Game Theory without Argmax [Part 1]
Written during the SERI MATS program under the joint mentorship of John Wentworth, Nicholas Kees, and Janus.
Preface
In classical game theory, we characterise agents by a utility function and assume that agents choose options which cause maximal utility. This is a pretty good model, but it has some conceptual and empirical limitations which are particularly troublesome for AI safety.
Higher-order game theory (HOGT) is an attempt to rebuild game theory without appealing to either utility functions or maximisation. I think Higher-Order Game Theory is cool so I’m writing a sequence on it.
I’ll try to summarise the relevant bits of the literature, present my own minor extensions, and apply HOGT to problems in AI safety
You’re reading the first post! Let’s get into it.
The role of argmax
For each set , let be the familiar function which receives a function and produces the set of element which maximise . A function like is sometimes called a higher-order function or functional, because it receives another function as input.
Explicitly, .^{[1]}
As you all surely know, plays a central role in classical game theory. Typically we interpret the set as the agent’s options,^{[2]} and the function as the agent’s task, which assigns a payoff to each option . We say an option is optimal to the agent for the task whenever . Classical game theory is governed by the assumption that agents choose optimal options in whatever task they face, where optimality strictly means utility-maximisation.
Definition 1 (provisional): Let be any set of options. A task is any function . An option is optimal for a task if and only if .
Due to the presence of the powerset operator in , this model of the agent is possibilistic — for each task , our model says which options are possibly chosen by agent. The model doesn’t say which options are probably chosen by the agent — for that we’d need a function . Nor does the model say which options are actually chosen by the agent — for that we’d need a function .^{[3]}
How many options are optimal for the task? | Example |
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Typically there’ll be a unique optimal option. | |
Perhaps multiple options will be optimal. | |
Perhaps no options are optimal, i.e. every option is strictly dominated by another. | |
Perhaps every option is optimal, i.e. the task is a constant function. |
Exercise 1: Find a set such that for every function .
Generalising the functional
The function is a particular way to turn tasks into sets of options, i.e. it has the type-signature . But there are many functions with the same type-signature (see the table below), so a natural question to ask is… What if we replace in classical game theory with an arbitrary functional ?
What we get is higher-order game theory.^{[4]} Surprisingly, we can recover many game-theoretic concepts in this more general setting. We can typically recover the original classical concepts from the more general higher-order concepts by restricting our attention to either or .
So let’s revise our definition —
Definition 2 (provisional): Let be any set of options. An optimiser is any functional . A -task is any function . An option is -optimal for a task if and only if .
When clear from context, I’ll just say task and optimal.
In higher-order game theory, we model the agents options with a set and model their task with a function . But (unlike in classical game theory) we’re free to model the agent’s optimisation with any functional . I hope to persuade you that this additional degree of freedom is actually quite handy.^{[5]}
Higher-order game theory is governed by the central assumption that agents choose -optimal options in whatever -tasks they face, where is our model of the agent’s optimisation. If we observe the agent choosing an option then that would be consistent with our model, and any observation of a choice would falsify our model.^{[6]}
Anyway, here is a table of some functionals and their game-theoretic interpretation —
Remarks | |
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This agent will choose an option which minimises . In classical game theory, this type of optimiser is typically used to model the adversary to the agent modelled by . | |
This agent will choose an option which dominates some fixed option . The option is called the anchor point. It might represent the “default” option, or the “do nothing” option, or the “human-approved” option. According to Herbert Simon, satisficing is an accurate model of human and institutional decision-making. Utility-satisficers are a kind of mild optimiser, which might be a safer way to build AI than a full-blown utility-maximiser. | |
This agent will chooses an option which maximises the function up to some fixed slack . Such agents behave like except that their utility is measured with finite precision. | |
This agent will choose an option which scores better than the average option, given that the option space is equipped with a distribution . | |
This agent will choose an option in a fixed subset , regardless of the task, e.g. DefectBot and CooperateBot.^{[7]} Using , we can model a non-agents as a special (degenerate) case of agents. | |
This agent will choose an option in the top quantile, given that the option space is equipped with a distribution . This is the possibilistic version of Jessica Taylor’s quantiliser. Her original probabilistic version is a function , and so wouldn’t count as an optimiser according to Definition 2.^{[8]} | |
This agent will choose an option which dominates every anchor point in . As special cases, when we get , and when is a singleton set we get Simon’s satisficer mentioned before. When the set of anchor points is smaller, then the resulting optimiser is less selective, i.e. more options will be optimal. | |
Exercise 2 | |
Exercise 3 | This agent will choose an option in the largest equivalence class, where two options are equivalent if they result in the same payoff. |
Generalising the payoff space.
