Mathematical Definition of Love

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Adamah
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Mathematical Definition of Love

Postby Adamah » Wed Jun 30, 2010 10:58 pm UTC

My friend and I were discussing if love could be defined mathematically, and we agreed on this definition (requires some knowledge of game theory) -

A loves B iff A would rationally choose to never betray B in any prisoner's dilemma. Or, depending on your definition, that no prisoner's dilemma can be formed involving A and B.

The idea being, A would value B's utility as greater than or equal to his own utility, and thus would never pick an option that sacrifices summed utility for his own. In an ideal situation, if A and B loved each other, then they would always cooperate.

I feel like this could be made into an xkcd comic pretty easily - Girl: "I love you", Guy: "I would never betray you in the prisoner's dilemma"

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Re: Mathematical Definition of Love

Postby skeptical scientist » Wed Jun 30, 2010 11:11 pm UTC

I think it's clear that people value others' outcomes as well as their own - all else being equal, most people will make the choice which most benefits others, even if they are affected the same either way. Similarly, most people would happily give up a dollar so that a friend could get a hundred bucks. So your idea is to define love based on how much an individual's utility function is affected by outcomes to others. This seems reasonable - I'm much more likely to do little things to make a significant other's life more comfortable, compared to, say, a roommate - but probably not sufficiently subtle to really capture love. For example, a really generous person and a really selfish person may both be capable of love, and both be more generous towards their loved ones than friends or strangers, but the baseline generosity is so different that the generous individual may be more generous to strangers than the selfish one is to loved ones. Also, there's a lot more to love than simply valuing another's happiness.
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Re: Mathematical Definition of Love

Postby Adamah » Wed Jun 30, 2010 11:34 pm UTC

skeptical scientist wrote:Also, there's a lot more to love than simply valuing another's happiness.


I actually don't think there is. There might be more to a relationship, like finding someone that fits you well, but I define "love" as simply valuing someone else's happiness as greater than or equal to your own.

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Re: Mathematical Definition of Love

Postby Cleverbeans » Thu Jul 01, 2010 12:10 am UTC

Adamah wrote:I define "love" as simply valuing someone else's happiness as greater than or equal to your own.


Between that and the game theory, I see a career with Halmark in your future.
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Re: Mathematical Definition of Love

Postby ++$_ » Thu Jul 01, 2010 3:44 am UTC

I think the prisoner's dilemma actually does a pretty good job of defining what love is, but I'd quibble with your definition.

First of all, you used the term "rational." Very often, love is not at all rational. I think love is better seen as something that causes one to ignore or suppress the payoff matrix than as something that modifies it.

Second, I disagree that love is a binary situation. For example, let's say the Temptation is -3.9, the Reward is -4, the Punishment is -4.1, and the Sucker's payoff is -4.2, where the unit involved is "years of freedom". (It's a one-shot game.) Then it wouldn't take too much to get me to cooperate. In fact, I might even cooperate with a total stranger, just out of a sense of it being the decent thing to do. If you change the numbers to T = 0, R = -1, P = -2, S = -5, it would take a pretty good degree of trust for me to cooperate. With T = 0, R = -1, P = -1.01, S = -5, there would be only a couple of people with whom I would cooperate. And if the game were T = 0, R = -5, P = -5.0000001, S = -100, it might be enough to break up an awful lot of committed relationships! (At least in the sense that one or more participants would defect.) So there are various strengths of love that can be distinguished in this way (there might be more than one dimension too).

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Re: Mathematical Definition of Love

Postby xkcdfan » Thu Jul 01, 2010 5:33 am UTC

WHAT IS WRONG WITH YOU PEOPLE.

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Re: Mathematical Definition of Love

Postby poxic » Thu Jul 01, 2010 5:48 am UTC

What, you want to leave it all anecdotal and ill-defined? That just can't be allowed.

After all, love is widely acknowledged to be less than three, so we know already that it's not imaginary (does not have i as a factor) and (if an integer) is not a positive square, cube, or any other (integer) exponent product. Aside from 1, which can lead us to all sorts of potentially autophilic conclusions. Or zero, which is not by definition positive, and does not lend itself well to multiplication.

