A projection game consists of a bipartite graph 
, finite label sets 
and 
, and, for every edge 
, a function 
. A labeling assigns a label to every
vertex and satisfies 
when the label of 
equals the image under 
of the label of 
.
The value of the game is the largest fraction of edges that can be satisfied simultaneously.
Projection games are also called label-cover instances. They are central constraint systems in hardness-of-approximation reductions and in formulations of the PCP theorem.
-Approximating Set-Cover." Theory Comput. 11,
221-235, 2015.