The Category of Node-and-Choice Forms, with Subcategories for Choice-Sequence Forms and Choice-Set Forms

. The literature speciﬁes extensive-form games in several styles, and eventually I hope to formally translate games across those styles. Toward that end, this paper deﬁnes NCF , the category of node-and-choice forms. The category’s objects are game forms in any style, and the category’s isomorphisms are made to accord with the literature’s small handful of ad hoc style equivalences.Morespeciﬁcally, the paper develops two full subcategories: CsqF for forms whose nodes are choice-sequences, and CsetF for forms whose nodes are choice-sets. I show that NCF is “isomorphically enclosed” in CsqF in the sense that each NCF form is isomorphic to a CsqF form. Similarly, I show that CsqF ˜a is isomorphically enclosed in CsetF in the sense that each CsqF form with no-absentmindedness is isomorphic to a CsetF form. The converses are found to be almost immediate, and the resulting equivalences unify and simplify two ad hoc style equivalences in Kline and Luckraz 2016 and Streufert 2018a.Aside from the larger agenda, this paper makes three practical contributions. Style equivalences are made easier to derive by [1] a natural concept of isomorphic invariance and [2] the composability of isomorphic enclosures. In addition, [3] some new consequences of equivalence are systematically deduced.


General Motivation
To (Identical to Western University Department of Economics Research Report 2018-6.)set the stage, this paragraph recalls that there are many styles in which to specify an extensive-form game.Streufert 2018a (henceforth "S5") lists five of them.[1] Nodes and choices 1 can be specified abstractly without restriction, as in Kuhn 1953 andSelten 1975. [2] Nodes can be specified as sequences of choices as in Osborne and Rubinstein 1994. [3] Nodes can be specified as sets of choices as in S5 itself.[4] Choices can be specified as sets of nodes, as in Alós-Ferrer and Ritzberger 2016 Section 6.3.[5] Both nodes and choices can be expressed as sets of outcomes, as in von Neumann and Morgenstern 1944, andAlós-Ferrer andRitzberger 2016 Section 6.2.S5 Section 1 provides examples of the five styles, and S5 Section 7 explains how each style has its own advantages and disadvantages.
It is difficult to formally compare the different styles.Indeed, the first such results have only recently appeared in Alós-Ferrer and Ritzberger 2016 Section 6.3, in Kline and Luckraz 2016, and in S5 (whose Figure 1.2 provides an overview).These contributions show, by ad hoc constructions, that the five styles in the previous paragraph are of roughly equal generality.To be somewhat more precise, these papers argue that one style is at least as general as another style, by showing that each game in the first style can be reasonably mapped to a game in the second style.Then two styles are regarded as equivalent if such an argument can be made in both directions.Notice that each such argument hinges upon an ad hoc mapping linking games in one style to games in another style.Lacking is a way to compare styles that is based on a systematic way of comparing games.I hope to provide that systematization in a fashion that is compatible with the prior style equivalences.
Further, I have a larger agenda in mind.Suppose that two styles have been compared and found to be equivalent.Then I hope to do more than merely translate each game in one style to an equivalent game in the other style.I hope to translate properties, defined for games, from one style to the other.I hope to translate equilibrium concepts from one style to the other.And ultimately, I hope to translate theorems from one style to the other.In other words, I hope to formally translate game theory from one style to another.
I believe this would be useful.Since each game-theory paper typically uses just one style, one can sort the game-theory literature into separate subliteratures corresponding to the different styles.Then, the present lack of formal translation across styles implies that communication across the subliteratures is informal.So, unavoidably, a considerable amount of game-theoretic understanding is informal.Formally defragmenting the literature would increase communication, reduce the likelihood of laboriously reproducing past results, reduce editorial effort, and facilitate teaching.In addition, the ability to translate would help researchers to freely choose the style that is most insightful and/or most convenient for the particular task at hand.
Formal translation is a daunting task.Fortunately, category theory promises to be a natural and powerful tool.In order to gain access to this tool, my intermediaterange objective has been to construct a category [a] whose objects are extensiveform games in any style, and [b] whose isomorphisms accord with the handful of style equivalences already in the literature.My first step was Streufert 2018b (henceforth "SP").That paper defined NCP, which is the category of node-andchoice "preforms", where a preform is a rooted tree with choices and information sets.My second step is the present paper.Here I will define NCF, which is the category of node-and-choice "forms", where a form augments a preform with players.Later, a third paper will augment NCF forms with preferences in order to define extensive-form games.
Little categorical work on game theory has been done elsewhere. 2Lapitsky 1999 and Jiménez 2014 define categories of simultaneous-move games.Machover and Terrington 2014 defines a category of simple voting games.Finally, Vannucci 2007 defines categories of various games, but in its category of extensive-form games, every morphism merely maps a game to itself.

