  Bell inequalities are important tools in contrasting classical and quantum
behaviors. To date, most Bell inequalities are linear combinations of
statistical correlations between remote parties. Nevertheless, finding the
classical and quantum mechanical (Tsirelson) bounds for a given Bell inequality
in a general scenario is a difficult task which rarely leads to closed-form
solutions. Here we introduce a new class of Bell inequalities based on products
of correlators that alleviate these issues. Each such Bell inequality is
associated with a unique coordination game. In the simplest case, Alice and
Bob, each having two random variables, attempt to maximize the area of a
rectangle and the rectangle's area is represented by a certain parameter. This
parameter, which is a function of the correlations between their random
variables, is shown to be a Bell parameter, i.e. the achievable bound using
only classical correlations is strictly smaller than the achievable bound using
non-local quantum correlations We continue by generalizing to the case in which
Alice and Bob, each having now n random variables, wish to maximize a certain
volume in n-dimensional space. We term this parameter a multiplicative Bell
parameter and prove its Tsirelson bound. Finally, we investigate the case of
local hidden variables and show that for any deterministic strategy of one of
the players the Bell parameter is a harmonic function whose maximum approaches
the Tsirelson bound as the number of measurement devices increases. Some
theoretical and experimental implications of these results are discussed.
