For t=1 we get the usual maximin (“pessimism”), for t=0 we get maximax (“optimism”) and for other values of t we get something in the middle (we can call “t-mism”).
It turns out that, in some sense, this new decision rule is actually reducible to ordinary maximin! Indeed, set
μ∗t:=argmaxμEμ[U(a∗t)]
Θt:=tΘ+(1−t)μ∗t
Then we get
a∗(Θt)=a∗t(Θ)
More precisely, any pessimistically optimal action for Θt is t-mistically optimal for Θ (the converse need not be true in general, thanks to the arbitrary choice involved in μ∗t).
To first approximation it means we don’t need to consider t-mistic agents since they are just special cases of “pessimistic” agents. To second approximation, we need to look at what the transformation of Θ to Θt does to the prior. If we start with a simplicity prior then the result is still a simplicity prior. If U has low description complexity and t is not too small then essentially we get full equivalence between “pessimism” and t-mism. If tis small then we get a strictly “narrower” prior (for t=0 we are back at ordinary Bayesianism). However, if U has high description complexity then we get a rather biased simplicity prior. Maybe the latter sort of prior is worth considering.
One of the postulates of infra-Bayesianism is the maximin decision rule. Given a crisp infradistribution Θ, it defines the optimal action to be:
a∗(Θ):=argmaxaminμ∈ΘEμ[U(a)]
Here U is the utility function.
What if we use a different decision rule? Let t∈[0,1] and consider the decision rule
a∗t(Θ):=argmaxa(tminμ∈ΘEμ[U(a)]+(1−t)maxμ∈ΘEμ[U(a)])
For t=1 we get the usual maximin (“pessimism”), for t=0 we get maximax (“optimism”) and for other values of t we get something in the middle (we can call “t-mism”).
It turns out that, in some sense, this new decision rule is actually reducible to ordinary maximin! Indeed, set
μ∗t:=argmaxμEμ[U(a∗t)]
Θt:=tΘ+(1−t)μ∗t
Then we get
a∗(Θt)=a∗t(Θ)
More precisely, any pessimistically optimal action for Θt is t-mistically optimal for Θ (the converse need not be true in general, thanks to the arbitrary choice involved in μ∗t).
To first approximation it means we don’t need to consider t-mistic agents since they are just special cases of “pessimistic” agents. To second approximation, we need to look at what the transformation of Θ to Θt does to the prior. If we start with a simplicity prior then the result is still a simplicity prior. If U has low description complexity and t is not too small then essentially we get full equivalence between “pessimism” and t-mism. If t is small then we get a strictly “narrower” prior (for t=0 we are back at ordinary Bayesianism). However, if U has high description complexity then we get a rather biased simplicity prior. Maybe the latter sort of prior is worth considering.