The reading that is almost right

Two equilibria and no way to choose states the central difficulty and leaves it open. Repetition makes it worse: repetition turns two answers into a region of them.

Both of those results, and everything else about simultaneous choice, rest on a hypothesis that is never written in a payoff matrix. Every chooser knows the payoffs, knows the other knows them, knows the other knows they know, and so on without end. Take the smallest possible bite out of that tower and the two answers collapse to one.

The band in the upper bar is the whole difficulty. The bar underneath has no band. The difference between them is that the second gives each chooser a reading of $\theta$ that is very nearly right, and the point the band collapses to is a half — not the middle of the band by symmetry, but the point that a specific and checkable argument singles out.

The game, and where the band comes from #

The stage game is the smallest coordination problem with a parameter in it. Each of two choosers picks invest or stay out. Staying out pays nothing. Investing pays $\theta$ if the other also invests, and $\theta - 1$ if the other stays out — so investing is a bet on the other’s participation, and $1$ is what being left alone costs. It is the stag hunt with a dial on it: the same two-equilibrium shape at every value inside the band, and no ambiguity at all outside it.

Three regions follow immediately.

Above $\theta = 1$, investing pays even against a certain refusal, so it is dominant and there is nothing to discuss. Below $\theta = 0$, staying out pays even against certain participation, so it is dominant the other way. Between nought and one, both outcomes are equilibria: if the other invests, investing is better, and if the other stays out, staying out is better.

That band is where the failure to choose lives. Nothing in the payoffs distinguishes the two, and the two standard tie-breakers disagree about which to prefer — payoff dominance picks the better one and risk dominance picks the safer one, and in a band this wide they point in different directions for most of it.

The change, which is very small #

Replace the choosers know $\theta$ with: $\theta$ is drawn from a wide prior, and each chooser sees a private reading $x_i = \theta + \varepsilon\nu_i$, with the noises independent and $\varepsilon$ small. A chooser’s rule is now a function of their own reading, and the natural shape for one is a cutoff: invest when the reading is above some number.

The tower of mutual knowledge is gone. A chooser knows their own reading, has a belief about $\theta$, has a belief about the other’s reading, has a belief about the other’s belief — and none of these is certainty about anything.

What replaces it is one computation. With a diffuse prior and independent readings uniform on $\pm\varepsilon$, the other chooser’s reading given one’s own is triangular on $\pm 2\varepsilon$, being the difference of two uniform errors. That distribution is symmetric about one’s own reading, and its symmetry is the whole of the argument that follows.

At the cutoff, a chooser believes the other is above the cutoff with probability exactly a half — whatever $\varepsilon$ is, however small. So a chooser sitting exactly at their own cutoff faces a fifty-fifty bet on participation, and the expected payoff to investing is $\theta - \tfrac12$. Setting that to nought gives

$$\theta^\ast = \tfrac12,$$

and the number has nothing to do with the size of the noise.

The deletion, and that it reaches the same place #

The symmetry argument above finds the symmetric equilibrium. It does not by itself show that no other rule survives, and that is the part that makes the result a selection rather than a candidate.

Start with the crudest thing that can be said. Above $\theta = 1$ investing is dominant, so any rule invests there; below $\theta = 0$ staying out is dominant, so any rule stays out there. Those are the two extremes: a chooser who believes the other invests only above $1$, and one who believes the other invests above $0$.

Now best-reply to each. A chooser who believes the other uses a high cutoff will use a cutoff slightly lower — because near the other’s cutoff there is some chance the other invests, which makes investing worth it a little sooner. A chooser who believes the other uses a low cutoff will use one slightly higher, by the same argument run the other way. Each round the two bounds move inward, and every rule outside them has been eliminated as never a best reply. That is iterated elimination of dominated strategies run over rules instead of over actions, and it inherits the property that makes the procedure trustworthy there: the order does not matter, because what is removed at each stage is removed for a reason no later stage can undo.

The figure computes the two sequences separately, requires that neither ever crosses the other, and requires that they bracket a half at every stage — checked against the exact rational the symmetry argument gives rather than against the last iterate, since a loop converging to the wrong number converges just as smoothly as one converging to the right one.

That second figure is where the result stops being a curiosity. Smaller noise does not move the answer. It makes the deletion slower, which matters if anybody is counting rounds of reasoning, and it leaves the surviving rule exactly where it was.

