Level Three in a Three-Player Beauty Contest

Three players each name a real number in [0, 100]. The target is 2/3 of the average of all three numbers, including your own, and the player closest to the target wins. A level-0 player names 50. For k ≥ 1, a level-k player believes both opponents are level-(k−1) and names the number that equals the target exactly (such a number beats both opponents outright, since they name the same number). What number does a level-3 player name? Give an exact fraction.

• nash equilibrium

Source: Beauty-contest game after Nagel (1995) and Keynes; level-k reasoning after Stahl and Wilson, Nagel. Three-player variant with own guess included written for AxiomIQ.