Meet in the Middle: What Love Does

Your friend has driven you to the airport four times.

You've driven her once.

She has never mentioned it. You have never mentioned it. Every time you text "any chance you can drive me Thursday?" you add "no worries if not!!", and you both know what those two exclamation points are doing.

Nobody is keeping score, and both of you know the score.

Every friendship has a version of this. So does every marriage, every pair of siblings, every two people who share a kitchen. One side carries more. The other side feels the weight, even when nobody says a word.

So what is that weight? Does it have a size? And when something finally fixes it, what exactly did the fixing?

The short answer

A lopsided bond carries a real cost, with an exact size. Love is the cheapest way to repair it from inside the pair, and nobody outside the pair pays for it.

Here's the whole answer in five steps. The rest of this post takes them one at a time and shows how we know each one.

1. A lopsided bond costs. Every bond between two people carries a number, and 1 is level. Any other number costs something, and the cost has one exact formula. A bond at 2 costs 1/4. A bond at 4 costs 9/8. Only a level bond costs nothing.
2. Love meets in the middle. Love takes two lopsided bonds and brings both to their geometric mean, the number you reach by multiplying instead of adding. A bond at 4 and a bond at 1 meet at 2 and 2, and the cost falls from 9/8 to 1/2. Of every way to ease the pair while keeping their product the same, this is the single cheapest, and it changes nobody else's bonds.
3. Giving pays more than it costs. Start level and take on part of someone's load until you meet in the middle, and their relief is always more than three times what it costs you. In our example you carry 1/4 more and they carry 7/8 less.
4. Taking is surcharged. Push a level pair apart, doubling one side and halving the other, and a pair that cost nothing now costs 1/2.
5. Love goes as far as you can carry. If the middle is more than either of you can carry, love can't reach it. It stops at the closest point you can both hold and still keep going.

Why this matters: love has an exact shape. It has an endpoint, a price and a limit, and you can calculate all three. A lopsided bond is a cost that both people carry, and there's exactly one cheapest way to repair it from inside the pair. The price tag is the same one that prices every mismatch between two amounts, so you'll see it again.

Here's the route. I'll stop after each step to zoom out, so you always know where we are, what we've got and why the next step comes next. The Ledger will keep the map. If you haven't met it: the Ledger is the permanent record of everything that has ever happened, and it keeps a number for every bond between people. We draw it as a book in reading glasses. It never erases a line, and it has one response to all of it: "Noted." The face is a drawing. The record is real. It comes from Part 1 of The Universe Writes in Pen, but you don't need that post to follow this one.

Step 1: A lopsided bond costs

Every bond between two people carries a number. Any number but 1 costs something, and the cost has one exact formula.

Start with what a bond is. A bond joins two people and compares two amounts of the same kind. Its number is one amount divided by the other. When the two amounts match, the number is 1. I'll call that level. When one side has twice as much as the other, the number is 2, or 1/2 if you read the same bond from the other end.

I'll draw the airport as a bond running from her to you, with the score on it as a number. She's driven four times for your one, so it reads 4. The bond is real: it's what the Ledger keeps for a relationship between two people. The rides are the drawing. What a real friendship's bond compares is still being worked out, and everything below uses only the number on the bond, so it holds whatever the bond compares.

What does a bond at 4 cost? Here's the price list. We call the cost J.

You can get any row yourself. Add the number to its flip (1 divided by the number), halve the total, and subtract 1. For 4: 4 plus 1/4 is 4 and 1/4. Half of that is 2 and 1/8. Minus 1 is 1 and 1/8, which is 9/8.

I'll count how far a bond sits from level in doublings and call that its tilt. A bond at 2 is one doubling up: tilt +1. A bond at 4 is two doublings up: tilt +2. A bond at 1/2 is one halving down: tilt -1. Level has tilt 0. In the drawings, one block is one doubling. The blocks are a drawing of tilt, and tilt is a real property of the bond.

Four things to notice.

Only level is free. The cost also works out as the number minus 1, squared, divided by twice the number. For 4 that's 3 squared over 8, which is 9/8 again. A square is never negative, and a bond's number is always above zero, so the cost is never negative. It reaches zero only when the number minus 1 is zero, which happens at 1 and nowhere else.