/backing carefully out of the numbers board now
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Re: Mathematical Definition of Love

Postby gmalivuk » Thu Jul 01, 2010 3:11 pm UTC

xkcdfan wrote:WHAT IS WRONG WITH YOU PEOPLE.
We're discussing math in the math forum. What's *your* problem?
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Re: Mathematical Definition of Love

Postby RabbitWho » Thu Jul 01, 2010 4:37 pm UTC

He's in love.

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Re: Mathematical Definition of Love

Postby Kurushimi » Thu Jul 01, 2010 5:12 pm UTC

poxic wrote:What, you want to leave it all anecdotal and ill-defined? That just can't be allowed.

After all, love is widely acknowledged to be less than three, so we know already that it's not imaginary (does not have i as a factor) and (if an integer) is not a positive square, cube, or any other (integer) exponent product. Aside from 1, which can lead us to all sorts of potentially autophilic conclusions. Or zero, which is not by definition positive, and does not lend itself well to multiplication.

/backing carefully out of the numbers board now
//sorry for the arts-student tangent


Not only do we know that love is less than 3, we also know that [imath]\varepsilon[/imath] is greater than love. And, this being true for an arbitrary [imath]\varepsilon[/imath] > 0, we are forced to conclude that love = 0, that is, love is nothing at all or love is less than 0 and is actually, contrary to popular, quite negative.

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Re: Mathematical Definition of Love

Postby PM 2Ring » Thu Jul 01, 2010 6:09 pm UTC

Kurushimi wrote:
poxic wrote:What, you want to leave it all anecdotal and ill-defined? That just can't be allowed.

After all, love is widely acknowledged to be less than three, so we know already that it's not imaginary (does not have i as a factor) and (if an integer) is not a positive square, cube, or any other (integer) exponent product. Aside from 1, which can lead us to all sorts of potentially autophilic conclusions. Or zero, which is not by definition positive, and does not lend itself well to multiplication.


Not only do we know that love is less than 3, we also know that [imath]\varepsilon[/imath] is greater than love. And, this being true for an arbitrary [imath]\varepsilon[/imath] > 0, we are forced to conclude that love = 0, that is, love is nothing at all or love is less than 0 and is actually, contrary to popular, quite negative.

I love your work, poxic.

You provide a convincing argument, Kurushimi, but bear in mind, the normal approach is useless here.

:)

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Re: Mathematical Definition of Love

Postby graatz » Thu Jul 01, 2010 7:05 pm UTC

poxic wrote:After all, love is widely acknowledged to be less than three


We also know 1 to be the loneliest number. Because love is greater than loneliness, we can thus conclude that 1<love<3.

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Re: Mathematical Definition of Love

Postby ++$_ » Thu Jul 01, 2010 7:07 pm UTC

graatz wrote:
poxic wrote:After all, love is widely acknowledged to be less than three


We also know 1 to be the loneliest number. Because love is greater than loneliness, we can thus conclude that 1<love<3.
So the odds are that love = 2.

Take that, polyamorists!

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Re: Mathematical Definition of Love

Postby graatz » Thu Jul 01, 2010 7:27 pm UTC

++$_ wrote:
graatz wrote:
poxic wrote:After all, love is widely acknowledged to be less than three


We also know 1 to be the loneliest number. Because love is greater than loneliness, we can thus conclude that 1<love<3.
So the odds are that love = 2.


Actually, the odds are stacked pretty well against that :( It's almost certain that 1<love<2 or 2<love<3 unless we can deduce more about it.

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Re: Mathematical Definition of Love

Postby skeptical scientist » Thu Jul 01, 2010 7:31 pm UTC

graatz wrote:Actually, the odds are stacked pretty well against that :( It's almost certain that 1<love<2 or 2<love<3 unless we can deduce more about it.

Like, perhaps, the fact that it's natural? :)
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Re: Mathematical Definition of Love

Postby ++$_ » Thu Jul 01, 2010 9:38 pm UTC

It occurs to me that in many situations, [imath]love = {1 \over 1}[/imath].