Why the answer is the risk-dominant one #

The number a half is not the middle of the band by accident of symmetry, and identifying what it is makes the result portable to games that are not symmetric.

An action is risk-dominant when it is the better reply against an opponent believed equally likely to do either thing. In the band, investing beats staying out under a fifty-fifty belief exactly when $\theta > \tfrac12$. So the surviving rule is invest exactly where investing is risk-dominant, and the noise has selected the risk-dominant equilibrium at every value of $\theta$ in the band.

That is not the equilibrium anybody would choose. For $\theta$ between $\tfrac12$ and $1$ both investing and staying out are equilibria, investing pays both choosers more, and the noisy game still produces investment — so there the two criteria agree. For $\theta$ between $0$ and $\tfrac12$ the efficient outcome is still mutual investment when $\theta > 0$, and the noisy game produces refusal. The selection is by safety, not by payoff, and it overrides efficiency across half the band.

The stag hunt is the standard test and it fails the optimistic reading badly. Its efficient equilibrium requires believing the other will commit with probability above three-quarters, so under a fifty-fifty belief the safe action wins, and a global-game version of it selects the safe equilibrium — the outcome nobody wants, chosen by an argument about how the payoffs are learned rather than about what they are.

What the discontinuity means #

The uncomfortable part of the result is not the answer but its behaviour as the noise shrinks.

At $\varepsilon = 0$ the game has two equilibria throughout the band. At every $\varepsilon > 0$, however small, it has exactly one. The limit as the noise goes to nothing is not the game with no noise, and a modelling assumption that looks like an idealisation — suppose everybody knows the payoff exactly — turns out to be a singular point rather than the end of a smooth family.

Two readings of that are available and both are defensible.

As an argument for the selection. Common knowledge is an idealisation that no real situation satisfies, so the answer at $\varepsilon = 0$ is the artefact and the answer for small $\varepsilon$ is the robust one. A prediction that survives every small perturbation of the information structure is worth more than one that holds only at a measure-zero assumption.

As an argument against reading too much into it. The selection depends on the shape of the perturbation, not merely on its size. Uniform noise about a diffuse prior gives the symmetric belief that produces the half. A prior with real curvature, or noise whose distribution is skewed, or readings that are correlated between the choosers, each move the surviving cutoff — and in some arrangements the uniqueness fails entirely. So the result is not noise selects risk dominance but this family of noise selects risk dominance, and the family was chosen for its tractability.

The honest statement is the conditional one, and it is still substantial: under a private-value perturbation with a diffuse prior, iterated deletion leaves one rule, and the rule is risk-dominant. What is not established is that every reasonable perturbation does the same.

Where the argument is put to work #

The construction’s origin is a currency problem, and the shape of that problem is the reason the model looks the way it does.

A government holding a fixed exchange rate can defend it while its reserves last. Whether an attack succeeds depends on how many others attack, and each speculator’s payoff has exactly the structure above: attacking is profitable if enough others attack and costly if not. Under common knowledge of the government’s strength there is a wide band of fundamentals in which both attack and do not are equilibria, which is the standard second-generation model of a currency crisis — and it predicts nothing inside the band, since a crisis happens exactly when everybody believes it will.

Morris and Shin’s 1998 paper applied the global-game argument to it: give each speculator a private estimate of the reserves, and the multiplicity disappears. There is a unique threshold of fundamentals below which the attack happens and above which it does not. A model that explained crises by saying anything could happen became a model with a computable trigger, which is the whole appeal.

The same structure has been used for bank runs, for debt rollover, for regime change and for investment cascades — anywhere the payoff to acting depends on how many others act. Carlsson and van Damme’s 1993 paper is the general result underneath all of them and is where the term global game comes from: a game in which the payoff matrix is itself drawn, so that the players face a whole family of games rather than one.

What the applications inherit is the qualification of the last section. The uniqueness is a property of the information structure, and a public announcement — a statement by the central bank, a rating, a widely-read forecast — reintroduces correlation and can bring the multiplicity straight back. That is a real and slightly perverse consequence: more public information can make an outcome less predictable, because it restores the common knowledge the argument was exploiting the absence of.

An announcement can put the band back #

The last section’s remark deserves its own paragraph, because it is the result’s least intuitive consequence and the one with the most direct practical reading.

