If you are trying to keep the number of turns per player close to equal while making the order random, you could try a couple things:
1. Randomize the order of turns each round.
For each round of play, create a random ordering for the group of players you have. They all play through, and then for the next round you create another random ordering. So each player plays once each round, but you don't when in the round you'll play. You might get a player playing twice in a row, but only if they happen to be at the end of one round and the beginning of the next.
In Python, you could do that like this:
import random
random.seed()
player_list = ['Daniel Casares', 'pladams9', 'Mario', 'Computer 1', 'Computer 2']
while(!game_over):
# Runs each ROUND
random.shuffle(player_list)
print(player_list)
# Possible output: ['pladams9', 'Computer 1', 'Daniel Casares', 'Computer 2', 'Mario']
2. Randomize the choice of next player in an inversely proportional way.
For each turn, pick a random next player, but make their chance of going inversely proportional to the number of turns they've already had. By simply making a random choice, each player's expected number of turns would already be equal, but this method discourages runaway streaks where one person gets several turns in a row. Streaks are still possible though, so a single player could get multiple turns in a row.
One way in Python:
import random
random.seed()
player_list = ['Daniel Casares', 'pladams9', 'Mario', 'Computer 1', 'Computer 2']
player_turns = {player:1 for player in player_list} # Starting at 1 to avoid division by 0
while(!game_over):
# Runs each TURN
next_player = random.choices(
population=list(player_turns.keys()), # All our players
weights=[1/turns for turns in player_turns.values()], # 1 divided by the number of turns each player has had
k=1)[0] # We only want one choice, so k=1; choices() returns a list, so we just take the first value
player_turns[next_player] += 1 # Increment that player's turn count
print(next_player)
# Possible output: 'Computer 2'
For more information on random.shuffle and random.choices, you can check out: https://docs.python.org/3/library/random.html