Categorical Investments
As suggested, this paper constructs a category of forms [a] whose objects are forms in any style, and [b] whose isomorphisms accord with the style equivalences already in the literature.Goals [a] and [b] are discussed in the next two paragraphs.
Section 2 introduces NCF, which is the category of node-and-choice forms, in which both nodes and choices are specified abstractly without restriction.Thereby goal [a] is achieved.Further, one special kind of node-and-choice form is a choicesequence form, in which nodes are choice-sequences.Correspondingly, Section 3 introduces CsqF, which is the full NCF subcategory for choice-sequence forms.Similarly, another special kind of node-and-choice form is a choice-set form, in which nodes are choice-sets.Correspondingly, Section 4 introduces CsetF, which is the full NCF subcategory for choice-set forms.Finally, consider again the five styles in the paper's opening paragraph.NCF itself corresponds to style [1], CsqF corresponds to style [2], and CsetF corresponds to style [3].Left for future research are style [4] with its node-set choices, and style [5] with its outcome-set nodes and outcome-set choices.These two additional styles will correspond to two additional subcategories of NCF, as suggested in Section 5.2's discussion of future research.
To achieve goal [b], Section 2 defines NCF's morphisms in such a way that the category's isomorphisms accord with the style equivalences in the literature.Since this paper does not build subcategories for the node-set and outcome-set styles, only two of the literature's style equivalences remain: [i] Kline and Luckraz 2016 Theorems 1 and 2, which are essentially an equivalence between node-andchoice forms and choice-sequence forms, and [ii] S5 Theorems 3.1 and 3.2, which are essentially an equivalence between (no-absentminded) choice-sequence forms and choice-set forms.As discussed earlier, each of these two equivalences is a matching pair of results, in which each result states that each form in one style can be reasonably mapped to a form in the other style.Section 3.2 proposes to strengthen each such result by requiring that each form in one style is NCF isomorphic to a form in the other style.This new kind of result is called an "isomorphic enclosure", and a matching pair of isomorphic enclosures is called an "isomorphic equivalence".Equivalence [i] accords with Corollary 3.3(b), which states that NCF and CsqF are isomorphically equivalent.Similarly, equivalence [ii] accords with Corollary 4.3(b), which states that CsqF ã and CsetF are isomorphically equivalent.The paragraphs after these two corollaries provide historical context, more details, and more senses in which the two corollaries accord with literature's equivalences [i] and [ii].
Other results show that NCF is pleasant in other ways.Theorem 2.3 shows that NCF is a well-defined category.Theorem 2. 4 shows that an NCF isomorphism can be characterized by bijections for nodes, choices, and players.Theorem 2.7 shows that there is a forgetful functor from NCF to NCP, which is SP's category of node-and-choice preforms.In addition, various results in Sections 2.1-2.3 show that the category interacts naturally with game-theoretic concepts like the assignment of information sets to players.Also, Section 2.4 shows that the properties of no-absentmindedness and perfect-information are invariant to NCF isomorphisms.
2 Extensive-form games are not readily comparable to the games defined in the theoretical computer-science literature.Categories of such games are developed by Abramsky, Jagadeesan, and Malacaria 2000, Hyland and Ong 2000, McCusker 2000, and Hedges 2017.The last seems the most relevant, but does not seem to accommodate players with different information.
Finally, the paragraph after Corollary 3.5 shows how the negation of isomorphic enclosure formalizes the notion that a property is truly "restrictive" and "substantial" as opposed to merely "notational".

Categorical Dividends
Section 1.2 above argues that NCF systematizes prior style equivalences and that it is a pleasant category in a variety of other ways.Also, Sections 1.1 and 5.2 argue that NCF promises to be of practical importance in the larger agenda of translating game theory across styles.Further, the following three paragraphs identify three practical ways that NCF directly contributes to game theory.
First, isomorphic invariance is a natural and powerful concept.For example, two elementary propositions in Section 3.3 use isomorphic invariance to find [1] general circumstances in which one subcategory is strictly isomorphically enclosed by another and [2] general circumstances in which an isomorphic enclosure can be restricted to smaller subcategories.The latter proposition is used by Corollary 3.7(b) to easily construct an isomorphic enclosure for the proof highlighted in the next paragraph.Further, both propositions are used by Section 4.3 to easily derive new results about perfect-information.
Second, isomorphic enclosures can be composed (note 15).Such compositions can make it much easier to derive other isomorphic enclosures.For example, the proof of Corollary 4.3(b)'s reverse direction is just six lines long, and the third paragraph following the corollary's proof explains how this simple argument replaces six difficult pages in S5's proof of its Theorem 3.2.Thus the isomorphic equivalence of Corollary 4.3(b) is much easier to prove than the corresponding ad hoc equivalence of S5 Theorems 3.1 and 3.2 (this was called equivalence [ii] above).
Third, isomorphic enclosures have consequences for form derivatives, and Section 5.1 deduces them simultaneously for all isomorphic enclosures.More specifically, each isomorphic enclosure is defined via isomorphisms, and Proposition 2.6 implies that each such isomorphism has consequences not only for form components (such as nodes, choices, and players) but also for form derivatives (such as the precedence relation among nodes, and each player's collection of information sets).In contrast, the literature's ad hoc style equivalences concern only form components.

Organization
Section 2 develops NCF, the category of node-and-choice forms.Less generally, Section 3 develops the subcategory CsqF for choice-sequence forms, and Section 4 develops the subcategory CsetF for choice-set forms.Sections 3.2 and 3.3 use the context of CsqF to introduce the general concept of isomorphic enclosure, and to introduce general propositions about isomorphic invariance.Further, Section 5.1 uses parts of Sections 3 and 4 to illustrate some general consequences of isomorphic enclosure.Finally, Section 5.2 discusses future research.
Although many proofs appear within the text, twelve lengthy proofs and their associated lemmas are relegated to the appendices.Appendix A concerns NCF, Appendix B concerns CsqF, and Appendix C concerns CsetF.

