The Poker Delusion

The Poker Delusion

On the Irrationality of Poker Players and Mark Up

Andrew Slack

Take a second the next time you are in a poker tournament and ask yourself how many people in the room actually have a positive expectation to make money.  Now ask yourself how many players think they have a positive expectation to make money.  Finally ask yourself, “Do I have a positive expectation to make money in this tournament?”  

We already know the answer to the third question, of course you think you have a positive expectation (to forthwith be referred to as being +ev).  You ponied up the money to enter the tournament and it’s doubtful you did it out of the goodness of your heart to pad the bankrolls of those players that are better than you.  Following this line of logic we have the answer to our second question, nearly every single person in the room with you thinks they are truly +ev (with rare exception for those that admit they are not, and just want the experience or to learn from other players).  The answer to the first question is actually the most obvious, but disregarded by nearly every poker player I have ever met: if the tournament is paying out 10% of its participants (a fairly standard payout rate throughout tournament history) than 10% of the field actually has a positive expectation, and 90% do not (they are, in fact, -ev).

So why do we have a room of 1,000 people who all think they have a rightful claim to the prize money, when in fact only 10% of them do?  Psychology tells us that learning at its most basic level requires two things: frequent practice and immediate feedback.  Frequent practice is not a problem in poker, even with the de facto closing of online poker for Americans most can still find places to play with relative frequency.  The main issue is that poker, by nature, masks feedback to the point that it is almost useless to amateur players.  Making a patently incorrect play is often rewarded and the opposite holds true; time and time again you will make the correct play and not only find a lack of reward, but an immediate punishment.  Take the following hand as an extremely basic example.  You are dealt an ace and a king (AK) and have the opportunity to put your entire stack on the line against a player who accidently went all in then exposed their cards as have a jack and a queen (QJ).  In theory you have 64% equity in the pot (your opponent has 36%) and you would correctly be happy to find yourself in this position, quickly calling the all-in wager.  But 36% of the time you will lose or tie (usually losing) and the feedback your brain will receive is negative, in spite of the fact you made an obviously +ev move.  This example is extraordinarily basic, most of poker is far more nuanced and receiving “incorrect” feedback on decisions that are not obvious makes learning to play poker correctly difficult.  

Showing up to the casino for your first ever session of poker, against a table in which every player is better than you, and still walking away a winner is quite common.  You play how you think you should (but is theoretically incorrect in many ways) and are rewarded.  It is now easy to see why so many young men and women truly believe that they are +ev when many are in a habit of making plays that are laughably -ev to experts.  

A second feedback problem that poker offers is an immediate out to disregard all the correct information you receive by blaming it on luck.  When a player makes obviously incorrect choices and loses a pot they still have the opportunity to blame it on bad luck.  This is especially true for recreational players who don’t have the capability yet to understand that the plays they are making are -ev and often think that losing plays are actually winning plays, attributing their loss to unlucky cards.

The third feedback issue is that of sample size.  In poker it takes an incredibly large amount of hands played to come to any accurate assessment of your actual skill level.  Playing 5,000 hands of poker sounds like a large amount to anyone outside of the poker world, but you will quickly be laughed out of the room if you cite your last 5,000 hands as proof that you are a huge winner with a high winrate that you can “prove”.  All players, even good ones, underestimate the role that variance plays in poker results by a large degree and are quick to come to the conclusion the are +ev when they are not (or at least do not have nearly enough played hands to prove it).

So this brings us back to the main point, when you are sitting in a tournament room the level of delusion in the crowd is immeasurable; comical to those with a dark sense of humor.  Thousands of hours and dollars are being wasted around you by those under the false pretense that they have a reasonable expectation in the tournament when in fact only one out of ten does.  

This on its own is distressing, but a required part of poker.  Losing players are needed for there to be winning players, and most losing players do not understand the extent of their situation (in fact, as previously discussed, they don’t know they are losing players at all).  It’s less distressing when you understand the fact that everyone has the opportunity to become better.  Poker books still line the shelves of your local Barne’s and Noble and poker tutorial websites are easy to find with a quick google search (in fact, the best information available is often free, if you frequent the correct poker forums and take the time to find out who you should actually listen to).  So all hope is not lost, and I don’t think there is anything immoral about playing poker as a winning player, but it is a disturbing look into the human condition no matter what way you cut it.

