Coordination Game

What makes a game a coordination game

A coordination game is a non-cooperative game whose strategic structure has at least two pure-strategy Nash equilibria, and in each of them the players' actions are aligned. The simplest example is also the most concrete. Two drivers approach each other on an empty road. Each can drive on the left or on the right. If they pick the same side, both pass safely; if they pick different sides, they crash. The payoff matrix is symmetric and trivial: both prefer to match, neither cares which way they match. There are two pure-strategy Nash equilibria — (left, left) and (right, right) — and the formal theory has nothing to say about which one prevails.

The British drive on the left because medieval right-handed riders kept their sword-hand free for an oncoming stranger. The French (and most of the world) drive on the right after Napoleon. Neither equilibrium is intrinsically better. Both are stable. Which one a country uses is purely an accident of history that gets locked in once enough infrastructure — road markings, steering wheel placement, manufacturing tooling — has accumulated. This is the pattern coordination games keep reproducing across the economy: arbitrary equilibrium selection, followed by enormous switching costs that make a one-off historical choice durable.

Three canonical coordination games

Game theorists have isolated three canonical 2 × 2 coordination structures. They differ in how the equilibria compare to each other.

Game	Equilibria	Are they symmetric?	Tension
Pure coordination	Two, equally good	Yes	None — purely about agreement
Stag hunt	Two, one Pareto-dominates the other	No (efficient vs safe)	Trust vs risk
Battle of the sexes	Two, players prefer different ones	No (distributive)	Coordination + bargaining

The intellectual move when you encounter a strategic situation is to ask which of these three structures it most resembles. A traffic convention is pure coordination. A team production task is a stag hunt. A married couple choosing between football and ballet is the battle of the sexes. Once you know the structure, you know which equilibrium-selection theory applies.

Pure coordination: drive on the left, drive on the right

Pure coordination is the simplest case. Both equilibria yield identical payoffs, and there is no intrinsic reason to prefer one to the other. The 2 × 2 normal form is:

                          Driver B
                       Left      Right
              Left    (1, 1)    (0, 0)
   Driver A
              Right   (0, 0)    (1, 1)

Both (Left, Left) and (Right, Right) are Nash equilibria. Neither player wants to deviate alone, because deviating means a head-on collision. There is also a third, mixed-strategy equilibrium in which each player picks Left with probability 1/2 — but it gives expected payoff 1/2 to each player, worse than either pure equilibrium, and so it has no claim to be the outcome.

What real players do in this situation depends entirely on context outside the game. If they have a country, they use its convention. If they share language or culture, they exploit any focal cue. If they have neither, they may simply each pick the side that is locally most prominent — and that is exactly where Schelling's focal points enter.

Stag hunt: efficiency versus risk

Rousseau introduced the stag hunt in his 1755 Discourse on Inequality. Two hunters stalk a stag. The stag is valuable enough that two hunters working together can corner it; alone, neither can. Each hunter also has the option of dropping out to chase a hare — small, certain, and catchable alone. A canonical numerical version:

                          Hunter B
                        Stag      Hare
              Stag    (4, 4)    (0, 3)
   Hunter A
              Hare    (3, 0)    (3, 3)

Both (Stag, Stag) and (Hare, Hare) are Nash equilibria. (Stag, Stag) Pareto-dominates: both players prefer it to (Hare, Hare). But stag-hunting is risky in a sharp asymmetric way: if your partner peels off to chase a hare, you get zero, whereas the hare-hunter still gets three. So stag-hunting is a best response only if you trust your partner to also pick stag. Hare-hunting is a best response no matter what — it is the risk-dominant strategy.

Empirically, two-player stag hunts in the lab often coordinate on the payoff-dominant (Stag, Stag) outcome, especially with cheap-talk pre-play communication. Larger-group stag hunts tend to collapse to the inefficient (Hare, …) outcome: the more partners you need to trust simultaneously, the smaller the probability that all of them pick stag, and at some group size the risk-dominant strategy wins. This is the formal mechanism behind Robert Putnam's social-capital story, and it is why building functioning institutions is harder than it looks.

