player observes opponent’s action with two periods delay, the game does not have proper subgames. Infinite games with Borel winning sets have recently been used in economics literature on testing the quality of probabilistic predictions. Consider a forecaster who claims to know the probability distribution that governs some stochastic process. To prove his claim, the forecaster provides probabilistic predictions about the process. An inspector tests the forecaster’s reliability using the infinite sequence of predictions provided by the forecaster and the observed realization of the process. Using Martin’s Theorem about the determinacy of Blackwell games, I proved [13] that any inspection which is based on predictions about the next-day realization of the process is manipulable, i.e., it can be strategically passed by a charlatan. Theorem 1.3 of this paper can be used to prove that inspections based on daily predictions about an arbitrarily long finite horizon are also manipulable [13, Section 5]. On the other hand, Olszewski and Sandroni [10] gave an example for a non-manipulable inspection which is based on prediction about the infinite future. Such an inspection relies on a game without eventual perfect monitoring. In Section 2 I give some examples of games with and without eventual perfect monitoring. In Section 3 I prove the determinacy of infinite games with perfect recall and a compact winning set; this result is used in the proof of Theorem 1.3. The proof of the theorem is in Section 4. Section 5 discusses the role of eventual perfect monitoring in the proof. Section 6 discusses some open questions. Martin’s Theorem is reviewed in the appendix. All the examples in this section have the same action set and the same winning set. The action set is A = { S , L } : At every stage, each player decides whether to Stay or Leave the game. Once a player leaves, his future actions do not affect the outcome of the game. For an infinite history u = ( a$_{0}$, a$_{1}$, . . . ), let n $^{1}$( u ) = min { n even | a$_{n}$ = L } be the (possibly infinite) first stage in which player 1 left the game, and let n $^{2}$( u ) be the first stage in which player 2 5 2. Endurance games