Here is a quick heuristic to use: if you are not the best player at your 9 or 10 handed table, you probably are not +ev in the tournament ( not always true by a large stretch due to the variance of table draws, but a decent starting point).  

So what can you do to make sure you are not one of the 90%?   At the risk of sounding cliche, the first step is to admit you have a problem.  The vast majority of players will never admit that they are probably -ev, a side effect of conflated ego’s brought on by pots won that should have been lost.  If you sit to yourself and think with honesty “I may not be +ev after all” the truth is you probably aren’t.  In fact if you answer that question with “I’m certainly +ev” you probably still are not, it’s hard to overcome the base rate of 90% of players being losers  To be in the top 10% you need to work harder and play better than the other 90% (and a high level of intelligence and self control is probably required as well).   A chinese proverb goes as such “The best time to plant a tree was 30 years ago, the second best time is today.”  If you have been playing for a few years, assuming you were +ev, but not doing much studying, you need to start today (or quit poker as a source of income; playing for entertainment with the understanding you may be a losing player is fine).

But things get even worse for the poker community.  In recent years the practice of charging markup to sell action has become commonplace.  Markup in and of itself is not a problem, it only becomes problematic when investors are paying a huge premium to invest in something that is -ev.  Markup works like such: a player wants to sell pieces of their action in a tournament that they believe they have a substantial edge in.  They deem their return on investment (ROI) to be +30% so they ask that investors pay them an amount above $1 for every dollar in action the investor receives.  If their true ROI is 30% then an investor paying $1.30 for $1 in action is making a breakeven proposition.  If they are able to pay only $1.20 (or any number under $1.30) they are making a +ev bet, and anything above $1.30 is -ev.  If you could accurately ascertain a winning players ROI in a tournament, this would be simple math and investors would not have many problems.  This is simply not the case, sample sizes are way too small, and the overall skill level of the average tournament player is rising too quickly, for us to get anything close to accurate predictors of true ROI for most players.  Yet players, many of them big names and relatively smart guys, solicit exorbitant markup and (usually unknowingly) shaft their friends and investors.

These two problems conflate each other very quickly, we have a massive amount of people asking for (and receiving) inappropriate markup that are not even +ev in the tournament to begin with.  So it is possible to find a player who is being invested in for 40% markup who is actually in the bottom 90% of the field, a double knock out punch to investors as a whole.

The defenders of markup as a commonplace practice cite that we are simply in a marketplace, in which the price of markup can just be decided by the market and selling at that number is ok.  The efficient market hypothesis stands on two legs: “The Price is Right” and “No Free Lunch”.  “The Price is Right” dictates that the price of a good is dictated by all information available and ends up with a correct price for that product.  Watch news break about any publicly traded company or commodity and the stock approaches a rational price almost instantaneously (with today’s technology, many reach this price in milliseconds).  The “No Free Lunch” portion dictates that noone can beat the market over the long run (mostly because of correct pricing).  For our discussion of markup in the poker marketplace we will focus on “The Price is Right”.  You can find countless examples of irrational prices if you look for them.  While browsing twitter today I read two tweets offering the sale of action within one week of each other.  The first tweet was asking for 3 to 1 on a tournament, and then a few days later the same player was soliciting sales at 4.4 to 1.  Both of these prices are laughable, investors would have to be sure he has a 300% and 440% ROI respectively to make it break even, but that’s not the point.  The point is that he was selling (and supposedly received action) on two different prices with very little change.  The tournaments had similar structures and were the same game format, I have a hard time believing this player added 110% to his true ROI in a few days.  This same player sold action to the same tournament at both 2.5 to 1 and 1.5 to 1.  Both of those prices cannot be simultaneously correct.

It would be nice if we had anything near an efficient market, but we don’t.  Most investors have nowhere near all the information available to them to come close to an appropriate price and are in general paying way too much.  There are many other issues with markup as a concept, namely how a player who has sold a lot of action at a steep markup has a different set of incentives on how to play in certain stages of a tournament that are directly opposite to their investors well being.  We also have an idea from economic theory that as stakes rise people should act more rationally, selling action inherently lowers the stakes for those selling and would theoretically dictate less rational play.  I won’t get into all the issues with markup here, but ask professional poker player and economics student Andrew Barber about it and he will probably be more than happy to share.

