tomorrow

Richard Thaler and Cass Sunstein describe a way to leverage our laziness for good.  The idea of a "nudge" is to set a steady-state which is likely optimal for the individual.  In order to move away from this optimal steady state, it requires work; which our lazy selves are reluctant to do.  For example, many people find it hard to have the self-control to save money for retirement.  A study published in 2011 by the National Bureau of Economics[2] found that nearly half of US citizens could not cope with an unexpected cost of $2000, even if they were given one month to do so... WTF!  To help mitigate this terrifying vulnerability, Thaler and Sunstein recommend that employers create expertly designed default retirement plans for their employees[3].  These retirement plans would be near optimal for a majority of the employees and would require some extra work on the employees behalf to opt out for a different plan.

Keep reading, you've already come this far...

The sunk cost fallacy is the propensity for humans to allow their past investments to influence their future ones.  When investing in stocks, the amount of money you have lost [or gained] should not influence your current decision to buy or sell.  Our decision should only be affected by our prediction of the future price, yet we all have a little voice rationalizing, "I've already invested so much, I should continue investing to see it out".  However, the sunk cost fallacy can sometimes work in your favor.  Exercise is an investment of time and effort.  What you eat and future investments in exercise should be made independent of the cost you've already sunk.  However, the sunk cost fallacy will make it more likely that you will eat healthy and continue exercising so that you can reinforce the idea that your previous investment was not wasted.

Pain in the Butt

Your past reality is what you remember it to be.  We're more or less likely to perform the same task based on how pleasant or painful we remember that task to be.  Multiple experiments have found that your memory of a certain task is independent of the duration of the task.  What matters most in the way we remember an experience is the peak pain and the very last moments of the experience (sometimes this is referred to as the peak-end rule).  A study by Kahneman, Redelmeier, and Katz found that colonoscopy patients rated the experience as less painful and were more likely to return simply by leaving the scope in an extra three, unnecessary minutes[4]!  This extra three minutes minutes of doing literally nothing made the end of the experience less painful, causing the memory of the event to be regarded as more pleasant.  

Often we choose the lazy route because we dread a task due to its inherent unpleasantness.  The duration neglect bias would tell us that when performing such a task, we should extend the duration as long as possible to get the most done, while trying to evenly distribute the discomfort and minimize the displeasure at the end of the task.

Your turn

Have other nifty examples on how to trick your brain into submission?  Add them to the comments.

See Also

Save Me From Myself: A New Freakonomics Radio Podcast

Schopenhauer's Philosophy
Escalation of Commitment


[0] https://www.researchgate.net/publication/223679624_The_resolution_solution_Longitudinal_examination_of_New_Year's_change_attempts
[1] http://home.uchicago.edu/~xyang4/papers/Idleness%20Aversion.pdf
[2] http://www.nber.org/papers/w17072
[3] https://en.wikipedia.org/wiki/Nudge_(book)#Retirement_saving
[4] https://www.amherst.edu/system/files/media/0678/colonoscopy%25202.pdf
Bootstrapping is the solution to the paradox it is named after.  It got its name from the impossible scenario of pulling oneself over a wall using the straps of one's own boots.  Practically, it refers to the process of kick starting a cycle that seemingly has no logical beginning.  For example, the chicken-egg paradox is solved using the bootstrapping technique: starting with a chicken-precursor and the right evolutionary conditions, the percentage of chicken genes will slowly increase until the first 100% chicken egg is laid.

Booting up your computer
Ever wonder why it's called booting up?  Software in your computer is loaded by and runs on other software.  For example, this website is a script that is interpreted by your browser which is running on your operating system.  There is nothing that tells your computer how to run software aside from other software.  If only software can run software, then how does the first piece of software run?  When you push the button to turn on your computer, it provides electricity to the CPU and initializes a hardware-based program.  The hardware-based program gives instructions to the computer on how to read basic software.  This process snowballs into loading more and more complex software until eventually your operating system is loaded.

