Ann, Brian and Carla decide to bring some excitement into their lives with
the help of their uncle Sam. Each of them is going to secretly write a number
chosen from the set 2,3,4 on a piece of paper and hand it to Sam. Sam
computes half the average of the three numbers, call it a, and determines the
winners as those whose number is closest to a. For example, suppose that
Ann writes 2, Brian writes 2 and Carla writes 4. Then a = 1/2 * (( 2+2+4)/3) = 1.33
and the winners are Ann and Brian. Sam then hands out money as follows:
nothing to the loser(s), $2 to the winner if there are no ties and $1 to each of
the winners if there are ties. Thus in the above example, Ann and Brian get
$1 each and Carla gets nothing.
(a) Represent this game as a strategic-form game (let Ann choose the rows, Brian the columns and Carla the matrices). Assume that each player is selfish and greedy (i.e. only cares about how much money he/she gets and prefers more money to less).
(b) For every pair of strategies of Brian, x and y, establish whether x dominates y (specify whether it is weak or strict dominance), y dominates x
or neither.
(c) Does Brian have a (weakly or strictly?) dominant strategy?
(d) Is there a dominant-strategy equilibrium?