I will offer two caveats to this issue that I think are important:

1. Charging markup in and of itself is not immoral, and those that offer incorrect prices are usually not doing it maliciously.  Ignorance isn’t a good excuse, and it should be shameful to sell at exorbitant prices regardless of the reason, but it’s important to distinguish malicious behavior (of which there is little) from ignorance (of which there is much).
2. Buying pieces at markup, just as playing at a table in which you know you think you may be -ev, is fine if you are ok with paying a premium for the entertainment value and are cognizant of what you are getting yourself into.  The point of this paper is simply to make people aware.

I buy small pieces of freinds frequently and at times for small amounts of mark up.  While I would never buy at 300% markup, I have and will continue to buy pieces at rates that I feel are at least in the ballpark of reality (for microscopic portions of my bankroll).  It is a huge mistake to pay 300% to a guy whose true ROI is 20% and only a small mistake to pay 20% to a guy whose ROI is 10%. The small amount of ev I may pay is made up in life value.  So buying action is ok, but buyer beware.  

If there are two things to take away from reading this they are as follows: don’t overvalue yourself (everyone else is doing it) and put yourself into -ev spots.  And don’t fall for the trap of others overvaluing themselves, before you buy action for any significant portion of your bankroll you need to do serious due diligence and be aware of the situation in every knowable way.

My Ten Million Dollar Robot was written on May 18th, 2016, it can be found it's entirety below.

My Ten Million Dollar Robot

Thoughts on expected value and probabilistic thinking.

Andrew Slack

I began to explore the idea of expected value in 2007 when I began a relatively short foray into online poker (Black Friday in April 2011 was the day the U.S. government shutdown the three largest poker websites, sidelining any pipe dreams of playing cards for a living I had once had).  Many people assume that poker is a game of psychological warfare, with a player’s poker face, or lack thereof, being the key to his success or downfall.  While poker does dictate a need for a steady demeanor and a lack of physical tells, it is a player’s ability to calculate expected value and make decisions based on well forecasted ideas of how their opponents play that actually decides who goes home with the money and who leaves the table broke over the long run.  If you were given the choice to invest in a player that has a knack for picking up physical tells or to back a player with a background in game theory, probability, or economics, your money should be given with haste to the intellectual. In poker, and in much of life, math reigns supreme.

Expected value is a term used a lot in the poker world and at this point it’s important we give it a definition.  In the most basic sense, “expected value is a predicted value of a variable, calculated as the sum of all possible values each multiplied by the probability of its occurrence”1.  Let’s look at an example, I offer to flip a fair coin with you and the following rules apply:

1. If the coin lands on heads I will give you 110 dollars
2. If the coin lands on tails you will give me 100 dollars

To calculate your expected value (EV) we take the sum of all possibilities values.  Since a fair coin offers a 50% chance of landing on either heads or tails, we assign (.5) as the probability for each result and input the value that each result has been assigned (+110 for heads, -100 for tails) and end up with an equation like this:

(.5)(110) + (.5)(-100)= 5

Our expected value of this experimental coin flip is a net gain of five dollars.  In simpler terms we may write EV = 5.  

The coin flip is a simplistic example of expected value, most ev calculations in life will be much more nuanced and will require accounting for extraneous factors.  Most situations will not offer a 50/50 shot, and there may be an overlay of money or value at stake.  Let’s look at a slightly more complicated example, a poker hand in which we are drawing to a flush with one card to come.  We will assume the following:

1. There is already 200 dollars in the pot
2. We are being asked to call 50 dollars to see the final card
3. We have 9 cards that will give us a winning hand, this has a probability of roughly 18%
4. If we do not catch one of those cards we will lose the hand, an 82% chance of loss

The calculation looks very similar to the coin flip example, when we call 50 we have an 18% chance to win 200 dollars, and an 82% chance to lose 50 dollars.

(.18)(200) + (.82)(-50) = -5

This wager would net us a theoretical loss of 5 dollars and would be what we deem -EV, a wager to be avoided.

Calculating expected value on coin flips and poker hands is interesting, and if done with enough mindful practice could turn you into a winning poker player, but this paper aims to focus on EV as a broader concept that can be applied to all aspects of your life.  Whether to spank your child or not, buy new tires, or eat the candy bar in your desk drawer can all be put through EV calculations.