Cell Differentiation
When new cells are created in your body, each cell is specialized to serve a certain function.  Despite each cell containing the same DNA, the cells in your hand differ from cells in your blood differ from cells in your bones.  Each specialized type of cell has a different shape and protein production depending on its function.  It's easy to imagine how a fully grown organism can create these differentiated cells.  What's more intriguing is how a single-celled zygote (the sperm-egg combo) can have enough variance to eventually transform into a complex organism with many uniquely differentiated cells.  Imagine a tiny, symmetrical looking blob developing into the vast asymmetry of a human body, using nothing more than chemical processes.  It starts with minor variance in chemical concentrations during a cell division.  This compounds after many divisions to form more and more complex and varied cells.  Not only is the process of creating a human bootstrapped, but the blueprints of this bootstrapped process are they themselves bootstrapped and baked over billions of years of evolution.  By building layer upon layer of abstraction, evolution can design something incredibly complex starting with only humble scraps.

Technological Singularity
This is the hypothetical birth of Artificial Intelligence; not through ingenious software design, but by iteratively building upon itself through small recursive improvements.  You don't have to design a system to be as smart as you, rather one that can build a system that is slightly more intelligent than itself.  Although the improvements would be small, they would happen quickly, resulting in an exponential explosion of intelligence like man-kind has never before witnessed.

This is the first part of a series that analyzes our behavior from an outsider's perspective, without any of the assumptions or dogma that we as a society have picked up over the generations.

Is voting an economically logical act?

In this post I'll attempt to quantify the cost and benefit of voting, as well as hypothesize where the stigma for the act of not voting came from.  If you are rusty on expected value, you may want to do a quick brush up so that it's easier to follow along.. I'll be waiting here until you're done.

Many people feel strongly about the necessity to vote.  Not voting is often looked down upon as the irresponsible act of neglecting your civic duty.  Surely, since many people have such strong feelings about voting, we have all asked the simple question of whether it makes sense to vote or not.  The argument against voting is simple: the cost outweighs the benefit.  The time that it requires for an individual to do her research on each politician and policy and then go to a website and fill out a form... wait what!? you can't vote online? You mean that all my money, proprietary documents, and dick pics are accessible on the internet, but it's impossible to secure a voting system online?  As I was saying, the individual must go to the store and buy stamps and scribble little marks on a piece of paper and place that in a mailbox for someone to physically carry to a center where they can efficiently aggregate.. er.. painstakingly feed each ballot into machine for digitization.  Arguably, some people get their jollies from politics, making this process less of a chore and more of pleasure.  But for the rest of us who find it a chore, should it be considered irresponsible to not vote?

It's difficult for people to admit, but for many, the reason of voting is that it makes people feel like they're a better person.  This is the reason that when you vote, just like when donating blood, you get a little sticker.  This sticker is your badge to tell others how good of a person you are.  In 1971 Richard Titmuss hypothesized that providing a monetary incentive to donate blood would decrease the likelihood of people to donate.  Since then, hundreds of studies have been performed that support this hypothesis.  The explanation of course being that people donate blood not to help others (the benefit to others remained constant), but so that they feel good about themselves.  Giving people money takes away that warm, fuzzy feeling and makes the act feel less altruistic.  Putting the pretentious fuzzies aside, let's see if there's any tangible, economic benefit to voting.

