  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.
