There is a well known notion of Strategic Dominance in Game Theory.
I am interested in elimination of strictly dominated strategies by Linear Programming and in LP for definition of never best response, which is equivalent to strictly dominated strategy.
Strictly Dominated Strategies.The action $a_i \in A_i$ of player $i$ in the strategic game is strictly dominated if there is a mixed strategy $\alpha_i$ of player $i$ such that $U_i(a_{-i},\alpha_i) > u_i(a_{-i},a_i)$, for all $a_{-i} \in A_{-i}$, where $U_i(a_{-i},\alpha_i)$ is the playoff of player $i$ if he uses the mixed strategy $\alpha_i$ and the other players vector of actions is $a_i$.
The problem is to form a LP to find strictly dominated strategies
LP:
$ U_i(a_{-i},\alpha_i) > u_i(a_{-i},a_i), \forall a_{-i} \in A_{-i}$.
Let's consider a kind of equivalent notion to strictly dominated strategy
Never Best Response. A pure strategy $a'_i$ is never best response if it's not best response to any mixed strategy of the opponent $\alpha_{-i}$.
LP:
$ u_i(\alpha_{-i},a_i) > u_i(\alpha_{-i},a'_i), \forall \alpha_{-i} \in A_{-i}, \forall a_i \in A_i$.
For me the definition seems not obvious, why don't compare $a'_i$ to any mixed strategy $\alpha_i$, is it enough to check just over all other pure strategies $a_i$?
The problem is to define a Linear Program for strictly dominated strategy and never best response. The main point I cannot get is how to iterate through infinite number of mixed strategies and how to convert the definition with sign equal for LP.