If I was doing that project I would have the Master poll each of the slaves in turn at regular intervals - maybe 10 times per second. That avoids transmission collisions.
The master could include its own millis() value in each message and the slave can use that to get a near universal time for all slaves. Then the slave records the time (with correction) when its own button was pressed and sends that to the master when next requested. The master can then review all the messages and decide which button was pressed first.
See the seconds example in this Simple nRF24L01+ Tutorial
...R
PS ... it is important to remember that the people playing the game will have no data about who pressed his/her button first with which to contradict the Arduino.