Expected value is only one of the keys you will need to create these calculations. Learning to think probabilistically is just as important.  Probabilistic thinking is simply using all the information you know in a rational way to come up with the probabilities of various results to a variable.  A probabilistic thinker would not tell her spouse “It is going to rain today” because they saw a forecast that said there was a 75% chance of rain.  Instead they would understand that three out of every four days that the region is under these conditions we can expect rain, and one out of four we will not.  A probabilistic thinker would not be all that surprised to look outside and find that it never began to rain, though he expected it was much more likely that it would.  Thinking in terms of “It will rain/it won’t rain” when given a probabilistic scenario is far too simple.  Probabilistic thinkers are not ashamed of uncertainty, they simply use the probabilities of various results to come to a conclusion that offers them the most value given the circumstance.

Homo Economicus (Econ) is the idea of a person who acts completely rationally, without any bias, when making decisions.  By definition a true Homo Economicus is completely rational and also narrowly self-interested2, but for the purpose of this experiment we will focus on their completely rational nature.  When coming up with calculations that will affect the value of your life, it may behoove you to learn about the various biases that affect human thought processes and attempt to counteract them, to think more like an Econ would.

To demonstrate the idea of EV and probabilistic thinking on a larger scale I have proposed the idea of the ten million dollar robot which may or may not require a one million dollar software patch. This robot is the product of a technology company and is worth exactly 10 million dollars.  Recently it has come to the attention of the lead engineer that there is the possibility of a catastrophic malfunction that would destroy the robot entirely, creating a 10 million dollar loss for the company.  The probability of this malfunction is currently being assessed by a group of analysts.  He has also discovered that this problem can be resolved with 100% certainty for one million dollars.  Should the company invest one million dollars to ensure their robot does not have an irreparable meltdown?  

I hope at this point you realize you don’t have nearly enough information to make that decision.  If the probability of the malfunction is deemed to be less than 1% then committing 1 million dollars to patching it would be an incredible blunder.  On the other hand if it is discovered that this malfunction will take place with 99% certainty than not patching it immediately would be a much larger blunder, by a magnitude of nearly ten.  It’s now time to calculate the EV of patching our robot under a variety of different malfunction probabilities.  To do so we will use the following rules:

1. If we do not patch the robot and it malfunctions it will cost our company exactly 10 million dollars
2. If we decide to patch the robot it costs exactly 1 million dollars
3. We are not considering any extraneous factors (loss aversion, likelihood of job loss for each decision, etc.)  We are looking at pure EV as an Econ.

As an example let’s assume our engineers analyze the data and tell us that their best forecasting model calculates that there is a 40% chance of the malfunction occurring.  We now have two choices: pay one million to patch the robot, or do nothing.  

Our expected value of doing nothing looks like this:

(.4)(-10,000,000) + (.6)(0) = -4,000,000

While our cost of fixing the robot looks like this:

(1)(-1,000,000) = -1,000,000

Neglecting to patch our robot would theoretically cost us the difference between these two values, or $3,000,000.  Making our decision to invest in the one million dollar patch incredibly easy, even though the chance of the malfunction was less than 50%.

But what if the malfunction is relatively rare, say somewhere between .5% and 15%? Table 1 calculates all the expected values for every probability between those two values to show us exactly at what point we should shift from a preference to forego the patch to a preference to invest in it.

Table 1

Probability	Cost of Robot Malfunction	Expected Value Calculation	Cost of Patch	Difference
.5%	-$10,000,000	-$50,000	-$1,000,000	$950,000
1%	-$10,000,000	-$100,000	-$1,000,000	$900,000
2%	-$10,000,000	-$200,000	-$1,000,000	$800,000
3%	-$10,000,000	-$300,000	-$1,000,000	$700,000
4%	-$10,000,000	-$400,000	-$1,000,000	$600,000
5%	-$10,000,000	-$500,000	-$1,000,000	$500,000
6%	-$10,000,000	-$600,000	-$1,000,000	$400,000
7%	-$10,000,000	-$700,000	-$1,000,000	$300,000
8%	-$10,000,000	-$800,000	-$1,000,000	$200,000
9%	-$10,000,000	-$900,000	-$1,000,000	$100,000
10%	-$10,000,000	-$1,00,000	-$1,000,000	$0
11%	-$10,000,000	-$1,100,000	-$1,000,000	-$100,000
12%	-$10,000,000	-$1,200,000	-$1,000,000	-$200,000
13%	-$10,000,000	-$1,300,000	-$1,000,000	-$300,000
14%	-$10,000,000	-$1,400,000	-$1,000,000	-$400,000
15%	-$10,000,000	-$1,500,000	-$1,000,000	-$500,000

The Difference column displays the cost of patching the robot given every probability listed.  In the first row, with a malfunction probability of .5% the patch theoretically costs our company $950,000.  At the bottom of the table, when our probability of malfunction reaches 15% the patch theoretically saves our company $500,000.  As the probability of disaster rises, neglecting the patch costs our company $100,000 for every rise in probability of 1% (and can be calculated for any value, i.e. a probability of rise of .1% offering $10,000 of value to the patch).

