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The St. Petersburg Paradox

Thinking about Risk and Uncertainty¶

This series of posts will be an introduction to the economics of risk and uncertainty as well as some finance. To make it easy to follow along, everything will be recorded in Python and uploaded to my Github as an iPython notebook.

If you were offered a bet that you were nearly certain to win, would you take it? Consider the following scenario:

You have \$100. In a lottery, you have an 80% chance of earning \$120 (\$100+20) and a 20% chance of losing your \$100. Would you choose to take such a bet? Would you play it the other way around and bet on the 20% chance to win \$180? Would you even play this bet at all? During the 2016 presidential election, several thousands of people made this very bet using political gambling websites (and not everyone was happy with the results).

From assessing stocks to picking insurance plans, risk assessment is incredibly valuable skill with a lot of hidden complexity. Furthermore, learning to understand risk better helps us engage with problems of judicial importance such as legal negligence standards and political importance such as climate change regulations.

Even Batman V Superman makes a nod to the idea of existential risk calculus:

“He has the power to wipe out the entire human race and if we believe there’s even if it’s a 1% chance that he is our enemy we have to take it as an absolute certainty!”

The 2 characteristics of uncertain environments¶

Given a choice made by a decision maker, an outcome is any possible occurrence that affects someone in some way. Mathematically speaking, an outcome (the first characteristic) is a parameter that affects the well-being (utility) of the decision maker.

For simplicity’s sake, we assume that the number of possible outcomes is always finite:

In [16]:

#First, get the import statements out of the way from __future__ import division import math import numpy as np from IPython.display import display, Image, HTML, Latex, Math import pandas as pd import matplotlib.pyplot as plt %matplotlib inline

Outcomes are always finite in terms of some arbitrary N. This lottery is of size N = 5 and has no outcomes.

In [2]:

#Outcomes are always finite in terms of some arbitrary N N=5 #This lottery is of size N and has no outcomes outcomes = [None]*N print(outcomes)

[None, None, None, None, None]

For example, a coin flip would have 2 possible outcomes: Heads or Tails. But knowing the outcomes alone isn’t particularly interesting. What we’re interested in is the outcomes and their corresponding probabilities. So, the vector of probabilities (the second characteristic) is the list of probabilities corresponding to a list of outcomes. Each probability must be $\ge$ 0 and the sum of the probabilities must equal 1.

If we assume that a coin is fair, then the probability of Heads is equal to 0.50, and the probability of Tails is also equal to 0.50.

In [3]:

#The probabilities in an event where each outcome is equally likely probabilities = [1/N for i in range(0,N)] print(probabilities)

[0.2, 0.2, 0.2, 0.2, 0.2]

Together, the corresponding lists of outcomes and probabilities forms a lottery.

In [4]:

#A lottery is the list of outcomes and their corresponding probabilities def lottery(outcomes, probs):     assert len(outcomes)==len(probs)     assert sum(probs)==1.0      lotto= zip(outcomes, probs)     return lotto example_lottery = lottery(outcomes,probabilities) display("Example:", example_lottery)

'Example:'

[(None, 0.2), (None, 0.2), (None, 0.2), (None, 0.2), (None, 0.2)]

In [5]:

#fair coin sides = ['H','T'] coin_probs = [0.5,0.5] coin_lotto = lottery(sides, coin_probs) display("Fair coin flip:",coin_lotto) #ex2 ex2_outcomes = ["$1000", "$4", "$-300"] ex2_probs = [0.1, .85, .05] ex2_lotto = lottery(ex2_outcomes,ex2_probs) display("Another example:",ex2_lotto)

'Fair coin flip:'

[('H', 0.5), ('T', 0.5)]

'Another example:'

[('$1000', 0.1), ('$4', 0.85), ('$-300', 0.05)]

Furthermore, a lottery may also contain lotteries. This is called a compound lottery.

In [6]:

#A compound lottery is a lottery that may contain other lotteries ex3_outcomes = [10, coin_lotto] ex3_probs = [0.5,0.5] ex3_lotto = lottery(ex3_outcomes,ex3_probs) display("Compound Lottery:", ex3_lotto)

'Compound Lottery:'

[(10, 0.5), ([('H', 0.5), ('T', 0.5)], 0.5)]

Just like with a probability mass function (PMF), we can calculate the expected value of a lottery by finding the sum of the products of each outcome and its probability.

