Here we explore an adversary-flavored alternative to Bayesian decision making. It's not supposed to supplant Bayes; it's just an interestingly different formulation of what it means to act well and of what data we need to determine good action. An optimistic outcome would be that with further study we find intuitive resonance with this framing in some of our daily decision making.
The idea is simply to specify world-salience in units of dollars, rather than of probability. Whereas Wald's minimax framework constructs a Bayes prior from a given cost function, our formulation eschews Bayes priors altogether, using a different concept, one not depending on the cost function, to express world-salience.
I thank Greg Wornell, Jimmy Koppel, and others for inspiring this post.
This aggregation does not need to treat the worlds symmetrically: intuitively, some worlds are more salient to our decision problem than others. A Bayesian models salience via a prior \(p:\Delta(W)\subset{}W\to\RR\); then an action's aggregated cost is \[ C_p(a) \triangleq \sum_{w:W} c(a)(w) \cdot p(w) \] We'll today study a different procedure --- call it the method of the Azidians --- that models salience via an offset \(o:W\to\RR\) and defines an action's aggregated cost as \[ C^o(a) \triangleq \bigvee_{w:W} c(a)(w) - o(w) \] Up to a merely conventional sign on \(o\), we've simply translated a formula for the semiring \((\RR, +, \cdot)\) to a formula for the semiring \((\RR, \vee, +)\). ●● Pure mathematicians call such translations (usually with opposite sign convention so that we use min instead of max) tropical.
Azidians make decisions in minimax fashion. They pick an action based on which an adversary chooses a world; the catch is that when the adversary chooses world \(w\), they must pay \(o(w)\) dollars to the Azidian. Thus, the adversary is disincentivized from picking large-\(o\) worlds. Large-\(o\) worlds are less salient to decision-making.
Wald --- Statistical Decision Functions, \(\S1.4\) --- 1950