With a vote you have a small say in the outcome of the future; a future that will impact you.  It's a really cool idea.  But how small is this "small"?  In an effort to quantify this answer, let's look at the presidential election in a vacuum where we hold everything constant but your vote (after all, you only have control over your choice to vote).  In the current election process of the U.S. each candidate gets all or nothing; even if you win by a single vote you get to be the one and only president.  In an alternate universe we could imagine that many candidates can win an election and the percentage of citizens that voted for a given candidate gives them that percentage of power; this would encourage voter turnout since every vote counts.  This would also prevent the pesky split vote effect; a paradoxical scenario where the least popular candidate can win an election.  In the current system, for your vote to make any difference there must be an equal vote between two parties in absence of your vote.  What's the probability of this occurring?  Each voting distribution is not equally likely to occur (ex. it is much more likely that there 49 vs 51 outcome over 99 vs 1).  From historical presidential elections I've estimated that 42% of elections are a close call: voting differences fall within +/-5% of the 50% mark.  Assuming that each percentage difference within this close call range is equally likely (roughly true), and a standard voter turnout is over 120M, the chance of having a dead even vote is:

.42 * 1 / 12M = 0.0000035%
(likelihood to land in close call territory) * (number of possibilities we're interested in) / (number of possibilities in close call territory)
Hey, don't despair, the odds that your vote matters are still better than winning the lottery... well barely.

This gives us the probability of having impact but not the magnitude that the impact will have on our life.  How can you possibly quantify how much impact each presidential candidate will have?  There are too many subjective and indirect relationships that it would be impossible to measure.  Instead, I would rather ask an individual a simpler question: How much would you pay to have your preferred candidate win over his/her opposition?  If our gut (the intelligence organ) tells us that we would pay $100 for our candidate to win over the other, then it implies that we think we would have slightly more than $100 worth of grievances to suffer if the opponent candidate won.  Keep in mind that the dollar here is being used as a universal value unit; it doesn't literally translate into a dollar in your bank account.  On the other hand, an economist would quantify our cost of voting by measuring the time taken to vote and the amount of money we could have made during same time; this also varies by individual.  So whether you should vote is a function of both your hourly income and how much you would be willing to pay for your candidate to win.  This calculator takes the amount you would be willing to pay for your candidate to win (opposed to the opposition) and calculates how little you would have to make per hour for it to make sense to take the time to vote.  It's a simple estimated value problem.

How much would you be willing to pay for your preferred presidential candidate to win?

In other words, if it cost $1 more than you would be willing to pay, then the opponent would be guaranteed to win.

$ Population Size:

Using this strategy, sadly only the unemployed can logically justify taking the time to vote, given the current voting population.  If you were following this economic model for voting, the voting population size would have to shrink to an incredibly small range (<10,000) for it to make sense again.

Some people look at voting as a community service, where those that don't vote are freeloading off of the time and effort of those that do.  If one is optimizing for the greater good of the community, the question then becomes, how much impact can I have on the community by voting compared to spending those 2 hours elsewhere.  Estimating the impact of 2 hours spent working in the community is easy to quantify; the community has already told us what our time is worth in the form of a paycheck.  More difficult to quantify is the impact that a certain candidate winning the election compared to his or her opposition will have on the community.  If we conclude that every citizen will suffer an average of $2 worth of grievances if the opposing candidate won, this implies a total impact of $319M with a probability of (.42*1/12M) which is an estimated value gain of $22.  If it took 2 hours of your time to vote (it should if you're doing your research), then you should make less than $11/hour for voting to have more impact on the community than donating your time.  Again, we see that it's dependent upon your hourly income and your estimation of per citizen grievances.

So does it make economic sense to vote?  The answer is as it is to most questions in life: it depends.   If we're optimizing for our own economic gain, the answer for any sane citizen is no.  If we're optimizing for the greater good of the community, then the answer is maybe; but we should seriously consider donating that time directly to the community for greater impact.  No motivation is any more correct than the other.  What matters more is that you can logically justify your actions, given your stated motivation.