The argument works because each chooser’s information is private. A chooser at their own cutoff is genuinely uncertain about the other’s reading, and the symmetry of that uncertainty is what pins the belief at a half. Add a signal that everybody sees — a published figure, a statement, a widely-followed forecast — and that uncertainty shrinks in a particular way: the choosers’ beliefs about $\theta$ become correlated, so a chooser near the cutoff no longer thinks the other is above it with probability a half, but with a probability that depends on what everybody saw.

Push that far enough and the uniqueness fails. A sufficiently precise public signal restores something close to common knowledge, the band comes back, and with it the multiplicity. So the model contains a statement that looks perverse and is a theorem within it: more public information can make the outcome less determinate, and a central bank that announces its reserves precisely may be reintroducing the ambiguity the private estimates had removed.

Two qualifications keep that from being advice. The effect is about correlation rather than about accuracy — a public signal that is precise and a public signal that is vague both correlate beliefs, and it is the correlation that does the work. And the model’s welfare reading is contested: whether the determinate outcome is the better one depends on which outcome it is, and the argument above has already shown that the selected equilibrium is often the inefficient one.

What survives without qualification is the negative half. A model whose conclusion depends on the absence of common knowledge will be sensitive to anything that manufactures it, and public announcements are exactly that. The same observation applies to a device that whispers a recommendation from the other direction: there, correlation is what buys the outcomes private mixing cannot reach, and here it is what takes the unique answer away.

Set against what repetition does #

Repetition and noise pull in opposite directions, and the contrast is the reason they belong together.

Repetition takes a game with one equilibrium and gives it a continuum. Noise takes a game with a continuum of equilibria — or two — and gives it one. Both are changes to the setting rather than to the payoffs; neither alters a single number in the matrix; and they move the answer in opposite directions by more than any change to the payoffs could.

So the number of equilibria a game has is not a property of the game. It is a property of the game together with a statement about who knows what and about whether there is a tomorrow, and those statements are usually not written down at all. That is the standing lesson of the three results together, and it is worth more than either on its own: an announcement that binds, a future to lose, and a doubt about the payoffs are three assumptions with nothing in common, and each of them changes the answer completely.

A belief has no picture #

Nothing here draws a belief. The whole argument turns on what a chooser thinks the other’s reading is, which is a distribution over a quantity nobody observes, and no figure on this page contains one. What is drawn is the consequence — a cutoff, and a sequence of cutoffs — and the symmetry that produces the half is checked where the figure is computed and argued in the prose.

The deletion is drawn as two sequences of numbers and is a statement about rules. Each point of the two curves is a whole strategy, a function from readings to actions, compressed to the one number that describes it because the rules that survive are all cutoffs. That they are all cutoffs is itself a theorem — a best reply to a cutoff rule is a cutoff rule, because the incentive to invest rises with one’s own reading — and the figure assumes it rather than showing it.

And the noise is drawn nowhere. The bars show what a chooser does as a function of $\theta$, and a chooser never sees $\theta$. A figure of what actually happens would show the two readings scattered about the true value and the two decisions made from them, which for small $\varepsilon$ is a picture of two dots close together and carries no information.

Still open: which perturbations behave this way #

The band closes under this family of noise and the argument for it is exact. Whether it closes under every reasonable family is not settled, and the counterexamples are not exotic: correlated readings, a prior concentrated where the band is, and a public signal alongside the private ones all break the uniqueness at some parameter values. The general question — which perturbations of common knowledge select, and what they select — is the open half of the subject the two 1990s papers opened.

The other direction is the one none of this has taken at all. Every one of these arguments, this one included, computes what is consistent with choosers who reason perfectly about each other. The deletion above takes fourteen rounds to close its bracket at one noise level and thirty at another, and every round is one more level of she thinks that he thinks. Whether anybody performs more than two or three such rounds is a question about people rather than about games, and the answer has been measured repeatedly and is small. The gap between what a construction assumes about reasoning and what anybody does is the same gap a perfectly informed follower opens, and neither account has an answer to it.

An assumption that is not in the matrix #

The habit is to ask, of any model that has produced an answer, what it assumed about knowledge.

A payoff matrix looks complete. It names the choosers, the actions and the consequences, and nothing appears to be missing. What is missing is the entire epistemic structure — who knows the matrix, who knows that, and to what depth — and here it has turned out to matter more than any entry in the matrix does.

The test is to perturb the assumption rather than the numbers. Perturbing a payoff by a per cent and finding the answer unchanged is the usual robustness check and it is the easy one. Perturbing common knowledge by a per cent and finding that two equilibria become one is the check that says the model’s conclusion was resting on something nobody wrote down.