Deception, Delay, and Detection of Strategies

Homology generators in a relation offer individuals the ability to delay identification, by guiding the order via which the individuals reveal their attributes (see arXiv:1712.04130). This perspective applies as well to the identification of goal-attaining strategies in systems with errorful control, since the strategy complex of a fully controllable nondeterministic or stochastic graph is homotopic to a sphere. Specifically, such a graph contains for each state $v$ a maximal strategy $\sigma_v$ that converges to state $v$ from all other states in the graph and whose identity may be shrouded in the following sense: One may reveal certain actions of $\sigma_v$ in a particular order so that the full strategy becomes known only after at least $n-1$ of these actions have been revealed, with none of the actions revealed definitively inferable from those previously revealed. Here $n$ is the number of states in the graph. Moreover, the strategy contains at least $(n-1)!$ such informative action release sequences, each of length at least $n-1$. The earlier work described above sketched a proof that every maximal strategy in a pure nondeterministic or pure stochastic graph contains at least one informative action release sequence of length at least $n-1$. The primary purpose of the current report is to fill in the details of that sketch. To build intuition, the report first discusses several simpler examples. These examples suggest an underlying structure for hiding capabilities or bluffing capabilities, as well as for detecting such deceit.