While writing this post, it occurred to me that I was doing myself a disservice by broadcasting this idea.  Arguably those who are likely to agree with this post and be discouraged to vote, are likely to have a similar belief system as me.  The fewer like-minded people to me that are voting, the less likely I am to have an outcome that is optimized for myself.  Therefore, I should logically try to both encourage like-minded people to vote and discourage people with antithetical views from voting.  Both have equal effect and interestingly, both are actively practiced.  Campaigns to encourage voting is usually performed by targeting populaces known to support a given candidate; see canvassing and GOTV.  Perhaps we in general feel so strongly about voting because we are influenced by these pro-voting campaigns.  While non-partisan pro-voting campaigns exist, they still target a specific sub-populace (women, youth, etc).  On the other hand, voter suppression is the (often deceitful) act of discouraging voter turnout from supporters of the opposition.  In the case of the 2011 Canadian election, automated telephone messages were used to trick voters into thinking that their poll location had changed to a location often hours away from their actual voting booth.

Why do you vote?  Let us know in the comments.

Do you think it makes sense to vote?

I refuse to vote in this poll
Poll Maker

If you've never had the luck of taking a statistics course or just are looking for a refresher, here's a little intuitive introduction into expected value.

Problem 1

You are given $10 and a proposition: if you give back the $10 you are put into a lottery with a 10% chance of winning $100.  Should you take the bet or keep your $10?  If you keep the $10 there is no risk and therefore you're guaranteed to have +$10 at the end of the day.  If you take the risk you have 1/10th chance of winning and 9/10th chance of losing.  If we simulate the outcome 10 times, one of the times is expected to produce a win (+$100) and nine of the times are expected to produce a loss (+$0 from when we started).  To get the expected value, we simply average all possible outcomes ($100*1+$0*9)/10 = +$10.  In this case, taking the bet gives us the same expected value as keeping the $10 so it doesn't matter which you choose.

Problem 2

Every day it takes an hour to walk to work.  One day we found a really sweet shortcut that saves us 50 minutes of travel time.  The only downside to the shortcut is that it forces us to walk across the freeway.  The probability of being killed by a car while we cross is 1%.  Assuming we're optimizing for getting to work as soon as possible (we have to beat Becky who is always waiting in the break room to show us pictures of her cat) which route should we take?  If we die we never show up to work so the time that it takes is infinite.  The safe way gives us an expected travel time of 60 minutes.  Averaging the different outcomes for taking the dangerous route gives us ((10 minutes * 99%) + ( minutes * 1%))/100 = ∞ minutes.  60 minutes is less than eternity, so we should play it safe and take the less sweet route and endure Becky's cat one more day.  On second thought, the freeway doesn't sound so bad...

Problem 3

What's the expected value of a single die roll?
Since each side of the die is equally likely to be rolled, we can add up all combinations and divide by the number of combinations.
Using the arithmetic series formula to calculate the sum of numbers from 1 to 6 gives us 6*7/2=21.  Divided by the number of combinations gives us an expected value of 3.5. But wait, 3.5 is not a value on any die! It's important to note that expected value does not necessarily mean that we would ever expect to see this value.  Think of it as the average of all possible outcomes if we had the ability to run the experiment an infinite number of times.

What's the expected sum of two dice rolls?
Although each side is equally likely, each outcome is not.  For example, the outcome of the dice adding up to 7 is the most likely result:
 2 (1+1)
 3 (1+2) (2+1)
 4 (1+3) (2+2) (3+1)
 5 (1+4) (2+3) (3+ 2) (4+1)
 6 (1+5) (2+4) (3+3) (4+2) (5+1)
 7 (1+6) (2+5) (3+4) (4+3) (5+2) (6+1) <- Peak number of combinations 
 8 (2+6) (3+5) (4+4) (5+3) (6+2)
 9 (3+6) (4+5) (5+4) (6+3)
10 (4+6) (5+5) (6+4)
11 (5+6) (6+5)
12 (6+6)
An easy way to sum the sums is by multiplying each sum by the number of combinations which lead to that sum:
(2+12)*1+(3+11)*2+(4+10)*3+(5+9)*4+(6+8)*5+(7)*6 = 252
and the total number of combinations is 6*6 = 36
Expected value for sum of 2 dice: 252/36 = 7