Cambridge Economics Interview: The Ultimatum Game

Economics

Matin K. · Economics (Cambridge)

Hi, I'm Matin, and I'm a second-year Economics student at Cambridge. I want to walk through a question you might meet in an Oxbridge economics interview. It sounds simple and has a neat textbook answer, but the interesting part is what happens when you test that answer against real people.

The question

I give you £100, and you must offer part of it to someone else. If they reject your offer, neither of you gets anything. How much do you offer?

If you've done some game theory, you'll recognise this as the ultimatum game. If you haven't, don't worry: careful reading and common sense will get you most of the way. The core tension is reward against risk. Every pound you offer is a pound you lose, but every pound you hold back makes a rejection more likely.

Setting up the model

The standard starting point is rational choice theory: both players are perfectly rational and care only about their own money. You also need a smallest unit of money. I'll use a penny, though an interviewer might ask you to work in whole pounds.

You move first and they respond, so this is a dynamic game, and I draw it as a tree. At the top is your decision, an offer x anywhere from 1p to £100, which I draw as a cone of possible offers. Below each offer, the other player either accepts, giving you £100 minus x and them x, or rejects, leaving you both with nothing.

Solving it backwards

Dynamic games are solved by working backwards, so start with the responder. If they care only about money, any positive amount beats nothing, so they accept any offer above zero. Knowing that, you offer the smallest amount possible: one penny. You keep £99.99.

That is a subgame perfect Nash equilibrium, also called a backward-induction equilibrium. Working backwards matters because it filters out empty threats. A responder might announce "I'll reject anything under £50", but once £20 is actually on the table, rejecting it means choosing nothing over £20. A purely money-driven responder would never carry the threat out.

Notice how much the small assumptions matter. At an offer of exactly zero the responder is indifferent, which is why the safe answer is the smallest positive amount rather than nothing. Stating details like this out loud is exactly what interviewers want to hear.

Why real people don't play it that way

You could stop there, but imagine doing it for real. Would you hand a stranger 1p and keep £99.99? Most people wouldn't, and the evidence says they'd be right. In experiments, average offers are typically 40 to 50 per cent of the pot, the most common offer among students is an even split, and offers of 20 per cent are rejected about half the time.

A responder who rejects £20 is paying £20 to make sure you get nothing, so the "empty" threat turns out to be real. The model's logic is fine, which means one of its assumptions must be wrong.

The assumption that breaks is not rationality but selfishness. Rationality means pursuing your goals consistently; selfishness is a claim about what those goals are. Economists call goals beyond your own payoff social preferences, and two kinds explain this game well:

• inequity aversion: people care how their share compares with yours, not just how big it is, so an insulting split can be worth rejecting (Ernst Fehr and Klaus Schmidt's well-known model formalises this)
• reciprocity: people expect a basic norm of fairness, close to 50:50, and will punish you for breaking it, even at a cost to themselves

Build these into the responder's payoff and rejecting a low offer becomes perfectly rational. It changes your side of the game too. You can't tell whether the other player has these preferences, but you know they might, so even a coldly selfish proposer should offer more to cut the risk of rejection. Fairness and fear can produce exactly the same generous offer.

How it would actually run in the room

A question like this usually starts simple and then develops. The interviewer might extend the game, bring in experimental evidence, or move you on to related games. They rarely do all of these, but expect at least one.

A natural follow-up is the dictator game, where the other player has no veto and must accept whatever you give. It separates fairness from fear. Among students, the most common dictator offer is nothing at all, so fear was doing much of the work, yet average gifts still come to around 20 to 30 per cent, so fairness is real too.

Another is the pirate game, where five pirates vote on how to split 100 gold coins and a rejected proposer goes overboard. Solved backwards, the most senior pirate keeps 98 coins, buying the two votes he needs with one coin each. It's a striking example of how far backward induction can take you from intuition.

You might even be asked whether "fair" means the same thing everywhere. In a well-known study led by Joseph Henrich across 15 small-scale societies, the purely self-interested model failed in every one, but average offers ranged from 26 to 58 per cent. Societies whose daily life depended more on markets and cooperation tended to play more generously, which suggests fairness is a norm shaped by the world people live in.

What the interviewers are looking for

A few things earn credit here:

• reading the question carefully and working out what is actually being asked
• stating assumptions before solving, including the smallest unit of money
• solving the game cleanly by working backwards, and owning the conclusion even when it looks strange
• testing the answer against how real people behave
• questioning the assumptions rather than the logic, and explaining which one fails and why

Common mistakes to avoid

The usual slips are jumping straight to "offer half" without modelling the game, stopping at one penny as if the textbook answer were the whole story, dismissing real behaviour as irrational instead of asking which assumption failed, and taking the question at face value. It asked "How much do you offer?", not "How much do you offer to maximise your money?"

Final thoughts

The best answers build a simple model, follow it honestly to its conclusion, and then have the confidence to question it. If you'd genuinely split 50:50 because a lowball offer would make you feel bad, say so and explain why; it opens up the most interesting part of the problem. If you'd like help preparing for Cambridge economics interviews, Oxbridge Solution can help.