This paper introduces state abstraction for two-player zero-sum Markov games (TZMGs) where the payoffs for the two players are determined by the state representing the environment and their respective actions, with state transitions following a Markov decision processes. For example, in games like soccer, the value of actions changes according to the state of play, we should describe them as Markov games. In TZMGs, the more the number of states becomes, the more difficult computing the equilibrium becomes. Therefore, we abstract the states of TZMGs and examine the performance. State abstraction reduces the number of states by treating multiple different states as a single state, and there is a substantial body of research on finding optimal policies for Markov decision processes using state abstraction. This study extends the state abstraction for MDPs to Markov games. In this case, the game with state abstraction may yield different equilibrium solutions from those of the ground game. To evaluate the equilibrium solutions of the game with state abstraction, we derived bounds on duality gap, which represents the distance from the equilibrium solutions of the ground game. Finally, we demonstrate our state abstraction with Markov Soccer, compute equilibrium policies, and examine the results.

This paper proposes a new coherent detector that exploits the real zero (RZ) concept for signal recovery. The detector recovers the complex envelope consisting of in-phase (I) and quadrature (Q) components of linear quadrature amplitude modulation (QAM). The received signal before amplification is processed by RZ conversion, after which the converted signal can be amplified by a severely distorted amplifier. The RZ conversion combines the modulated received signal with a sinusoidal wave whose frequency is slightly outside the modulated frequency band, and the resultant signal is amplified to a saturated level. Next, the IQ detection applies the RZ concept to the IQ signal recovery. It utilizes a time to digital converter (TDC) for RZ timing estimation. A linear interpolation type TDC is introduced. Computer simulations demonstrate the effectiveness of the proposed detector. It is shown that error vector magnitude (EVM) of the recovered complex envelope is around -30 to -40 dB for a narrowband signal.