One doubling up costs the same as one doubling down. 2 and 1/2 both cost 1/4. 4 and 1/4 both cost 9/8. Being twice as far ahead costs exactly what being twice as far behind costs, because it's the same bond read from the other end.

The cost climbs fast. A bond at 4 has twice the tilt of a bond at 2, and it costs four and a half times as much.

There is no top. Gary, the drummer, has driven anyone anywhere exactly once, and he still brings it up. His score with the friend who drives him is 10 to 1. That bond costs 4.05. Push the score further and the cost keeps climbing. No ratio is too big to have a price.

Where does the formula come from? We didn't pick it. When comparisons combine, their costs have to combine in one consistent way. Take two comparisons, 2 and 3. Multiply them and you get 6. Divide them and you get 2/3. The costs of 6 and 2/3, added together, have to come out of the costs of 2 and 3 by one exact rule: multiply the two costs together, add both costs, and double the total. Check it. 2 costs 1/4 and 3 costs 2/3, so that's 1/6 + 1/4 + 2/3 = 13/12, doubled to 13/6. And 6 costs 25/12, and 2/3 costs 1/12 like its flip 3/2. Together, 13/6. Add one choice of scale at level, which fixes how sharply the cost curves up from 1, and exactly one formula is left. It's proved, and published.¹ The Ledger prices every bond with that same J.

Zoom out. Step 1 is done. A bond has a number, 1 is level, and every other number costs: 2 and 1/2 cost 1/4, 4 costs 9/8, and the cost has no top. Next, and why: the fix. If lopsidedness costs, any move that lowers the cost is a repair, and we can ask which repair is cheapest.

Step 2: Love meets in the middle

Love takes two lopsided bonds and brings both to their geometric mean. Of every way to ease the pair while keeping its total tilt, it's the single cheapest, and it changes nobody else.

I'll draw the friendship as two bonds, one running each way. Say hers to you reads 4, as before, and yours to her reads 1. Together they cost 0 + 9/8 = 9/8.

A bond has two ends, and both ends carry all of it. Neither end carries half. So both of you carry that 9/8.

Here's the rule for the move I'm about to name. It can move tilt from one bond to the other, but it can't make any tilt and it can't lose any. Right now the pair's total tilt is 2: two doublings on her bond, none on yours. After the move it has to still be 2. Keeping the total tilt is the same as keeping the product of the two numbers: 4 times 1 is 4. That's what makes it a repair inside the pair. Nothing is added from outside, and nothing is carried off.

Move one doubling from her bond to yours. Hers comes down from 4 to 2. Yours goes up from 1 to 2. The total tilt is still 2, and the product is still 4. The cost is 1/4 + 1/4 = 1/2. It was 9/8, so the move saved 5/8. Both of you carry both bonds, so each of you now carries 1/2 instead of 9/8, and each of you is relieved by 5/8.

The Ledger still has every ride. The move doesn't erase a line. It changes what the bonds read from now on.

That's the geometric mean of 4 and 1: the number that, multiplied by itself, gives the same answer as multiplying the two together. 2 times 2 is 4, and so is 1 times 4. Another way to see it: 1, 2 and 4 are three evenly spaced steps, each one a doubling, and 2 is the middle step.

The middle isn't 2.5. Put both bonds at 2.5 and their product is 6.25, not 4. The total tilt would be more than the 2 you started with, and you'd have made tilt out of nothing.

How we know it's the cheapest. Check every pair that keeps the total tilt at 2. Those are all the pairs of numbers that multiply to 4. Walk along them.

At 1 and 4 the pair costs 9/8. At 4/3 and 3 it costs 17/24. At 2 and 2 it costs 1/2. On this trail the cost works out to 5/8 of the sum of the two numbers, minus 2. Check it at 2 and 2: the sum is 4, 5/8 of 4 is 2 and 1/2, and minus 2 leaves 1/2. And for numbers that multiply to 4, the sum is smallest when they're equal: 1 + 4 is 5, 4/3 + 3 is 4 and 1/3, and 2 + 2 is 4. Equal wins, and only equal.

