On the main page, two shapes sit on the payoff chart: a dashed hull through the Nash points, and a solid body of correlated equilibria. They look like the same kind of object. They are not. One is built from points you already have; the other is the shadow of a body you never enumerate, traced by solving a linear program once per direction. Everything below is draggable — move the points and watch which method survives.
Start with what a public coin flip can do. If the players can all watch the same coin, they can agree in advance: heads we play this equilibrium, tails that one. Averaging payoffs that way reaches every point in the convex hull of the Nash payoffs — and nothing outside it.
Drag the hollow magenta point too: it is a target payoff, and the panel shows which coin flip produces it. Drag it outside the shaded hull and no coin flip can.
The algorithm is almost embarrassingly simple. Sort the points left to right, then walk them keeping a stack, and at every new point ask one question: did I just turn the wrong way? If the last three points make a right turn, the middle one cannot be a corner, so throw it away and ask again. That is the whole of Andrew's monotone chain.
The test is one cross product. For three points o, a, b, the sign of
(a−o)×(b−o) says which way the path bent: positive is a left turn and the point stays,
negative or zero is a right turn or a straight line and the middle point is dropped. No trigonometry,
no angles, no division — which is why it is hard to get wrong.
Now the hard one. The correlated equilibria are not a list of points — they are every joint distribution satisfying the obedience inequalities, an infinite set described by constraints rather than by corners. You cannot take the hull of something you cannot enumerate. So you ask a different question: pick a direction, and find the furthest point of the body that way. That is a linear program, and the answer is always a corner. Ask enough directions and the corners you collect are the region.
This is Aumann's game, solved live — the same linear program the main page runs 160 times to draw its magenta region. The direction you are dragging is the objective; the obedience inequalities are the constraints; the dot that lights up is the optimum the simplex method lands on.
Two facts, and the containment follows with no geometry at all. First, every Nash equilibrium is a correlated equilibrium: independent dice are a mediator who happens not to correlate anything. Second, the correlated set is convex, because it is cut out by linear inequalities and an intersection of half-spaces is always convex. Put them together: mix any two Nash equilibria and the mixture must still satisfy every inequality, so it is still correlated. The Nash hull cannot escape.
Built from a finite list of points. Needs every equilibrium found first — and finding them is not a linear program.
Built from inequalities. Never enumerates anything, is solved by linear programming, and stays one program as players are added.
The gap between the dashed hull and the magenta boundary is the mediator's advantage — payoff reachable only when the advice is private and correlated.