Mathematics of Love Final PDF

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Group 16 — David Lu (z5113515), David Wassef (z5113080) Emily Chen (z5098910), Garox Lee (z5118852), Ryan Murray (z5115108)

The Mathematics of Love INTRODUCTION Love is often praised as the epitome of the inexplicable. When talking about love, many aphorisms come to mind, such as “the heart wants what it wants”. This is supposed to imply that the mechanisms of love and attraction are unknown and random. Thus, it can be argued that love asserts itself as a mysterious force. On the other end of the spectrum, the field of mathematics contends that it can model all phenomena. With its system of axiomatic proofs and plethora of theorems, maths has demonstrated itself to be capable of modelling extremely complex phenomena, such as the behaviour of free markets. Thus, love is no exception and can also be modelled using mathematics. Typically, when one thinks of the role of maths in love, the first thing that comes to mind is statistics. For example, a quick look on the website for the Australian Bureau of Statistics returns several data points: the number of marriages increased 1.9% from 2013 to 2014, 72.5% of the marriages in 2014 were a first marriage, and the number of divorces decreased by 2.4% from 2013 to 2014 (Australian Bureau of Statistics, 2015). These statistics, while interesting, are limited in what they reveal about the mechanisms of love. Is this really all that maths has to offer? The answer, of course, is no. Several theorems exist that are able to shed more light on the mechanisms of love, rather than these statistics which merely present indicators. More importantly, we will discuss how these theorems can be applied to life as a student at UNSW and our own quests for partnership. As a snapshot of the models available, this paper will investigate how even the most nuanced and elusive elements of love can be ascertained using the Fermi estimation theory, game theory, geometric means, and optimal stopping theory.

FERMI ESTIMATION What would you do if you were asked to solve the following problems?

1. How many piano tuners are there in Chicago? 2. Has humanity produced enough paint to cover the entire surface area of the Earth? 3. How likely is it that intelligent alien civilization exists in our galaxy? 4. How many people in the class will look at their cell phone during this presentation? 5. How much wood would a woodchuck chuck if a wood chuck could chuck wood? 6. What are the chances of finding love at UNSW? Fortunately, there is a very effective technique for calculating these seemingly impossible questions with little or no data. That technique is Fermi Estimation. Developed by physicist Enrico Fermi, this

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Group 16 — David Lu (z5113515), David Wassef (z5113080) Emily Chen (z5098910), Garox Lee (z5118852), Ryan Murray (z5115108)

estimation technique relies on the powers of ten and is capable of making estimations within one order of magnitude of the actual answer. Historically, Fermi used this technique to estimate the strength of the atomic bomb to be 10 kilotons of TNT, which was remarkably close to the actual value of around 20 kilotons (84 Terajoules). In a previously top-secret eyewitness account of the Trinity Test, Fermi wrote the following: “About 40 seconds after the explosion the air blast reached me. I tried to estimate its strength by dropping from about six feet small pieces of paper before, during, and after the passage of the blast wave. Since, at the time, there was no wind I could observe very distinctly and actually measure the displacement of the pieces of paper that were in the process of falling while the blast was passing. The shift was about 2 1/2 meters, which, at the time, I estimated to correspond to the blast that would be produced by ten thousand tons of T.N.T.” (Fermi, 1945)

APPLICATIONS OF FERMI ESTIMATION Fermi estimation is a quantitative tool that produces a quick, rough estimate of a quantity that is either difficult or inconvenient to measure directly. In general, it is necessary to make quick estimates during the planning or overview process that can then be generalised to estimate something more abstract. Nowadays, Fermi estimations are applicable to a number of various fields, including science, engineering, medicine and business. Please see an outline of examples below.

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Science — calculating quantum probabilities and the number of alien civilizations in the universe

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Engineering — calculating the amount of material and time required for certain projects

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Medicine — calculating germ/disease spread and patient recovery time



Business — calculating market size and expected sales

HOW DOES FERMI ESTIMATION ACTUALLY WORK? During the early 1900s, many physicists were obsessed with accuracy and perfect formulas, such as Einstein’s theory of relativity. However, Fermi, being a main contributor to quantum theory, realised the inherent uncertainty in reality and had no problem embracing imprecision. As a result, Fermi was renowned for being able to form surprisingly accurate estimates using little or no information to produce a quantitative result. His technique of taking small successive approximations instead of attempting to make a single large educated guess is far more effective at estimating the actual answer for two main reasons.

