Reputation mechanisms of a network or p2p applications can be compromised by a large number of fake or stolen identities recommending higher trust values of a malicious node. A high trust value classify such malicious or selfish nodes as legitimate nodes and help them to gain more access to network resource. It is detrimental to network security and makes a network vulnerable to Sybil attacks. In this paper, a game theoretical defense strategy based on zero sum and imperfect information game is proposed to discourage sybil attacks which exploit underlying reputation mechanism of a network. An attacker tries to maintain an optimum level of trusts of its sybil identities in the network in an intention to launch a successful sybil attack and obtain an attack benefit. Simultaneously, the attacker has to incur some cost to maintain trustworthiness of its identities. A rational attacker analyses associated costs and benefits prior to launch an attack. The proposed mechanism uses this idea and makes the attack costly by setting a global trust threshold for nodes or identities to remain trustworthy and active in network. In this work, utility functions of attacker and defender are defined. Nash Equilibrium strategies of the game are discussed and experiments are carried out which suggest the effectiveness of proposed mechanism to enhance network security against sybil attacks.