Now let’s generalise the payoff space to any set , not only . We will think of the elements of as payoffs in a general sense, relaxing the assumption that the payoffs assigned to the options are real numbers. The function describes which payoff would result from the agent choosing the option .
Definition 3 (provisional): Let be any set of options and be any set of payoffs. An optimiser is any functional . A -task is any function . An option is -optimal for a task if and only if .
This is the final version to this definition today.
This is significantly more expressive! When we are tasked with modelling a game-theoretic situation, we are can pick any set to model the agent’s payoffs!^{[9]}
I’ll use the notation to denote the set of functionals , e.g. .
Anyway, here is a table of some functionals and their game-theoretic interpretation —
Payoff space | Remarks |
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An optimiser only needs to be well-defined for bounded utility functions . | |
An optimiser only needs to be well-defined for utility functions . | |
An optimiser must be well-defined for infinatary utility functions . For example, might be the expected total winnings of a gambler employing the gambling strategy . The gambler themselves are modelled by an optimiser . This functional is characterised by its attitude towards infinite expected profit/loss. | |
, where is the distribution monad on . | An optimiser will choose options given a stochastic task . This is the type-signature of risk-averse and risk-seeking maximisers studied in behavioural microeconomics. The field of Portfolio Theory is (largely) the comparison of rival optimisers in . |
is the Levi-Civita Field, an extension of the reals with infinitesimals. | The Levi-Civita field contains infinitesimals like , as well as infinite values like . In infinite ethics, we encounter tasks , and we can model the infinitary ethicist by an optimiser . Exercise 4: Solve infinite ethics. |
An optimiser can model different multi-objective optimisers. For example, there’s an optimiser which, given multi-objective task , returns those options which are maximal according to the lexicographic ordering, and there’s another optimiser which uses the product ordering instead.^{[10]} Later in this post, we’ll encounter an optimiser in which returns the nash equilibria of -player games, where a task for this optimiser is an -by-payoff matrix . The field of cooperative bargaining is concerned with different optimisers . A bargaining task parameterises both the feasibility set and the disagreement point where is the singleton set. | |
Any preorder | Suppose that is any set equipped with a preorder .^{[11]} Then a function will induce a preorder on via . Let be the optimiser which chooses the maximal points of , i.e. options which aren’t strictly dominated by any other options. Explicitly If isn’t total (i.e. the agent has incomplete preferences over the payoffs) then the resulting optimiser is less selective (i.e. more options are optimal). In the extreme case, where no options are comparable, then might choose any option.^{[12]} Exercise 5: Which optimisers defined in the previous table can be generalised to any preorder ? |
, where is the option space of the agent. | Occasionally the same set will serve as both the option space and the payoff space. In this case, a task represents some transformation of the underlying option space. There’s an optimiser , which choices options which are fixed-points of . That is, . We can use to model a conservative agent who chooses options which remain untransformed by the task. Note that this optimiser is not consequentialist, because the optimality of an option is not determined by its payoff alone. For example, but , despite the fact that . |
Subjective vs objective optimisers
It’s standard practice, when modelling agents and their environments, to use payoff spaces like , , , etc, but I think this can be misleading.
Consider the following situation —
A robot is choosing an option from a set . There’s a function such that, were the robot to choose the option , then the world would end up in state , where is something like the set of all configurations of the future light-cone.
