Figure 2 | Scientific Reports

Figure 2

From: Quantum stochastic walks on networks for decision-making

Figure 2

From the tree to the network.

A first example step by step. We define an arbitrary toy-problem in which a decision-maker (the column-agent) chooses one option among S = {L, M, R}, with the payoff being a function of the state of the world Ω W = {U, D}. The numbers in panel (a) represent the profit for the column-agent given her choice of action and the state of the world that is realized, in the form of a payoff matrix. In panel (b) we show the normative representation of the sequential decision-making process as a tree. In this setting, the column-agent first makes her own belief about the state of the world and then, she optimizes her action as a response to her belief. In panel (c) we model the same problem with a networked topology. The numbers in the links represent the entries {πij} of the matrix Π(λ). They are weighted according to Eq. (3), with λ = 1 and using the information in panel (a). These two connected components define the dynamical comparison between alternatives, for each possible state of the world. This process happens simultaneously with the formation of beliefs through the matrix B (green connections), as stated in Eq. (4). For basic illustrative purposes, we do not need to specify if the state of the world is a random variable, or the choice of a row-player whose payoff rule is unknown for the column-player. We elaborate further on this issue and its influence on the definition of the matrix B in the main text.

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