The Address Resolution Protocol (ARP) spoofing is a form of attack typically used by attackers to cause a denial of service or man-in-the-middle attacks. This attack comes from ARP weaknesses and aims to compromise victims' ARP caches by sending ARP packets containing fake IP-MAC pairs. To overcome ARP spoofing attacks, several approaches leverage the advantages of software-defined networking (SDN) to detect malicious users in the network. To achieve their goal, the SDN controller consecutively checks the characteristics of each ARP packet to ensure its correctness. However, this verification method can lead to latency and congestion at the controller level or render the system unusable for large-scale networks. To address this drawback, we propose a game-theoretic approach to provide an optimal verification method that considers the intelligent attackers' decision-making process during an ARP cache poisoning attempt. This approach is a zero-sum game between the attacker who wants to poison the victims' ARP caches and the defender who must avoid this poisoning. The game model results in mixed-strategy Nash equilibria that identify optimal verification methods to prevent control plane latency and congestion during attacker detection. The results show that an intelligent attacker will refrain from poisoning ARP caches with a high-impact strategy because the defender frequently checks such a strategy. In addition, the attacker's penalty value can deter both rational and irrational attackers from poisoning ARP caches. Simulations in the Mininet simulator have shown that the proposed approach can significantly mitigate control plane latency and congestion during the attacker's characteristic checking.