Battle of the sexes: coordination plus conflict

The battle of the sexes — modern textbooks use "battle of the partners" or just BoS — is the canonical mixed-motive coordination game. Two partners must choose between two activities (Schelling's original was a prizefight vs the ballet; modern versions use football vs ballet, restaurant A vs restaurant B). Both prefer being together to being apart, but each prefers a different activity:

                          Partner B
                      Football    Ballet
   Partner A    Football  (2, 1)   (0, 0)
                Ballet    (0, 0)   (1, 2)

(Football, Football) and (Ballet, Ballet) are both Nash equilibria, but they distribute payoffs unequally. There is a third, mixed-strategy equilibrium with worse expected payoffs for both. The interesting feature of BoS is that the equilibria are not interchangeable: each player would like the partner to concede. This makes BoS a hybrid of coordination and bargaining, and the relevant equilibrium selection mechanisms include commitment devices, social conventions, and asymmetric information about preferences. Real couples solve it with rules like "we alternate" or with one partner having a publicly known stronger preference.

Focal points: how players coordinate without communicating

The most important contribution to equilibrium selection is Thomas Schelling's notion of focal points, introduced in The Strategy of Conflict (1960). Schelling asked Yale undergraduates: You and a stranger must meet in New York tomorrow, but you cannot communicate. Where and when? A large majority answered "Grand Central Terminal at noon". The choice of location was overdetermined by cultural salience — Grand Central was, for that population at that time, the prominent meeting place. Noon was the obvious time because of its symmetry around 12. Neither was forced by any payoff calculation: the focal point lives in shared knowledge, not in the payoff matrix.

Focal points work for the same reason language works: they exploit shared salience to make one option uniquely conspicuous. Some focal points are cultural (Grand Central, Times Square on New Year's Eve), some are mathematical (round numbers, midpoints), some are linguistic (which list item is "first"). Once a focal point exists, it self-reinforces: people coordinate on it because they expect others to expect them to. It does not need any formal payoff support.

This is also why focal points are fragile. A coordination convention can vanish in a generation if the cultural cue that supported it disappears, and a new focal point can emerge unexpectedly. The shift from VHS to DVD was not driven by a payoff difference between equilibria — it was driven by retailers and movie studios converging on the new standard, which made it conspicuous to consumers, which converted the convention.

Risk dominance versus payoff dominance

Harsanyi and Selten, in their 1988 book A General Theory of Equilibrium Selection in Games, formalised two equilibrium selection refinements that frequently disagree.

• Payoff dominance. An equilibrium is payoff-dominant if it Pareto-dominates every other equilibrium — every player gets at least as much, and at least one gets more. The (Stag, Stag) outcome in the stag hunt is payoff-dominant.
• Risk dominance. An equilibrium is risk-dominant if it maximises the product of deviation losses (the Nash product). Equivalently, it is the equilibrium you would play if you assigned equal probability to your opponent picking each strategy. In the stag hunt with payoffs (4, 4) for stag and (3, 3) for hare, the (Hare, Hare) equilibrium is risk-dominant because the loss from miscoordinating (4 → 0) is much larger than the gain from coordinating on stag (3 → 4).

When the two refinements disagree — as they routinely do in stag-hunt structures — experimental subjects often pick the risk-dominant equilibrium. The evolutionary biology literature reaches the same conclusion: in stochastic best-response dynamics with mutation, the risk-dominant equilibrium has a larger basin of attraction and is selected in the long run. This is a surprisingly pessimistic result. Even when a Pareto-superior outcome is available, populations may converge to the worse equilibrium because it is safer to play under uncertainty. Coordination failure is not a marginal pathology; it is the default for risky cooperation games.

Coordination failures in the wild

Once you know the structure, you start seeing coordination games everywhere — and you start noticing the cases where the bad equilibrium prevails.

