Question

There is 3 people who want to eat the only one cookies left in the package. No one wants to split the cookie, so Uday proposes the following solution to who gets to eat it. On the count of 3 each one will show either one or two fingers, they will add up all the fingers shown and divide by 3. If the remainder is 0, Joseph gets the cookie, if the remainder is 1, Uday gets it, and if the remainder is 2, Vanya gets the cookie. Whoever gets the cookie gets a payoff of 1, while the others get payoffs of 0.

a) Represent this game in normal form.

b) Solve for all pure strategy Nash equilibrium for this game.

c) Is this game zero sum? Explain.

d) Is this game Paretto optimal? Explain.

Answer 1

a.

Assuming Joseph as the row player, Uday as column player, and Vanya as page player.

Three players normal form of the game is as follows:

When the remainder is 1:

		Uday
		1	2
Joseph	1	(1,0,0)	(0,1,0)
2	(0,1,0)	(0,0,1)

When the remainder is 2:

		Uday
		1	2
Joseph	1	(0,1,0)	(0,0,1)
2	(0,0,1)	(1,0,0)

So, when the remainder is 0, then Joseph's payoff is 1 taking 1st place in the cell; if the remainder is the 1, then Uday will get the payoff as 1, which represented by 2nd place in the cell, and similarly, if the remainder is 2 then Vanya's payoff is 1 taking 3rd place in the cell.

b.

Pure strategy in this game is (0,1,0) and (0,0,1) where Uday and Vanya both have an equal chance of winning, whereas there are only two possible when Joseph has a chance to win. So, Joseph has less chance of winning. Vanya has a dominant strategy of showing one finger, and Sara has a dominant strategy of showing two finger

c.

Yes, the game is a zero-sum game as there is only one cookie, whoever wins or losses. Another two's loss is one's gain.

d.

Pareto optimal solution refers to when there is no way to improve any objective without lowering at least one other objective. So, this game is Pareto optimal, as, to have a higher chance for Vanya or Uday to win, the chance of Joseph should have to be less.

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