In [7]:

def expected_value(lottery):     mean = 0     for outcome, probability in lottery:      mean+= outcome*probability     return mean

Here, the expected value is 0

In [8]:

ex3_lotto = lottery([1,-1],[.5,.5]) display(expected_value(ex3_lotto))

0.0

Now, given this framework, we can calculate the value of a real lottery such as a powerball lottery.

In [9]:

#Odds of winning a jackpot (a rough guess) powerball_prob = 1.0/300000000 #Odds of losing loss_prob = 1-powerball_prob #Jackpot jackpot = 10000000 ticket_price = -1 powerball = lottery([jackpot,ticket_price],[powerball_prob,loss_prob]) display("The expected value of your ticket:",expected_value(powerball))

'The expected value of your ticket:'

-0.9666666633333333

The expected value is a powerful idea because it helps us order our preferences between lotteries. When using expectations, we make three assumptions:

• When the expected value of a lottery is greater than a guaranteed option, a decision maker will choose the lottery.

• Conversely, if the expected value of a lottery is lower than a guaranteed option, a decision maker will always choose the guaranteed option.

• Finally, if the expected value of a lottery is equal to the value of a guaranteed option, the decision maker will be indifferent between the two.

Simply put, For all lotteries $L1,L2$ either $L1 \succsim L2$ or $L2 \succsim L1$ where $\succsim$ indicates a preference relationship. For more information, check out the Debreu theorems

The St. Petersburg Paradox¶

Finding the expectation of a lottery may seem like a useful tool in most decision-making situations, but consider the following gamble:

Suppose you were made an offer. A fair coin would be tossed continuously until it turned up tails. If the coin came up tails on the $n^{th}$ toss, you would receive $\$2^n$ i.e. if it came up tails on the 5th toss, you would receive $\$2^5 = \$32$. How much would you be willing to pay to participate in this lottery?

Intuitively, we already know that not many people would spend much to play this game. But why? To visualize the paradox, we can re-write the game in Python.

Game: Get payoff that depends on when heads comes up first on toss of fair win.

Payoff: $2^{i}$ where $i$ = toss number of first H

In [10]:

def game():     i = 0 #number of heads     while(True):      flip = (np.random.randint(low=0,high=2))      #If the flip is a tails, return the payoff      if flip == 0:      return 2**(i-1)      #If the flip is a heads, flip again      else:      i += 1

In [11]:

trials = [game() for x in xrange(0,10000)] plt.figure(figsize=(15,5)) plt.plot(trials) plt.xlabel("Trial number") plt.ylabel("Payoff") plt.title("St. Petersburg Lottery") print("Expected value of sample:", np.mean(trials))

('Expected value of sample:', 6.10215)

This question is asking for the amount someone would be willing to pay for such a lottery, so we can consider this question as asking for the certainty equivalent of the lottery. The certainty equivalent is the guaranteed amount of money that an individual would consider just as desirable as a risky asset (lottery). We calculate the certainty equivalent by taking the expected value of the lottery. But given exponential payoff of this function, we encounter a problem: This gamble has an infinite expected value!

Expected payoff = $(\frac{1}{2}^0 *2^0) + (\frac{1}{2}*2^1) + (\frac{1}{2}^2 *2^2) + (\frac{1}{2}^3 *2^3) + … + (\frac{1}{2}^i *2^i) =\infty $

Despite the fact that the expected payoff is $∞$, only a few people are willing to pay much for this lottery.

This problem is known as the St. Petersburg Paradox. To understand why decision makers are not willing to purchase this lottery, it is important to understand that decision makers are utility maximizers not outcome maximizers. Instead of calculating the expected value of the lottery, we instead calculate the expected utility of the lottery.

Mathematician Daniel Bernoulli, suggested that when we take utility into account, we can develop a solution. He states, “The determination of the value of an item must not be based on the price, but rather on the utility it yields…. There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man though both gain the same amount”

This theory of utility maximization is explained by the law of diminishing marginal utility which states that when a decision maker consumes a specific good, the next good consumed will be less enjoyable than the previous. So, the marginal utility, the difference of utility between the consumption of a good and the consumption of a consecutive good, will decrease.