• Bank runs (Diamond-Dybvig, 1983). A bank funds long-term illiquid assets with short-term demand deposits. There are two pure-strategy equilibria: the good one in which depositors expect each other to wait and so no one withdraws early; and the bad one in which everyone expects everyone to run, so everyone races to the front of the queue. The bad equilibrium is self-fulfilling: the bank cannot honour all early demands, and so being first matters. Nothing about the bank's fundamentals determines which equilibrium occurs. Deposit insurance was designed precisely to eliminate the bad equilibrium.
• Currency attacks. A government with a fixed exchange rate has limited foreign reserves. If speculators believe the peg will hold, no one attacks and the peg holds. If they believe the peg will break, everyone attacks and the peg breaks. The classic models of Krugman (1979) and Obstfeld (1986) are coordination structures.
• Technology standards. VHS vs Betamax in the 1980s. Blu-ray vs HD-DVD in the 2000s. Once enough retailers, content owners and consumers tip toward one standard, the other dies — even if it is technically superior. The infamous arguments about whether Betamax was actually better than VHS are beside the point: in a coordination game, the better-engineered equilibrium does not necessarily win.
• QWERTY lock-in. The QWERTY layout was designed for mechanical typewriters in the 1870s and was not optimised for typing speed. Dvorak (1936) and Colemak (2006) are arguably superior. Yet QWERTY persists because the equilibrium "everyone learns QWERTY because keyboards and training are made for QWERTY" is locally stable: switching imposes individual cost without coordinating benefit. Liebowitz and Margolis (1990) argued the case for Dvorak's superiority is weaker than commonly claimed — but the structural point about lock-in does not depend on which alternative wins the empirical battle.
• Network effects in platforms. Social networks, payment systems, programming languages, file formats. Network goods are coordination goods almost by definition: their value to me increases with the number of other users. Once a platform tips, the alternative collapses; before it tips, both equilibria are live.

The common thread is that coordination failures are not market failures in the conventional sense — there is no missing market, no externality, no information asymmetry. They are failures of expectations to converge on the better of multiple stable points. Policy interventions for coordination problems therefore look different from interventions for ordinary externalities: deposit insurance, public commitments, focal-point engineering, sunset clauses.

Global games: forcing a unique equilibrium

The most influential modern attack on the multiplicity problem is the global-games approach, introduced by Carlsson and van Damme in 1993. Their idea: take a coordination game and perturb it by giving each player a private noisy signal about the true payoffs. Even with arbitrarily small noise, the players can no longer treat the payoff structure as common knowledge — and that breakdown of common knowledge has dramatic implications.

Iterated dominance, applied to strategies that condition on the private signal, eliminates all but one equilibrium. The unique equilibrium has a cut-off form: each player attacks the currency (or runs the bank, or invests in the new technology) if and only if their private signal exceeds a threshold. Morris and Shin (1998) used this technique to study currency crises: instead of a multiplicity of self-fulfilling crisis equilibria, the global-game refinement predicts a unique fundamentals threshold below which the currency falls.

The global-games approach has been criticised — its predictions depend on the structure of the private noise, which is hard to estimate empirically — but it transformed how applied theorists handle coordination problems. It made them tractable. You can now ask: what is the threshold? How does it move with policy? Are deposit insurance or capital requirements equivalent at reducing the probability of a crisis? These are quantitative questions, not just qualitative ones.

How equilibria are selected over time

Outside of one-shot experiments, coordination games are typically played repeatedly, by many overlapping pairs of agents, in a population that learns. The evolutionary game theory literature, starting from Maynard Smith and Price's 1973 paper, asks which equilibrium is selected when players adjust strategies in response to recent payoffs.

Kandori, Mailath and Rob (1993) and Young (1993) showed that in stochastic best-response dynamics with mutation — small random perturbations — the long-run distribution is dominated by the risk-dominant equilibrium. The mechanism is intuitive: the risk-dominant equilibrium has a larger basin of attraction in the best-response dynamics, so mutations are more likely to push the population into its basin than out of it. Over long time horizons, you wait at the risk-dominant outcome.

This result is one reason why historical lock-in is real. A small advantage of one equilibrium over another, accumulated over millions of small bilateral encounters and amplified by network effects, can dominate even large payoff differences. The QWERTY story, the VHS story, and the dollar-as-reserve-currency story are all instances of this same selection logic.

Coordination games versus other game-theoretic structures

Game type	Equilibria	Players' preferences	Canonical example
Pure coordination	Multiple, symmetric	Same direction	Drive left vs right
Stag hunt	Multiple, Pareto-ranked	Same direction, but risk	Joint hunting / team production
Battle of the sexes	Multiple, distributively asymmetric	Want to coordinate, disagree how	Couples choosing activities
Prisoner's dilemma	One, inefficient	Opposed (free-ride)	Arms race, public goods
Zero-sum	Mixed-strategy, unique value	Strictly opposed	Matching pennies, poker
Chicken (hawk-dove)	Two pure asymmetric + mixed	Coordination of who concedes	Crisis bargaining, animal contests

Coordination games sit between the cooperative end (pure coordination, where preferences align entirely) and the conflict end (zero-sum, where they oppose entirely). The stag hunt is interesting precisely because it occupies the boundary where the payoff structure rewards cooperation but the risk structure punishes unilateral cooperation.

