  Optimizations premised on open-loop metrics such as Age of Information (AoI)
indirectly enhance the system's decision-making utility. We therefore propose a
novel closed-loop metric named Goal-oriented Tensor (GoT) to directly quantify
the impact of semantic mismatches on goal-oriented decision-making utility.
Leveraging the GoT, we consider a sampler & decision-maker pair that works
collaboratively and distributively to achieve a shared goal of communications.
We formulate a two-agent infinite-horizon Decentralized Partially Observable
Markov Decision Process (Dec-POMDP) to conjointly deduce the optimal
deterministic sampling policy and decision-making policy. To circumvent the
curse of dimensionality in obtaining an optimal deterministic joint policy
through Brute-Force-Search, a sub-optimal yet computationally efficient
algorithm is developed. This algorithm is predicated on the search for a Nash
Equilibrium between the sampler and the decision-maker. Simulation results
reveal that the proposed sampler & decision-maker co-design surpasses the
current literature on AoI and its variants in terms of both goal achievement
utility and sparse sampling rate, signifying progress in the
semantics-conscious, goal-driven sparse sampling design.
