This felt to me like it was something that could be reframed through category theory but I’m not the best at explaining it so here’s my Terence Tao Claude giving it’s best shot?:
Nostalgebraist has identified a fundamental confusion that elegantly reveals itself through category theory. What’s happening is that discussions about preference orderings conflate two distinct mathematical structures without acknowledging the switch. (It’s using Steven Byrnes comment and going through a similar comparison from the perspective of category theory to more cleanly restate the confusion.)
Claude Tao:
Two Distinct Categories
Category theory offers a clear framework for understanding this confusion. A category consists of objects, morphisms (arrows) between them, and rules for composition. The crucial insight is that preference orderings can be modeled as two different categories:
The Decision Theory Category (DT)
Objects: World states (A, B, C)
Morphisms: Pure preference relations where A→B means “B is preferred over A”
Composition Rule: If A→B and B→C, then A→C (transitivity)
Key Property: Arrows represent what would be chosen if all transitions were available
The Reinforcement Learning Category (RL)
Objects: Same world states (A, B, C)
Morphisms: Optimal policies where A→B means “move from A to B is the optimal action”
Composition Rule: Multi-step planning that maximizes value
Key Property: Arrows incorporate both preference and availability information
Exactly Matching Nostalgebraist’s “Two Readings”
These categories precisely correspond to nostalgebraist’s two interpretations:
“Give up on the inference” = Use the DT category, where arrows represent pure preferences that don’t dictate actions
“Refuse to pick arrow orientations until...” = Use the RL category, where arrows represent planned behavior
The Money Pump Confusion
The money pump argument inadvertently mixes these categories in exactly the way nostalgebraist describes. It:
Starts by defining preference cycles in the DT category (pure preference relations)
Then claims these are problematic by applying reasoning from the RL category (assuming agents follow arrows)
But as nostalgebraist correctly points out, an agent capable of planning won’t blindly follow preference arrows when doing so leads to a worse outcome. In the RL category, such cycles simply wouldn’t exist in the optimal policy graph if the state space properly includes resources like money.
The Example Problem
Consider nostalgebraist’s example where A→C and C→B→A forms a cycle:
In the DT category: This represents cyclical pure preferences, which may be mathematically problematic but don’t automatically lead to money pumps
In the RL category: If planning is allowed, the agent would see the resource loss and avoid completing the cycle
The confusion arises when we use DT-category arrows but assume RL-category behaviors. As nostalgebraist notes, this implicitly requires “the agent is kind of stupid (i.e. it can’t plan)” as a side assumption.
Mathematical Clarity
This categorical distinction explains why formal RL treatments avoid this confusion. They cleanly separate:
Reward functions (belonging to the DT category)
Transition functions (describing available state transitions)
Value functions and policies (belonging to the RL category)
The problem occurs when we try to reason about “rational preferences” without explicitly stating which category we’re working in.
Conclusion
Nostalgebraist’s insight is precisely correct: the confusion about preference orderings stems from mixing two different mathematical structures. By maintaining categorical discipline – being clear about whether our arrows represent pure preferences or optimal actions – we can avoid the apparent paradoxes in money pump arguments and similar discussions.
The question isn’t which category is “right,” but recognizing that they model different aspects of decision-making and being explicit about which one we’re using at any given time.
This felt to me like it was something that could be reframed through category theory but I’m not the best at explaining it so here’s my Terence Tao Claude giving it’s best shot?:
Nostalgebraist has identified a fundamental confusion that elegantly reveals itself through category theory. What’s happening is that discussions about preference orderings conflate two distinct mathematical structures without acknowledging the switch. (It’s using Steven Byrnes comment and going through a similar comparison from the perspective of category theory to more cleanly restate the confusion.)
Claude Tao:
Two Distinct Categories
Category theory offers a clear framework for understanding this confusion. A category consists of objects, morphisms (arrows) between them, and rules for composition. The crucial insight is that preference orderings can be modeled as two different categories:
The Decision Theory Category (DT)
Objects: World states (A, B, C)
Morphisms: Pure preference relations where A→B means “B is preferred over A”
Composition Rule: If A→B and B→C, then A→C (transitivity)
Key Property: Arrows represent what would be chosen if all transitions were available
The Reinforcement Learning Category (RL)
Objects: Same world states (A, B, C)
Morphisms: Optimal policies where A→B means “move from A to B is the optimal action”
Composition Rule: Multi-step planning that maximizes value
Key Property: Arrows incorporate both preference and availability information
Exactly Matching Nostalgebraist’s “Two Readings”
These categories precisely correspond to nostalgebraist’s two interpretations:
“Give up on the inference” = Use the DT category, where arrows represent pure preferences that don’t dictate actions
“Refuse to pick arrow orientations until...” = Use the RL category, where arrows represent planned behavior
The Money Pump Confusion
The money pump argument inadvertently mixes these categories in exactly the way nostalgebraist describes. It:
Starts by defining preference cycles in the DT category (pure preference relations)
Then claims these are problematic by applying reasoning from the RL category (assuming agents follow arrows)
But as nostalgebraist correctly points out, an agent capable of planning won’t blindly follow preference arrows when doing so leads to a worse outcome. In the RL category, such cycles simply wouldn’t exist in the optimal policy graph if the state space properly includes resources like money.
The Example Problem
Consider nostalgebraist’s example where A→C and C→B→A forms a cycle:
In the DT category: This represents cyclical pure preferences, which may be mathematically problematic but don’t automatically lead to money pumps
In the RL category: If planning is allowed, the agent would see the resource loss and avoid completing the cycle
The confusion arises when we use DT-category arrows but assume RL-category behaviors. As nostalgebraist notes, this implicitly requires “the agent is kind of stupid (i.e. it can’t plan)” as a side assumption.
Mathematical Clarity
This categorical distinction explains why formal RL treatments avoid this confusion. They cleanly separate:
Reward functions (belonging to the DT category)
Transition functions (describing available state transitions)
Value functions and policies (belonging to the RL category)
The problem occurs when we try to reason about “rational preferences” without explicitly stating which category we’re working in.
Conclusion
Nostalgebraist’s insight is precisely correct: the confusion about preference orderings stems from mixing two different mathematical structures. By maintaining categorical discipline – being clear about whether our arrows represent pure preferences or optimal actions – we can avoid the apparent paradoxes in money pump arguments and similar discussions.
The question isn’t which category is “right,” but recognizing that they model different aspects of decision-making and being explicit about which one we’re using at any given time.