In [12]:

def utility_table(size):     n = np.arange(0, size, 1) p = pd.DataFrame()     p['n'] = n     payoff = 2**n     p['Payoff'] = payoff     p['Probability'] = 0.5**n     p['Expected Payoff'] = p['Payoff']*p['Probability']     p['Bernoulli Utility Function'] = np.log(payoff)     p['Expected Utility'] = p['Bernoulli Utility Function']*p['Probability']     return p table = utility_table(size=10) display(HTML(table.to_html(index=False)))

n Payoff Probability Expected Payoff Bernoulli Utility Function Expected Utility 0 1 1.000000 1 0.000000 0.000000 1 2 0.500000 1 0.693147 0.346574 2 4 0.250000 1 1.386294 0.346574 3 8 0.125000 1 2.079442 0.259930 4 16 0.062500 1 2.772589 0.173287 5 32 0.031250 1 3.465736 0.108304 6 64 0.015625 1 4.158883 0.064983 7 128 0.007812 1 4.852030 0.037906 8 256 0.003906 1 5.545177 0.021661 9 512 0.001953 1 6.238325 0.012184

Given Bernoulli’s hypothesis of diminishing marginal utility, we can show that although the outcomes increase with each bet, the utility gained from each consecutive outcome will be lower than the last. Bernoulli suggests that we can model the decision maker’s utility using a natural log function. The natural log function is useful since it is monotonic (each additional unit provides additional gains in utility) yet diminishing such that the gains converge to 0 as x approaches $\infty$. $$u (x) = ln(x) $$ $$ u'(x) = 1 / x$$

In [13]:

t = np.arange(1, 10, 0.2) constant = plt.plot(t, t, 'r', label="Constant returns", linewidth=3) diminishing = plt.plot(t, np.log(t), 'b', label="Diminishing Returns",linewidth=3) plt.legend(loc="center right",) plt.xlabel("Quanity consumed") plt.ylabel("Total Utility") plt.title("Constant marginal returns vs diminishing marginal returns") plt.plot()

Out[13]:

[]

Graphically, we can see utility uniformly increases as the payoff exponentially increases. However, the expected utility decreases since the probability exponentially decreases.

In [14]:

table = utility_table(size=5) plt.figure("Bernoulli Utility Function") expected_payoff= plt.plot(table['n'], table['Expected Payoff'], 'r', label="Expected Payoff",linewidth=3) diminishing = plt.plot(table['n'], table['Bernoulli Utility Function'], 'g', label="Utility",linewidth=3) expected_utility = plt.plot(table['n'], table['Expected Utility'], 'b', label="Expected Utility",linewidth=3) plt.legend(loc="center right",) plt.xlabel("Number of heads") plt.ylabel("Returns") plt.title("Utility vs Payoff") plt.plot() plt.figure("Comparison") constant = plt.plot(table['n'], table['Payoff'], 'r', label="Payoff", linewidth=3) diminishing = plt.plot(table['n'], table['Bernoulli Utility Function'], 'g', label="Utility",linewidth=3) expected = plt.plot(table['n'], table['Expected Utility'], 'b', label="Expected Utility",linewidth=3) plt.legend(loc="center right",) plt.xlabel("Number of heads") plt.ylabel("Returns") plt.title("Utility vs Payoff") plt.plot()

Out[14]:

[]

For calculating the certainty equivalent, we need to take the sum of the expected utilities for an infinite number of trials. Since the function converges, we can reach an estimate as n approaches $\infty$ $$E(L) = \sum_{n=1}^{\infty} \frac{1}{2}^x * ln(x) \\ = \sum_{n=1}^{\infty} 2^{-x} * ln(x)$$

So, we can estimate the certainty equivalent as:

In [15]:

table = utility_table(size=1000) util_lottery = np.sum(table['Expected Utility']) print("The expected utility of the lottery is ",util_lottery) print("The certainty equivalent of the lottery is ",math.exp(util_lottery))

('The expected utility of the lottery is ', 1.3862943404624948) ('The certainty equivalent of the lottery is ', 3.9999999173704177)

Since we are still dealing with a utility function, the certainty equivalent will not be 1.386. Instead, it will be the antilog of 1.386. In other words, we are trying to find the parameter that makes our utility function = 1.386. So, we can estimate the certainty equivalent as:

$$CE = e^{E(L)} = e^{1.386} = 4$$

A final note¶

The logarithmic utility function that Bernoulli proposes is case specific to the $2^n$ lottery. By adjusting the values in the lottery, we can again make it a paradox with an infinite expected value. The St. Petersburg paradox is used to demonstrate that there are diminishing marginal returns to utility and we can take further steps to model choices under uncertainty.

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More choices, more problems? Exploring the paradox of choice

Suppose you need a new pair of jeans. You head over to the clothing store expecting an easy find, and you instead collide with a wall brimming with blue jeans. There are hundreds of them. These are jeans with varying shapes, sizes, tears, patterns, fits, shades, and colors.

