Nash equilibrium and coordination

A merchant, two QR codes, and no dominant move

Walk into a small shop in Nairobi, Accra or Kampala and you will often find two or three payment stickers taped to the counter — one for M-Pesa, one for MTN Mobile Money, perhaps one for a bank wallet — each with its own QR code. A customer on the wrong network cannot scan the wrong sticker, so the merchant hedges by displaying them all. Everyone here — merchant, customer, the operators, the central bank above them — would gain if a single QR standard let any wallet pay any merchant, yet the fragmented counter persists. In the previous module you solved games by dominance: delete the strategies no rational player would use and read off what survives. That logic says nothing here, because no operator has a dominant strategy — the best QR format for one firm depends entirely on the format the others pick. When your optimal move tracks everyone else's, you need a solution concept built for interdependence. That concept is the Nash equilibrium, and coordination problems like this counter are where it earns its keep.

The best-response correspondence

Fix a finite game: players i ∈ {1, …, n}, a finite strategy set Sᵢ for each, and a payoff function uᵢ giving player i a real number at every strategy profile. Write a profile as s = (sᵢ, s₋ᵢ), separating player i's own strategy sᵢ from the strategies s₋ᵢ of everyone else. Player i's best-response correspondence collects the strategies that maximise her payoff against a given s₋ᵢ: BRᵢ(s₋ᵢ) = { sᵢ ∈ Sᵢ : uᵢ(sᵢ, s₋ᵢ) ≥ uᵢ(sᵢ′, s₋ᵢ) for every sᵢ′ ∈ Sᵢ }. It is a correspondence, not a function, because ties are allowed: against some s₋ᵢ two strategies may both maximise your payoff, so BRᵢ returns a set rather than a single point. When the maximiser is unique, that set is a singleton.

Nash equilibrium: a profile of mutual best responses

A pure-strategy Nash equilibrium is a profile s* = (s₁*, …, sₙ*) in which every player is simultaneously best-responding to everyone else: sᵢ* ∈ BRᵢ(s*₋ᵢ) for every i. Equivalently, for each player i and every alternative sᵢ ∈ Sᵢ, uᵢ(sᵢ*, s*₋ᵢ) ≥ uᵢ(sᵢ, s*₋ᵢ). Read the condition slowly: no player can raise her own payoff by changing her strategy alone, holding the others fixed. An equilibrium is thus a mutually consistent set of plans — a rest point at which everyone's expectations about everyone else are confirmed and no one regrets her choice. This generalises the dominance reasoning of the previous module rather than replacing it. If every player has a strictly dominant strategy, the profile in which each plays it is the unique Nash equilibrium, since a dominant strategy is a best response to anything the others might do. More generally, any strategy used in a Nash equilibrium survives iterated elimination of strictly dominated strategies. But the converse fails: a game can survive elimination with many strategies intact and pin down its stable outcomes only through the equilibrium condition. Nash equilibrium is the weaker, more widely applicable requirement — it asks for mutual consistency, not for strategies that are good regardless of what others do.

Finding every pure equilibrium by underlining

For a two-player game written as a bimatrix — the row player picks a row, the column player a column, and each cell lists (row payoff, column payoff) — a mechanical procedure finds every pure-strategy equilibrium at once. The idea is to mark, or underline, each player's best responses and see where they coincide.

1. Go down each column and underline the row player's largest payoff in that column: that row is the row player's best response to that column.
2. Go across each row and underline the column player's largest payoff in that row: that column is the column player's best response to that row.
3. Any cell in which both payoffs are underlined is a pure-strategy Nash equilibrium — each player is best-responding to the other. Cells with one underline or none are not equilibria.

Take the Stag Hunt, the classic parable of cooperation under risk. Two hunters can jointly track a stag — a large prize that needs both of them — or separately snare a hare, which either can catch alone. If one hunts stag while the other slips off after the hare, the stag escapes and the lone stag-hunter goes home empty-handed. One natural set of payoffs, in illustrative units, is as follows.

                    Stag         Hare
Stag                4, 4         0, 3
Hare                3, 0         3, 3

Underline the best responses. In the Stag column the row player compares 4 against 3 and prefers Stag, so underline the 4; in the Hare column she compares 0 against 3 and prefers Hare, so underline the 3. By symmetry the column player underlines the same diagonal payoffs. Two cells end up with both payoffs underlined, so there are two pure-strategy equilibria, (Stag, Stag) and (Hare, Hare); the off-diagonal profiles are not equilibria, since whoever hunts stag alone earns 0 and would switch. The two equilibria differ along distinct axes. (Stag, Stag) is payoff-dominant: it gives both players strictly more than any other equilibrium, 4 against 3. Yet (Hare, Hare) is risk-dominant, the safer choice when you are unsure of your partner. To see this, let p be the probability you assign to your partner hunting stag; hunting stag is your best reply exactly when p·4 + (1 − p)·0 ≥ p·3 + (1 − p)·3, which reduces to 4p ≥ 3, i.e. p ≥ 3/4. You must be at least 75% confident your partner will commit before you dare; at the neutral belief p = 1/2 you take the hare. Equivalently, the loss from a unilateral deviation is larger at (Hare, Hare), 3 − 0 = 3, than at (Stag, Stag), 4 − 3 = 1, and the larger-loss equilibrium is the risk-dominant one. Strategic risk, not the size of the prize, keeps cautious hunters on the hare.