Objects
Let T be a set of elements t called nodes.As in SP Section 2.1 (where "SP" abbreviates Streufert 2018b), a pair (T, p) is a functioned tree iff there are t o ∈ T and X ⊆ T such that [T1] p is a nonempty function from T {t o } onto X and [T2] (∀t∈T {t o })(∃m∈N 1 ) p m (t) = t o . 3Call p the (immediate) predecessor function.
A functioned tree (uniquely) determines many entities beyond T and p.First, it determines its root node t o and its set X of decision nodes.Second, it determines its ). Relatedly, it determines its weak precedence relation on T by (∀t 1 ∈T, t 2 ∈T ) t 1 t 2 iff (∃m∈N 0 ) t 1 = p m (t 2 ).Finally, it determines the set Z of maximal chains in (T, ).This can be split into the set Z ft of finite maximal chains and the (possibly empty) set Z inft of infinite maximal chains.These derived entities and their basic properties are developed in SP Sections 2.1 and 2.2.
Let C be a set of elements c called choices.A triple Π = (T, C, ⊗) is a (nodeand-choice) preform (SP Section 3.1) iff [P1] there is a correspondence 4 F :T C and a t o ∈T such that ⊗ is a bijection from 5 F gr onto T {t o }, Call ⊗ the node-and-choice operator, and let t⊗c denote its value at (t, c) ∈ F gr .Call F the feasibility correspondence, call t o the root node, call p the immediatepredecessor function, and call H the collection of information sets.In addition, let X equal F −1 (C) (inconsequentially, SP uses F −1 (C) rather than X).Call X the decision-node set. 7 A node-and-choice preform Π (uniquely) determines many entities.First, it determines its components T , C, and ⊗.Second, it determines its F , t o , p, H, and X, as discussed in the previous paragraph.Third, [P2] determines the functioned tree (T, p), which in turn determines k, ≺, , Z ft , and Z inft , as discussed in the second-previous paragraph.Finally, define the preform's previous-choice function 3 I adopt the conventions that N 0 is {0, 1, 2, ...}, that N 1 is {1, 2, ...}, and that, for any function f , f 0 is the identity function. 4To be clear, let 5 In contrast to SP, the present paper notationally distinguishes between a correspondence and its graph, between a function and its graph, and between a binary relation and its graph.Thus [P1] distinguishes between the correspondence F and its graph F gr ⊆ T ×C.Also, [P2] distinguishes between the function p and its graph p gr ⊆ T ×T , and between the function ⊗ and its graph ⊗ gr ⊆ T ×C×T .Also, for example, Proposition 2.2(o) distinguishes between a relation ≺ and its graph ≺ gr ⊆ T ×T .
6 SP Lemma C.1(a) shows that [P1] implies the well-definition and surjectivity of p.
7 SP Lemma C.1(b,c) implies that a preform's t o and X coincide with the underlying tree's t o and X. Hence the symbols t o and X are unambiguous.
q:T {t o }→C by q gr = {(t , c)∈T ×C|(∃t∈T )(t, c, t )∈⊗ gr }.All these entities and their basic properties are developed in SP Sections 3.1 and 3.2.Among the basic properties is the convenient fact that (p, q) = ⊗ −1 .Further properties appear in SP Lemmas A.1, C.1, and C.2, and also in Lemma A.1 here.
Let I be a set of elements i called players.A quadruple Φ = (I, T, Each C i is the set of choices that are assigned to player i.The definitions in this paragraph are new to this paper (and an earlier version, Streufert 2016).
A node-and-choice form Φ (uniquely) determines many entities.First, it determines its components I, T , (C i ) i∈I , and ⊗.Second, [F1] determines C and the preform (T, C, ⊗), which in turn determines F , t o , p, q, H, X, k, ≺, , Z ft , and Z inft , as discussed in the second-previous paragraph.In addition, define (X i ) i∈I at each i by X i = ∪ c∈Ci F −1 (c).X i is the set of decision nodes that are assigned to player i.Further, define (H i ) i∈I at each i by H i = {F −1 (c)|c∈C i }.H i is the collection of information sets that are assigned to player i.
Proposition 2.1.Suppose (I, T, (C i ) i , ⊗) is a node-and-choice form with its X, H, (X i ) i∈I , and (H i ) i∈I .Then the following hold.(a) ∪ i∈I X i = X and (∀i∈I, j∈I {i}) Here are two minor remarks.[1] A preform can be understood as a one-player form.Specifically, (T, C, ⊗) is a preform iff ({1}, T, (C), ⊗) is a form, where (C i ) i = (C) is taken to mean C 1 = C. [2] A player i in a form is said to be vacuous iff C i = ∅.A vacuous player i necessarily has X i = ∅ and H i = ∅.Vacuous players can be convenient.For example, one can posit the existence of a chance player, and yet create a game without chance nodes by letting the chance player be vacuous.

Morphisms
A (node-and-choice) preform morphism SP Propositions 3.3 and 3.4 give two characterizations of preform morphisms which feel more category-theoretic.A (node-and-choice) form morphism is a quintuple ι:I→I , and The first paragraph of Proposition 2.2 rearranges the definition of a morphism.Meanwhile, the second and third paragraphs concern the many derivatives which can be constructed, via Section 2.1, from the source and target forms.Parts (k) and (m) are new, while the remainder are obtained by combining [FM1] with various SP results for preforms and trees.

The category NCF
This paragraph and Theorem 2.3 define the category NCF, which is called the category of node-and-choice forms.Let an object be a (node-and-choice) form Φ = (I, T, (C i ) i∈I , ⊗).Let an arrow be a (node-and-choice) form morphism β = [Φ, Φ , ι, τ, δ].Let source, target, identity, and composition be and where id I , id T , and id ∪ i∈I Ci are identities in Set.
To address a minor technical issue, note that many of the proposition's consequences are formulated by restricting functions.In each case, the codomain of the restriction is defined so that the restriction is surjective.Some other minor technical issues are discussed in notes 5, 9, and 10.
is a bijection from p gr onto p gr .(i) (τ, δ)| q gr is a bijection from q gr onto q gr .(j) τ | X is a bijection from X onto X .(k) (∀i∈I) τ | Xi is a bijection from X i onto X ι(i) . 98 The proposition's list of consequences is far from exhaustive.For example, in the notation of the proposition's second paragraph, Lemma A.2(b) deduces that (∀c∈C) τ (F −1 (c)) = (F ) −1 (δ(c)).
9 To be clear, parts (d), (k), and (m) do hold when there is a vacuous player i.In this case, C i is empty, and thus, δ| and H ι(i) are all empty as well. ( Finally, derive k, ≺, , Z ft , and Z inft from (T, p).Also, derive k , ≺ , , Z ft , Z inft from (T , p ). Then the following hold.
As already noted, the definition of a form incorporates a preform, and the definition of a form morphism incorporates a preform morphism.Correspondingly, Theorem 2.7 shows there is a "forgetful" functor P from NCF to NCP.Incidentally, SP Theorem 3.9 shows there is a similar functor F from NCP to Tree.Hence F•P is a functor from NCF to Tree.
Theorem 2.7.Define P from NCF to NCP by Then P is a well-defined functor.(Proof A.10.)

