Your friends matter. So do their friends.

In 2016, Facebook calculated that the typical pair of its users was separated by just 3.57 people, down from 3.74 in 2011.

Network economics studies how the pattern of connections between people shapes what they gain, and which connections they choose to make.

Some people sit in the middle of networks, linking everyone else together.

But is it worth being the person in the middle?

Three students, Asha (A), Ben (B) and Cara (C), have just arrived at university.

Two rules decide which friendships exist:

We’ll call each possible pattern of friendships a network, including the one where nobody is friends with anybody.

Q1: How many networks?

How many different patterns of friendship are possible among the three students?

How many genuinely different shapes are there, if we don’t care who is who?

Q2: Which network is best for the group?

Work out each student’s payoff in each shape of network.

Which shape gives the highest total payoff? In that network, who does best, and who does worst?

Q3: Will it actually happen?

Take the network you chose in Q2.

Would any student want to end one of their friendships? Would any two students who aren’t friends want to become friends?

Is the network that’s best for the group also stable when friendships are formed or ended by individual choice?

Q4: Friendship gets more expensive

Now suppose each friendship costs each friend 0.55 instead of 0.4. The benefits are unchanged.

Which network is now best for the group? Which network will the students actually end up in? What has gone wrong?

Q5: A bigger group

Now there are n students. Each direct friend is worth δ, each friend-of-a-friend is worth δ² (where 0 < δ < 1), and each friendship costs each friend c. (Notice that 0.25 = 0.5².)

Compare two networks:

Find the total payoff in each network. Show that the star gives a higher total payoff than the complete network if and only if c > δ − δ².

What is the highest cost of friendship at which the complete network could ever be the best for the group?

What is the smallest fee that keeps the hub willing to stay friends with everyone? Can the students on the edge afford it?

Can you think of real-world hubs that get paid like this? And if we ran this with real students in a classroom, would they form the star?

P.S. Roll up your sleeves and explore the questions. Once you’ve grappled with them for a while, if you’re still stuck, there are graduated hints below, but challenge yourself first! (The answer comes in the next issue. Subscribe to get it straight to your inbox.)

That's the proportion of available airline seat capacity filled by paying passengers worldwide in 2025, according to the International Air Transport Association (IATA). A record high.

The economics is straightforward. Flying a plane involves substantial fixed costs: the aircraft, crew, airport charges and maintenance. Once a flight is scheduled, the marginal cost of carrying one more passenger is relatively small. An empty seat represents revenue that can never be recovered once the plane takes off.

But there's another reason airlines can fill their planes: networks.

Instead of flying passengers directly between every possible pair of cities, airlines often use a hub-and-spoke system. Passengers from different destinations converge at a central airport before connecting onwards.

This creates economies of density. By combining passengers from several routes onto the same flight, airlines can fill more seats, spread costs across more customers and make routes viable that might otherwise be unprofitable.

But hubs also create dependencies. Delays at a major airport can ripple through the entire network, while airlines and passengers become increasingly reliant on a relatively small number of connections.

The lesson? The value of a network isn't just in the connections it creates. It's also in how efficiently those connections are used.

Last time, a town of 90 voters had to choose between a park (A), a library (B) and a sports centre (C), and we asked: can the majority be wrong?

Count the votes in each pairing:

So a majority prefers A to B, a majority prefers B to C, and a majority prefers C to A. The town's preferences go round in a circle: A beats B, B beats C, C beats A.

There is no clear majority preference. Every option loses to another option by 60 votes to 30.

What has gone wrong? Not the voters. Each individual has perfectly consistent preferences: someone who prefers A to B and B to C also prefers A to C. The inconsistency appears only when we add the votes together.

This is the Condorcet paradox, identified by the French mathematician the Marquis de Condorcet in 1785: individuals can have consistent preferences while the majority's preferences cycle.

Try each possible order:

The option left out of the first vote always wins, because whichever option survives the first round then loses to it.

So, as long as everyone votes sincerely, the mayor can determine the winner. Without changing anyone's preferences, they can deliver any of the three outcomes simply by choosing the order.

