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Volunteer computing has been used for solving important problems for a few decades and its potential is growing due to increasing number of computers and their growing performance. An important challenge is possible activity of adversaries who try to spoil the computing process on purpose by returning wrong answers. We propose a simple model of such a grid considered from the point of view of the game theory. Players are the project organizer and the adversary who controls many computing nodes. The organizer replicates the tasks sacrificing performance for reliability. The adversary gains some profit by fooling the organizer but loses reputation if revealed. We obtain the optimal mixed strategies and show that the mean gain of the players depend only on the server loss unit, reputation of the nodes, and the size of the grid. Provided that both sides act in the optimal way: organizer's expenses under optimal behaviopr of both sides are not less than in the absense of aversaries, no matter how much work there computers do in order to improve their reputation; adversaries do not earn anything on the average; both gains do not depend on adversaries' gain unit; strategies depend on the penalty and gain value per a node (including fair ones); organizer's strategy depends only on the quantities available to him (in particular, not on the amount of intruded nodes). Finally, using this reputation technique reduces the organizer's losses so they are similar to those in case the number of intruders is known.