If our analysts came to us with a malfunction probability of 10% we would be completely indifferent as to whether to invest in the patch or not.  A truly rational thinker, looking only at the numbers, would in fact support the patch fully at a 10.1% chance of failure and vehemently oppose it at 9.9%

This chart looks nice and our robot issue seems to be fairly easily solved, but in the real world losses hurt much more than saving money, by a factor of two or more3.  Kahneman, Knetsch, and Thaler conducted an experiment in which half the students in a group are given a mug with their university’s logo on it, and half the students are given nothing.  The two groups of students then take a survey in which they indicate how much money they would need to be given to sell the mug (for the mug havers) or how much they would be willing to pay to attain the mug (for the mug wanters).  This experiment can be easily replicated at your local college’s psych department and the results are pretty clear: people require close to twice as much money to lose something than they are willing to pay to gain the same thing.  We can see the same phenomenon when asking people how much they would have to stand to win to risk $100 on a coin flip.  How much would you require?  If you answered a number close to $200 you are not alone, in fact you are in the vast majority.  An Econ would happily take this bet being offered $101 to their $100 assuming they had no risk of ruin and were not incurring an opportunity cost to take the wager.  They would be overjoyed to be offered $110 (as in our previous example, this would theoretically be worth five dollars to them).  Yet real people are not Econs, and they require substantially more money to be willing to risk $100, this is the Loss Aversion Bias in it’s most basic form.

How can we account for the pangs of loss in our robot experiment?  If we create a hypothetical loss aversion value (2), or LAC (loss aversion coefficient) to account for this phenomenon we could recreate the equations so that they look something like this

(%)(-$10,000,000)(2) - (-$1,000,000)

Plugging any probability value into this equation simply doubles the EV calculated value for the catastrophic malfunction.  Some new values are listed in table 2 below:

Table 2

Probability	Cost of Robot Malfunction	Loss Averse Expected Value Calculation	Cost of Patch	Difference
3%	-$10,000,000	-$600,000	-$1,000,000	-$400,000
4%	-$10,000,000	-$800,000	-$1,000,000	-$200,000
5%	-$10,000,000	-$1,000,000	-$1,000,000	0
6%	-$10,000,000	-$1,200,000	-$1,000,000	$200,000
7%	-$10,000,000	-$1,400,000	-$1,000,000	$400,000

In a company with unlimited funding, being ran by completely rational thinkers, this table would never be used, it gives up way too much value and tells us that we should be happy to implement the patch at a 6% probability of malfunction, even though in the real world this would have a theoretical cost of $400,000.  Recreating the equation with a loss aversion value does have it’s merits though, a company can place any value in that spot after coming to their own conclusion about how loss averse they want to be.

There is also a theoretical scenario in which you may want to give up some EV and place a value less than one in the risk aversion value place.  If patching the robot will take up time or resources from other projects it is possible that an engineering group would want to give up some theoretical value to avoid patching the robot.  Opportunity costs are the theoretical losses we take by investing funds or time into one project or opportunity instead of another.  If we invest one million dollars to patch the robot that could have been invested into a financial instrument with a return on investment of 10% over the same time period we are really paying $1,100,000 to patch the robot and thus our breakeven point would be 11% chance of malfunction.  

In the real world we have to account for more unknown variables than a simple  monetary calculation provides.  A chief engineer would be far more likely to be fired from his job (and possibly blackballed from his industry) by a catastrophic failure of the robot than if he happened to spend $1,000,000 fixing it even if it didn’t really need it.  How much value he would find in “overpaying” for the patch to ensure he didn’t have to face the board of directors with a $10,000,000 paperweight would depend on his real world circumstances, but I imagine many real world, naturally irrational, engineers calling for the patch at a probability of 1%, and as much as I’d like to, I really can’t blame them.