You leave the store one and a half hours longer than you originally planned, but you now have the best fitting pair of jeans you've ever worn. However, with all of those options, how do you know you got the right pair? After all, the benefits of declined opportunities are unknown. The question is whether all of those extra choices actually make us better off. Think about it like this: If there are that many jeans in the pile, there has to be the perfect one out there for you, so you raise your expectations. If we consider all of the time wasted browsing jean brands, trying on duds, and internally calculating which pair you can work in your budget, there’s quite a bit of stress involved. Do more choices make us better off, or are we just wasting time struggling to pick out the things we want?

This is the paradox of choice. Psychologist Barry Schwartz identifies this problem in his aptly named book The Paradox of Choice. Schwartz suggests that too many choices and freedoms result in choice “paralysis”, high levels of regret after good consumption, and relative disappointment due to higher expectations.

I think Schwartz has a point. There was probably a point in your life where you went to an ice cream shop, and when you went to order your flavor, you find yourself staring through the glass for a solid amount of time because there were so many flavors to choose from. Only when you notice the line of mildly annoyed customers behind you do you make a choice, and it probably wasn't one you put much thought into picking. You probably wouldn't have chosen pistachio-choco-mango if there were only a couple of options to choose from. 

Way too many flavors!

In other words, making the right decision is actually pretty hard. With a decision, there is both a known cost to deciding what you want, and an unknown opportunity cost of the items you did not choose. This may lead people to “freeze” in their decision making, make poor choices, or plainly opt out of deciding (“I’ll get what he’s getting”).

Although these examples may seem trivial, underestimating the paradox of choice can dramatically impact a business’s sales. 

Schwartz notes a study that shows offering too many choices of jam may dramatically reduce its sales. On one day, researchers Sheena Iyengar and Mark Lepper set up a jam display in a gourmet grocery store displaying 24 different varieties of gourmet jam. Furthermore, customers were offered a coupon $1 off of any jam. On another day, the researchers set up a similar table, but instead, only 6 different varieties of jam were displayed.

Although more customers visited the large display, only 3% of visiting customers made a purchase. On the other hand, 30% of customers who visited the small display made a purchase.

Although the paradox of choice seems plausible in practice, there is a lot of conflicting evidence. For instance, psychologist Benjamin Scheibehenne designed 10 experiments similar to the jam study and found little evidence that variety had much of an effect on purchases. Furthermore, Scheibehenne conducted a meta-study of several similar experiments and found no meaningful impacts either way.

So is the paradox of choice a meaningful theory? Schwartz makes several important insights regarding patient choice (patients are very bad at making choices when presented with options by medical professionals). But the evidence is mixed. However, as Richard Thaler points out in his book Nudge, the way you frame choices for consumers as a non-negligible impact on their selection. Either way, before you start your jam enterprise, consult the evidence first. 

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How to think about happiness

“Happiness” can be a somewhat vague term, and it is often difficult to understand its proper meaning. From Ancient Greece to contemporary times, many have philosophized about the definition of happiness, but we rarely find much of a conclusive answer. 

Interestingly enough, in 2012 the kingdom of Bhutan declared that they would no longer use gross national product (GNP) as their measure of success and instead opt for gross national happiness. 

“Gross National Happiness is more important than Gross National Product”

– His Majesty Jigme Singye Wangchuck, the Fourth King of Bhutan

The definition of happiness has become an interesting task for sociologists and behavioral economists. Happiness studies such as the World Happiness Report have gained a lot of public attention recently. So what exactly is happiness anyway?

Philosopher Robert Nozick offers three categories of happiness that should be taken into account when evaluating the term. Nozick identifies the first type of happiness as being a response to some particular thing occurring. The second is identified as a blissful and joyous state of mind; it is the feeling of having a “good life.” The third category of happiness, however, is much more abstract and philosophical. Nozick writes that this third category is a holistic evaluation of satisfaction with one’s life. This approach complicates the matter because it tries to objectify happiness in life as a whole. This approach also creates a dichotomy between itself and the first two categories (particular instances of happiness). 

Nozick explains that “happiness can occur at the metalevel as an evaluation of one’s life as a whole, and at the object level as a feeling within the life; it can be at both places at once.” (For those familiar with Greek philosophy, this echoes the distinction between Eudaimonic happiness and Hedonistic pleasures). Although an individual may entirely maximize object happiness, he may not necessarily achieve metalevel happiness. Metalevel happiness often requires goal fulfillment and a sense of purpose, and object level happiness does not fit into this category. 

