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# System prepended metadata

title: Johnson & Busemeyer (2016)

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# Johnson & Busemeyer (2016)

risky choice $(x_1,p_1;x_2,p_2;x_3,p_3)$

1. prior attention vetor $z=[z_1,z_2,z_3]^T$
2. transient matrix (3×3) $Q$
3. absorbing matrix $R$
$$
R = 
\left[ 
\begin{matrix}{}
p_1 & 0 & 0 \\
0 & p_2 & 0 \\
0 & 0 & p_3
\end{matrix}
\right]
$$

Attention $W = z(I-Q)^{-1}R$

> Absorbing Markov chain:
> 1) there exists at least an state $j$ that $p_{jj}=1$ (absorbing state) 
> 2) From every state it is possible to reach an absorbing state (no necessarily in one step)
>
> Let $b_{ij}$ be the probability that an absorbing chain will be absorbed in the absorbing state $j$ if it starts in the transient state $i$. Then:
> $$b_{ij}=p_{ij}+\sum_{k} p_{ik}b_{ik} \Longrightarrow B = R + QB \Longrightarrow B = (I-Q)^{-1}R$$ 
> 

