Spies in the minority game

We study the effects of the existence of another type of agents, called spies, in the minority game (MG). Unlike the normal agents in the MG, the spies do not carry any strategy. Instead, they decide their action by scouting some normal agents and take the minority action of the spied group. For a few spies and when there is useful information in the normal agents' actions, the spies can avoid the crowd effect of the normal agents and win more readily. When information becomes less useful and when more spies are present, the spies' crowd effect hurts the success rate of the spies themselves, and the normal agents could have a higher success rate than the spies. More spies actually assist more normal agents to win, as the spies also provide more winning quotas. This leads to a nonmonotonic behavior in the total success rate of the population as a function of the fraction of spies.