The Minority Game with a deterministic choosing rule of the strategy to play

GABRIEL ACOSTA; INÉS CARIDI; SEBASTIÁN GUALA; JAVIER MARENCO

Villa Carlos Paz, Córdoba

Congreso; XIII LATIN AMERICAN WORKSHOP ON NONLINEAR PHENOMENA; 2013

In this work we propose the MGprior, namely an MG with a new choosing rule of the strategy to play, which takes into account both prior preferences and game information [1]. MGprior clarifies the quasi-periods observed in the sequence of minority sides of the MG [2]. In this new model, agents use their favourite strategy in case of tie, thus generating a deterministic  execution. We have shown that in the Strict Period Two Dynamics (SPTD) regime for even occurrences of the states, the outcomes of the MGprior are a periodic sequence and, moreover, the decisions of the agents are also periodic (strong periodicity). Furthermore, we propose the FSMGprior, a maximal instance of the MGprior in which all the potential agents are present (in the same way that the FSMG was defined in [3]). By exploiting the symmetry of the FSMGprior, we show that the FSMGprior necessarily verifies the Stricted Period Two Dynamics. We prove some general theorems applicable for sequences which meet periodicity and SPTD for even and/or odd occurrences of the states (i.e., not necessarily coming from a minority game). These theorems imply that in the regime in which SPTD is met for even occurences of the states, the sequence of minority sides of the MGprior results periodic with length L = 2kH, and k = 1 when the SPTD is met for both even and odd occurrences. In these cases, we showed that the periodic sequences for the MGprior with parameter m are obtained as the eulerian cycles in the De Bruijn graph of order m. For example, when m = 2, there are two eulerian cycles associated with the periodic sequences 11100010 and 00011101. We have characterized the quasi-periods of the MG for m = 2 as deviations from these eulerian cycles. These deviations sometimes generate inner paths which end in the same eulerian cycle, and sometimes generate outer paths which end in the other eulerian cycle. Finally, we conclude that the fact that the sequence of outcomes is not periodic in the MG is generated by the random breaking of tied strategies, following the original choosing rule of strategies of the MG.[1] G. Acosta, I. Caridi, S. Guala, J. Marenco, Physica A 392, 4450-4465 (2013).[2] S. Liaw, C.Liu, Physica A 351, 571 (2005).[3] G. Acosta, I. Caridi, S. Guala, J. Marenco, Physica A 391, 217-230 (2012).