1. The problem is broken down into more specific factors that are much simpler to estimate, significantly reducing the margin of error.

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Group 16 — David Lu (z5113515), David Wassef (z5113080) Emily Chen (z5098910), Garox Lee (z5118852), Ryan Murray (z5115108)

2. He deliberately overestimates some of the values while underestimating others. As a result, the overall estimate is effectively balanced out by these over- and underestimates. This all produces a result within at least one order of magnitude.

AN EXAMPLE OF FERMI ESTIMATION The Fermi problems listed previously are quite well-known, and a simple google search would direct you towards the answer. So, we’ve come up with our own Fermi problem to illustrate that this technique can indeed be used to quantitatively estimate anything as long as we can identify the key components to the problem.

How many potatoes can fit into a Boeing 747? (We shall only consider the aircraft cabin where the passengers are seated)

Using Fermi Estimation The radius of an average potato is between 1cm – 10 cm We are unable to decide whether we should take 1cm or 10cm as our estimate so we simply take the geometric mean, which is √ 1×10 ≈ 3.16227766 cm Volume(potato) =

4 π r3 3

= 132.4611769 cm 3 Length of a 747 = 100 m Width of a 747 is between 1m – 10m Thus we take the geometric mean, which is 3.16227766…m Height of a 747 is between 1m – 10m Thus we take the geometric mean, which is 3.16227766…m Volume(747) = 100 ×3.16 ×3.16 = 1000 m 3

= 1,000,000,000 cm3

Volume(747) ÷ Volume(potato) =

1,000,000,000 cm3 132.4611769 cm3 = 7,549,382 potatoes

Literature values Volume(747) = 876,000,000 cm3 Volume(potato) =

(Boeing, 2010)

4 π (3.43)( 3.43)3 (Northern Plains Potatoes, n.d.) 3

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Group 16 — David Lu (z5113515), David Wassef (z5113080) Emily Chen (z5098910), Garox Lee (z5118852), Ryan Murray (z5115108)

cm3

=169.0327937 Volume(747) ÷ Volume(potato) =

876,000,000 cm3 169.0327937 cm 3

= 5,182,426 potatoes Through a series of logical estimates, our Fermi estimation is well within a factor of ten. In fact, our answer is within a factor of 2!

HOW CAN FERMI ESTIMATION CALCULATE THE CHANCES OF FINDING LOVE AT UNSW? First, we shall examine Peter Backus’ 2010 paper “Why I don’t have a girlfriend” where he uses Fermi estimation and adapts the Drake formula in order to estimate his chances at love.

THE DRAKE FORMULA (DRAKE, 1961) In 1961, Dr Frank Drake used Fermi estimation to calculate how many intelligent civilizations there are in our galaxy by making many little educated guesses rather than a big one.

Variable N R¿ fp ne fl fi fc

L

Factors Number of intelligent civilisations in the Milky Way The average rate of star formation in our galaxy The fraction of those stars that have planets The average number of planets that can potentially support life per star that has planets The fraction of planets that could support life that actually develop life at some point The fraction of planets with life that actually go on to develop intelligent life (civilizations) The fraction of civilizations that develop a technology that releases detectable signs of their existence into space The length of time for which such civilizations release detectable signals into space

Values −12

4 × 10

to 3.64 × 107

7/year 0.00001 to 1 0.00001 to 0.2 0.00001 to 0.13 0.000,000,001 to 1 0.2

304 to 1,000,000,000 years

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Group 16 — David Lu (z5113515), David Wassef (z5113080) Emily Chen (z5098910), Garox Lee (z5118852), Ryan Murray (z5115108)

By considering the Rare Earth Hypothesis and substituting the minimum values into the equation, we −12 get a minimum N of 4 × 10 , ultimately suggesting that we are alone in the observable universe. The Rare Earth Hypothesis states that the origin of life and the evolution of complex multicellular organisms required a highly improbable combination of astrophysical and geological events and circumstances. The hypothesis argues that complex extra-terrestrial life forms are likely to be an extremely rare phenomenon. Similarly, by using anthropic and mathematical reasoning and substituting the maximum values, we get a maximum of 3.64 ×107 , suggesting that our Milky Way should be teeming with advanced alien civilisations. Anthropic and mathematical state that it is illogical to assume that complex life is rare and that we are so special in the Universe. By considering the billions of habitable plants in the Milky Way alone, it seems mathematically impossible for extra-terrestrial life to not exist.

