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Let's say I have small $2 \times 2$ games. How would I go finding the optimal social welfare by hand?

This would mean I have to find the probability matrix (of dimension $2 \times 2$) that maximizes social welfare.

Mathematically, this is the game matrix with rewards:

\begin{matrix}&C&D\\A&a_1,a_2&a_3,a_4\\B&a_5,a_6&a_7,a_8\end{matrix}

Below is the mediator matrix (probability of drawing which moves will be taken by each player):

\begin{matrix}&C&D\\A&p_1&p_2\\B&p_3&p_4\end{matrix}

where $p_1 + p_2 + p_3 + p_4 = 1$.

To maximize social welfare is to find utilitarian equilibrium.

Function which has to be maximized: $$p_1 (a_1 + a_2) + p_2 (a_3 + a_4) + p_3 (a_5 + a_6) + p_4 (a_7 + a_8)$$

Correlated equilibrium inequalities also have to be satisfied. This is a linear program but I'd like to know if there are some regularities one can use (for example if one knows all pure Nash equilibria maybe it's easy to find correlated etc.) to find the maximum easily.

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    For a $2\times 2$ game you can assign probabilities $p,$ $1-p$ $q$, $1-q$ to each player for each option, then compute the expected total payoff (i.e. the payoff for player $1$ plus that of player $2.$) Then maximize over $p$ and $q.$ If one of the four squares has a higher total payoff than any other, then the optimum will be for both players to play that with probability $1.$ – 2017-02-01
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    @spaceisdarkgreen I've explained a bit more what the problem is. I think I have to maximize over 3 variables, not two. – 2017-02-01

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