The Full Strategy Minority Game

Abstract

The Full Strategy Minority Game (FSMG) is an instance of the Minority Game (MG) which includes a single copy of every potential agent. In this work, we explicitly solve the FSMG thanks to certain symmetries of this game. Furthermore, by considering the MG as a statistical sample of the FSMG, we compute approximated values of the key variable {\sigma}2/N in the symmetric phase for different versions of the MG. As another application we prove that our results can be easily modified in order to handle certain kind of initial biased strategies scores, in particular when the bias is introduced at the agents' level. We also show that the FSMG verifies a strict period two dynamics (i.e., period two dynamics satisfied with probability 1) giving, to the best of our knowledge, the first example of an instance of the MG for which this feature can be analytically proved. Thanks to this property, it is possible to compute in a simple way the probability that a general instance of the MG breaks the period two dynamics for the first time in a given simulation.Comment: To appear in Physica

Properties of interaction networks underlying the minority game

“…Ho et al redefined a Gini index for the M G [19] which showed that whenever the reduced variance takes its minimum value, the inequality among the agents is maximized. In a recent work [15], we studied crowd region and presented a calculation of σ that matched the simulation results. This calculation was based on the FSMG (Full Strategy Minority Game), an imaginary instance of the MG where all the possible agents (let call N ) participate in the game.…”

“…In [20] we showed that a MG with a deterministic rule to play in case of a tie of the strategies for which the SPTD (Strict Period Two Dynamics, the P T D with probability equal to 1) for even occurrences of the states is met, is a periodic game. For example, it applies to the M G prior where agents have an a priori favorite strategy to use in case of tie.…”

“…, N , the pair (e i 1 , e i 2 ) represents the set of strategies assigned to the agent i. We define a realization E of the game as a pair E = {S E , I}, where S E = {μ 1 ,μ 2 , • • • } is a sequence of states (generated by any updating rule), and I is an instance of the M G. We will follow this notation proposed in previous works ( [15] for example).…”

“…Period Two Dynamics (PTD) in the sequence of the minority sides was first observed within this region by Savit et al [16] . The dynamics established in this region enables the number of virtual points assigned to the strategies to be reduced to a limited set of cases [15,18], which facilitates the analytical treatment of the model. One of the reasons why the Minority Game has attracted attention is that in certain region of the parameters, the reduced variance is smaller than that obtained for a game in which each of the agents randomly chooses between the two sides (see Figure 1), meaning a better use of the resources by the population.…”

“…Once the FSMG was solved, we obtained results for σ for the MG (in the region in which PTD is valid) by sampling N agents from the N agents of the FSMG. Additionally, in a more recent work, we analyzed two well-known properties: quasiperiodicity of the sequence of outcomes of the game and Period Two Dynamics, by studying the sequence of minority sides through a network and by using the FSMG to shed light on these properties [20].…”