PETER BACKUS’ MODEL FOR FINDING LOVE These were Backus’s criteria adapted from the Drake equation. 1. How many women are there who live near me? (In London -> 4 million women) 2. How many are likely to be of the right age range? (20% -> 800,000 women) 3. How many are likely to be single? (50% -> 400,000 women) 4. How many are likely to have a university degree? (26% -> 104,000 women) 5. How many are likely to be attractive? (5% -> 5,200 women) 6. How many are likely to find me attractive? (5% -> 260 women) 7. How many am I likely to get along well with? (10% -> 26 women) Leaving him with just 26 women in the whole world he would be willing to date. Just to put that into perspective, that means there are around four hundred times more intelligent civilizations living on other planets than potential partners for Peter Backus!

OUR MODEL FOR FINDING LOVE AT UNSW We can adapt the Drake equation and redefine the parameters of Backus’ model to answer our own question.

G=S × f s × f p ×f i × f y × f m

Variable

G

Factors Potential girlfriends

Values TBA

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Group 16 — David Lu (z5113515), David Wassef (z5113080) Emily Chen (z5098910), Garox Lee (z5118852), Ryan Murray (z5115108)

S

Number of students at UNSW

54,517

fa

age appropriate (first year)

fp

fy

How many are a potential partners? Single, opposite gender, heterosexual, monogamous… How likely are we to interact? (Same class, common interests, matching timetable…) How many do I find attractive?

0.18 (first year) (UNSW, 2013, p. 6) 0.46 are female and 0.7826 are single

fm

How many find me attractive?

fr

How likely are we able to get along and sustain a relationship?

fi

0.032 (geometric mean of 0.01 and 0.1) 0.32 (geometric mean of 0.1 and 1) 0.10 (I’m being optimistic) 0.32 (geometric mean of 0.1 and 1)

Possible girlfriends = 54,517 × 0.18 × 0.36 × 0.10 × 0.032 ×0.10 ×0.32 = 0.36174864384 possible girlfriends at UNSW Our percentage chance =

0.36174864384 ×100 54,517 = 0.000663552%

Based on our model, I only have a 0.0007% chance of finding a girlfriend at UNSW. That’s depressingly low.

HOW CAN WE INCREASE OUR CHANCES AT FINDING LOVE? 1. The easiest way to increase our chances is to relax our criteria. For example, in the previous calculation, we can instantly triple our chances if we choose to date 2nd and 3rd year students. Similarly, we can increase our chances by tenfold if we interact with every potential partner we encounter. On top of that, we can also choose to look for partners outside of UNSW to further increase our chances. 2. Another way to increase our chances is to be realistic and to not prepare an extensive list of pre-emptive dating criteria. If you attempt to find a perfect partner that matches everything on our checklist, you are essentially setting yourself an impossible challenge. Instead, we should only pick a few qualities that are important to us and be generous and accepting with everything else.

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Group 16 — David Lu (z5113515), David Wassef (z5113080) Emily Chen (z5098910), Garox Lee (z5118852), Ryan Murray (z5115108)

For example, we shouldn’t be too picky when it comes to: the intelligence of our partner, their socio-economical background, their weight and height, religion, politics and hobbies. We also should not have a long list of deal-breakers such as: a. I will not date you if: b. You drink and go clubbing every night c. You update your facebook every 5 minutes d. You splurge on items that you absolutely do not need

As tempting as it is to limit your search to a sober, fiscally-responsible, social media hater, the more deal-breakers you have, the less likely you are to find love. When you add dozens of these additional factors into Backus’s equation – or even our version – you’ll most likely get an answer close to zero potential partners! 3. The last thing we can do to increase our chances it to not give up! Peter Backus, who used this method to estimate his chances of love, found out that he would only be willing to date 26 women in the entire world! However, he managed to beat his own odds by being persistent and more flexible with his dating criteria. He has been happily married since 2014.