Existence, multiplicity, and why culture enters

The Stag Hunt already shows the two features that make equilibrium analysis subtle. The first is existence: does a game even have one? John Nash proved in 1950 that every finite game — any finite number of players, each with finitely many strategies — has at least one equilibrium, provided we allow mixed strategies, in which a player randomises over her pure strategies with fixed probabilities. Pure-strategy equilibria can fail to exist — think of a penalty kick, where any predictable choice is punished — but a mixed equilibrium always does. We treat mixing formally in a later module; for now, read Nash's theorem as a licence to always look for equilibrium, and a reminder that a game with no pure equilibrium is not broken, merely one whose equilibrium is randomised.

The second feature is multiplicity: the Stag Hunt has two pure equilibria, and nothing in the definition ranks them as a prediction. This is not a flaw in the example. A coordination game — any game whose players want to align their choices, whether or not they agree on which alignment is best — generically has several strict equilibria. Africa's integration agenda is thick with them. Should East Africa's new railways run on standard gauge or on the older metre gauge inherited from colonial lines, so that rolling stock crosses borders unbroken? Should the African Union's continental business run in English, French or Kiswahili — adopted in 2022 as an official working language of the AU and long the lingua franca of the East African Community? In each case any common choice beats fragmentation, several common choices are equilibria, and equilibrium analysis alone cannot say which prevails. Something outside the payoff matrix must select.

Extended example: a shared merchant QR standard

Return to the counter and model it as two mobile-money operators — picture a market with an incumbent like Safaricom's M-Pesa and a challenger like MTN Mobile Money — each choosing which QR standard to deploy at merchants. Operator A has built Format A; Operator B has built Format B. Both strongly prefer a single interoperable standard to today's fragmentation, because interoperability enlarges the whole network: a merchant reachable by every wallet is worth far more than one reachable by half. But each would rather the common standard be its own format, sparing itself migration costs and keeping its technology at the centre. That is exactly the Battle of the Sexes — shared interest in coordinating, opposed interests over which equilibrium to coordinate on.

                    B: Format A     B: Format B
A: Format A         3, 2            0, 0
A: Format B         0, 0            2, 3

Underline the best responses. If B deploys Format A, A earns 3 by matching against 0 by clashing, so underline A's 3; if B deploys Format B, A earns 2 by matching against 0, so underline A's 2. The column operator's best responses fall on the same diagonal, so both diagonal cells carry two underlines. There are two pure-strategy equilibria — (Format A, Format A) and (Format B, Format B) — and they conflict: A prefers the first (3 > 2), B the second (3 > 2). Both dominate the off-diagonal mismatches, where incompatible standards strand customers and each earns 0. That mismatch is the fragmented counter: not an accident but a coordination failure, a non-equilibrium profile that lingers because neither firm will unilaterally abandon its format without assurance the other will meet it on a shared one. By Nash's theorem there is also a third, mixed equilibrium in which each operator randomises; it carries a positive probability of costly mismatch and pays less than either coordinated outcome. Its lesson, even before we compute it, is that an unmanaged market can hover in recurrent miscoordination.

Focal points: how the tie gets broken

If equilibrium analysis cannot choose between (Format A, Format A) and (Format B, Format B), what does? Thomas Schelling's answer, from The Strategy of Conflict (1960), is the focal point: among multiple equilibria, players converge on the one made salient by something they share — a convention, a precedent, a label, a common culture, an authority everyone is watching. Salience does the work payoffs cannot, because coordination requires not merely that you pick an equilibrium but that you pick the same one the others expect, and expectations settle on whatever is conspicuous. This is how QR fragmentation is actually resolved across the continent. In Ghana, GhIPSS — the central-bank-owned payment-system operator — launched GhQR, a single universal QR standard that every bank and wallet is directed to adopt; the regulator manufactures a focal point the operators could not reach alone. Nigeria's NIBSS did the same with NQR, and the African Union's Pan-African Payment and Settlement System plays the analogous role across borders. A regulator's economic function here is not to compute the efficient standard but to make one standard the obvious expectation, tipping every operator's best response onto the same equilibrium. Where no regulator acts, culture supplies the focal point: across much of West Africa, rotating market calendars — the Igbo four-day cycle of Eke, Orie, Afọr and Nkwọ, and the staggered market days that let neighbouring towns share traders without clashing — are coordination equilibria selected and stabilised by custom over generations. The market meets on the customary day because it always has, and everyone knows that everyone knows it.

What equilibrium does and does not deliver

Equilibrium is a consistency condition, not an optimality one, and three caveats follow. First, multiplicity means the theory often underdetermines the outcome: locating the equilibria is the analysis, but predicting which one is reached needs the extra-strategic information a focal point supplies. Second, the selected equilibrium need not be efficient — a coordination game can settle on the risk-dominant but payoff-inferior outcome, which is why beneficial upgrades such as a common rail gauge or a shared payment standard stall even when everyone agrees they would help. Third, conflict over which equilibrium can itself block coordination. Kenya's LAPSSET corridor — the Lamu Port–South Sudan–Ethiopia Transport project — is a Battle of the Sexes at national scale: neighbouring regions and states all gain from a shared corridor, yet each prefers the alignment that routes the ports, pipeline and railway through its own territory, and it is that distributional conflict, not any doubt about the value of a corridor, that slows agreement.

Exercises

                 F2: X       F2: Y       F2: Z
F1: X            4, 3        3, 0        0, 0
F1: Y            0, 0        2, 2        0, 0
F1: Z            0, 0        0, 3        3, 4

                 A2: Standard    A2: Legacy
A1: Standard     6, 6            0, 5
A1: Legacy       5, 0            4, 4