No-absentmindedness and perfect-information
Consider an arbitrary category Z, and a property which is defined for the objects of Z.The property is said to be isomorphically invariant iff, for each object, the object satisfies the property iff all of its isomorphs satisfy the property.This section explores two isomorphically invariant properties: [1] no-absentmindedness and [2] perfect-information.Both properties restrict information sets.
No-absentmindedness is a standard property which is widely regarded as being very weak (see, for example, Alós-Ferrer and Ritzberger 2016 Section 4.2.3).To define this property in NCP, consider an NCP preform with its ≺ and H. Then the preform is said to have no-absentmindedness iff (/ ∃H∈H, t A ∈H, t B ∈H) t A ≺ t B . 11Further, consider an NCF form with its preform.Then the form is said to have no-absentmindedness iff its preform has no-absentmindedness. 10 In parts (l), (m), (q), and (r), τ is understood to be the function 11 Piccione and Rubinstein 1997 Figure 1  Let NCP ã be the full subcategory of NCP whose objects are preforms with no-absentmindedness. (I am endeavouring to use subscripts for isomorphically invariant properties.)Similarly, let NCF ã be the full subcategory of NCF whose objects are forms with no-absentmindedness.No-absentmindedness will appear again in Section 3.3.Perfect-information is another standard property.It is restrictive, and at the same time, there are many interesting games which satisfy it (see, for example, Osborne and Rubinstein 1994 Part II).As in SP Section 3.5, an NCP preform, with its collection H of information sets H, is said to have perfect-information iff (∀H∈H) |H| = 1.Perfect-information is strictly stronger than no-absentmindedness. 12,13 Further, an NCF form is said to have perfect-information iff the form's preform has perfect-information.(In spite of Proposition 2.9, the existence of a morphism does not lead to any logical relationship between the source's perfect-information and the target's perfect-information.) Proposition 2.9.(a) Perfect-information is isomorphically invariant in NCP.
(b) Perfect-information is isomorphically invariant in NCF.(Proof A.12.) Let NCP p be the full subcategory of NCP whose objects are preforms with perfect-information.(The subscript ãp would be equivalent to the subscript p , because no-absentmindedness is implied by perfect-information, as shown in note 12.) Further, let NCF p be the full subcategory of NCF whose objects are forms with perfect-information.Perfect-information will appear again in Section 4.3.

Objects
Let a (finite) sequence be a function from {1, 2, ... m} for some nonnegative integer m (to be clear, the empty sequence 14 with empty domain is admitted by m = 0).I will regard a sequence as a set of ordered pairs.For example, t * = {(1, g), (2, f), (3, f)} is a sequence with domain {1, 2, 3}.An alternative notation for the same entity is t * = (g, f, f).Yet another is t * = (t * n ) 3 n=1 where t * 1 = g and t * 2 = t * 3 = f.Let the length of a sequence t be |t|.For instance, the length of the example sequence is 12 To see that perfect-information implies no-absentmindedness, assume no-absentmindedness is violated.Then there is H ∈ H, t A ∈ H, and t B ∈ H such that t A ≺ t B .Thus t A = t B .So |H| > 1 and perfect-information is violated. 13The "horse game" of Selten 1975  let the range of a sequence t be R Let the concatenation t⊕s of two sequences t and s be {(1, t 1 ), ... (|t|, t |t| ), (|t|+1, s 1 ), ... (|t|+|s|, s |s| )}.Thus the concatenation of a sequence t = (t 1 , t 2 , ... t |t| ) with a one-element sequence (c) is t⊕(c) = (t 1 , t 2 , ... t |t| , c).Next, for any sequence t and any ∈ {0, 1, 2, ... |t|}, let 1 t denote the initial segment (t 1 , t 2 , ... t ).Thus for any sequence t, Let CsqP be the full subcategory of NCP whose objects are choice-sequence preforms.Proposition 3.1 lists some of the special properties of CsqP preforms.Incidentally, property (h) and assumption [Csq1] together imply that each node in a CsqP preform is actually a choice sequence, as the terminology suggests.
) Finally, let a choice-sequence NCF form be an NCF form whose preform is a CsqP preform.Then let CsqF be the full subcategory of NCF whose objects are choice-sequence NCF forms.

Isomorphic Enclosure
Consider two full subcategories A and B of some overarching category Z. Say that A is isomorphically enclosed in B (in symbols, A → .B) iff every object of A is isomorphic to an object of B. Note that A → .B concerns not only the subcategories A and B but also, implicitly, the overarching category Z within which isomorphisms are defined.Further note that isomorphic enclosures can be composed in the sense that A → .B and B → .C imply A → .C. 15 Finally, let A ↔ .B mean that both A → .B and A ← .B hold.Call ↔ .isomorphic equivalence.Isomorphic equivalence implies the standard categorical concept of equivalence in MacLane 1998 page 18.
Theorem 3.2.(a) NCP → .CsqP.In particular, suppose Π = (T, C, ⊗) is an NCP preform with its p, q, and k.Define 2 This equivalence has a long history.In the more distant past, it was informally understood that game trees could be specified in terms of either [i] a collection of nodes and a collection of edges or [ii] a collection of sequences.Harris 1985 page 617 provides an example of this informal understanding.Specification [i] uses the nomenclature of graph theory (e.g., Tutte 1984), and specification-[i] trees were the basis on which Kuhn 1953 andSelten 1975 built game forms.Later, specification-[ii] trees became the basis on which Osborne and Rubinstein 1994 built game forms.
Kline and Luckraz 2016 17 (henceforth "KL16") develop this equivalence by a pair of theorems.In recognition of the above authors, they call specification-[i] forms "KS forms" and call specification-[ii] forms "OR forms".Then, one of their theorems (their Theorem 2) shows that a KS form can be derived from each OR form, while the other theorem (their Theorem 1) shows that each KS form can be mapped to an OR form. 18These two theorems are depicted by the two arrows in Figure 3.1(a).The arrows are dashed to convey that the equivalence is ad hoc. 16Theorems 3.2 and 4.2 draw upon Lemmas A.14 and A.15.These nontrivial lemmas show how to construct isomorphisms in NCP and NCF from bijections for nodes, choices, and players.These lemmas appear to have application beyond this paper. 17The terms "choice", "action", and "alternative" are fundamentally synonymous.However, the literature tends to use "choice" when it is assumed that information sets do not share alternatives, and conversely, to use "action" when the assumption is relaxed.The assumption itself is insubstantial in the sense that one can always introduce more alternatives until each information set has its own alternatives (see S5 Section 5.2, first paragraph, for more discussion).This paper makes the assumption for notational convenience, and correspondingly, uses "choice" (see SP Proposition 3.2(16b) and the paragraphs beforehand).In contrast, KL16 relaxes the assumption and uses "action". 18S5 Theorems 3.2 and 3.1 adapt and slightly extend KL16 Theorems 2 and 1.