GAME THEORY Game theory is the branch of mathematics concerned with the analysis of strategies for dealing with competitive situations in which the outcome of a participant's action depends critically on the actions of other participants. Game theory has been utilized to explore the outcomes in various areas, including war, business, and even biology. In addition, this complex mathematical theory can also be applied to love. There are quite a few terms that must be understood about game theory before it is possible to use its models to explore the mathematics of love.

1.

DOMINANT STRATEGY:

A dominant strategy is the condition in which a player or agent will receive the best possible outcome, regardless of what the other player’s actions. To demonstrate, see the payoff matrix below modelling a sample Prisoner’s dilemma, in which two people have the option to either confess or not confess to a crime. First, assume that person B confesses. Person A will then either confess and get 8 years of prison time or not confess and get 20 years. Clearly, confessing gives person A the best pay off. Now, assume that Person B does not confess. Person A can then either confess and get 1 year of prison time or not confess and get 3 years. It is evident that confessing in both situations provides player A with the best payoff. Hence, confessing is the dominant strategy for person A. The same result can be achieved for person B using the line of reasoning. Person B Person A

Confess

Don’t confess

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Group 16 — David Lu (z5113515), David Wassef (z5113080) Emily Chen (z5098910), Garox Lee (z5118852), Ryan Murray (z5115108)

Person A: 8 years

Person A: 1 year

Person B: 8 years

Person B: 20 years

Person A: 20 years

Person A: 3 years

Person B: 1 year

Person B: 3 years

Confess

Don’t confess

2.

NASH EQUILIBRIUM

A nash equilibrium is a system involving the interaction of different participants, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged (Oxford Dictionaries, 2016). In simple terms, an agent cannot gain a greater pay off by changing their own decision, it must be other agent who should change their action for the possibility of a higher pay off. This is pervasive in the dating world, especially when discussing an individual’s aspirations during a night out.

3.

PARETO EQUILIBRIUM

A pareto equilibrium is a state of interaction between two or more people during which no person can be made better off in terms of money, output of products, or potential partners without making someone else worse off. In the dating world, this is particularly evident when a group of one sex is seeking to interact with another group. A small pay off matrix can help to explain this concept. The clearest demonstration of a pareto equilibria is seen in the movie “A Beautiful Mind” (Ron Howard, 2001). In the movie, John Nash and his three male friends are in a bar and see a group of five women, four brunettes and a blonde. In that situation, all the males are more attracted to the blonde more than the brunettes. However, before each guy attempts to talk to the blonde, Nash poses a counter-proposal. He suggests that no one should go for the blonde, for if everyone did they would all block each other. Consequently, when trying to approach one of the other women they would all give the men the cold shoulder as no one likes being a second choice. He suggests that everyone should approach one of the brunettes. Thus, all of the men have the possibility of gaining a partner, as opposed to the previous outcome of no one gaining a partner. If the groups’ intentions are to maximise the groups’ payoff as a whole, then all participants will receive a payoff. However, the payoff will not be at its maximum potential relative to each individual participant. Person B

Person B

c

d

A

1, 3

2, 2

B

2, 1

1, 2

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Group 16 — David Lu (z5113515), David Wassef (z5113080) Emily Chen (z5098910), Garox Lee (z5118852), Ryan Murray (z5115108)

In this example payoff matrix above, no matter what outcome one chooses, if an agent attempts to improve their own pay off, they can only do so by making someone worse off. In this example all four results are pareto efficient.

THE COURNOT COMPETITION MODEL This form of game theory is traditionally used in economic situations where two companies can collude to both cut supply, thus both increasing profits (Elvis Picardo, May 1 st 2015). However, if one company chooses to cooperate while the other defects and secretly doesn’t lower its supply, then the defective company will gain all the rewards. A Cournot Competition Model is a form of pareto equilibria in three out of its four results. The only non-pareto efficiency point is where both companies cooperate.

Company B Cooperate

Defect

Company A: 5

Company A: 0

Company B: 5

Company B: 7

Company A: 7

Company A: 3

Company B:...