Two equilibria and no way to choose

Two choosers can each work alone for a certain three, or together for four apiece. Working together pays more and needs both: a chooser who commits to the joint effort while the other works alone gets nothing.

Both states are equilibria in the sense the first rung of this anchor established and the second pressed on: at both together, neither improves by defecting — four beats three; at both alone, neither improves by committing — three beats nothing. The definition is satisfied twice, it prefers neither, and one of them pays four apiece while the other pays three.

Three-quarters of the possible beliefs lead to the worse equilibrium. That number is not a matter of taste; it is read off an axis, and it is the whole of this rung.

Where the crossing comes from #

Write $q$ for the probability that the column chooser commits. The row chooser’s expected payoff from committing is $4q$ — four when the other commits, nothing otherwise. From working alone it is $3$, whatever the other does.

Those are two straight lines and they cross where $4q = 3$, at $q = \tfrac34$. Above the crossing committing is the better reply and below it working alone is, and at the crossing the row chooser is indifferent — which is exactly the condition that makes $\tfrac34$ the mixed equilibrium, since a mixture is only worth playing when the alternatives it mixes are equally good.

So the crossing is two things at once and the coincidence is not one. The mixed equilibrium is the point where the best reply changes, which is why it always sits between the two pure equilibria and why it is always unstable: a belief a hair above it pushes toward one, a hair below toward the other.

The number the crossing supplies is the interesting one. Under any rule that plays a best reply to a belief about the other, the joint effort is reached from beliefs above $\tfrac34$ and the safe option from beliefs below — so the safe equilibrium’s basin is three times the size of the other’s, on a game where it pays less.

Two tests, and they disagree #

Two standard tie-breakers are applied to a situation like this, and they are not different formalisations of one idea.

Payoff dominance prefers the equilibrium paying more. Four beats three; the joint effort wins. This is the reading everybody has on first sight and it is the one that makes the game look like a puzzle about coordination.

Risk dominance prefers the equilibrium that is worth playing against an opponent whose behaviour is unknown. It is measured by how much each chooser loses by choosing an option the other does not match: committing alone loses $4 - 0 = 4$ against the joint equilibrium, and working alone while the other commits loses $3 - 3 = 0$. Multiply each side’s loss and compare the products — the safe equilibrium wins, and it wins by a wide margin.

The two tests disagree, and the disagreement is not a coincidence of these numbers. Risk dominance and the basin measurement are the same statement. The equilibrium whose product of deviation losses is larger is exactly the one whose basin is bigger, on a two-by-two game, and the threshold $\tfrac34$ is the arithmetic of both.

For contrast, here is the case in which eliminating the dominated options settles everything by itself, so the selection question never arises and no tie-breaker is needed at all.

Where the disagreement does not happen #

It is worth drawing a case where the two tests agree rather than disagree, because what the agreement rests on is exactly what the disagreement above rested on, read the other way round.

When both equilibria pay the same total, payoff dominance has no opinion and only risk dominance says anything. When one pays more and is safer, the two agree and there is no selection problem worth the name.

The interesting case is the one at the top of this page, and what makes it interesting is a specific arrangement: the better equilibrium requires both to move, and moving alone is expensive. The cost of a failed attempt is what makes the safer option safe, and it is what makes the better outcome hard to reach even though everybody prefers it and everybody knows everybody prefers it.

That is not a failure of information. Both choosers know the whole matrix, know the other knows it, and know the other prefers the joint effort. Knowledge is not the binding constraint; the binding constraint is that preferring an outcome is not the same as being willing to bet on it.

What the potential says #

The previous rung’s landscape has something to say here, and what it says is uncomfortable.

A two-by-two coordination game is a potential game, and its potential has two local minima — one at each equilibrium. Which one a descent reaches depends on where it starts, exactly as the basins say. But the potential also has a global minimum, one of the two is deeper, and the deeper one is the risk-dominant equilibrium rather than the payoff-dominant one.

So on this game the descent’s preferred destination is the worse outcome, and it is preferred in a precise sense: the deeper valley is larger, and any process that occasionally makes a mistake and then descends again will spend most of its time in the deeper one. That is the mechanism behind the standard result in this area — under a dynamic with rare random errors, the long-run distribution concentrates on the risk-dominant equilibrium as the error rate goes to nought, whatever the payoffs of the other.