You know the robot is maximising over all their options, but you aren’t sure what the robot is maximising for exactly — perhaps for paperclips, perhaps for happy humans.
Now, let be the function which counts the number of paperclips in a light-cone, and let be the function which counts the number of happy humans.
Here’s what classical game theory says about your predicament —
The payoff space is . You know that the robot applies the optimiser , but you don’t know whether the robot faces the task or the task , and hence you don’t know whether the robot will choose an option or .
I call this a subjective account, because the robot’s task depends on the robot’s preferences. Were the robot to have difference preferences, then they would’ve faced a different task, and because you don’t know the robot’s preferences you don’t know their task.
However, by exploiting the expressivity of higher-order game theory, we can offer an objective account which rivals the subjective account. In the objective account, the task that the robot faces doesn’t depend on the robots preferences —
The payoff space is itself. You know that the robot faces the task but you don’t know whether the robot applies the optimiser or the optimiser , and hence you don’t know whether the robot will choose an option or .
Notice that both accounts yield the same solution! Nonetheless, I think the objective account is nicer for four reasons. (Feel free to skip if you’re convinced.)
Disclaimer: Admittedly, the distinction between subjective accounts — where payoff spaces are stuff like , , , , , e.t.c. — and objective accounts — where payoff spaces are stuff like future light-cones, or brain states, or pixel configurations, e.t.c — is an informal (and somewhat metaphysical) distinction, but hopefully you can see what I’m pointing at.
(1) Carve nature at its joints.
The objective account, where , bares a closer structural resemblance to the physical reality. The physical robot corresponds to the functional and the physical environment corresponds to the function . Notably, all the information about the robot’s idiosyncratic preferences is bundled up inside the functional .
In contrast, in the subjective account, where , the functional contains almost no substantial information about the agent itself. It suggests (if read too literally) that all agents are basically indiscernible, and they behave differently because they face different environments.
(2) Moral antirealism.
The subjective account (again, read too literally) suggests that values are out there in the world, that the environment contains entities called utilities which all rational agents seek, that all conflict is disagreement, that correctness is a property of pebble heaps, that microeconomics is normative, and (most concerning of all) that the primary obstacle to building a safe superintelligence is writing down a utility function.
The objective account, I think, is more moral antirealist. It says, “The world contains only paperclips and happy humans, never utilities! The world contains only paperclip-maximisers and happy-human-maximisers, never utility-maximisers!”
(3) Experimental independence
In the objective account, the task and the optimiser have independent semantic meaning. At least in principle, I know how to find independently of — namely by inspecting the physical dynamics of the robot’s environment or inspecting the robot’s world-model. And I know how to find independently of — namely by placing the robot in different physical environments and observing their choices.
By contrast, in the subjective account, the task and the optimiser have no independent meaning — they are merely exist to compress the optimality condition . What would it even mean for the robot to possess the utility function without the presumption that they maximise utility? I’ve honestly no clue. And without the task , how would I determine the robot’s optimiser experimentally? Presumably I should vary the task , however I can’t do this experimentally because contains the robot’s preferences which is a variable outside my control.
Granted, for most historic applications of classical game theory, we do know the preferences of the agent — we already know that White wants to checkmate Black, and the consumer wants cheaper goods, and the statistician wants to accurate predictions, e.t.c — so it doesn’t matter whether one sticks those preferences in the task or the optimiser. But in AI safety, a big chunk of our perplexity comes from the preferences of the agents. So it matters more that we stick those preferences in the right part of our model.
(4) No spooky reals.
The subjective account seems to rely on the elements of a mysterious set called “” which is extraneous to the phenomenon under consideration. By contrast, the objective account refers only to the sets and , where the elements of and are physical stuff intrinsic to the situation being modelled. Hence, higher-order game theory promises to dispense with from game theory, along with argmax and utility functions, ensuring the weirdness of doesn’t contaminate our game theory.^{[13]}
This has a computational upshot as well.
Supposes that and are small finite sets. A task can be implemented as dictionary whose keys lie in and whose values lie in , which uses bits. The functional can be implemented as a program which receives input of type and returns output of type . Easy!