NCF CsqF
Corollary 3.3(b) develops the equivalence further.Specification-[i] forms are written as NCF forms, and specification-[ii] forms are written as CsqF forms.Corollary 3.3(b) is then a pair of results: one half (the very easy half) shows that an NCF form is isomorphic to each CsqF form, while the other half (Theorem 3.2) shows that each NCF form is isomorphic to a CsqF form.Thus the corollary's isomorphic equivalence strengthens the KL16 equivalence by introducing isomorphisms.
There are further senses in which the corollary's isomorphic equivalence accords with the KL16 equivalence.In the backward direction, KL16 Theorem 2 is appealing because the nodes in the constructed KS form are identical to the sequences in the given OR form.This is possible because KS nodes admit OR sequences as special cases.Nonetheless KL16 Theorem 2 is nontrivial because KS forms do not admit OR forms as special cases.Here the analogous result is cleaner: NCF forms have been defined so that NCF forms admit CsqF forms as special cases.In the forward direction, KL16 Theorem 1 is made appealing by KL16 Lemma 2, which shows that there is a bijection α from the "vertex histories" in the given KS form to the nodes in the constructed OR form.That bijection is closely related to Theorem 3.2's bijection τ , which maps from the nodes of the given NCF form to the nodes in the constructed CsqF form.

More about No-absentmindedness
3.3.1.Proposition 3.4 describes a general situation in which one subcategory strictly isomorphically encloses another.In the proposition, w and s are two properties defined for the objects of Z. Further, w ⇐ ⇒ s means that w is strictly weaker than s.In other words, w ⇐ ⇒ s means that [a] each object of Z satisfies w if it satisfies s, and [b] there is an object of Z that satisfies w but not s.Corollary 3.5 applies Proposition 3.4 to the nonvacuous property of no-absentmindedness.Proposition 3.4.Suppose w and s are properties defined for the objects of Z, and that s is isomorphically invariant.Let Z w be the full subcategory of Z whose objects satisfy w, and let Z s be the full subcategory of Z whose objects satisfy s.Then w ⇐ ⇒ s implies Z w ← .→ .Z s .(Proof here.)Proof.Suppose w ⇐ ⇒ s.To see Z w ← .Z s , take an object of Z s .Since w ⇐ s, the object is also an object of Z w .Thus (trivially) the object is isomorphic to an object of Z w .To see Z w → .Z s , note the assumption w ⇐ ⇒ s implies that there is an object of Z that satisfies w and violates s.Thus there is an object of Z w that violates s.Thus since s is isomorphically invariant, this object does not have an isomorph that satisfies s.Thus the object does not have an isomorph in Z s . 2 Proof.(a).Consider Proposition 3.4 at Z equal to NCP, when w is the vacuous property satisfied by all objects of NCP, and s is the property of noabsentmindedness.No-absentmindedness is invariant by Proposition 2.8(a).Further the vacuous property is strictly weaker than no-absentmindedness because there exists an absentminded preform (recall note 11).Thus Proposition 3.4 implies that NCP w = NCP strictly isomorphically encloses NCP s = NCP ã. (b).This is very similar to (a).Change "preform" to "form", P to F, and (a) to (b). 2 To better interpret Corollary 3.5, recall Theorem 3.2(b) which states NCF → .CsqF.Formally, this means each NCF form is isomorphic to a CsqF form.This can be interpreted to mean that the property of having choice-sequence nodes is not "restrictive".In contrast, Corollary 3.5(b) implies NCF → .NCF ã.Formally, this means there is at least one NCF form (such as the one in note 11) that is not isomorphic to an NCF ã form.This can be interpreted to mean that the property of no-absentmindedness is "restrictive".Informally, the first result states that choice-sequence-ness is "purely notational".In contrast, the second result states that no-absentmindedness is "substantial", "significant", and "real", and that it "limits the range of decision processes and social interactions that can be modelled".The categorical concept of isomorphic enclosure (→ .) serves to formalize and to standardize these important terms.Note that both an isomorphic enclosure, and the negation of an isomorphic enclosure, are meaningful.
3.3.2.Next, Proposition 3.6 shows that an isomorphic enclosure can be restricted by any isomorphically invariant property.Corollary 3.7 uses this result to restrict Corollary 3.3 by no-absentmindedness.Corollary 3.7 will in turn be used in the remarkably quick proof of Corollary 4.3.
Proposition 3.6.Suppose that A and B are full subcategories of Z, and that w is an isomorphically invariant property defined for the objects of Z.Let A w be the full subcategory of A whose objects satisfy w, and let B w be the full subcategory of B whose objects satisfy w. (∀(t, c, t )∈⊗ gr ) t∪{c} = t .
Then let CsetP be the full subcategory of NCP whose objects are choice-set NCP preforms.Proposition 4.1 lists some of the special properties of CsetP preforms. 19 Incidentally, property (f) and assumption [Cset1] together imply that each node in a CsetP preform is actually a choice set, in accord with the terminology.More significantly, property (g) shows that every CsetP preform has no-absentmindedness.In this sense the combination of [Cset1] and [Cset2] is restrictive.
Proposition 4.1.Suppose (T, C, ⊗) is a CsetP preform with its F , t o , p, q, k, ≺, , and H. Then the following hold.(a) Finally, let a choice-set NCF form be an NCF form whose preform is a CsetP preform.Then let CsetF be the full subcategory of NCF whose objects are choiceset NCF forms.Corollary 4.3(b) is analogous to an ad hoc style equivalence in S5.There, a pair of results argues that no-absentminded OR forms ("ORā forms" in this subsection) are equivalent to S5-choice-set forms ("S5cs forms" in this subsection).One of the results (S5 Theorem 3.2) shows that an ORā form can be reasonably derived from each S5cs form, and the other result (S5 Theorem 3.1) shows that each ORā form can be reasonably mapped to an S5cs form.These two theorems are depicted by the two dashed arrows in Figure 4.1(a).Corollary 4.3(b) strengthens this equivalence.CsqF ã forms are like ORā forms in that both specify nodes as choice-sequences, and CsetF forms are like S5cs forms in that both specify nodes as choice-sets.Then, Corollary 4.3(b)'s isomorphic equivalence is a matching pair of results: one half (labelled "easy" in Figure 4.1(b)) shows that a CsqF ã form is isomorphic to each CsetF form, while the other half (Theorem 4.2) shows that each CsqF ã form is isomorphic to a CsetF form.Thus   Proof.(a).Consider Proposition 3.4 at Z equal to NCP, when w is the property of no-absentmindedness ã, and s is the property of perfect-information p. Perfect-information is isomorphically invariant by Proposition 2.9(a).Further noabsentmindedness is strictly weaker than perfect-information by notes 12 and 13.Thus Proposition 3.4 implies that NCP ã strictly isomorphically encloses NCP p .(b).This is very similar to (a).Change P to F, and (a) to (b). 2 CsetP.Thus, Propositions 3.6 and 2.9(a) imply that NCP ãp ↔ .CsqP ãp ↔ .CsetP p , where NCP ãp is the full subcategory of NCP consisting of those objects that satisfy both no-absentmindedness and perfect-information, and where similarly CsqP ãp is the full subcategory of CsqP consisting of those objects that satisfy both no-absentmindedness and perfect-information.Since no-absentmindedness is weaker than perfect-information (note 12), NCP ãp = NCP p and CsqP ãp = CsqP p .(b).This is very similar to (a).Change P to F, and (a) to (b). 2 Incidentally, since isomorphic equivalence implies categorical equivalence, Corollary 4.5(a) implies NCP p , CsqP p , and CsetP p are categorically equivalent.Further, SP Theorem 3.13 and Corollary 3.14 show that NCP p , Tree, and Grph ca are categorically equivalent, where Tree is the category of functioned trees which SP uses in its development of NCP, and where Grph ca is the full subcategory of Grph whose objects are converging arborescences.Thus, NCP p , CsqP p , CsetP p , Tree, and Grph ca are categorically equivalent.CsqF p ←→ .

