Mean

We also consider a mean prior. The mean prior uses the average marginal action probability from all previous timesteps. This corresponds to *pt*[*a*] = ∑*<sup>t</sup>*−<sup>1</sup> *t*=0 *ft* [*a*] *t* , for *t* > 0, and *pt*[*a*] = 0.5 for *t* = 0. This can be seen as belief evolution, where over time, the previous decisions help build the current prior (modulated by *T*) at each stage.

Extreme Priors

As two further examples, we introduce extreme priors (more for visualisation/discussion sake as opposed to being particularly useful). The extreme buy prior corresponds to a strong prior preference for the buy action, *pt*[buy] = 0.99, *pt*[sell] = 0.01, for all *t*. Likewise, the extreme sell case is simply the inverse of the buy case, a strong prior preference for selling, i.e., *pt*[sell] = 0.99, *pt*[buy] = 0.01, for all *t*.

However, the formulations provided above by no means represent an exhaustive set of possible priors. For example, Genewein et al. [56] discuss "optimal" priors, which draws parallels with rate-distortion theory and can be seen as building abstractions of decisions (see Appendix C). Adaptive expectations [57] are discussed in [58–60], where priors could be partially adjusted based on some strength term (*λE*), where the strength term adjusts the contribution from some error. For example, an adaptive prior could be represented as *pt* = *pt*−<sup>1</sup> + *<sup>λ</sup>*(*pt*−<sup>1</sup> − *p*<sup>ˆ</sup>*t*−<sup>1</sup>), where *p*<sup>ˆ</sup>*t*−<sup>1</sup> is the actual known likelihood of actions from the previous time period. With our specific housing market data, we do not have *p*ˆ, i.e., we do not have the true buying and selling likelihoods, but if known, such information could be used to adjust future beliefs, i.e., over time the adaptive priors would adjust decisions based on the previously observed likelihoods (controlled by *λ*). The proposed approach makes no assumption about the forms of prior beliefs, so the ideas outlined above can be incorporated into the method outlined here by adjusting the definition of *pt*.