In the subjective account, by contrast, the task requires infinite bits to specify, and the functional must somehow accept a representation of an arbitrary function . Oh no! This is especially troubling for embedded agency, where the agent’s decision theory must run on a physical substrate.
Recovering utility functions
According to the objective account, what is fundamental about an agent is the functional where is some objective payoff, and the claim that the agent has a utility function is understood as the claim that can be approximately decomposed into . Hence, the existence of a utility-decomposition of is an additional fact about the agent to be discovered, rather than an assumption that should be baked into the formalism itself.
Utility functions are an emergent property of the underlying functional.
One clue that utility functions are emergent properties is that they aren’t unique! It’s well-known that a utility function for an agent is only well-defined modulo positive-affine transformation, i.e. there is no meaningful distinction between and whenever for some . This fact falls immediately from the objective-first view, because and are equal functionals whenever
Now, if we were dealing with or — instead of — then there would be a meaningful difference between some utility functions which are equivalent modulo positive-affine transformation.
Let’s make this notion precise —
Definition 4: Let be an optimiser. We say that is a (classical) utility function of if and only if . In general, for any , we say that is a -utility function of if and only if .
Typically, is some objective optimiser and is some subjective optimiser. When then we obtain the classical utility functions of an objective optimiser , and we may obtain non-classical utility functions of the same optimiser by considering (e.g.) or or whatever.
Classical game theory is the study of optimisers with classical utility functions. There are some theoretical and empirical arguments for restricting only to such optimisers but these arguments are probably overrated. In any case, I suspect that unifying deep learning and classical game theory will require studying non-classical agents. Here’s why — in the deep learning paradigm, we build agents by training a large neural network with stochastic gradient descent on tasks which fortify agentic-like behaviour. At initialisation, these neural networks aren’t classical agents, and classicality emerges incrementally, probably after passing through phases of nonclassical agency. Therefore, if we want to account for the emergence of agency (classical or otherwise), then we need to account for the loss gradient over the entire space of optimisers , not merely over the subspace of corresponding to classical optimisers.
Some properties of optimisers
We can define formalise various properties and operations of optimisation using arbitrary functional .
I’ve included the list of examples below for illustrative purposes only —
Property | Remarks |
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Totality | We’ll say that an optimiser is total iff for all . Informally, this condition states that our model of the agent never “breaks down”, i.e. regardless of the task the agent faces, there’s always some optimal choice. |
Selectivity | We’ll say that an optimiser is (weakly) more selective than if and only if . This relation defines a partial order on . For example, is more selective than whenever . Mild optimization, an approach for mitigating Goodhart’s law, replaces with less selective optimisers. |
Consequentialism | We’ll say that an optimiser is consequentialist if for some .^{[14]} In other words, for any task , if then . This condition says that, once we know the agent’s task, then the only thing relevant to the optimality of a particular choice is its payoff. For example, and are consequentialist, but and are not. This function says, for each task , which payoffs would be acceptable to the agent, so is the quantifier for . |
Context-independence | We’ll say that a consequentialist optimiser is context-independent if . Context-independence is a stronger condition than consequentialism — this condition says that, once we know which payoffs are achievable in the agent’s task, then the only thing relevant to the optimality of a particular option is its payoff. For example, is context-independent, but is not. |
Filtered optimisation | Suppose that is an optimiser, and is a subset of the options which are safe/valid/legal. Then we can define another optimiser who always chooses options in . This operation captures the notion of filtering options after the agent has applied the optimisation. For example, if and then , so if then . |
Constrained optimisation | The filtering operation doesn’t capture what we typically mean by optimisation within side-constraints. For example, if we change to , then we would like the optimiser to choose as this maximises subject to the constraint . The filtered optimisation would produce the emptyset, as none of the options optimal to are legal. Let’s define constrained optimisation for an optimiser . Suppose that there is an element such that . Informally, this means that detests the payoff and would never chooses an option resulting in unless it must. For example, detests the payoff . We can defined constrained optimisation as follows: where Exercise 6: How might we define constrained optimisation if there is no such ? |
Supervision | Maybe the set of legal options depends on the task itself. This dependency itself corresponds to an optimiser . We can define the filtered optimiser as . We can define the constrained optimiser where The optimiser behaves like one agent being supervised by another agent where the mode of supervision is filtering and constraining respectively. |
Unanimity | For a collection of optimisers , we can define a single optimiser who will only choose an option if each constituent optimiser would also choose the option, forming a unanimous coalition of . Dually, we can define the optimiser forming a unilateral coalition. |
Shards | Suppose that assigns an optimiser to each task . In terms of , we can define the optimiser by diagonalising — i.e. will match the optimiser in each task . This operation captures (somewhat) the notion of shards, or context-activated optimisation. For example, imagine an agent who “plays it safe” by satisficing unless they can achieve sufficiently high payoff, i.e. Or imagine an agent who desires cheese whenever they are sufficiently close to cheese, but otherwise desires to run around aimlessly. |
Recap
In classical game theory, agents maximise their utility functions , i.e. they might choose any option .