Worked example: which equilibrium is risk-dominant?

Take a generic symmetric 2 × 2 coordination game:

                       B chooses
                       X          Y
              X      (a, a)     (b, c)
   A chooses
              Y      (c, b)     (d, d)

Suppose a > c (so X is a best response to X) and d > b (so Y is a best response to Y). Both (X, X) and (Y, Y) are Nash equilibria.

The Harsanyi-Selten risk-dominance condition picks (X, X) when

(a − c)(a − c) ≥ (d − b)(d − b)
  ⇔  a − c ≥ d − b
  ⇔  a + b ≥ c + d

That is, X is risk-dominant if its "row sum" (a + b) exceeds the row sum (c + d) of Y. Plugging in the stag hunt values a = 4, b = 0, c = 3, d = 3: a + b = 4, c + d = 6. So 4 < 6, meaning Y (Hare) is risk-dominant. The (Stag, Stag) equilibrium is payoff-dominant (a = 4 > d = 3) but the (Hare, Hare) equilibrium is risk-dominant. In experiments with this payoff structure, large fractions of subjects do indeed pick Hare, illustrating the gap between payoff dominance and the equilibrium people actually play.

The lesson generalises: whenever you set up a coordination problem with a tempting-but-risky efficient equilibrium and a safe-but-inefficient one, the population can get stuck at the safe one. This is the formal theoretical case for why coordination is harder than the payoff structure alone suggests.

Variants and refinements

• Pre-play communication (cheap talk). If players can talk before they choose, coordination on the payoff-dominant equilibrium becomes much more likely (Cooper, DeJong, Forsythe and Ross 1989). Cheap talk has no formal commitment value but works because it creates a focal point.
• Repeated play and learning. Subjects in repeated stag-hunt experiments converge over time, often to the risk-dominant equilibrium in larger groups (Van Huyck, Battalio and Beil 1990 — the canonical "minimum effort" experiment).
• Heterogeneous types. If players have different preferences and some of these differences are publicly known, coordination on a particular equilibrium is easier — the type structure is itself a focal cue.
• Local interaction networks. When players interact only with neighbours in a network, the risk-dominant equilibrium can spread contagiously through the network even from a small seed (Morris 2000). Network structure matters for which equilibrium wins.
• Quantal response equilibrium (McKelvey-Palfrey 1995). Players make small mistakes in best-responding, which smooths the equilibrium correspondence and breaks ties in favour of robust equilibria.

Common pitfalls

• Confusing coordination with cooperation. The prisoner's dilemma is a cooperation problem — defection is dominant. The stag hunt is a coordination problem — cooperation is a best response to cooperation. The fix for a coordination failure is to align expectations; the fix for a cooperation failure is to change payoffs (punishment, repeated play, side contracts).
• Treating multiplicity as a flaw. Multiple equilibria are not a defect of the model. They are a feature of the underlying strategic situation — multiplicity is exactly what makes path dependence, focal points, and lock-in possible.
• Ignoring mixed-strategy equilibria. Almost every coordination game has a mixed-strategy equilibrium in addition to the pure ones. The mixed equilibrium is usually unstable and Pareto-dominated, but it shows up in dynamic models and as a stepping stone between pure-strategy basins.
• Reading focal points off the payoff matrix. Focal points exploit information outside the formal game — culture, history, salience. A "rational" agent staring only at the payoff matrix cannot identify them. Modelling them requires bringing in context.
• Assuming the payoff-dominant equilibrium wins. Real subjects, especially under uncertainty or in large groups, often choose the risk-dominant equilibrium. Predictions that assume Pareto-dominant outcomes are systematically optimistic.
• Confusing the global-games refinement with an empirical claim. Global games select a unique equilibrium given a particular structure of private noise. The noise structure matters and is often unobserved; treating the global-games threshold as a hard prediction is over-confident.