  We introduce a discrete-time search game, in which two players compete to
find an object first. The object moves according to a time-varying Markov chain
on finitely many states. The players know the Markov chain and the initial
probability distribution of the object, but do not observe the current state of
the object. The players are active in turns. The active player chooses a state,
and this choice is observed by the other player. If the object is in the chosen
state, this player wins and the game ends. Otherwise, the object moves
according to the Markov chain and the game continues at the next period.
  We show that this game admits a value, and for any error-term $\veps>0$, each
player has a pure (subgame-perfect) $\veps$-optimal strategy. Interestingly, a
0-optimal strategy does not always exist. The $\veps$-optimal strategies are
robust in the sense that they are $2\veps$-optimal on all finite but
sufficiently long horizons, and also $2\veps$-optimal in the discounted version
of the game provided that the discount factor is close to 1. We derive results
on the analytic and structural properties of the value and the $\veps$-optimal
strategies. Moreover, we examine the performance of the finite truncation
strategies, which are easy to calculate and to implement. We devote special
attention to the important time-homogeneous case, where additional results
hold.