In higher-order game theory, we replace the utility functions with an arbitrary function , called a “task”, and replace with an arbitrary functional called the “optimiser”. They might choose any option .
This additional expressivity lets us include mild optimisers and multi-objective optimisers which don’t crudely maximise utility.
It also lets us include objective optimisers, which strive for particular physical configurations, which dispenses with the concept of utility altogether.
We can recover utility functions as an emergent property of an objective optimiser, relative to any choice of subjective optimiser, not only to .
Finally, we can define some interesting properties and operations on the optimisers which correspond (loosely) to things that AI safety researchers care about.
Next time...
The next post will answer the age-old question, “What happens when two optimisers and play the simultaneous game ?” We know what happens when and are both utility-maximisers — the possible option-profiles are pairs in nash equilibrium.
Can we really generalise the nash equilibrium to any pair of optimisers?
Further reading
Selection Functions, Bar Recursion, and Backward Induction (2009)
A generalisation of Nash’s theorem with higher-order functionals (2013)
A Higher-order Framework for Decision Problems and Games (2018)
Coalgebraic Analysis of Subgame-perfect Equilibria in Infinite Games without Discounting (2018)
Sequential games and nondeterministic selection
functions (2019)Translating Extensive Form Games to Open Games with Agency (2021)
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The little is from (simply-typed) lambda notion for introducing functions. If is the familiar sine function, then is the function which sends each to , so . We can also use lambda notation to define higher-order functions, e.g. if then .
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Synonyms: actions, alternatives, behaviours, configurations, coordinates, hypotheses, moves, options, outputs, parameters, plays, policies, positions, strategies — or whatever else is being chosen due to the payoffs assigned to it.
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Scott Garrabrant has called the the type-signature of agency, and you can interpret this sequence as an expansion on his remark.
In response to Garrabrant, MrMind raises the objection that no map can be parametric in both and because there is no map . Fortunately, this objection doesn’t apply here, because we’re looking for functions , and there is a function .
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The term higher-order game theory is coined in [Hedges 2015], presumably because functions of the form are often called higher-order functions. Throughout the article, I will only use the term in this sense.
Warning: The term higher-order game theory is also used in [Nisan 2021] to refer to “the study of agents thinking about agents thinking about agents...” — and [Diffractor 2021] calls this higher-level game theory. I will never use the term in this sense.
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Spoilers: As you’ll see later, this is the definition of optimisation you need to ensure that the nash equilibrium between two optimisers is also an optimiser.
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Of course, if , then this model of the agent and their task would be falsified by any observed option, and can be eliminated a priori.
We can also interpret as the statement that the computer program implementing the agent’s decision theory would throw an exception, or fail to halt, when given the input .
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Discussed in Program Equilibrium in the Prisoner’s Dilemma via Lob’s Theorem and elsewhere.
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In tomorrow’s post, will be generalised to an arbitrary commutative monad . When we let be the probability monad , then we get optimisers of type which would include Taylor’s original probabilistic quantiliser. But don’t worry about that now!
