Table 1 Winning model parameters and their role in the model

From: D2/D3 dopamine supports the precision of mental state inferences and self-relevance of joint social outcomes

Parameter

Generative purpose

pHI0

Magnitude of the prior that the actions of others are generally motivated by harmful intent (HI) towards the self, p(HI)t=0. Increasing this parameter increases the belief that a partner is motived by harmful intent before any actions are observed.

pSI0

Magnitude of the prior that the actions of others are generally motivated by self-interest (SI) irrespective of the self, p(SI)t=0. Increasing this parameter increases the belief that a partner is motived by self-interest before any actions are observed.

uPri

Uncertainty over priors. Increasing this parameter broadens the prior distribution of both p(HI)t=0 and p(SI)t=0.

Prior

p(HI)t=0 ≈ Bin(HI; pHI0, uPri, NB)

p(SI)t=0 ≈ Bin(SI; pSI0, uPri, NB)

p(HI,SI)t=0 = p(HI)t=0 p(SI)t=0

NB = 9

w0

Intercept of the likelihood matrix, πgen, which calibrates the magnitude of attributional change when a fair or unfair action is made by a partner.

wHI

Impact on beliefs that an outcome (rew) is motivated by harmful intent. Increasing this parameter leads to greater influence of a partner’s behaviour on attributions of harmful intent (belief flexibility).

wSI

Impact on beliefs that an outcome (rew) is motivated by self-interest. Increasing this parameter leads to greater influence of a partner’s behaviour on attributions of self-interest (belief flexibility).

Likelihood

πgen(rew = 0; HI, SI) = σ(w0 + [wHI × HI-δ] + [wSI × SI-δ])

πgen (rew = 0.5; HI, SI) = 1 − πgen (rew = 0;HI, SI)

\(\delta =\frac{\rm{NB}+1}{2}\)

\(\sigma (x)=\frac{1}{1+{e}^{-x}}\)

Update

\({\widehat{p({\rm{HI}},{\rm{SI}})}}^{t}=\frac{{\uppi }_{\rm{gen}}\left({{\mathrm{rew}};\;{\rm{HI}}},{\rm{SI}}\right){p({\rm{HI}},{\rm{SI}})}^{t-1}}{\sum _{{\rm{HI}}^{\prime} ,{\rm{SI}}^{\prime} }{\uppi }_{\rm{gen}}\left({{\mathrm{rew}};\;{\rm{HI}}}^{\prime} ,{\rm{SI}}^{\prime} \right){p({\rm{HI}}^{\prime} ,{\rm{SI}}^{\prime} )}^{t-1}}\,\)

uπ

The consistency with which partners were believed to act in accordance with their character. Higher values reduce consistency, causing a partner’s behaviour to have less impact on beliefs.

Consistency rule

\({p\left({\rm{HI},{SI}}\right)}^{t}\propto {\widehat{p\left({\rm{HI},{SI}}\right)}}^{t\frac{1}{{\textbf{u}}{{\uppi }}}}+\xi\)

\(\xi =0.02/{\rm{NB}}^{2}\,\)

η

Controls the mixture of prior and posterior beliefs used as a starting point for each new encounter. Higher values indicate more reliance on information gathered from the last encounter, rather than reverting to prior beliefs. The product from the below equation, \({\overline{p({\rm{HI},{SI})}}}^{\;t=C}\) replaces p(HI,SI)t−1 when beginning a new encounter.

Change point

\({\overline{p({\rm{HI},{SI}})}}^{\;t=C}={p({\rm{HI},{SI}})}^{t=0} \times \left[1-{\eta }\right]+{p\left({\rm{HI},{SI}}\right)}^{t=C} \times {\eta }\)

C = final action of an other in an interaction

  1. By using model fitting procedures modellers can invert the model to approximate the parameter values that may give rise to the observed data. This includes the hidden, prior beliefs of each participant given the variance and magnitude of observed attributions. Using fitted parameter values to simulate each participant allows for generation of pseudo-experimental data—in this case, an agent’s reported intentional attributions, which we can directly compare with the real data. This also approximates the prior beliefs of each participant given the variance and magnitude of observed attributions. NB, number of bins discretizing the variable represents each attribution (in this case each distribution comprises nine bins); Bin, binomial distribution with an added precision parameter, that is, in the case of HI: p(HI)t=0 ≈ Bin(HI; pHI0, uPri, NB) = p(HI)t=0 ≈ B(HI; pHI0, NB)\({\,}^{1/{u}_{\mathrm{Pri}}}\).
  2. The bold text indicates the free parameters of interest that contribute to the equations.
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