Controlling the agenda isn't the same as controlling the votes, but here it's just as powerful. That's why who sets the agenda matters so much in real politics: committee chairs influence which proposals are considered, the Speaker of the House of Commons selects amendments, and whoever designs a referendum determines which alternatives appear on the ballot.

With A against B first, sincere voting gives:

C wins: the worst possible result for your group (A > B > C).

Now suppose your group votes for B in the first round:

B wins: your second choice instead of your last. By voting against your first preference, your group gets a better result.

How many of you need to switch? If x members switch to B, B gets 30 + x votes and A gets 60 − x. B wins once x > 15, so at least 16 of the 30 must switch (15 produces a 45–45 tie).

This is called strategic (or tactical) voting. It works by thinking ahead: work out what will happen in the final, then vote in the first round to shape it, just as Country A did in #014 by thinking about how the firms would respond.

Does strategic voting make majority voting more or less representative? Less, in one important sense: votes no longer necessarily reveal people's genuine first preferences. The outcome now depends on who has information, who coordinates, and who thinks ahead. Here, B wins, but there's no single option that the town's majority prefers to every alternative.

And this isn't a quirk of our example. The Gibbard–Satterthwaite theorem shows that strategic manipulation is unavoidable in a broad class of voting systems with three or more options. Tactical voting in real elections illustrates the same general incentive: people may support a less-preferred candidate to prevent an even less-desired outcome.

Expected benefit of voting:

(1 / 10,000,000) × £1,000,000 = £0.10

Cost of voting: £10.

On these numbers, voting costs 100 times more than its expected benefit. A purely self-interested voter wouldn't bother. This is the paradox of voting, associated with the economist Anthony Downs: if voting is so unlikely to make a difference, why does anyone vote?

So why do millions vote?

So is not voting irrational? Not necessarily. For someone who expects no personal benefit and feels no duty, staying at home can be a reasonable response to the incentives.

But turnout differs systematically between groups. Younger and older voters, for example, may participate at different rates. Politicians seeking electoral support have an incentive to pay attention to groups that reliably turn out. This can influence which policy concerns receive political attention.

The policy is a net loss to society: benefits of £10 million, costs of £20 million. So why might it pass?

Look at the benefits and costs per person:

So the small group has a strong incentive to organise, while the large group may remain silent. This is the logic of collective action, associated with the economist Mancur Olson: small groups with a lot at stake per member can find it easier to organise than large groups whose members each have little at stake, especially when everyone hopes someone else will do the work (free riding).

Taxpayers may not even know about the policy. When finding out costs time and the personal stakes are tiny, it can be rational to remain uninformed: economists call this rational ignorance.

The incentives may line up:

So democracy doesn't automatically choose policies that maximise social welfare. Just as markets can fail, so can governments: economists call this government failure. Agricultural subsidies and protective tariffs are often examined through this lens (as are the strategic subsidies in #014).

Try some alternatives on our town:

So is there a perfect system? No, and there's a theorem explaining why. In 1951, the economist Kenneth Arrow proved that with three or more options, no system based on rankings can satisfy a particular set of reasonable fairness conditions and always produce a consistent social ranking, unless it gives decisive power to one individual: a dictator.

This is Arrow's impossibility theorem. Arrow subsequently received the Nobel Prize in Economics in 1972.

There is an important qualification. If preferences are “single-peaked” — for example, if everyone ranks options along one scale, such as spending from low to high, and prefers options closer to their favourite — majority voting produces consistent comparisons, and the median voter's favourite wins.

Our town's preferences aren't like that: no single ordering of the park, library and sports centre makes all three groups' rankings single-peaked.

The majority can be inconsistent. Even when every voter has perfectly consistent preferences, majority preferences can go round in circles, so the outcome may depend on who sets the agenda, who votes strategically, who turns up, and who lobbies.

“Let the people decide” is a powerful idea. But how the people decide can matter as much as what they want.

Gaming the Vote (William Poundstone, 2008): a lively, readable account of why voting systems can produce surprising results, including the Condorcet paradox and Arrow's impossibility theorem.

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