C4.5(b)
CsetF p  arrows.They are derived by composing arrows as in the proof of Corollary 3.8.Many diagonal arrows could be similarly derived.

Deducing consequences from an isomorphic enclosure
Consider this paper's first isomorphic enclosure.Theorem 3.2 shows that each NCF form Φ is isomorphic to a CsqF form Φ by means of an isomorphism which transforms nodes via the bijection τ .Proposition 2.6 deduces many consequences from such an isomorphism.For example, its part (o) implies that (∀t 1 ∈T, t 2 ∈T ) t 1 ≺ t 2 iff τ (t 1 ) ≺ τ (t 2 ), where T is the node set of Φ, ≺ is derived from Φ, and ≺ is derived from Φ.Although such consequences about form derivatives like ≺ and ≺ are tantalizingly natural, the consequences about form derivatives in Proposition 2.6(f)-(r) take about 10 pages to prove.That work is important because such consequences are fundamental to drawing more conclusions from the isomorphic enclosure of NCF in CsqF.
As Section 3.2 explained, the isomorphic enclosure of NCF in CsqF is analogous to KL16 Theorem 1.No consequences about form derivatives have been deduced from that ad hoc theorem, and an analog of Proposition 2.6(f)-(r) would likely require about 10 pages to prove.Moreover, like KL16 Theorem 1, no consequences about form derivatives have been deduced from KL16 Theorem 2 or from S5 Theorems 3.1 and 3.2.Each of these ad hoc theorems has its own formulation, so deriving analogs of Proposition 2.6(f)-(r) for the three of them would likely require another 3×10 = 30 pages.
In contrast, Proposition 2.6(f)-(r) applies not only to the isomorphic enclosure of NCF in CsqF.It applies to any isomorphic enclosure.Thus it applies to all the arrows in Figure 4.2, as well as to all isomorphic enclosures in the future.

Future research
As discussed in Section 1.1, this paper is part of a larger agenda to translate game theory across specification styles.In this larger context, isomorphic enclosures can be seen as a way to translate form components from one style to another, and on the basis of these isomorphic enclosures, Proposition 2.6(f)-(r) (discussed just above) can be seen as a way of translating form derivatives from one style to another.The

Further Remarks
results of this paper wait to be expanded in three orthogonal directions.
[1] There is more to translate beyond forms and their derivatives.This would include properties that forms might satisfy, and theorems that might relate these properties to one another.(This paper makes some limited progress in this direction by exploring the isomorphically invariant properties of no-absentmindedness and perfect-information, and by identifying some special properties of CsqF forms and CsetF forms via Propositions 3.1 and 4.1.)Expanding in this direction would correspond to expanding the three substantive sections of this paper.
[2] This paper concerns only three styles: NCF, CsqF, and CsetF.There are other styles to explore, including the two neglected styles mentioned at the start of this paper, namely, the "node-set" style of Alós-Ferrer and Ritzberger 2016 Section 6.3, and the "outcome-set" style of von Neumann and Morgenstern 1944 and Alós-Ferrer and Ritzberger 2016 Section 6.2.Expanding in this direction will require defining new NCF subcategories for "node-set" forms and "outcome-set" forms, and will correspond to adding, to the present paper, two new sections for the two new subcategories.
[3] This paper concerns only forms, which need to be augmented with preferences in order to define games.At the higher level of games, many more issues emerge.To return to [1], there is more to translate, including equilibrium concepts and the theorems which might relate one equilibrium concept to another.To return to [2], there will be more than five specifications because there are alternative ways to specify preferences over the same form.Expanding in this third direction will require building a new category for games that incorporates this paper's category for forms.
Appendix A. NCF Lemma A.1.Suppose (T, C, ⊗) is an NCP preform with its F , t o , p, q, and H. Then the following hold.
Proof.(a).In the paragraph after SP equation (1), remark [ii] shows that (/ ∃t∈T ) p(t) = t.Thus, since p is nonempty by [T1], there are distinct t 1 ∈ T and t 2 ∈ T such that t 1 = p(t 2 ).Thus, by the definition of p in
2 (c); which by the definition of C equals ∪ c∈C F −1 (c); which by definition equals F −1 (C); which by definition (in Section 2.1) equals X.Thus it remains to show that (∀i∈I, j∈I {i}) X i ∩X j = ∅.Toward that end, suppose there are i 1 ∈ I and i 2 ∈ I such that [2] i 1 = i 2 and X i 1 ∩X i 2 = ∅.This nonemptiness and [1] imply there is part of SP Proof C.12's argument for SP Proposition 3.5, and the proof of the part (b) rearranges part of the argument for SP Lemma C.17(e).and thus [6 1 ] and [F2] imply [8 1 and  thus [6 2 ] and [F2] imply [8 2