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… and any set to model the agent’s options, any function to model the agent’s task, and any functional to model the agent’s optimisation.
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When is equipped with the product ordering, the optimiser is the called the pareto optimiser which maps each multi-objective task to the pareto-frontier. This is the least selective optimiser which cares about each objective .
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A partial order is a relation on satisfying three rules:
(1) Reflectivity: .
(2) Transitivity: If and then .
(3) Anti-symmetry: If and then .
A preorder is a relation satisfying only reflectivity and transitivity. A preorder is the non-evil version of a partial order, because it’s agnostic about “”identity”″ between equivalent objects.
Now, we say an agent’s preferences are incomplete when neither nor , and an agent’s preferences are indifferent when both and but not . So a partial order represents preferences which might be incomplete but never indifferent, while a preorder represents preferences which might be both incomplete and indifferent. Agents with incomplete preferences have recently been proposed as a solution to the Shutdown Problem.
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Warning: There’re another optimiser in who chooses the maximum points of , i.e. options which weakly dominate all other options. Explicitly, . This optimiser is qualitatively different, being more selective when its preferences are incomplete.
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In Science without Numbers (1980), Hartry Field attempted to reformulate Newtonian physics without quantification over mathematical objects. Instead, the formulae of his theory would quantify only over physical objects and regions of spacetime. He called this normalisation of Newtonian physics.
Likewise, higher-order game theory promises a normalisation of game theory.
I don’t know why, but this smells correct to me.
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This property is called closedness in [Hedges 2015] because the set is closed under the equivalence relation for every .
A function is called a quantifier in the HOGT literature.
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argmaxX is not context-independent according to the given definition.
Consider tasks u,u′:{0,1}→{0,1}, with u(x)=x and u′(x)=1−x. Then argmaxu={1}≠argmaxu′={0}, despite Imageu=Imageu′.
I guess the correct definition says q(u)=q(u′) instead of ψ(u)=ψ(u′).
Thanks v much! Can’t believe this sneaked through.
Wow! Happy to see this exposition. I’ve been a fan of this framework for a while now. Happy to see it is finally percolating to the alignment community!
Typo: This should be argmaxR(sin)={2πk+π2|k∈Z}.
The power set part seems sus. Have you considered something more continuous?
Yes!
In a subsequent post, everything will be internalised to an arbitrary category C with enough structure to define everything. The words set and function will be replaced by object and morphism. When we do this, P will be replaced by an arbitrary commutative monad M.
In particular, we can internalise everything to the category Top. That is, we assume the option space X and the payoff R are equipped with topologies, and the tasks will be continuous functions u:X→R, and optimisers will be continuous functions u:(X→R)→M(X) where (X→R) is the function space equipped with pointwise topology, and M is a monad on Top.
In the literature, everything is done with galaxy-brained category theory, but I decided to postpone that in the sequence for pedagogical reasons.
I think this part uses an unfair comparison:
If X and W+ are small finite sets, then any behavior can be described with a utility function requiring only a finite number of bits to specify. You only need to use R as the domain when W+ is infinite, such as when outcomes are continuous, in which case the dictionaries require infinite bits to specify too.
I think this is representative of an unease I have with the framing of this sequence. It seems to be saying that the more general formulation allows for agents that behave in ways that utility maximizers cannot, but most of these behaviors exist for maximizers of certain utility functions. I’m still waiting for the punchline of what AI safety relevant aspect requires higher order game theory rather than just maximizing agents, particularly if you allow for informational constraints.
I can’t find this term anywhere else, and “infinitary” means something else so I think it’s not a typo, is it your own neologism? If not, where is it from?
Exercise 6:
I repeat the construction using each element of R in turn in place of ⊥.
For each such u′, ψ(u′) spits out a set. I pick the u′ producing the set which is closer to Xlegal.
For finite sets, I could look at the size of the intersection; more in general I’ll need some metric on P(X) to say what it means to be close to Xlegal.
Does that make sense?