Three friends (Julie, Kristin, and Larissa) independently go shopping for dresses for their high-school prom. Upon reaching the store, each girl sees only three dresses worth considering: one black, one lavender, and one yellow. Each girl furthermore can tell that her two friends would consider the same set of three dresses, because all three have somewhat similar tastes.
Each girl would prefer to have a unique dress, so a girl's utility is zero if she ends up purchasing the same dress as at least one of her friends. All three know that Julie strongly prefers black to both lavender and yellow, so she would get a utility of 3 if she were the only one wearing the black dress, and a utility of 1 if she were either the only one wearing the lavender dress or the only one wearing or the yellow dress. Similarly, all know that Kristin prefers lavender and secondarily prefers yellow, so her utility would be 3 for uniquely wearing lavender, 2 for uniquely wearing yellow, and 1 for uniquely wearing black. Finally, all know that Larissa prefers yellow and secondarily prefers black, so she would get 3 for uniquely wearing yellow, 2 for uniquely wearing black, and 1 for uniquely wearing lavender.
Provide the game table for this three-player game. To make your work easier to grade, please make Julie the row player, Kristin the column player, and Larissa the page player. Identify any dominated strategies in this game, or explain why there are none. What are the pure-strategy Nash equilibrium to this game?