Equivalences among five game specifications, including a new specification whose nodes are sets of past choices

The current literature formally links “OR forms” (named after Osborne and Rubinstein, in A course in game theory. MIT, London, 1994) with “KS forms” (named after Kuhn and Selten by Kline and Luckraz, in Econ Theory Bull 4:85–94, 2016). It also formally links “simple forms” with “AR forms” (the former is less prominent than the latter, and both are from Alós-Ferrer and Ritzberger, in The theory of extensive form games. Springer, Berlin, 2016a). This paper makes three contributions. First, it introduces a fifth game form whose nodes are sets of past choices. Second, it formally links these new “choice-set forms” with OR forms. Third, it formally links KS forms with simple forms. The result is a formal five-way equivalence which provides game theorists with a broad spectrum of alternative game specifications.


Introducing choice-set forms
The first contribution of this paper is to introduce a new game specification in which nodes are sets of past choices. An example choice-set form appears in Fig. 1. This is similar but not identical to the well-known Osborne-Rubinstein (OR) game specification in which nodes are sequences of past choices. An example OR form also appears in Fig. 1. The new specification has an apparent advantage in the sense that sets are simpler mathematical objects than sequences. Section 7 carefully discusses B Peter A. Streufert pstreuf@uwo.ca http://economics.uwo.ca/people/faculty/streufert.html 1 Economics Department, Western University, London, ON N6A 5C2, Canada the advantages and disadvantages of the new specification relative to the OR form, and also relative to the remaining three specifications in this paper.

Linking choice-set forms and OR forms
The second contribution of this paper is to build a formal connection between choiceset forms and OR forms. This formal equivalence will require two qualifications, because choice-set forms are slightly less general than OR forms in two regards. First, choice-set forms implicitly impose no absentmindedness in the sense of Piccione and Rubinstein (1997). Second, they implicitly impose no shared alternatives in the sense that two information sets are not allowed to share the same (feasible) alternatives (i.e., choices).
Theorem 3.1 shows that every OR form with no absentmindedness and no shared alternatives is equivalent to a choice-set form. Conversely, Theorem 3.2 shows that every choice-set form is equivalent to an OR form with no absentmindedness and no shared alternatives. Thus the theorems show (given no absentmindedness and no shared alternatives) that there is a logical redundancy at the heart of the OR specification: sets of past choices can unambiguously replace sequences of past choices. All five specifications are essentially equivalent, even though they are fundamentally different in how they specify nodes and choices. (A minor issue is that only OR forms and KS forms allow absentmindedness. Section 7.1 discusses absentmindedness, shared alternatives, and two other minor issues.)

Linking OR forms, KS forms, simple forms, and AR forms
Let an "AR form" be a discrete extensive form as defined by Alós-Ferrer and Ritzberger (2016a) (henceforth AR16). An example AR form appears in Fig. 1. Such forms extend the specification of von Neumann and Morgenstern (1944). Incidentally, AR16 also defines non-discrete extensive forms for differential and yet more general games. Such non-discrete forms are beyond the scope of this paper.
The current literature 1 does not provide a formal connection between OR forms and AR forms. This may be surprising because some prominent and valuable articles 2 have tangentially and informally suggested otherwise. It seems that the misstep was caused by improperly combining the following two (correct) results from AR16. [a] AR16 Example 6.5 (p. 145) essentially says that OR trees constitute a special case of "simple trees". This accords with Fig. 2's top line, which shows that OR trees specify nodes as choice sequences while simple trees specify nodes as abstract entities.
[b] AR16 Theorems 6.2 and 6.4 (pp. 139 and 147) say that "simple forms" are equivalent to AR forms. This equivalence is repeated here as Theorems 6.1 and 6.2, and is shown between the last two columns of Fig. 2. Together, [a] and [b] might seem to suggest that OR forms are special cases of AR forms. But this logic would be faulty because [a] concerns only trees while [b] concerns entire forms. 3 Thus the reasoning does not provide a formal link between OR forms and AR forms.
Fortunately, it only remains to link OR forms and simple forms because [b] links simple forms and AR forms. Further, part of this remaining gap is bridged by Kline and Luckraz (2016) (henceforth KL16). They essentially show that OR forms are equivalent to "KS forms", where the initials K and S correspond to Kuhn (1953) and Selten (1975). This equivalence is repeated here as Theorems 4.1 and 4.2, and is shown between the OR and KS columns of Fig. 2. 4 Thus it yet remains to link KS forms and simple forms. KS forms and simple forms are similar in that they both specify nodes as abstract entities (as shown in the top row of Fig. 2). However, KS forms and simple forms differ in that KS forms specify their choices as abstract entities, while simple forms specify their choices as sets of nodes (as shown in the second row of Fig. 2).
The third contribution of this paper is to bridge this gap between KS forms and simple forms. This new equivalence appears as Theorems 5. 1 and 5.2, and is shown between the KS and simple columns of Fig. 2. With this missing piece in place, it emerges that all five specifications are essentially equivalent. This provides game theorists with a wide spectrum of equivalent specifications.

