This is a variant of the Nash equilibrium. Let's say that there are 3 prizes: A Ferrari, a diamond watch, and a new boat. There are 6 players. 3 players with a motive while 3 players with another motive, but all 6 players are playing for themselves: 3 players, for example, are wearing red shirts. The people wearing red shirts have an objective: to pick the prize that they think no red shirt or purple shirt will pick. The people wearing purple shirts have a different objective: to pick the same prize as someone wearing a red shirt so they can claim their chosen prize

The Scenarios:

- If a Purple Shirt picks a prize (Ex. diamond watch) no one else picks, they can't claim the prize (Ex. diamond watch)
- If a Purple Shirt picks a prize (Ferrari, for example) that at least one other Purple Shirt picks (Ferrari, for example), they collide and the Purple Shirts that picked the prize(Ferrari. for example) go home with nothing, regardless of how many Red Shirts picked the prize (Ferrari, for example).
- If a Purple Shirt picks a prize (Boat, for example) that at least one Red Shirt picks(Boat, for example), then that means the Purple Shirt wins the prize assuming that the Purple Shirt is the only Purple Shirt picks the prize(Boat, for example).
- If a Red Shirt picks a prize (Ferrari, for example) picks a prize that no other Red Shirt chooses(Ferrari, for example), and assuming that the amount of Purple Shirts that choose the prize(Ferrari, for example) follows the interval (2, ∞), then the Red Shirt rightfully claims the prize(Ferrari, for example). This means that if at least one other Red Shirt picks the same prize as the Red Shirt did, the Red Shirt goes home with nothing, and regardless of how many Purple Shirts picked the same prize.
- When it comes to collisions, purple shirts collide first(Which means that if 2 or more Purple Shirts pick the same prize, they go home with nothing), then Red Shirts(Which means that if 2 or more Red Shirts picked the same prize, that means that the Red Shirts come home with nothing.) We also assume that we're using whole non-negative numbers and not decimals

We assume that they have folders and write their answers down on paper, and can't communicate with anyone during the game. They hand the paper to the judge/referee. The following questions are: The formula for increasing the amount of players? The formula for increasing the amount of prizes(How many prizes, not their value)? The formula for both increasing the amount of players and prizes?

This is my first question on mathoverflow.net, so it may contain errors. Citations Nash Equilibrium(mathoverflow.net) Monty Hall Problem(Einstein's Riddle by Jeremy Stangroom) Simple(?) game theory