That's true for any starting pair, not just 4 and 1. Fix the total tilt, and the pair is cheapest exactly when both numbers are equal. Of all the endless ways to ease the pair while keeping its total, exactly one is cheapest. I find that hard to get over.

The move changes only the two bonds it moves. Carol, who lives three towns over and has never met either of you, carries exactly what she carried before.

Yeah-But, the voice in the back of your head that has never once said wow, clears his throat. "Why not set both bonds to 1? Zero cost!"

You can. It also changes the total tilt, from 2 to nothing, and that isn't a move inside the pair. It's a different act, called restoration, and who pays for it is its own question. Meeting in the middle is the best the pair can do while keeping its total.

Gary has a different idea. Raise your own bond to match hers, so both read 4. Equal! And the pair now costs 9/4, twice what it started at, because Gary has made two doublings of tilt out of nothing. "Noted. 9/4."

This move is what we call love. The name is ours. The arithmetic isn't, and here's why the name fits.

• It's the cheapest move there is that keeps the pair's total tilt.
• It's the only one of those moves that treats both bonds alike.
• It touches nobody outside the pair.
• Both people who carry the pair come out lighter.²

Zoom out. Step 2 is done. Love is the move to the geometric mean. It keeps the pair's total tilt, it's the one cheapest move that does, and nobody outside the pair is touched. Next, and why: who carries the weight. With a bond running each way, both of you carry both bonds, so both of you get relief. But sometimes the two bonds are two people's own loads, and then somebody has to pick part of the weight up.

Step 3: Giving pays more than it costs

Start level and take on part of someone's load until you meet in the middle, and you carry a little more while they carry a lot less. Their relief is always more than three times what it costs you.

Everyone also has a bond of their own. It reads how heavy their load is, and 1 means level. Now the same friend is moving apartments, alone. I'll draw her load as a bond at 4. Yours is level, at 1. You show up with a van.

When you take part of her load onto yours, her bond comes down a doubling and yours goes up a doubling. They meet at 2 and 2. It's the same move as Step 2, and the two bonds' combined cost falls by the same 5/8, from 9/8 to 1/2. But now each bond belongs to one person, and each person carries only their own. So who carries what has changed.

You carried nothing, and now you carry 1/4. She carried 9/8, and now she carries 1/4. Carol, three towns over, carries what she always carried. Her relief is 7/8, and your added cost is 1/4. 7/8 divided by 1/4 is exactly three and a half.

Giving isn't free. You now carry 1/4 you didn't carry before. But her relief arrives at three and a half times the price.

Is that the luck of the number 4? No. Try a heavier load.

The pattern is exact. Take what you carry, double it, and add 3. Multiply that by what you carry and you have her relief. So her relief is always more than three times your cost, and the heavier the load, the better the ratio, though your bill grows too.³

Zoom out. Step 3 is done. Starting level, when you take on part of someone's load and meet in the middle, you carry the cost of your new number, and they're relieved by more than three times that. Next, and why: the opposite move. If easing a lopsided pair pays this well, what does it cost to push a level pair apart?

Step 4: Taking is surcharged

Push a level pair apart and a pair that cost nothing starts to cost something. The bigger the push, the bigger the bill.

In the Ledger, taking is a move that tips a level pair apart. Start with two level bonds, 1 and 1. The pair costs 0. Now take: push one bond up a doubling and the other down a halving. They read 2 and 1/2. The total tilt is +1 and -1, which is still 0. Nothing made, nothing lost.

The pair now costs 1/4 + 1/4 = 1/2. From nothing, to 1/2.

Push harder and it climbs. Push by a factor of 3, to 3 and 1/3, and the pair costs 4/3. Push by a factor of 4, to 4 and 1/4, and it costs 9/4.

It's love run backward. Love took 4 and 1 to 2 and 2 and saved 5/8. Push 2 and 2 apart to 4 and 1 and it costs 5/8 right back.

Yeah-But has been waiting for this one. "Yeah, but if I just take, I come out ahead."