Organization
Sections 2, 3, 4, 5, and 6 of this paper move from left to right across the five specifications in Fig. 2. Section 7 considers the minor qualifications in the theorems, and discusses the advantages and disadvantages of each game specification. Finally, the four pairs of equivalence theorems in this paper suggest four equivalences between subcategories of a category of game forms. First steps in this direction are Streufert (2018a, b).

Choice-set Forms
The following game specification is new.
Let N be a set of nodes n, and let C be a set of choices c. By assumption, each node is a set of choices. In other words, each node n satisfies n ⊆ C. A node can be either a finite set or an infinite set. Let T be the set of finite nodes t. In other words, let T = { n∈N | n is a finite set }. A choice-set tree is a pair (C, N ) such that [cs1] N is a nonempty collection of subsets of C, To understand [cs2] 5 , let a last choice of a finite node t be any choice c∈t such that t {c} is also a node. In other words, let a last choice of a node be any choice in the node whose removal results in another node. To understand [cs3], note that this equation relates the infinite nodes (that is, the members of N T ) to the finite nodes (that is, the members of T ). By definition, a chain in T is a subcollection T * ⊆ T such that any two distinct nodes t and t in T * satisfy t ⊂ t or t ⊃ t . The union of an infinite chain of finite nodes is obviously an infinite set.
∈t and t∪{c}∈T }. Thus each C t is the set of choices that are feasible at the node t. Second, define X = { t∈T | C t =∅ }. Call its members the decision nodes. Then assume [cs4] considers a collection of sets. Call each of its member sets an information set. Note each {t∈X |c∈C t } is the set of decision nodes from which the choice c is feasible. Routinely, the same set is generated by multiple choices. That set is the information set from which those choices are feasible.
[cs4] requires that these constructed information sets cannot intersect. The familiar properties of information sets then follow. In particular, [a] the collection of information sets partitions the decision-node set X , 7 and [b] two nodes in the same information set have the same set of feasible choices. 8 For example, consider the choice-set tree defined by C = {a,b,e,f} and N = {{}, {a}, {b}, {a,e}, {a,f}, {b,e}} = {a,b,g,d,e,f} and N = {{}, {a}, {a,g}, {b}, {a,d}, {b,e}, {b,f}, {a,d,e}, {a,d,f}} Further, let I be a set of players i, and let (C i ) i∈I assign a set C i of choices to each [cs5] requires that each choice is assigned to exactly one player. Thus [cs6] implies that all a decision node's choices are assigned to exactly one player. Thus [b] (two paragraphs above) implies that all an information set's choices are assigned to exactly one player.
Incidentally, the set of outcomes is N X = (N T )∪(T X ). The set N T consists of the infinite nodes, and the set T X consists of the finite nodes that are not decision nodes. Later, preferences can be assigned over N X , or more generally if appropriate assumptions are introduced, over some space of probability distributions over N X . 7 To prove this, first note that each information set is nonempty by inspection (empty sets of the form {t∈X |c∈C t } arise from choices c that are never feasible, and [cs4]'s construction simply discards them). Second, the information sets are disjoint by [cs4] itself. Third, the union of the information sets is a subset of X because each information set is a subset of X by inspection (incidentally (∀c∈C) {t∈X |c∈C t } = {t∈T |c∈C t }). To show the reverse inclusion, take any t * ∈ X . Its C t * is nonempty by the definition of X , and so, there exists c * ∈ C t * such that t * ∈ {t∈X |c * ∈C t }. 8 To prove this, suppose both t 1 and t 2 belong to the information set {t∈X |c A ∈C t } and yet C t 1 = C t 2 .
Without loss of generality, the inequality implies that there exists c B ∈ C t 1 C t 2 . Thus t 1 ∈ {t∈X |c B ∈C t } and t 2 / ∈ {t∈X |c B ∈C t }. Then {t∈X |c A ∈C t } and {t∈T |c B ∈C t } intersect because they both contain t 1 , and yet they are unequal because the first contains t 2 and the second does not. This contradicts [cs4].