Both forms of happiness are worth pursuing, and Nozick offers an interesting thought experiment to prove this point. Suppose psychologists were able to create a machine that is able to deliver any desirable or pleasurable feeling imaginable. And this machine is created in such a way that experiences in the machine could not be distinguished from those in reality. Would you plug yourself into the machine?

If we choose to plug ourselves into this machine, then we would experience more pleasure than if we didn’t. So if object level happiness is all that matters to us, then we should have no reason to not plug ourselves into this experience machine. 

But Nozick argues that we do have a reason not to plug ourselves into the machine. Having happy experiences is different than wanting to achieve those feelings. While the machine may be able to stimulate pleasures, it does not stimulate the feeling of wanting to do something. Nozick contends, "It is only because we first want to do the actions that we want the experiences of doing them." Furthermore, Nozick argues that we find innate value in being a person with agency. There seems to be something inherently valuable about having the freedom to be someone. Therefore, there is something more to life than just continuously experiencing pleasures, and we can pursue that in the reality outside of the machine’s false reality.

According to Nozick, we choose not to enter this machine because we value being more than we value our experiences. Happiness is therefore not just the surface-level pleasures we experience; it must also comprise of feelings of wanting to be and wanting to achieve. Finding happiness means having both pleasureful experiences as well as formative ones. As such, it is important that we do not only seek out only object pleasures but also a sense of being and purpose. 

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The lunch money heist

A few years ago, I was on an extra tight college budget. So before I discovered the existence of Soylent, I put a lot of effort into maximizing my nutritional intake per day in terms of dollars. University meal plans happened to be a great way to get a healthy, nutritious, and convenient meal at a reasonable price. However, I found that most student customers were continually ripped off because of the menu’s convoluted pricing scheme. And from my experience, I met quite a few students unnecessarily wasted hundreds of dollars every year.

Undergraduates are required to subscribe to a meal plan. There are a variety of meal plans, but some are just objectively more valuable than others. 

A chart of the University’s available meal plans. Plan K is for Kosher meals, so I choose to ignore this.

Freshmen are required to enroll in Plan A, but upperclassmen are allowed to select any meal plan. There are two ways to spend your dining money on campus: Meal swipes at the cafeteria or Dooley Dollars. 

Meal swipes can be used at the cafeteria at any time regardless of the meal price. Dooley Dollars are tax-free store credits with a 12% discount on every food purchase. Dooley Dollars can be spent at any dining hall or restaurant on campus so, it gives you the most freedom to eat. Most people I know either enroll in Plan F and get $500 or Plan B for 100 swipes and $650.

And they’re total ripoffs.

First, here is the price schedule for cafeteria meals:

I also included the price of eating meals over the semester. The prices listed are cafeteria prices after Dooley Dollar discounts. 

Now, we break down the costs of the most popular plans:

At first glance, we see that if you plan on eating 2 cafeteria meals per day, you will lose $458 with Plan A. If you only eat 1 dinner per day, you lose $1340.

I calculate the Maximum value of the plan assuming you use each meal swipe on a $10.50 dinner. The average value is spending half of your swipes on lunches and half of your swipes on dinners. It turns out, that if you opt in for plans B or C, you lose quite a bit of money. Yet, these seem to be some of the most popular meal plans.

The best plan is Plan G. You essentially get $105 free if you spend all of your swipes on dinner. In fact, this is the only plan that offers a real discount. Given the fact that you can add Dooley Dollars to your account at any time, there’s no reason not to get Plan G to begin with. 

So if there’s a free lunch to be had (well, a discounted one), why are so many students being duped into spending more? I think there’s a simple explanation for this. When math is involved, people get lazy.

Israeli-American psychologist Daniel Kahneman offers an explanation to this in his book Thinking Fast and Slow. Kahneman explains that there are two modes of thinking and decision making. First, there is heuristic-based thinking.This mode of thinking is mostly instinctual, or it follows a rule thumb. For instance, when you see a hungry lion behind you, know immediately to panic and run.

The other mode of thinking is much slower but more contemplative. This way of thinking requires using advanced decision-making techniques and rational effort to execute. While you will typically make better decisions with this method of thinking, your decisions will still slower than with the heuristic-based approach. 

Kahneman explains:

“A general ‘law of least effort’ applies to cognitive as well as physical exertion...The law asserts that if there are several ways of achieving the same goal, people will eventually gravitate to the least demanding course of action. In the economy of action, effort is a cost, and the acquisition of skill is driven by the balance of benefits and costs. Laziness is built deep into our nature.”

