Learning path
When Groups Decide Badly
Every person acts sensibly and the group still ends up somewhere none of them wanted. Five results showing that this is not a failure of character but a structural feature of putting preferences together.
The Prisoner's Dilemma
The atom of the whole subject. Two people, two choices, and a stable outcome that both of them dislike.
Two suspects are questioned in separate rooms. Each is offered a deal: betray the other and walk free. Each reasons impeccably, each betrays, and both end up worse off than if neither had said a word.
BeginnerCounterintuitive result18 min
The Tragedy of the Commons
The prisoner's dilemma with a field and many players — and, importantly, the entry where you learn why the pessimistic version of the story turned out to be empirically wrong.
A shared pasture, and every herder gains by adding one more animal while the cost of the extra grazing falls on everybody. Garrett Hardin called the ruin inevitable. Elinor Ostrom spent thirty years documenting communities where it was not.
IntermediateCounterintuitive result15 min
Braess's Paradox
The same structure in traffic, with a twist that closing a road can make everyone's journey faster. Cities have tested this and it works.
Open a fast new road between two points and every driver's journey gets longer. Close a busy street and the traffic speeds up. Dietrich Braess proved in 1968 that this is not bad luck — it is what selfish route choice does.
AdvancedCounterintuitive result19 min
The Condorcet Paradox
Now the problem moves from incentives to arithmetic. Majority preference can go round in a circle even when every individual is perfectly consistent.
Three people vote. A majority prefers A to B, a majority prefers B to C, and a majority prefers C to A. The group has no favourite at all — and whoever sets the order of the votes decides the winner.
BeginnerCounterintuitive result17 min
Arrow's Impossibility Theorem
Condorcet's cycle turned into a theorem. No ranked voting system can satisfy a short list of obviously reasonable conditions. This is where the path stops being about behaviour and starts being about mathematics.
Write down four things any fair voting system should obviously do. Arrow proved in 1951 that with three or more options, no method of combining ranked preferences can do all four — except one that always follows a single person.
ExpertImpossibility result25 min
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