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Optimal Hiding with Partial Information of the Seeker's Route

Prajakta Surve, Shaunak D. Bopardikar, Daigo Shishika, Dipankar Maity, Michael Dorothy

Abstract

We consider a hide-and-seek game between a Hider and a Seeker over a finite set of locations. The Hider chooses one location to conceal a stationary treasure, while the Seeker visits the locations sequentially along a route. As the search progresses, the Hider observes a prefix of the Seeker's route. After observing this information, the Hider has the option to relocate the treasure at most once to another unvisited location by paying a switching cost. We study two seeker models. In the first, the Seeker is unaware of the fact that the Hider can relocate. In the second, the Seeker select its route while accounting for the possibility that the Hider observes its path and reallocates. For the restricted case, we define the value-of-information created by the reveal and derive upper bounds in terms of the switching cost using a worst-case evaluation over routes. We also show that seeker awareness reduces the game value, with the difference between the restricted and feedback models bounded by the entry-wise gap between the corresponding payoff matrices. Numerical examples show how this benefit decreases as the switching cost increases and as the reveal occurs later along the route.

Optimal Hiding with Partial Information of the Seeker's Route

Abstract

We consider a hide-and-seek game between a Hider and a Seeker over a finite set of locations. The Hider chooses one location to conceal a stationary treasure, while the Seeker visits the locations sequentially along a route. As the search progresses, the Hider observes a prefix of the Seeker's route. After observing this information, the Hider has the option to relocate the treasure at most once to another unvisited location by paying a switching cost. We study two seeker models. In the first, the Seeker is unaware of the fact that the Hider can relocate. In the second, the Seeker select its route while accounting for the possibility that the Hider observes its path and reallocates. For the restricted case, we define the value-of-information created by the reveal and derive upper bounds in terms of the switching cost using a worst-case evaluation over routes. We also show that seeker awareness reduces the game value, with the difference between the restricted and feedback models bounded by the entry-wise gap between the corresponding payoff matrices. Numerical examples show how this benefit decreases as the switching cost increases and as the reveal occurs later along the route.

Paper Structure

This paper contains 22 sections, 5 theorems, 88 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Related work
  3. Contributions
  4. Problem Formulation
  5. Preliminary Results
  6. Main Results
  7. Analysis with restricted seeker model
  8. Analysis with seeker-aware model
  9. Feedback payoff matrix.
  10. Game value under feedback.
  11. Numerical Examples
  12. Base payoff matrix
  13. Restricted Seeker Model
  14. Hider's switching payoff matrix
  15. Critical switching cost
  16. ...and 7 more sections

Key Result

Lemma 1

Suppose the baseline game admits a unique pure saddle point $({r_j^\star},i^\star)$, that is, Then, even if the Hider observes partial route information at time $t_{\mathrm{reveal}}$ and is allowed to switch once to an unvisited location at a cost $c \ge 0$, there is no incentive to switch. Staying at $i^\star$ remains optimal.

Figures (6)

  • Figure 2: Game Tree for Restricted Seeker Model for $N=2$.
  • Figure 3: Game Tree for Seeker-Aware Model for $N=2$.
  • Figure 4: Restricted seeker model game tree for $N=3$ with $t_{\mathrm{reveal}}=1$ and $c=1$. After the first visit, $h_1=\{1\}$, so $\mathcal{U}(h_1)=\{2,3\}$. The Hider compares staying at 2 with switching to 3 (cost $c$), yielding $A^{\mathrm{switch}}(1,2)=3.4142$.
  • Figure 5: Six-location configuration used in the numerical example, with the Seeker starting at the origin.
  • Figure 6: Expected value-of-information $\mathbb{E}[\mathrm{VOI}]$ versus switching cost $c$ for different reveal times $t_{\mathrm{reveal}}$. The decay in $c$ and downward shift with increasing $t_{\mathrm{reveal}}$ are consistent with Theorem \ref{['lem:expvoi_bound']} and Lemma \ref{['lem:treveal_mono']}.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Lemma 1: Unique pure saddle point case
  • proof
  • Lemma 2: Probability of termination by reveal time
  • proof
  • Definition 1
  • Theorem 1: Upper bound on $\mathbb{E}\mathrm{[VOI]}$ as a function of $c$
  • proof
  • Proposition 1: Effect of reveal time on VOI
  • proof
  • Proposition 2: Effect of seeker awareness
  • ...and 3 more