If you assume that the robot also has some potential for future earnings or funding assuming it does not malfunction, then we can replace the ten million dollar cost with any reasonable number we come up with and change the equation.  If our potential earnings are forecasted to be $100,000,000 then patching the robot at a 1.01% chance of malfunction would be +EV.  If the robot is the cornerstone of a multibillion dollar round of funding for our company, then patching the robot at almost any non-zero value becomes a necessity.

I mentioned earlier you could start thinking probabilistically, and creating EV calculations for almost any scenario in life in which probabilities are involved and values differ for various outcomes.  EV does not always have to be put in monetary terms, if we can come up with a reasonable life expected value (lifeEV, or LEV) number we can calculate EV for everyday decisions.  This requires abstract and honest thought when assigning life value points to different decisions or outcomes.  Let’s look at a hypothetical scenario in which we are deciding whether or not to eat the candy bar in our desk drawer assuming the following conditions:

1. We create an assume a life value (LV) maximum of 100 in which a self reported 100 means a state of pure bliss in which a person could not possibly be any happier and a reported LV of 0 means complete and utter despair.
2. We come upon our candy bar decision when we are self-reporting a LV of 50.
3. We are on a calorie restricted diet in which we are not supposed to be eating junk food.  We feel guilty when breaking our diet and it has a negative effect on our LV.  How negative we feel depends on our current mood, if the calories for the candy still kept us within our diet or not, and any other factors affecting our guilt.
4. We enjoy candy and, at least momentarily, it adds value to our life.  Depending on how hungry, bored, or emotional we are it may add more or less value.

Now we simply need to calculate honestly and as rationally as possible a value for the negative impact the guilt will have and the positive value that the candy bar will have.  For our example let’s assume that eating a candy bar on average offers 5 LV points, and the guilt we feel from eating it costs us 15 LV points.  This by itself would not require an EV calculation, you could simply look at the difference between the two values and see that eating the candy bar would drop your self reported LV from 50 to 40, an obvious decision to avoid.  But what if we only feel guilt x% of the time we eat a candy bar, or the value candy adds to our life is dependent on extraneous factors that can be placed into a coefficient.  A life EV equations may look like this:

(% chance we feel guilty)(-15) + (Candy value coefficient)(5) =

If we come into work and eat sensibly throughout the day, even planning ahead to eat our candy bar, we will not feel guilty.  Our LEV calculation looks like this:

(.00)(-15) + (1)(5) = 5

Eating the candy has added 5 points of value to your life.  Your coworkers will thank you for taking a snack break as you become more prone to smile and your productivity rises.

Now let’s look at an example on the other end of the spectrum.  You’ve already blown your diet today and even though you are not having a particularly bad day at work you find yourself bored, yet unhungry.  Under these circumstances you find that the candy value coefficient is actually less than 0, it is not even serving the basic need of satiating hunger and we recognize with 100% certainty we will feel guilty, thus we find an equation like this:

(1)(-15) + (.5)(5) = -12.5

Eating the candy bar under these circumstances would be an egregious error, dropping your LV to 37.5.

Finally let’s look at circumstance in which we are not sure if we will feel guilty or not, and the candy bar will both give us a slight sugar rush and satiate some mild hunger.  We have stuck to our diet decently well this day but eaten most of the allotted calories we usually allow ourselves by this point, by eating the candy bar now we know we will have to eat less at dinner later in the day.  We also realize that while we are not starving, we are a little bit hungry and it’s possible it is affecting our mood and work.  We roughly calculate that we will feel -15 points of guilt 40% of the time and that eating the candy bar is 25% more valuable to us than usual given the circumstances.  Our equation looks like this:

(.40)(-15) + (1.25)(5) = .25

Eating the candy bar offers .25 of a point to your life value.  If you felt you had accounted for every factor in your decision and had an accurate idea of how often you would feel guilty, the Econ in you would be wise to eat the candy bar.  If you felt like there was a reasonable margin for error in that you undervalued your guilt percentage or overvalued the candy coefficient, you would be wise to save the calories.

Humans are naturally irrational, and that is ok.  The idea of Homo Economicus is purely theoretical (and thankfully so, a true Econ only cares about their personal EV with complete disregard for everyone else).  But understanding the importance of probabilistic thinking, and how your decisions should change given various probabilities of an event will help you to make more rational decisions in your life and should contribute to your overall Life Value.

1 Oxford English Dictionary

2 Rittenberg and Trigarthen. Principles of Microeconomics: Chapter 6. pp. 2 [1] Accessed June 20, 2012

3 Kahneman, Knetsch, and Thaler (1991).