by the definition of
Proof A.4 (for Proposition 2.2).The next two paragraphs prove the first paragraph of the proposition.In particular, the next two paragraphs show that [Φ, Φ , ι, τ, δ]  Henceforth assume that [Φ, Φ , ι, τ, δ] is a morphism.The remaining two paragraphs of the proposition follow from Claims 1, 2, 4, and 5 below.
Claim 2: (m) holds.Take i and H ∈ H i .By the definition of H i , there exists Claim 3: (a) holds.This follows from the bijectivity of δ and Claim 2.
Proof A.7 (for Theorem 2.4).Let the components of Φ be (I, T, The forward half of (a) and all of (b).Suppose that β is an isomorphism (Awodey (2010, page 12, Definition 1.3)).Recall that β = [Φ, Φ , ι, τ, δ] and let where the first equality in both lines holds by the definition of β −1 , and the second equality in both lines holds by the definition of id.The well definition of Similarly again, the fifth components of [1] and [2] imply δ is a bijection from C onto C and [e] δ * = δ −1 .To conclude, the previous three sentences have shown that ι, τ , and δ are bijections.Further, where the first equality follows from the definition of β −1 , and where the second equality follows from [a]-[e].
The reverse half of (a).Suppose that ι, τ , and δ are bijections.Define Derive Π from Φ and Π from Φ .The remainder of this paragraph will show that β * is a form morphism by showing that it satisfies To see [FM1 ], first note that [Π, Π , τ, δ] is a preform morphism by [FM1] for β.Thus the bijectivity of τ and δ, together with SP Theorem 3.7(a), imply that [Π , Π, τ −1 , δ −1 ] is an NCP isomorphism.Hence a fortiori, it is a preform morphism.To see [FM2 ], first note that ι:I→I by [FM2] for β.Thus the bijectivity of ι implies that ι −1 :I →I.Finally, to see [FM3 ], consider Lemma A.6.The lemma's assumptions are met because the theorem assumes that β = [Φ, Φ , ι, τ, δ] is a morphism and because the start of this paragraph assumes that ι and δ are bijections.
Thus the lemma's part (b) implies that (∀i To conclude, β * is a form morphism by the previous paragraph.Further, Hence β is an isomorphism (and incidentally, Proof.Derive F from Π, and Proof A.9 (for Proposition 2.6).The proposition follows from Claims 1-4 and 6-7.
Claim 3: (k) holds.Take i.Since τ is a bijection by Claim 1 (part (b)), it suffices to argue that The first equality holds by the definition of X i and a rearrangement.The second equality follows from Lemma A.2(b) because [Π, Π , τ, δ] is an isomorphism by Corollary 2.5.The third equality holds by Claim 2 (part (d)).The fourth equality holds by the definition of X ι(i) .
Claim 4: (m) holds.Take i.Since [Π, Π , τ, δ] is an isomorphism by Corollary 2.5, Lemma A.8 implies that τ | Hi is a well-defined function from 2 Proof A.10 (for Theorem 2.7).By [F1], P 0 maps any form into a preform.By [FM1], P 1 maps any form morphism into a preform morphism.Thus it suffices to show that P preserves source, target, identity, and composition (Mac Lane 1998 page 13).This is done in the following four claims.
Claim 3: P 1 (id Φ ) = id P0(Φ) .Take Φ = (I, T, (C i ) i∈I , ⊗) and let C = ∪ i C i .First I show [a] P 0 (Φ) = (T, C, ⊗) by arguing, in steps, that P 0 (Φ) by the definition of Φ is P 0 (I, T, (C i ) i∈I , ⊗), which by the definition of P 0 is (T, ∪ i∈I C i , ⊗), which by the definition of C is (T, C, ⊗).Then I argue, in steps, that P 1 (id Φ ) by the definition of id in NCF is equal to P 1 ([Φ, Φ, id I , id T , id C ]), which by the definition of P 1 is equal to [P 0 (Φ), P 0 (Φ), id T , id C ], which by [a] is equal to [(T, C, ⊗), (T, C, ⊗), id T , id C ], which by the definition of id in NCP is equal to id (T,C,⊗) , which by [a] is equal to id P0(Φ) ., ι , τ , δ ].First I note that, since P 1 is well-defined by the first paragraph, P 1 ([Φ, Φ , ι, τ, δ]) = [P 0 (Φ), P 0 (Φ ), τ, δ] and P 1 ([Φ , Φ , ι , τ , δ ]) = [P 0 (Φ ), P 0 (Φ ), τ , δ ] are preform morphisms.Then I argue that     By [T2] there exists m ≥ 1 such that t o = p m (t).On the one hand, suppose m = 1.Then the claim holds by the definition of m.On the other hand, suppose m ≥ 2. Then proving the claim requires several steps.First, I show Take any such n.Since τ is bijective, it suffices to show that the composition (τ   Claim 4: The first equality holds by the lemma's definition of ⊗ .The second holds by the definition of F , and the third by the definition of t o .The fourth holds by Claim 1. The fifth and sixth are rearrangements.The seventh holds by Claim 1. Toward that end, consider Lemma A.13. Lemma A.13's assumptions are met by [a] and the injectivity of τ .Thus Lemma A.13 implies that (T , p ) is a functioned tree, where the function p is defined by [d] p being surjective and [e] p gr = { (τ (t ), τ (t)) | (t , t)∈p gr }.Thus it suffices to show that p = p .Toward that end, note [c] and [d] imply that both p and p are surjective.Thus it suffices to show p gr = p gr .I argue The first equality holds by the definition of p two paragraphs ago, and the second equality holds by the definition of ⊗ in the lemma statement.The ⊆ direction of the third equality holds simply because the variable c does not appear in the right-hand side.The ⊇ direction follows from ⊗ gr ⊆ T ×C×T and δ:C→C .The fourth equality holds because the codomain of τ is T .The fifth equality follows from the definition of p two paragraphs ago, and the sixth equality follows from [e].
Claim 12: (∀t A ∈T, t B ∈T ) t A t B iff t A ⊆ t B .Take t A ∈ T and t B ∈ T .First, suppose t A t B .Then by the definition of , either t A ≺ t B or t A = t B .In the first case, Claim 7724o implies t A ⊂ t B .Thus t A ⊆ t B in both cases.Conversely, suppose t A ⊆ t B .Then either t A ⊂ t B or t A = t B .In the first case, Claim 11 implies t A ≺ t B .Thus the definition of implies t A t B in both cases.
Proof.Let α be the function from T to T , and conversely, let β be the function to T from T .

