[Math] Are pure Nash equilibria Pareto-optimal

game theory

I am new in game theory and trying to solve a game with $3$ players and $2$ choices. Below I constructed matrixes.

\begin{align}&\text{Matrix } A& \text{Matrix } B\\[0.1cm]&\begin{array}{|r|r|r|}\hline 9,8,6 &9,3,6 \\ \hline 5,8,6 & 8,6,4 \\\hline \end{array}&\qquad \begin{array}{|r|r|r|}\hline 9,8,6 & 7,6,9 \\ \hline 8,6,9 & 11,9,12 \\\hline \end{array}\end{align}
where P1 chooses line (A or B), P2 column (A or B) and P3 matrix (A or B).
But I have some troubles with following questions

  1. What are the pure Nash equilibria of this game? Are any of the pure Nash equilibria Pareto-optimal?

  2. What strategy profile has the highest social welfare?

My attempt:

  1. I found all Nash eq. They are $(A, A, A),(A, A, B), (B, B, B)$.
    Which of them are Pareto-optimal? Ok, from these $3$ strategies I should choose such strategy where increasing any of player's payoff attracts decreasing of others. For the first strategy it's false because if I try to increase the payoff of P1, there is only one strategy to do it $(B,B,B)$, others also increase. The same for the second strategy. For the last strategy $(B, B, B)$ I don't know the answer. All payoffs are max no way to increase them. Does it mean that it's pareto optimal? I think yes.

  2. No idea about second question. I read the def of social welfare function but didn't understand it. Could you give me a hint or rephrase the def to be more clearly to me.

Thanks.

Best Answer

Yes, it is possible for Nash equilibrium to be either Pareto optimal or not. One of the justifications economists present for game theory is that we can enact mechanisms that move us from one non-PO equilibrium to a PO one.

Check the definition of Pareto optimality as it relates to payoffs in a game (nobody can be made strictly better off without making someone else worse off), and look at the payoffs for a prisoner's dilemma. The interesting insight of a prisoner's dilemma is that the only Nash equilibrium is Pareto dominated by another possible outcome.

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