email to jam about dietrich-list [2022-03-21 Mon]

Dear Jam,

I hope the past 20 days have been good to you. How've you been?

Vipassana went well, and they did indeed feed me enough. (At any rate it was enough not to feel hungry or faint. The day does involve a lot of sitting still.)

Here is what happened in the past week, and in part in my brain in the ten days preceding:

misunderstanding and resolution

• I realized I'd misundestood part of the Dietrich-List paper: I'd taken the alphabet of properties as constituting outcomes, but DL constructs them in the reverse way, saying that alternatives/outcomes constitute properties. This is much nicer for treating properties as a structure-preserving "intermediate representation" when going from motivational states to outcomes or vice versa.

  • It also solves "for free" the issue I was facing in building a player's preferences over alternatives from their preferences over properties: I was assuming that preferences over properties would be acyclic, but this is the exact assumption the paper is trying to dispense with :facepalm:

• I reread the axiomatization to be sure I understood it correctly this time, and wrote a note and a distinguishing scenario:

  The model discussed by this paper is one where the outcomes of a game each have various properties; and player preferences operate over these properties, rather than the outcomes themselves.

  In addition, players filter their preferences on the basis of which properties they may consider relevant. That set of relevant properties is called a player's motivational state. Each player has a set of possible motivational states.

  Two axiom systems are presented: 1+2, and 1+3.

  1. motivational states that contain no properties that discriminate between two outcomes do not generate preferences that distinguish between the outcomes.

  2. for motivational states A,B, A \subset B, if B/A contains no properties that are present in two outcomes, then B and A agree with respect to those outcomes.

  3. for motivational states A,B, A \subset B, if B/A contains no properties that discriminate between two outcomes, then B and A agree with respect to those outcomes.

  A scenario that distinguishes the systems:

  Suppose motivational states A \subset B \subset C; properties p_1 \in A, p_2 \in B/A, p_3 \in C/B; outcomes x \in p_1, y \notin p_1, x,y \in p_2, x \in p_3, y \notin p_3.

  • Under 1+2 system, there can exist a player preference family such that x \preceq_{A} y \land y \preceq_{B} x.

  • Under 1+3 system this would not be possible; p_2 contains both outcomes, which mean it does not distinguish between them. But x \preceq_{B} y, y \preceq_{C} x is possible, because p_3 does distinguish between the outcomes.

  Two theorems are presented:

  1. If \mathcal{M} is intersection-closed, an agent's family of preference-orders satisfies 1+2 iff it is "property-based".

  2. If \mathcal{M} is subset-closed, an agent's family of preference-orders satisfies 1+2 iff it is "property-based in a separable way", i.e. the ranking over the powerset of all properties exhibits independence of irrelevant alternatives. S_1 \ge S_2 \text{ iff } S_1 \cup T \ge S_2\cup T

A topology over properties

We have been discussing making the set \mathcal{M} of a player's possible motivational states a topology, in order to use it as an "organizing principle" that lets us start engaging with attaching meanings to the properties that have some internal structure. I think I understand some of why now: a topology over properties might let us treat the properties as sentences in a language. The semantics of the language ought to follow from its motivating example, and then dictate what the structure even ought to be; but its at least allowing unions and intersections, and having a Top and a Bottom, seem like good starting assumptions.

mathcal{M} being a topology is ensured by axiom 3, and under axiom 2 it's actually a pi-system? Apparently? Which is to say, Wikipedia tells me that this is the name for a subset of a powerset that is intersection-closed but not union-closed.

The function to output the topological closure (is this valid phrasing?) of a given subset of a powerset is written. What relies on it holding? I remain unsure, and am going to try to approach the problem from the angle of the motivating examples instead of tooling around here without, well, motivation.

Ideas for semantics

Some ideas for the semantics we can assign to this system:

Properties of outcomes as intentions

I've been reading G.E.M. Anscombe's Intention, and thoguht of this.

Agents select for outcomes, operating from inside certain contexts. Properties that are derivable by applying DL's theorems can then be a construction of "what the agent was going for."

What is the organizing principle of the properties that then tells us the relationship between aim A and aim B? That structure is then the structure of intention.

We can take Anscombe as a first step to understanding how to assign semantics to motivational states, outcomes, and properties respectively; and what structure over properties to begin looking at (and what might follow as a result.)

Components of organizing principles that suggest themselves, from our understanding so far:

• epistemics. the motivational states are indications of what agents know about the alternatives. thus, the properties are factual assertions about the outcomes that matter to agents.

• the properties of outcomes can be factual assertions that describe the extensional game further

  • assertions about reachability.