19
Almost every CsetP property in Proposition 4.1 has a CsqP analog in Proposition 3.1.The properties are merely presented in different orders because they are proved in different orders.The exceptions are that properties (g)-(i) have no CsqP analogs in Proposition 3.1. 20Lemma C.1 shows the following are equivalent: [a] c / ∈t and t∪{c}=t .[b] t =t and t∪{c}=t .[c] t =t and t=t {c}.[d] t⊆t and {c}=t t.

Figure 4
Figure 3.2 Corollary 4.3(b)  strengthens the S5 equivalence by introducing isomorphisms. 21Corollary 4.3(b)'s proof highlights how useful it is to compose isomorphic enclosures.In particular, consider the reverse direction ofCorollary 4.3(b), which isCsqF ã ← .CsetF inFigure 4.1(b), and compare it with S5 Theorem 3.2, which is ORā S5cs in Figure 4.1(a).The lemmas and proof for S5 Theorem 3.2 span six difficult pages.In contrast, the reverse direction of Corollary 4.3(b) is proved in six lines by composing an easily-proved enclosure (CsetF → .NCF ã in part [1] of proof) with a previously-proved enclosure (NCF ã → .CsqF ã from the forward half of Corollary 3.7(b)).

Figure 4 .
1(b)  shows this composition as the curved arrow followed by the forward direction of Corollary 3.7(b).

Figure 4 .
Figure 4.2's arrow diagram illustrates most of the isomorphic-enclosure results from Sections 3.2 and following.In addition, the diagram has some unlabelled 21 There is also another sense in which Corollary 4.3(b) accords with the S5 equivalence.The forward half of the corollary is Theorem 4.2, and that theorem transforms choice-sequence nodes to choice-set nodes via the bijection R| T .That same bijection is used in S5 Theorem 3.1.

Figure 4 . 2 .
Figure 4.2.Most of the previous figure, augmented with some results about perfect-information.C = Corollary.

Claim 2 :
τ | X :X→X is a bijection.This follows from the bijectivity of τ and the definition of X .Claim 3: τ | X •p•(τ | T {t o } ) −1 is a nonempty function from T {t o } onto X .The claim follows from composition.In particular, (τ | T {t o } ) −1 :T {t o }→T {t o } is a bijection by Claim 1, p:T {t o }→X is nonempty and surjective by [T1], and τ | X :X→X is a bijection by Claim 2. These bijections appear on the bottom, left, and top of Figure A.1.
The first equality holds by the lemma's definition of p gr .The second holds since the domain of p is T {t o } by [T1].The third is a rearrangement.The fourth holds by Claim 1.The fifth and sixth are rearrangements.The last holds because the domain of (τ| T {t o } ) −1 is T {t o } by Claim 1. Claim 5: p = τ | X •p•(τ | t {t o } ) −1 , that is, Figure A.1 commutes.This follows from Claim 4 because [a] p is surjective by assumption and [b] τ | X •p•(τ | t {t o } ) −1 is surjective by Claim 3.Claim 6: [T1 ] holds.This follows from Claims 3 and 5.

Claim 2 :
τ | τ {t o } :T {t o }→T {t o } is a bijection.This follows from the bijectivity of τ and the definition of t o .Claim 3: τ | τ {t o } •⊗•[(τ, δ)| F gr ] −1 is a bijection from F gr onto T {t o }.The claim follows from composition.In particular, ((τ, δ)| F gr ) −1 :F gr →F gr is a bijection by Claim 1, ⊗:F gr →T {t o } is a bijection by the definitions of F and t o , and τ T {t o } :T {t o }→T {t o } is a bijection by Claim 2. These three functions appear on the top, left, and bottom of Figure A.2.

Figure A. 2 .
Figure A.2. Set diagram for Claims 3 and 5.
Figure A.2 commutes.This follows from Claim 4 because [a] ⊗ is surjective by definition and [b] τ | T {t o } is surjective by Claim 2. Claim 6: [P1 ] holds.This follows from Claims 3 and 5. Claim 7: [P2 ] holds.Define p by [P2].[P2] implies that [a] (T, p) is a functioned tree.Define p by [P2 ].Claim 6 and SP Lemma C.1(a) implies [b] p is well-defined and [c] p is surjective.Because of [b], it suffices to show that (T , p ) is a functioned tree.

Claim 7 :
Not (e) ⇒ not (d).Assume not (e).Then R| T is not injective.Then there are s and t in T such that [a] s = t and [b] R(s) = R(t).
b).Take i.First I show [1] H i ⊆ H.I do this by arguing, in steps, that H i by definition equals {F −1 (c)|c∈C i }; which by the definition of C is a subset of {F −1 (c)|c∈C}; which by definition (in [P3]) equals H. Since H is a partition by [P3], [1] implies that the elements of H i are nonempty and disjoint.Thus it remains to show that ∪H i = X i .I argue, in steps, that ∪H i by the definition of H i equals ∪{F −1 (c)|c∈C i }; which by the definition of X i equals X i .(c).First I show ∪ i∈I H i = H.I do this by arguing, in steps, that ∪ i∈I H i by the definition of (H i ) i∈I equals ∪ i∈I {F −1 (c)|c∈C i }; which by rearrangement equals {F −1 (c)|c∈∪ i∈I C i }; which by the definition of C equals {F −1 (c)|c∈C}; which by definition (in [P3]) equals H. Thus it remains to show (∀i∈I, j∈I {i}) H ) is well-defined.In other words, it suffices to show[i] that p n (t) ∈ X and [ii] that τ | X •p n (t) is in the domain of (τ | T {t o } ) −1 .[i]holdsbecause the codomain of p is X by [T1].To see[ii], note that t o = p m (t) and m−1 ≥ n ≥ 1 imply that p n (t) is in the domain of p. Thus, since the domain of p is T {t o } by Claim 1: (τ, δ)| F gr :F gr →F gr is a bijection.This follows from the bijectivity of τ , the bijectivity of δ, and the definition of F .