# Roxanne, Sara, and Ted all love to eat cookies, but there’s only one left in the package. No one…

Roxanne, Sara, and Ted all love to eat cookies, but
there’s only one left in the package. No one wants to split the cookie, so Sara
proposes the following extension of “Evens or Odds” (see Exercise S12) to
determine who gets to eat it. On the count of three, each of them will show one
or two fingers, they’ll add them up, and then divide the sum by 3. If the
remainder is 0, Roxanne gets the cookie, if the remainder is 1, Sara gets it,
and if it is 2, Ted gets it. Each of them receives a payoff of 1 for winning
(and eating the cookie) and 0 otherwise.

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(a) Represent this three-player game in normal form, with
Roxanne as the row player, Sara as the column player, and Ted as the page
player.

(b) Find all the pure-strategy Nash equilibria of this
game. Is this game a fair mechanism for allocating cookies? Explain why or why
not.