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deltaList(List) -- generate the list of Newton polytopes for a generic game

Description

For a generic $n$-player game where the $i$-th player has $d_i$ pure strategies, the maximum number of isolated totally mixed Nash equilibria is given by the mixed volume of the following list of polytopes:

\[ (\Delta^{(1)}, \cdots, \Delta^{(1)},\Delta^{(2)}, \cdots, \Delta^{(2)}, \cdots, \Delta^{(n)}, \cdots, \Delta^{(n)}),\]

where each $\Delta^{(i)}$ repeats itself $d_i - 1$ times, and is the product of simplices

\[ \Delta^{(i)} := \Delta_{d_{1}-1}\times \Delta_{d_{2}-1} \times \cdots \times \Delta_{d_{i-1}-1} \times \{0\} \times \Delta_{d_{i+1}-1} \times \cdots \times \Delta_{d_{n}-1}.\]

This function constructs and returns this list of polytopes. Each $\Delta^{(i)}$ is a polytope in an ambient vector space of dimension $d_1+d_2+\cdots+d_{n}-n$.

i1 : DL = deltaList {2,2,2}

o1 = {Polyhedron{...1...}, Polyhedron{...1...}, Polyhedron{...1...}}

o1 : List

Each entries of DL is a polytope of dimension 2, in a $2+2+2-3=3$ dimensional vector space.

i2 : apply(DL, p -> dim p)

o2 = {2, 2, 2}

o2 : List
i3 : apply(DL, p -> ambDim p)

o3 = {3, 3, 3}

o3 : List

See also

Ways to use this method:

  • deltaList(List) -- generate the list of Newton polytopes for a generic game

The source of this document is in GameTheory.m2:1413:0.