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Re: Mathematical Definition of Love

Postby Quaternia » Fri Jul 02, 2010 3:46 am UTC

"It's just a Relation", said the logician.
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Re: Mathematical Definition of Love

Postby Adamah » Fri Jul 02, 2010 3:58 am UTC

++$_ wrote:Second, I disagree that love is a binary situation. For example, let's say the Temptation is -3.9, the Reward is -4, the Punishment is -4.1, and the Sucker's payoff is -4.2, where the unit involved is "years of freedom". (It's a one-shot game.) Then it wouldn't take too much to get me to cooperate. In fact, I might even cooperate with a total stranger, just out of a sense of it being the decent thing to do. If you change the numbers to T = 0, R = -1, P = -2, S = -5, it would take a pretty good degree of trust for me to cooperate. With T = 0, R = -1, P = -1.01, S = -5, there would be only a couple of people with whom I would cooperate. And if the game were T = 0, R = -5, P = -5.0000001, S = -100, it might be enough to break up an awful lot of committed relationships! (At least in the sense that one or more participants would defect.) So there are various strengths of love that can be distinguished in this way (there might be more than one dimension too).


Ah, but that's the point of the word "any". Just because you'd cooperate with someone in a particular prisoner's dilemma (such as the first example you gave) doesn't mean you love them. To be in love with someone, you must choose to cooperate in ANY prisoner's dilemma, including the last one. If you don't, then it's something less than love. Maybe "friendship" perhaps.

The key is that instead of looking to maximize your utility, you're looking to maximize the sum of your utility and your partner's utility since you value your partner's happiness equal to your own. Thus, by definition of the prisoner's dilemma, you would always choose to cooperate since this would maximize your combined utility, regardless of what the individual payoffs might be.

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Re: Mathematical Definition of Love

Postby Fanway » Fri Jul 02, 2010 11:52 am UTC

graatz wrote:
++$_ wrote:
graatz wrote:
poxic wrote:After all, love is widely acknowledged to be less than three


We also know 1 to be the loneliest number. Because love is greater than loneliness, we can thus conclude that 1<love<3.
So the odds are that love = 2.


Actually, the odds are stacked pretty well against that :( It's almost certain that 1<love<2 or 2<love<3 unless we can deduce more about it.


I agree that love = 2, however ...

2<Love+Tequila<5 (on a good day)

Also ...

Love+TooMuchTime=Assets(.55)

Seriously, how would you write the equation that expresses the Net Present Value of love over a 13-year investment period? And, what is the unit of measure?

Thanks!

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Re: Mathematical Definition of Love

Postby Pvt. Parts » Sat Jul 03, 2010 9:16 am UTC

I suppose your definition works well enough for A and B, but have you thought about situations involving more than two people, for instance children?

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Re: Mathematical Definition of Love

Postby squareroot » Sun Jul 04, 2010 9:17 pm UTC

Back to the OP's thoughts -

If a person A's reason for living is Y(h,x) - where x is the happiness of person B, and h are other factors of the A's life - then A is in love with B if (dY/dx)/Y > k for all x and some constant k > 0.

In other words, you NEED the other person's love. A lot. If they get happier by some amount, then you get exponentially happier. The only problem is, if you have a negative reason for living, then their happiness makes you sadder. But we can safely assume that, once Y hits zero, you'll commit suicide - or shortly thereafter or before.
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Re: Mathematical Definition of Love

Postby Quaternia » Mon Jul 05, 2010 5:34 pm UTC

Pvt. Parts wrote:I suppose your definition works well enough for A and B, but have you thought about situations involving more than two people, for instance children?

I think it's safe to assume that your partner would factor that into their happiness, and since you are dependent upon that, you will then be dependent on those factors as well.
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Re: Mathematical Definition of Love

Postby Shokk » Mon Jul 05, 2010 5:43 pm UTC

I'd try not to think of love as a point that can be found, but rather a relation between points.
For example, I'm much of the opinion that 1 is in love with itself, because it always looks out for itself, and it's the multiplicative identity, so take that for what it's worth.
What the relation "love" would technically entail, I have no clue, are we going to keep it real and not look at those overly romantic or unnecessarily complex and fantastic numbers in the complex plane?
This relation, "love" must depend wholly on the points in question.
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Re: Mathematical Definition of Love

Postby TheAlmightyEgg » Wed Jul 14, 2010 3:36 am UTC

squareroot wrote:Back to the OP's thoughts -

If a person A's reason for living is Y(h,x) - where x is the happiness of person B, and h are other factors of the A's life - then A is in love with B if (dY/dx)/Y > k for all x and some constant k > 0.

In other words, you NEED the other person's love. A lot. If they get happier by some amount, then you get exponentially happier. The only problem is, if you have a negative reason for living, then their happiness makes you sadder. But we can safely assume that, once Y hits zero, you'll commit suicide - or shortly thereafter or before.