The tie-break that wins is the one nobody would choose, and it wins because it is the one an unreliable process falls into. That is a statement about basins rather than about rationality, and it is the reason risk dominance is taken seriously despite being the answer that pays less.

The device does not fix it either #

Correlation enlarged the set of stable outcomes and it does not settle this.

A public signal recommending the joint effort to both is a correlated equilibrium: told to commit, a chooser knows the other was told to commit, so committing is a best reply and the pair reaches four apiece. So the better outcome is reachable, and it is reachable without any enforcement.

What the device does not supply is a reason to believe it is being obeyed. Its recommendations are best replies given that the other obeys, and the whole difficulty here is that a chooser contemplating disobedience is contemplating a world in which the other might too. The correlated equilibrium is stable against a single unilateral deviation and so is the safe equilibrium, and the definition does not rank them.

Every solution concept in this field ranks outcomes by resistance to a single deviation, and the difficulty here is not a single deviation. It is a doubt about what the other will do, and a doubt is not a deviation.

Where the numbers come from #

The threshold moves with the payoffs and it moves in a way worth knowing.

Raising the joint payoff from four to six moves the crossing from $\tfrac34$ to $\tfrac12$, and the two basins become equal.

Lowering the cost of a failed attempt does the same thing and by the same arithmetic.

So there are two ways to make the good outcome likely and they are interchangeable at these numbers: raising the reward by two and cushioning the failure by two both move the threshold from $\tfrac34$ to $\tfrac12$. What the threshold responds to is the difference between the two lines, and either end of that difference will do — which is a conclusion about the shape of the arithmetic rather than a recommendation, and it is available only because the threshold is an explicit ratio rather than an intuition.

The interchangeability is worth one more sentence, because it is not obvious and it does not survive. It holds here because both lines are straight and the change is a translation of one of them; on a game with more than two options the best-reply boundary is a surface in a simplex and the two adjustments move it differently.

The same lines, settling a question completely #

There is a construction one anchor away that looks identical on the page and answers rather than opens, and the comparison says exactly what is different here.

The value from both sides draws a chooser’s expected payoff against a mixing probability, takes the lower envelope of those lines, and finds its highest point. That crossing is the answer: it is the value of the game, both sides name it, and neither can be argued out of it.

The lines on this page are the same kind of object and the crossing does the opposite. It is not a value anybody secures; it is the belief at which the better reply changes, and on either side of it a different pure equilibrium is reached. In a zero-sum game the crossing is where the two sides’ interests meet; here it is where two possible futures separate.

The structural reason is that a zero-sum game has one number and a coordination game has two. When one chooser’s gain is the other’s loss, the mixed equilibrium is the unique sensible outcome and the lines are computing it. When both prefer the same cells, the mixed equilibrium is a knife-edge nobody wants, the pure equilibria are what actually happens, and the lines are computing which one.

That is worth stating because the algebra gives no warning. Both figures solve $\text{line}_1(q) = \text{line}_2(q)$ for $q$ and report the answer, and whether the answer is a solution or a watershed depends on the sign structure of the matrix rather than on anything in the calculation.

Why the safe option is chosen even when everybody knows #

The uncomfortable feature of this game is that no amount of shared knowledge dissolves it, and the reason is worth separating from the arithmetic.

Suppose both choosers prefer the joint effort, both know the other prefers it, both know that the other knows, and so on without end. That state — common knowledge of the payoffs and of the preference — does not make the joint effort the unique equilibrium, because the safe option is still a best reply to the safe option. Knowing what somebody prefers settles nothing about what they will do, when what they do depends on what they think the other will do.

The regress is the whole difficulty and it does not terminate. Committing is right if the other commits; the other commits if they think this one will; this one is thought to commit if they are thought to think the other will. There is no first step, and both equilibria are consistent completions of the whole chain.

That is why the interesting repairs change the information rather than adding to it. Global games are the cleanest: give each chooser a slightly noisy private observation of the payoffs rather than an exact shared one, and the regress terminates — the noise means a chooser is never quite sure the other sees what they see, the induction runs, and exactly one equilibrium survives. It is the risk-dominant one.

