In many-participant evolutionary game theoretic models pair interactions are described by 
many-parameter matrix games. Recent studies have indicated that these matrices represent 
combinations of four basic types of interactions: self- and cross-dependent, 
coordination-type, and cyclic games. Games with cyclic components are not potential 
games. Strategy distributions of multiagent evolutionary potential games follow the 
Boltzmann distribution if the evolution is governed by a properly chosen rule, which means 
that the tools and methods of statistical physics can be used to investigate the 
macroscopic behaviour of such systems. 
We study simple multiagent logit-rule-driven evolutionary potential games on a square 
lattice, and apply some of these methods to investigate their equilibrium properties and their 
phase transitions. We find that in elementary coordination games, which closely resemble 
Ising-type models, the order of the phase transition is determined by the number of 
available strategies. The introduction of different self-dependent game components can be 
used to change the critical temperature and the order of the phase transition, and it can 
even abolish the transition.