Quoted from jake35:if each game pulls 2.5 amps, a 20 amp single pole should be sufficient shouldn't it?
Most late model and modern games do not pull 2.5 amps, not always even in attract mode. They average 3-3.5 amps.
Some games can reach over 7 amps in multiball.
Consider the age of the machines as well, as this can also make a difference in the "steady state" of electricity.
You have under gauge wall wire or an older home, if you are damaging breakers.
You are correct to call an electrician, because if your circuit breaker failed, there are more problems than running 4 pinball machines on a single 15 amp outlet.
There is not enough detail to provide a diagnosis, but the baseline is correct as you cannot increase breaker size while using the same wire gauge unless you want to start a wall fire in your home.