With many thanks to Beluga and lackofcheese.

When trying to decide between SIA and SSA, two anthropic probability theories, I concluded that the question of anthropic probability is badly posed and that it depends entirely on the values of the agents. When debating the issue of personal identity, I concluded that the question of personal identity is badly posed and depends entirely on the values of the agents. When the issue of selfishness in UDT came up recently, I concluded that the question of selfishness is...

But let's not get ahead of ourselves.

This post considers ambient control in a more abstract setting, where controlled structures are not restricted to being programs. It then introduces a notion of preference, as an axiomatic definition of constant (actual) utility. The notion of preference subsumes possible worlds and utility functions traditionally considered in decision theory.

Followup to: Controlling Constant Programs.

In the previous post I described the sense in which one program without parameters (the agent) can control the output of another program without parameters (the world program). These programs define (compute) constant values, respectively actual action and actual outcome. The agent decides on its action by trying to prove statements of a certain form, the moral arguments, such as [agent()=1 => world()=1000000]. When the time is up, the agent performs the action associated with the moral argument that promises the best outcome, thus making that outcome actual.

Let's now move this construction into a more rigorous setting. Consider a first-order language and a theory in that language (defining the way agent reasons, the kinds of concepts it can understand and the kinds of statements it can prove). This could be a set theory such as ZFC or a theory of arithmetic such as PA. The theory should provide sufficient tools to define recursive functions and/or other necessary concepts. Now, extend that theory by definitions of two constant symbols: A (the actual action) and O (the actual outcome). (The new symbols extend the language, while their definitions, obtained from agent and world programs respectively by standard methods of defining recursively enumerable functions, extend the theory.) With new definitions, moral arguments don't have to explicitly cite the code of corresponding programs, and look like this: [A=1 => O=10000].

This post explains the sense in which UDT and its descendants can control programs with no parameters, without using explicit control variables.

Related to: Towards a New Decision Theory, What a reduction of "could" could look like.

Usually, a control problem is given by an explicit (functional) dependence of outcome on control variables (together with a cost function over the possible outcomes). Solution then consists in finding the values of control variables that lead to the optimal outcome. On the face of it, if we are given no control variables, or no explicit dependence of the outcome on control variables, then the problem is meaningless and cannot be solved.

Consider what is being controlled in UDT and in the model of control described by Vladimir Slepnev. It might be counterintuitive, but in both cases the agent controls constant programs, in other words programs without explicit parameters. And for constant programs, their output is completely determined by their code, nothing else.

Let's take, for example, Vladimir Slepnev's model of Newcomb's problem, written as follows:

def world(): 
  box1 = 1000 
  box2 = (agent() == 2) ? 0 : 1000000 
  return box2 + ((agent() == 2) ? box1 : 0)

The control problem that the agent faces is to optimize the output of program world() that has no parameters. It might be tempting to say that there is a parameter, namely the sites where agent() is included in the program, but it's not really so: all these entries can be substituted with the code of program agent() (which is also a constant program), at which point there remains no one element in the program world() that can be called a control variable.