# Nash equilibrium at another level

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

With the given parameters, the Nash equilibria are exactly those situations where the three purple players pick each a different price. This ensures that the red players can't win anyway, so they'll do whatever.

To see this, we first observe that a purple player would never change to a prize another purple player already picked. If there is a price no purple player has chosen, but a red player, a purple player not currently winning would change to it. If there is a price no player has currently chosen, a red player not currently winning would change to it.

If we go for 4 prizes, and still 3 players of each colour, there is no Nash equilibrium. There is at least one prize not chosen by a purple player. Such prizes are the only options for a red player to win, so there will be at least one red player picking one of them. This in turn means at least one purple player is losing, and would thus switch over to the unclaimed-by-purple prize.

This generalizes as follows: As long as there are at least as many purple players as prizes, there is a Nash equilibrium, and every prize being chosen by at least one purple player suffices to be a Nash equilibrium. If there are at least as many red players as prizes, there also is a Nash equilibrium, and having red and purple players "spread out as much as possible" works. If there are fewer purple players than prizes, and at least one red player but fewer than prizes, there is no Nash equilibrium.

• Please do not answer off-topic questions. Commented Nov 25, 2022 at 13:46
• The third paragraph assumes that the purple players all pick different options. What if two purple players pick the same prize? Or all three of them pick the same prize? What are the odds of a Red Player winning in those scenarios? Overall, the purple players would have an advantage. We also assume that no player is allowed to change their answer. Plus, what if the collision rule was changed into: Red players collide first, then purples? For example, purple players need exactly one red and no other purples. Commented Nov 25, 2022 at 16:40
• @HassBoyouk No, the third paragraph makes no particular assumption. With 3 players each and 4 prizes, there is just always a player currently losing who could win instead, hence no Nash equilibrium.
– Arno
Commented Nov 26, 2022 at 23:24