Impossible. Mutual love does not appear to cause a constant (possibly wrong word, it's that or continuous, take your pick!) exponential increase in happiness, as would be expected were this true.

I'm pretty sure my definition (in terms of game theory) is somewhere along the lines of an individual's happiness becoming the mean of their own happiness and their partners happiness.

Essentially, using your notation
Y(h,x) = (h+x)/2
In cases of mutual love, Y(h,x)=Y(x,h) so feedback isn't a problem anyway.

I'd put friendship/general relationships as Y(h,x) = (h+nx)/n+1 for 0<n<1
with the strength of the friendship proportional to n
A rather interesting implication of this, is that it allows enemity to exist (-1<n<0) on the same spectrum as both love and friendship.

I think I've put too much thought into this, I already overthink emotions - I'd take this further (i.e. some kind of comment on happiness becoming undefined as an individual becomes an absolute enemy (n --> -1), ideas on involvement of multiple people, etc), but I'm gonna sleep instead.

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Re: Mathematical Definition of Love

Postby squareroot » Wed Jul 14, 2010 4:21 am UTC

I actually like the idea of it reaching singularity... maybe something like Y(h,x)=(h+nx)/(n+1)(1-n) would work well. Here, a good friend might be around n=0.3, someone you just married around n=0.95, someone whom you mildly dislike/are jealous of n=-0.2. If someone was at n=-1 (positive side), it would equate to someone who is so much worse than every other thing in your life that you want to kill yourself. At n=1, as long as they're happy, you have no need to do anything else in life. The problem with this expression is that it destroys the idea of the weighted arithmetic mean... but I think it might be more accurate, considering infinite happiness and all...

I just remembered - another good equation for love is (x^2 + y^2 - 1)^3 - x^2*y^3 = 0. This is to be interpreted totally different from the previous equations, though..
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Re: Mathematical Definition of Love

Postby Adamah » Thu Jul 15, 2010 2:57 am UTC

TheAlmightyEgg wrote:
squareroot wrote:Back to the OP's thoughts -

If a person A's reason for living is Y(h,x) - where x is the happiness of person B, and h are other factors of the A's life - then A is in love with B if (dY/dx)/Y > k for all x and some constant k > 0.

In other words, you NEED the other person's love. A lot. If they get happier by some amount, then you get exponentially happier. The only problem is, if you have a negative reason for living, then their happiness makes you sadder. But we can safely assume that, once Y hits zero, you'll commit suicide - or shortly thereafter or before.


Impossible. Mutual love does not appear to cause a constant (possibly wrong word, it's that or continuous, take your pick!) exponential increase in happiness, as would be expected were this true.

I'm pretty sure my definition (in terms of game theory) is somewhere along the lines of an individual's happiness becoming the mean of their own happiness and their partners happiness.

Essentially, using your notation
Y(h,x) = (h+x)/2
In cases of mutual love, Y(h,x)=Y(x,h) so feedback isn't a problem anyway.

I'd put friendship/general relationships as Y(h,x) = (h+nx)/n+1 for 0<n<1
with the strength of the friendship proportional to n
A rather interesting implication of this, is that it allows enemity to exist (-1<n<0) on the same spectrum as both love and friendship.

I think I've put too much thought into this, I already overthink emotions - I'd take this further (i.e. some kind of comment on happiness becoming undefined as an individual becomes an absolute enemy (n --> -1), ideas on involvement of multiple people, etc), but I'm gonna sleep instead.


Yeah, that's the definition I go with. An ideal love relationship would value each person's happiness equally (i.e, n = 1). An interesting thing to consider - what if n > 1? That is, you value the other person's happiness more than your own? And what if n > 1 for both people?

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Re: Mathematical Definition of Love

Postby squareroot » Thu Jul 15, 2010 5:01 am UTC

Well, it depends whether or not "n" in the formula is explicitly the other person's happiness (corresponding to your h), or their will to live (corresponding to your Y). If it's their will to live, with j= their base happiness, n1=your love for them, and n2=their love for you, it's an easy equation to solve:

Y=(h+n1 x)/(n1 + 1)
x=(j+n2 Y)/(n2 + 1)
Y=(h + n1*j/(n2 + 1))/(n1 + 1) + Y*n2/(n2 + 1)
(h + n1*j/(n2 + 1))/(n1 + 1) = Y*(1 - n2/(n2 + 1))
= Y*( 1/(n2 + 1) )
Y = (n2+1)*(h + n1*j/(n2 + 1))/(n1 + 1)
Y = (n2*h + n1*j + h)/(n1 + 1)

