Wireless sensor networks (WSNs) have become an important and promising technology owing to their wide range of applications in disaster response, battle field surveillance, wildfire monitoring, radioactivity monitoring, etc. In WSNs, routing plays a significant role in delivery latency, energy consumption, and packet delivery ratio. Furthermore, as these applications are used in critical operations with limited irreplaceable batteries, routing protocols are required to be flawless as well as energy efficient. The dynamic environment also requires intelligent and adaptive routing. Game theory is widely used for designing routing protocols in WSNs to achieve not only reduced energy consumption but also increased packet delivery ratio. The core features of efficiently designed game theory-based routing protocols include optimal cluster head selection in hierarchical routing, energy-efficient and delay-aware route discovery, fault-tolerant data delivery, and coalition forming and grouping among nodes for stringent data transfer. In this paper, different routing protocols based on various types of games are extensively reviewed, which have been reported so far for improving energy consumption, delay, route establishment time, packet delivery ratio, and network lifetime. The different game theory-based routing protocols are qualitatively compared with each other in terms of major features, advantages, limitations, and key characteristics. For each protocol, possible applications and future improvements are summarized. Certain important open concerns and challenges are also discussed, along with future research directions.