An extreme equilibrium $(x,y)$, where $x$ and $y$ are the mixed strategies of players 1 and 2, is a Nash equilibrium where for both players mixed strategies cannot be described as convex combinations of other mixed strategies that form equilibria.
Fact: There are always finitely many extreme equilibria.**
For a mixed strategy $x$, let the support of $x$ be defined as the set of pure strategies that $x$ uses with positive probability, i.e.,
$$\text{support}(x) := \{ i : x_i > 0 \}.$$
A game is called non-degenerate game if for all mixed strategies $x$, if $|\text{support}(x)| = k$ then the number of best responses against $x$ is at most $k$.
Fact: Every non-degenerate game has only extreme equilibria.
For a simple example of a game with non-extreme equilibria consider a $2\times 2$ game where both players get $0$ for all strategy profiles. In this case, all four pure profiles are extreme equilibria and all strict mixtures are non-extreme Nash equilibria. For a less trivial example, consider any $2 \times 2$ bimatrix game where there is no strict dominance for either player but exactly one of the players has a weakly dominant strategy. Then, the game is degenerate and the game will have infinitely many equilibria. As an example, consider the following game.
$$A = \left( \begin{array}{ccc}
1 & 0 \\
0 & 1 \\
\end{array} \right)
\quad
B = \left( \begin{array}{ccc}
1 & 1 \\
1 & 0 \\
\end{array} \right)
$$
The game has exactly two extreme equilibria. Here is the output from the online game solver http://banach.lse.ac.uk, which shows these two extreme equilibria:
2 x 2 Payoff matrix A:
1 0
0 1
2 x 2 Payoff matrix B:
1 1
1 0
EE = Extreme Equilibrium, EP = Expected Payoff
Decimal Output
EE 1 P1: (1) 1.0 0.0 EP= 1.0 P2: (1) 1.0 0.0 EP= 1.0
EE 2 P1: (1) 1.0 0.0 EP= 0.5 P2: (2) 0.5 0.5 EP= 1.0
Rational Output
EE 1 P1: (1) 1 0 EP= 1 P2: (1) 1 0 EP= 1
EE 2 P1: (1) 1 0 EP= 1/2 P2: (2) 1/2 1/2 EP= 1
Connected component 1:
{1} x {1, 2}
The final line of this output shows that against the unique equilibrium strategy of player 1 (top row), player 2 can play any convex combination of $(1,0)$ and $(1/2,1/2)$.
For a thorough technical exposition, which covers both finding extreme equilibria and then from these all equilibria of bimatrix games, see, e.g.,
David Avis, Gabriel D. Rosenberg, Rahul Savani, and Bernhard von Stengel (2010).
Enumeration of Nash equilibria for two-player games.
Economic Theory, 42:9–37.
On the relationship between correlated equilibria and Nash equilibria:
Fact:: The set of correlated equilibrium
distributions of an $n$-player noncooperative game is a convex
polytope that includes all the Nash equilibrium distributions. In fact, the Nash equilibria all lie on the boundary of the polytope.
See:
Robert Nau, Sabrina Gomez Canovas, and Pierre Hansen (2004).
On the geometry of Nash equilibria and correlated equilibria.
International Journal of Game Theory 32: 443-453.
Best Answer
In a mixed (strategy) Nash equilibrium, the players' actions (strategies) are independent random variables. In other words, if you know that player 1 (randomly) chose $x$, that doesn't give you any additional information about what player 2 might do.
In correlated equilibria the actions (strategies) need not be independent. Wikipedia gives an example in the setting of the game of chicken, showing that the expected value of a correlated equilibrium can be greater than that of the mixed strategy equilibrium.