Incomplete Information Games with Transcendental Values

In a repeated zero-sum two-person game with incomplete information on both sides, the asymptotic value is defined as v = lim_n→∞v_n, where v_n is the value of the game with n repetitions. It is shown here that v may be a transcendental number even for games in which all parameters defining the game are rational. This is in contrast to the situation in stochastic games where by the result of Bewley-Kohlberg (Bewley, T., E. Kohlberg. 1976. The asymptotic theory of stochastic games. Math. Oper. Res.1 197–208.) v is algebraic. This indicates a fundamental difference between stochastic games and repeated games with incomplete information.