The level sets of typical games

In a non-cooperative game, players do not communicate with each other. Their only feedback is the payoff they receive resulting from the strategies they execute. It is important to note that within each level set of the total payoff function the payoff to each player is unchanging, and therefore understanding the structure of these level sets plays a key role in understanding non-cooperative games. This note, intended for both experts and non-experts, not only introduces non-cooperative game theory but also shows its fundamental connection to real algebraic geometry. We prove here a general result about the structure of the level sets, which although likely to be known by experts, has interesting implications, including our recent application to provide a new mathematical explanation for the "paradox of the plankton." We hope to encourage communication between these interrelated areas and stimulate further work in similar directions.