In the standard model of dynamic interaction, players are assumed to receive public signals according to some exogenous distributions for free. We deviate from this assumption in two directions to consider an aspect of information structure in a more realistic way. We assume that signals are private rather than public and that each player needs to actively monitor the other player with some costs to observe the other player's behavior. In each stage, each player decides whether to monitor the other player with some costs in addition to which action to take. We first provide a class of strategies which approximate efficiency and describe some of its interesting properties, among them are (1) each player monitors the other player randomly like "random auditing" to reduce monitoring costs and (2) players cheat and monitor at the same time in their cooperative phase. In particular, this implies that cheating may happen (randomly) during collusion for efficiency reason. Then we discuss multi-task partnership games with endogenous monitoring, where two players play H games (tasks) instead of one. The additional twist is that we allow each player to choose freely which tasks to monitor. Our main result is that, how large the monitoring cost per task is, the efficient outcome can be approximated as players become patient when there is a large enough number of tasks. This result suggests that the size of a partnership tends to be large when active monitoring is important.
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