I don’t claim that this is canonical, but here’s the way David and I use the terms amongst ourselves.
First, we don’t usually use the term “representation theorem” at all, but if we did, that would naturally refer to a theorem saying that some preferences/behavior/etc can be represented in a particular way, like e.g. expected utility maximization over some particular states/actions/whatever. We would probably classify e.g. VNM as a representation theorem, though we basically-never think about VNM at all so we don’t really need a term for it.
Second, coherence. When we talk about coherence theorems, we usually don’t think about exploitability, but rather about pareto suboptimality. (Of course exploitability is a special case of pareto suboptimality, but the reverse doesn’t always apply easily.) Vibes-wise, we’re usually thinking about a system behaving pareto-optimally across different places—e.g. different parts of the system behaving jointly-pareto-optimally, or choices made across different inputs/worlds being jointly-pareto-optimal, or decisions at different times being jointly-pareto-optimal. The behavior is “coherent” in the sense that the system at all these different parts/places/times all “act in a consistent way”, such that they’re jointly pareto-optimal. That’s the sort of thing which “coherence” gestures at in our usage.
a theorem saying that some preferences/behavior/etc can be represented in a particular way, like e.g. expected utility maximization over some particular states/actions/whatever
So, I take it that Savage’s theorem is a representation theorem under your schema?
Of course exploitability is a special case of pareto suboptimality, but the reverse doesn’t always apply easily
Theoretically or practically? I.e. you can’t derive an exploitability result easily from a parto suboptimality? Or you’re IRL stuck in an (inadequate) equllibrium far from the pareto frontier but you can’t exploit this fact?
As an aside, the reason I like the exploitability framing is bc. coherence properties look to me like they’re downstream of some agent exploiting eating up some “wasted resources”. E.g. markets and arbitrage or probabilities and money pumping.
So, I take it that Savage’s theorem is a representation theorem under your schema?
Yes. Arguably it is also a coherence theorem, the two are not mutually exclusive, but it’s more unambiguously a representation theorem.
Theoretically or practically? I.e. you can’t derive an exploitability result easily from a parto suboptimality?
Practically. Consider e.g. applying coherence tools to an e coli. That thing is not capable of signing arbitrary contracts or meaningfully choosing between arbitrary trades, and insofar as it’s wasting resources those resources likely end up e.g. as waste heat. For another agent to “eat up” the wasted resources, it would need to e.g. restructure the e coli’s metabolic pathways; it’s not really something which can happen via things-which-look-like-trading-with-an-agent.
Arguably, evolutionary pressures driving E coli to reduce waste come from other agents exploiting e coli’s wastefulness. At least in part. Admittedly, that’s not the only thing making it hard for e coli to reproduce while being wasteful. But the upshot is that exploiting/arbitraging away predictable loss of resources may drive coherence across iterations of an agent design instead of within one design. Which is useful to note, though I admit that this comment kinda feels like a cope for the frame that exploitability is logically downstream of coherence.
I don’t claim that this is canonical, but here’s the way David and I use the terms amongst ourselves.
First, we don’t usually use the term “representation theorem” at all, but if we did, that would naturally refer to a theorem saying that some preferences/behavior/etc can be represented in a particular way, like e.g. expected utility maximization over some particular states/actions/whatever. We would probably classify e.g. VNM as a representation theorem, though we basically-never think about VNM at all so we don’t really need a term for it.
Second, coherence. When we talk about coherence theorems, we usually don’t think about exploitability, but rather about pareto suboptimality. (Of course exploitability is a special case of pareto suboptimality, but the reverse doesn’t always apply easily.) Vibes-wise, we’re usually thinking about a system behaving pareto-optimally across different places—e.g. different parts of the system behaving jointly-pareto-optimally, or choices made across different inputs/worlds being jointly-pareto-optimal, or decisions at different times being jointly-pareto-optimal. The behavior is “coherent” in the sense that the system at all these different parts/places/times all “act in a consistent way”, such that they’re jointly pareto-optimal. That’s the sort of thing which “coherence” gestures at in our usage.
So, I take it that Savage’s theorem is a representation theorem under your schema?
Theoretically or practically? I.e. you can’t derive an exploitability result easily from a parto suboptimality? Or you’re IRL stuck in an (inadequate) equllibrium far from the pareto frontier but you can’t exploit this fact?
As an aside, the reason I like the exploitability framing is bc. coherence properties look to me like they’re downstream of some agent exploiting eating up some “wasted resources”. E.g. markets and arbitrage or probabilities and money pumping.
Yes. Arguably it is also a coherence theorem, the two are not mutually exclusive, but it’s more unambiguously a representation theorem.
Practically. Consider e.g. applying coherence tools to an e coli. That thing is not capable of signing arbitrary contracts or meaningfully choosing between arbitrary trades, and insofar as it’s wasting resources those resources likely end up e.g. as waste heat. For another agent to “eat up” the wasted resources, it would need to e.g. restructure the e coli’s metabolic pathways; it’s not really something which can happen via things-which-look-like-trading-with-an-agent.
Arguably, evolutionary pressures driving E coli to reduce waste come from other agents exploiting e coli’s wastefulness. At least in part. Admittedly, that’s not the only thing making it hard for e coli to reproduce while being wasteful. But the upshot is that exploiting/arbitraging away predictable loss of resources may drive coherence across iterations of an agent design instead of within one design. Which is useful to note, though I admit that this comment kinda feels like a cope for the frame that exploitability is logically downstream of coherence.