Nothing in the Ledger stops you. The cost doesn't make anyone kind. People live under this same cost and still take. What the Ledger does is write down the price: 1/2 for a push of that size, and a bond's cost is carried by both of its ends. "Noted. 1/2, please."

Zoom out. Step 4 is done. Level is the only place that costs nothing, every step away from it costs, and a push apart is a step away. Next, and why: a limit on Step 3. Giving relieves three and a half times what it costs, but you're one person with a reserve, not a bottomless supply.

Step 5: Love goes as far as you can carry

Love goes to the middle if both of you can carry it. If you can't, it stops at the closest point to the middle that you can both carry and still keep going.

Step 3 left something out: how long you can keep it up. Each of us has a reserve, the capacity we can spend. I'll draw it as money in the bank. The bank is a drawing, and the reserve is real. Time passes in steps, and I'll call each step a day. Every day, the cost of the bonds you carry comes out of your reserve, and your budget, the amount that comes back each day, tops it up. That's the rule this example runs on.

If your budget covers what you carry, your reserve never falls and you keep going. If it doesn't, your reserve runs down a little each day, and when it reaches zero you can't keep carrying it.

Say your budget is 1/12 a day and hers is 2/3. Those numbers are chosen so the arithmetic comes out tidy. Each of you starts with 1 in the bank.

Look at the middle row. The pair's total cost went down, and you're sinking. You carry 1/4 a day and only 1/12 comes back, so you lose 1/6 a day, and a reserve of 1 is gone on day 6.

So here, love can't go all the way to the middle. It goes where both of you can keep going. Your budget holds a bond up to 3/2, because the price list says 3/2 costs 1/12. Hers holds a bond up to 3, because 3 costs 2/3. Walk your bond up from 1 toward 2 and stop at 3/2, your limit. Her bond has come down from 4 to 8/3, since 3/2 times 8/3 is still 4, and 8/3 is inside her limit of 3. She carries 25/48 a day, comfortably under her 2/3. (Try the recipe: 8/3 plus 3/8 is 73/24, half of that is 73/48, and minus 1 is 25/48.) You carry exactly your 1/12.

It keeps most of the benefit. The pair's cost falls from 9/8 to 29/48, which is five-sixths of the full saving of 5/8, and you carry a third of what the full middle would have asked of you.

So the rule is this. Meet in the middle if both of you can carry it. If you can't, go as close to the middle as both of you can carry. Of all the ways to meet that you can both hold, that one is the single cheapest. And if any way of meeting lets both of you keep going, that one does.⁴ If no way of meeting works for both of you, no move inside the pair lets you both keep going. Swap the two budgets, so yours is 2/3 and hers is 1/12, and the answer swaps with them: you end up carrying 25/48 and she carries 1/12. Love has no fixed fraction to give. Capacity decides.

Gary's idea of helping is to drum double time for the whole band. The band sounds great. Gary collapses at the bridge. The total went down, and Gary's reserve went to zero. The total looks great. Gary does not.

The best help is as close to the middle as both of you can carry and still keep going.

Zoom out. Step 5 is done. Love goes to the middle, or as close to it as both of you can carry. That's every step. Next: the whole trip, in one breath.

The whole trip, in one breath

Draw the weight at the airport as a lopsided bond and it has a size. Every bond between two people carries a number, 1 is level, and every other number costs, by one exact formula: 2 costs 1/4, 4 costs 9/8, and there's no top. Love is the move that brings two lopsided bonds to their geometric mean, so 4 and 1 meet at 2 and 2. Of every way to ease a pair while keeping its total tilt, it's the single cheapest, and nobody outside the pair is touched. Starting level, taking on part of someone's load until you meet in the middle relieves them by more than three times what it costs you. Pushing a level pair apart costs, and the cost grows with the push. And love goes to the middle, or as close to it as both of you can carry and still keep going.

We named the move. The arithmetic was already there. A lopsided bond has a size, a price, one cheapest repair and a limit, and every one of them is a number you can check by hand. That is how the bonds in reality's own record work.

(The Ledger would like you to know that it noted the airport. It notes everything. It has you three rides behind. Don't take it personally.)