The result is startling in the right direction. Adding a small amount of uncertainty, which ought to make a coordination problem harder, removes the multiplicity entirely. What was blocking the argument was not too little information; it was that perfectly shared information gives the regress nothing to bite on.

Where the account is incomplete #

Two-by-two is the only case where risk dominance is unambiguous. With three or more options the product of deviation losses generalises in several inequivalent ways, and no one of them has become standard. The basin measurement generalises but stops being a single number, since the belief space is a simplex and the basins are regions in it.

The basins depend on the dynamic. Best-reply dynamics gives $\tfrac34$; a rule that adjusts gradually rather than jumping gives the same boundary, since the boundary is where the best reply changes. A rule that imitates whoever is doing better, or that experiments occasionally, gives different regions and can give a different answer.

Nothing here is a prediction. The arithmetic says which equilibrium a particular dynamic reaches from a particular belief. What beliefs people actually hold, and whether they use any such rule, is a question no amount of this settles, and the experimental literature on this exact game does not agree with any single answer.

And the mixed equilibrium is not a candidate. It is an equilibrium, it exists — existence is what the fixed-point argument buys and it buys nothing else — and it is unstable in both directions, since any deviation from it grows. It is drawn here as a boundary rather than as a possible outcome, which is what it is.

Harsanyi and Selten, and a programme that did not finish #

Harsanyi and Selten published A General Theory of Equilibrium Selection in Games in 1988, and the ambition was in the title: a single procedure that names one equilibrium of any finite game.

Risk dominance is theirs, and it is the part of the book that survived. The general procedure did not: it is elaborate, it depends on choices inside it that are hard to defend, and it gives answers on small games that most readers find unconvincing. Selten himself later described the theory as unsatisfactory, and the field moved to the evolutionary approach — asking which equilibrium a stochastic dynamic settles on rather than which one a principle names.

The move is worth registering because it is a change in what counts as an answer. A selection principle says which equilibrium is right; a dynamic says which one happens. The second is a weaker kind of claim and it is checkable, and the two need not agree — on this game they agree with each other and disagree with the payoffs, which is why risk dominance survived its own book.

What the pictures cannot show #

The lines are the row chooser’s expected payoff against a belief, and a belief is not an observable. Nothing in the picture is a probability anybody holds; it is a parameter swept across an axis, and the reader is invited to imagine one value of it at a time.

The shaded basin is a statement about a dynamic that is not drawn. What the figure shows is which line is higher; that this makes the joint effort reachable from those beliefs and not from others is an inference about a process, and the process has no picture here.

And the two equilibria are two points on a picture with no time in it. Whether a pair arrives at one, and how, is the question the whole rung is about, and the figure shows the destinations and not the journey — which is exactly the limitation that made the field replace selection principles with dynamics.

The ladder from here #

Rungs above: stochastic stability, where rare mistakes make the long-run distribution concentrate on the risk-dominant equilibrium and the concentration is proved rather than argued. Global games, where a small amount of private noise about the payoffs removes the multiplicity entirely and selects the risk-dominant equilibrium as the unique answer. Evolutionary stability and the replicator flow, which asks the same question of a population rather than of two choosers and turns the crossing point into an unstable fixed point of an iteration. Cheap talk, where the choosers may speak before choosing and the question becomes whether a statement is credible. And the folk theorems for repeated play, where almost every outcome becomes an equilibrium and the selection problem becomes very much worse.

The definition that answers a smaller question #

The habit is about what a definition is entitled to.

An equilibrium is a state with no profitable unilateral deviation, and that is a complete and precise notion which answers exactly one question: can this arrangement be upset by one participant acting alone? It was never a definition of what happens, what is chosen, or what is good, and the rules-for-choosing that fail their own promises elsewhere in this field are the same shape of finding, and the two rungs below this one are two different demonstrations of the gap — an equilibrium worse for everybody than a state they left, and here two equilibria with nothing to separate them.

The right response to a definition that under-determines is to say what it under-determines, rather than to patch it. Risk dominance is a patch and it did not hold; the basins are a measurement and they did. The measurement is worth more because it is a statement about a stated process, and a reader can ask whether the process is the right one — which is a question with an answer, unlike whether a principle is the right principle.