Week 1-Part 6: Pareto Optimality

We’ve thought about how to play some canonical games. If you take a step back, and analyze the game as an outside observer, can we see that some outcomes of a game are better than others? It may be useful to pause and think about some of the examples we’ve covered to help form an answer.

It is hard to determine when one agent’s interests are more iportant than another. You also wouldn’t know what the scale is of the utilities (cannot assume a common scale). This idea can be shown by analyzing a game where the outcomes are in different currencies, and you don’t know what the currencies are.

You would be looking at a game where player 1’s outcome is in currency 1 and player 2’s outcome is in currency 2, with no idea what the exchange rate is.

Pause and think again, can there be scenarios where you’d prefer one outcome over another?

Definition (Pareto Optimality)

Sometimes, there is one outcome $o$ which is at least as good for every agent as another outcome $o'$, and there is some agent who strictly prefers $o$ to $o'$. e.g. if $o: (7,8)$ and $o':(7,2)$ then $o$ is at least as good for every agent, and it is strictly better for somebody (player 2).

In this case, it seems reasonable to say that $o$ is better than $o'$. In this case, $o$ Pareto Dominates $o'$.

We say that an outcome $o^*$ is Pareto Optimal if there is no other outcome that Pareto-dominates it.

Questions (Pareto Optimality)

• Can a game have more than one Pareto-optimal outcome?

  Of course, if all payoffs of a game are the same, then nothing dominates anything else (no strict preference)

• Does every game have at lease one Pareto outcome?

  Yes, every game has to have at least one Pareto-optimal outcome. This is easy to see because by definition, for something to not be Pareto optimal, it must be dominated by something else. Therefore, it would only work if one cell was Pareto dominated by another cell, and that cell was Pareto dominated by another cell, i.e. a cycle of Pareto dominance.

  However, such a cycle cannot exist because by definition, $o'$ must be at least preferred by everybody and strictly preferred by one agent.

Example (Sidewalk Co-ordination)

Consider the following matrix:

$\begin{array}{c|c|c|} & \text { Left } & \text { Right } \\ \hline \text { Left } & 1,1 & 0,0 \\ \hline \text { Right } & 0,0 & 1,1 \\ \hline\end{array}$

In this case, (Left, Left) and (Right, Right) are Pareto optimal.

Example (Battle of the Sexes)

Consider the following matrix:

$\begin{array}{c|c|c|} & \mathrm{B} & \mathrm{F} \\ \hline \mathrm{B} & 2,1 & 0,0 \\ \hline \mathrm{F} & 0,0 & 1,2 \\ \hline\end{array}$

In this case, (B,B) and (F,F) are Pareto optimal.

Example (Matching Pennies)

Consider the following matrix:

$\begin{array}{c|c|c|} & \text { Heads } & \text { Tails } \\ \hline \text{ Heads } & 1,-1 & -1,1\\ \hline \text { Tails } & -1,1 & 1,-1 \\ \hline\end{array}$

In this case, every outcome is Pareto optimal because there is no pair of outcomes where everybody likes the two outcomes equally well (generally true of zero-sum games).

Example (Prisoner’s Dilemma)

Consider the following matrix:

$\begin{array}{c|c|c|} & C & D \\ \hline C & -1,-1 & -4,0 \\ \hline D & 0,-4 & -3,-3 \\ \hline\end{array}$

All but one outcome are Pareto optimal, namely (C,C), (C,D), and (D,C). (D,D) is not Pareto optimal because it is Pareto dominated by (C,C).

Remark (Paradox of Prisoner’s Dilemma)

The Nash equilibrium of this game is the only non-Pareto optimal outcome!

Built with GatsbyJS