  • assertions stronger than reachability

  • conditions that help us answer the question "given what they did, what did they want?" ("can we derive it? can we prove it?")

  • conditions that help us assert (or, less likely, disprove) that "given that an agent did that, they must have wanted something that is congruent to the structure presented." This gives us a falsifiable assertion that I can throw a dataset at.

• relevance. the motivational states represent the playing-out of a resource bound on reasoning, which we can understand to be true based on how many properties are in each state - which is monotonic to how many outcomes the agent can distinguish between.

I will send my summary notes on Anscombe once I have made a fair copy. Their current form is, in several senses, unreadable chicken scratch.

reasons why players might move from one motivational state to the other:

The condition kept in mind while generating these: the reasoning must be expressed in terms already defined in the game structure, or derived from them.

• The motivstates represent what players know about the outcomes - updating to add a property is adding a fact to the universe of consideration.

• The motivstates represent "relevance" - updating to add a property necessitates dropping another one, and players optimize for reaching outcomes that would be preferred under a maximal motivstate that they can't actually ever hold.

• The motivstates represent information about the game itself:

  • Reachability - it would be a simplification of the game tree

  • Whether some future game is gluable onto an outcome.

  • A fact about the other players (what type would that fact have?)

    • We are trying to assert that motivstates are things that players can infer about other players. What would they be trying to find out?

      • Easy answer: what the other players are going to do. So, a partial strategy? Something of the form "p \in M_{Alice} iff Alice expects Bob to move left at some point in future play".

      • Another easy answer: what players want. So: "f(p) \in M_{Alice} iff Alice thinks Bob is ranking some property p higher than all other properties."

  • Verbs - intentions that connect moves to ends, that can be inferred and matched to signalling moves over the course of play.

• The motivstates represent concerns that are apt to some environmental fact: e.g. the season dictating the parameters for selecting fruit. In monsoon you must look for thick skins, in summer you must look for high water content, etc. Interesting, because it's a way to show that the preference cycle might be entrained by an environmental cycle, and therefore rational in context.

  • This is called a zeitgeber. Diseasonality - by Scott Alexander - Astral Codex Ten is where I first head about this; there's a LARGE body of literature attached to the idea of a zeitgeber, and I should read it if and when it seems like a good idea to. Modeling it using a similar kind of intransitivity to what we're playing with here would be pretty cool.

    • Spitball intuition: size of motivstate tracks how "tight" your curves can be on a cycle - shifting sands do not make for fine-grained preferences.

      • something something lotka-volterra

        • I do need to spend some time studying this magic when I can.

CONCRETE QUESTION: let properties be words in the builder-assistant game. How many rounds of play does it take before the epistemic game catches up to common knowledge of which properties constitute a player p's motivational state?

Let there exist a turn based two-player game.

• We begin at state R

  • At R, A, and B, one can either

  • play a, which takes one to state A

  • play b, which takes one to state B

  • play s, which takes one to state S, where both players must play either x or y

    • if both players select x or both select y, we go to state WIN; each get one point.

    • else we go back to R.

This can be extended over some arbitrary alphabet of states-and-moves. Call that alphabet the builder-assistant language.

I think there exist property-based agents who have a winning strategy at this game.

Once we have the collaborative picture, then we can try to break it.

• can we build a simulation in which siloing happens?

  • I'm defining "siloing" as "several coalitions achieving coordination internally takes fewer steps to reach than the grand coalition achieving coordination."

• can we define a complexity measure according to which a population speaking a language exhibits siloing, or other such effects, past a threshold in that measure?

  • My naive first guess for a complexity measure is "number of tokens in the motivstate."

  • My second guess is "minimum number of steps needed to achieve common knowledge of everybody's motivstate in a coalition."

• what is the "shape" of the game that exhibits the minimum number of steps needed to achieve common knowledge?

If I recall correctly you have mentioned you and Parkih 2004 as a referent for agents agreeing upon a protocol of further discourse. I will go read that this week.

the reason DL have presented an axiomatization is that axioms are falsifiable statements.

They serve as the condition A in the guarded statement "if A holds in universe U then model M holds in universe U".

We can check if A holds per falisfiability, and elevate the rest of model M to hold also if it does.

SO: I ought to find a dataset of preferences to test for the axioms suggested. I will go hunting. Time to strengthen our semiotics.

This has been last week and part of the ten days preceding, adapted from my notes, the taking of which I've reapplied myself to. I hope I'm on a useful track (or at least a few useful tracks out of the many I seem to be on - convergence seems nigh, but I can't be sure.) Let me know what you think, whether here or in call. Speaking of: is this Friday good for you or would an alternate time be better?

See you soon, and I hope to find you well.

Sahiti