Game Theory Foundations: Production-Possibility Frontiers

The next concept relating to efficiency is the idea of production-possibility frontiers. This is the idea that given a limited amount of resources and accounting for systematic diminishing returns, that there is some limit where you cannot create more of one product without sacrificing another. In the simplest case, this is described as a curve on a graph with two axis each reflecting one product (a more complicated model with more products can be represented with more dimensions in a multi-dimensional model, but is beyond the scope of this discussion).

The most popular form of this model includes the Guns or Butter model as well as the efficient frontier in market portfolio theory.

In the Guns or Butter model, a government has the choice of spending its resource on foreign or defense projects (Guns) versus domestic or civilian projects (Butter). With a limited budget, there is some limit where in order to get more of one, they have to sacrifice production of the other. Plotting the points along this curve produces the production-possibility frontier.
Excusing the violent association with the "Guns" component, it is possible to instead substitue the idea of exportable goods and globalization which affect the balance of trade between countries (a topic for another time).

An efficient frontier in market portfolio theory describes a securities portfolio which includes securities in the most efficient weights and correlations such that the portfolio exhibits no un-systematic risk and is composed entirely of non-diversifiable risk (Capital market line - CML). Here the frontier is presented a little differently with the resource simply being combinations of different securities betas versus their expected return to form the efficient frontier (which intersects with the investor's marginal utility curve to determine the efficient market portfolio).

It seems like everywhere we go we naturally run into these frontier limits. How can we describe them in game theory? As it turns out, there are some models which describe behaviour in different stages of game theory.

Positions on the efficient possibility frontiers can not produce more of one benefit without sacrificing another. In game theory, this manifests as a zero-sum game. That is to say for one participant to gain (show on the production possibility frontier as one axis), another has to lose out).

Points contained within the curve (but not on the curve) allow for gains of one benefit or the other without having to make any sacrifices. This is represented in game theory as a non-zero sum game. That is to say, that at the end of the day, it is possible for both parties to benefit positively without the expense of the other.

You'll notice that the theme here is one of efficiency. In an environment where no more benefit can be extracted from the system with out additional resources (maximum efficiency), it becomes a zero-sum model. However, if production is inefficient, there is potential for a non-zero sum model (recouping the inefficientcy as non-zero sum gains - getting more of A without comprimising B).

This is also an indicator of efficiency and competition. In an industry with a maturing life cycle, transactions, deals and strategy will slowly being to have less non-zero sum models.