Of course, there may be a litany of other reasons why students make their decisions, but Kahneman’s theory seems to hold in this context. Students are not looking to calculate expected values; they are more inclined to purchase what’s easy to understand. 

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Are third party authenticators useful? Not quite.

Recently, I was asked to authenticate one of my online accounts using Google Authenticator. Google Authenticator links a personal device of yours to an account, so if you lose your account, you can use your device to reclaim it. Seems like a reasonable level of security, but Google sort of dropped the ball on this one. 

One reason is because Google uses third party barcode scanning software in order to confirm your device. We don't know the security mechanisms used by the barcode scanner, so potentially information could be compromised.

Additionally, Google also stores plaintext passwords serverside... This isn't the greatest idea. There should at least be some sort of encryption method if you store text like that (or hashing and salting).

Read more here:

http://chrisdrake.com/safe-with-google-authenticator-think-again/

[reibsan]

The FREE Stuff Bluff

Recently, I had to read the opportunity to read Predictably Irrational by Dan Ariely. Predictably Irrational, as the title might suggest, explores the our flawed sense of rationality. As the animals at the top of the food chain, we think that we're smart, but we happen to be prone to flawed ways of thinking. Ariely explains why we tend to overspend for mediocre products, why we fail at dieting, and why wine tastes better only when we think it costs a lot.

If you've ever watched an infomercial, you probably weren't surprised when you heard the special deal at the end where you could double your offer "Absolutely Free!" By no means are these vendors offering you a special offer, and I think most of us realize that by now. But why bother including something like that when the cost of a TV ad can reach almost $150,000 per second (from the 2015 Superbowl)? It turns out that this buzzword "free" actually carries a lot of meaning.

Ariely explores the condition where consumers are misguided in decision making by slogans such as “Zero Cost” and “Free!” "Zero" is not just a price; it is an emotional influence on the buyer. Ariely posits that while price is an important factor, the number “Zero” and the slogan “Free!” have intrinsic powers of influencing costumers.

To show this, Ariely describes a series of social experiments he conducted. In one example, he sets up a small sale where he sells Swiss truffles and Hershey’s kisses. The truffles cost about 15 cents while the kisses went for a penny. Under these circumstances, the truffles always sold more. However, in the next step the Kisses are labeled as “Free.” Suddenly, the circumstances change and everyone wants kisses. This suggests that even though the customers were objectively losing value by accepting the free kisses over the truffles, the price of “Free” misdirects their better judgments. Ariely notes that “Free” often becomes an issue when the customer can only choose between free and another product. With the chocolate example, it would always be a better option to choose the truffle. Yet, the value of “Free” surprisingly made the ordinary kiss seem more valuable than it actually was. Lowering the prices generally did not have much of an effect compared to the value of free.

Ariely thinks that “Free” has an emotional charge that offers the illusion of being more valuable than what it actually is. Additionally, he thinks that “Free” eliminates the risk of loss when choosing an item with a price. At least when something’s free, you eliminate that possibility of a loss. He identifies this phenomenon as the zero price effect.

Surely, this seems like a silly pricing gimmick, but the idea is tremendously scalable. Amazon uses this ploy to convince costumers to buy in bulk. With orders that cost over a certain amount, Amazon offers free shipping, and most customers end up purchasing more because of this. Surely, it seems reasonable to Amazon to convince people to buy more per order, but we see that "Free!" actually means a lot. Originally, Amazon France did not have a free shipping policy; instead, they charged 1 Franc (about 20 cents) for bulk shipping. This shouldn't be much of a difference. But we see that once France transitioned to free shipping, their sales saw a dramatic increase. Even if companies can offer low-cost alternatives, marking something as “Free” will yield better sales.

"Where did I put free universal pre-k?"

Ariely sees a lot of opportunities with this clever ploy, and he even goes as far as to say that this could potentially be used to influence public policy. I think he happens to be right. This is perhaps why democrats are able to excite so many voters (think free healthcare, free pre-kindergarten, etc). Perhaps if republicans want to attract more voters, they too will employ this tactic. I think that "tax and tariff-free markets" could be an attractive phrase. But at the same time, they should be careful not to confuse it with "free markets" or "zero regulation." These phrases seem to get a lot of bad reputation.

Regardless of the implications, we should all be careful around "Free!" stuff so we don't get tricked into buying lousy products, or worse yet, lousy politicians.