So, it actually turns out you can still write out each person's will to live without it going insane for n>1. This is because it's taking the average, and not actually growing without bound. An interesting consequence of the formula is that, if they love you more than you love them by a large factor, (that is, n2>>n1) your will to live is mostly dependent on your happiness, growing without bound proportionally to their love for you. And if (n1 ~~ 1, n2 ~~ 1), so neither of your feelings is exceptional, your own happiness is still predominant.

Of course, if x is to be understood as their general happiness - corresponding to your h - there will be no feedback systems. Unless, of course, "unrequited love" factors into your general h variable.
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Re: Mathematical Definition of Love

Postby heelium » Thu Jul 15, 2010 7:57 am UTC

Adamah wrote:
skeptical scientist wrote:Also, there's a lot more to love than simply valuing another's happiness.


I actually don't think there is. There might be more to a relationship, like finding someone that fits you well, but I define "love" as simply valuing someone else's happiness as greater than or equal to your own.


I personally think that one could act in all prisoner's dilemmas in some way by principle. And one could have all those love-feelings without actually going in for whatever reason. It's a mixed complex.

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Re: Mathematical Definition of Love

Postby brötchen » Thu Jul 15, 2010 3:15 pm UTC

squareroot wrote:An interesting consequence of the formula is that, if they love you more than you love them by a large factor, (that is, n2>>n1) your will to live is mostly dependent on your happiness, growing without bound proportionally to their love for you

its almost creepy how well this coresponds to what is happpening to me rigth now .... it really sucks if n2>>n1(edit: I meant n1>>n2).
also it seems as if your happynes is in an invers relation to their love for you if n1<0 which would mean the best way to handle rejection is to get n1 below 0

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Re: Mathematical Definition of Love

Postby quintopia » Sun Sep 26, 2010 6:07 am UTC

I just want to say that there is no such thing in real life of either the type of love defined in the OP or in TheAlmightyEgg's recent post. There is always a Prisoner's Dilemma type game that will cause a defection in at least one person in the relationship, and the only time when two partners love each other equally is that brief period between when one's love is on the rise and the other's dominance is on the rise (and love on the decline). Actually, it is possible that the love is never equal if the love-over-time function is as discontinous as I believe it to be.

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Re: Mathematical Definition of Love

Postby gmalivuk » Sun Sep 26, 2010 9:04 pm UTC

Someone just have a messy break-up?
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Re: Mathematical Definition of Love

Postby wasapiguy2 » Sun Sep 26, 2010 11:19 pm UTC

quintopia wrote:...and the only time when two partners love each other equally is that brief period between when one's love is on the rise and the other's dominance is on the rise (and love on the decline).


This just made so many graphs-- all of which are based on qualitative data that I attempt to make quantitative-- pop into my head that would allow me to try to define a relationship, its strengths, and its weaknesses.

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Re: Mathematical Definition of Love

Postby Sagekilla » Mon Sep 27, 2010 5:51 pm UTC

So I was looking at the equation of love a bit, and I realized something. If you look at it normally, you just see this:
Spoiler:
Image


But zoom in a bit... ;)
Spoiler:
Image
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Re: Mathematical Definition of Love

Postby userxp » Wed Sep 29, 2010 5:34 pm UTC

I always assumed that, if both partners value the other person's happiness more than their own, then their relationship would become a war where each one attempts to make the other one happier against his or her will.
For example, suppose person A likes film X, and person B likes film Y (they both know what the other partner likes), and they are trying to choose which one too watch.
A will try to convince B that they should watch Y, and B will try to convince A to watch X. This means that in the end the smartest one will be the less happy, and that will make him/her happy.

That is only in cases where both happinesses are mutually incompatible. If both A and B like film Z, then they will cooperate to watch Z.

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Re: Mathematical Definition of Love

Postby KrO2 » Wed Oct 06, 2010 1:24 am UTC

Kurushimi's argument makes it seem likely that the numerical value of love is negative, which means that the various components of this statement could be true. It's an xkcd 55 wannabe that claims to have used the usual approach. Note that this claimed identity holds even if love=0 and is not negative, the final value is still simple.


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