📍

You have to meet a stranger in New York tomorrow. No phones, no plan, no way to talk. You each have to pick a place and a time and simply show up. Where do you go?

Schelling and focal points

A game can have several equilibria that pay exactly the same, and then the theory shrugs: it cannot say which one you will land on. Yet people coordinate anyway, and they do it well. The tie-breaker is not in the payoffs. It is in what stands out.

From earlier lessons

Two quick ones before we start.

In Battle of the Sexes, both attend Opera. If one switches to Football alone, what changes?

Battle of the Sexes has how many pure Nash equilibria?

Predict, then play

Here is the game stripped bare. You and a stranger each call Heads or Tails, with no way to talk. You both score only if you match. The two of you have no reason to prefer one over the other, so what do you do?

Most people say Heads, so most people match. Nothing in the payoffs points there. The word just comes first.

Why did that happen?

Take a guess, then reveal the reason

You did not reason about payoffs, because the payoffs were a tie. You reasoned about the other person: what will look obvious to them? Heads comes before Tails, so it feels like the natural call, and because it feels that way to almost everyone, choosing it is the smart move. A shared sense of what stands out does the coordinating that the payoffs cannot.

The math

A pure coordination game is a game where the players’ interests line up perfectly, but there is more than one way to line up. In the 2x2 case both matching cells are Nash equilibria and both pay the same:

u_1(H,H) = u_1(T,T) > u_1(H,T) = u_1(T,H),

and the same for player 2. Nash equilibrium narrows the answer to the two matching cells. It cannot go further, because the two are identical in every payoff. Any theory that looks only at payoffs is stuck at a tie.

Definition, focal point

A focal point (or Schelling point) is an equilibrium that players tend to select because something outside the payoffs, a label, a tradition, a roundness, a shared frame, makes it stand out as the obvious choice. It coordinates expectations without changing a single payoff.

The lesson generalizes. Meeting in New York, Schelling’s respondents converged on Grand Central at noon. Asked to split $100 by matching, people say 50/50. Asked to name a positive number, many say 1. In each case salience, not optimization, picks the equilibrium.

Work it out

Why theory cannot break the tie

Write the coordination game: matching pays $(1,1)$ , mismatching pays $(0,0)$ . Mark best responses. If the other player picks Heads, your best reply is Heads; if they pick Tails, yours is Tails. So $(H,H)$ and $(T,T)$ are both Nash equilibria.

Now try to rank them. Every payoff in $(H,H)$ equals the matching payoff in $(T,T)$ , so no efficiency or fairness argument distinguishes them. The selection has to come from outside the game: which label feels obvious. That is the focal point, and it is a rational response to needing to match, not a lapse in reasoning.

Key idea

When equilibria are tied on payoffs, players coordinate on whatever stands out. The focal point lives outside the math, but choosing it is the rational thing to do.

Check yourself

Check yourselfapply

Two drivers approach on a narrow road and must each pick a side. Both pass safely only if they choose opposite sides, and both sides are equally good. Why do they manage it?

Check yourselfanalyse

A friend says focal points prove people are irrational, since they pick on hunches instead of payoffs. What is the flaw?

Check yourselfbrings back: nash equilibrium

How many Nash equilibria does the pure Heads-or-Tails coordination game have, and what selects among them?

Sources

Schelling, T. C. (1960). The Strategy of Conflict. Harvard University Press.
Mehta, J., Starmer, C., and Sugden, R. (1994). The nature of salience: an experimental investigation of pure coordination games. American Economic Review, 84(3), 658-673.