Definition
An OR form here is identical to a game in Osborne and Rubinstein (1994) (p. 200) without the players' preferences. This specification is repeated in KL16 (p. 86). 9 Let C be a set of choices c, and letN be a set of nodesn. By assumption, each node is a sequence of choices. In other words, each node is of the formn This paragraph derives three entities from an OR tree. First, letT =N C ∞ be the collection of finite sequencest belonging toN . Second, define (Ct )t ∈T at eacht by Ct = { c∈C |t⊕(c)∈T }, wheret⊕(c) is the concatenation of the finite sequencet with the one-element sequence (c). Thus each Ct is the set of choices that are feasible at the nodet. Third, defineX = {t∈T | Ct =∅ }. CallX the set of decision nodes.
Next let I be a set of players i, and assign decision nodes to players by a playerassignment functionP. In brief, assume [OR4]P:X →I . Then associate with each player i a collectionH i of information setsH . Assume [OR5] (∀i∈I )H i partitions P −1 (i). Thus each player's information-set collectionH i partitions the player's set P −1 (i) of decision nodes. Further assume [OR6] (∀H ∈∪ i∈IHi ,t∈H ,t ∈H ) Ct = Ct . This is the usual assumption that two nodes in the same information set must have the same feasible choices. By definition, an OR form (C,N ), (P, (H i ) i∈I ) is an OR tree (C,N ) together with a player set I , a player-assignment functionP, and a list (H i ) i∈I of player information-set collections that satisfy [OR4]- [OR6].
Incidentally, the set of outcomes isN X = (N T )∪(T X ). The setN T consists of the infinite sequences, and the setT X consists of the nondecision finite sequences. Later, preferences can be defined overN X , or more generally if appropriate assumptions are introduced, over some space of probability distributions over N X .

The equivalence between choice-set forms and OR forms
[OR6] states that two nodes in the same information set have the same alternatives. 10 As usual, define CH at eachH ∈ ∪ i∈IHi by (∀t∈H ) CH = Ct . Thus CH is the set of alternatives at the information setH . (I i ) i∈N become i ∈ I and (H i ) i∈I . Sixth, P and P i becomeP andP −1 (i). 10 The terms "alternative", "action", and "choice" are fundamentally synonymous (see note 22).

Fig. 3
An OR form with no absentmindedness. In accord with Proposition 3.0, R|N is injective alternatives. On the one hand, this condition is vacuous in the sense that one can always introduce enough alternatives so that each information set has its own alternatives. On the other hand, it is natural to repeatedly use the same alternatives in a repeated game.
An OR form has no absentmindedness if and only if ( and Rubinstein 1997, p. 10). Hence a form has no absentmindedness if and only if none of its information sets contains both a node and a predecessor of that node. No-absentmindedness is regarded as a very weak assumption. It is explicitly incorporated into the game specification of Kuhn (1953) (p. 48, Definition 2(II)), and is defended at length by AR16 (Sect. 4.2.3). Further, it is weaker than perfect recall, and AR16 (p. 150) argues that a game without perfect recall "fails to capture rational behaviour".
No-absentmindedness plays a pivotal role in this section, as the following proposition demonstrates. In the proposition, the function R takes any sequence (c k ) K k=1 to its range. In particular, if K is finite, Consider the forward direction of the proposition. This paragraph notes how easy it is to derive injectivity when the form's information sets are ordered. Consider any nodē n. Since a choice determines its information set because of the no-shared-alternatives assumption, the choices in R(n) must be played in the order of their information sets. Hence the set R(n) determines the sequencen.
But the forward direction of Proposition 3.0 goes further. It shows that R|N is injective even when the form's information sets are unordered, provided only that no-absentmindedness holds. For example, consider Fig. 3, which replicates the classic example of unordered information sets from Kuhn (1953), Fig. 1;Gilboa (1997), Fig. 2;Ritzberger (1999), Fig. 1;and Ritzberger (2002), Fig. 3.8. Unordered information sets give rise to choices that can be played in different orders. Accordingly, the choices a 1 and a 2 in Fig. 3 have been played in different orders at the nodes (1 * , a 1 , a 2 ) and (2 * , a 2 , a 1 ). However, the choices in R((1 * , a 1 , a 2 )) = {1 * , a 1 , a 2 } can only be played in the order (1 * , a 1 , a 2 ), and the choices in R((2 * , a 2 , a 1 )) = {2 * , a 1 , a 2 } can only be played in the order (2 * , a 2 , a 1 ). Intuitively, this happens because the set {1 * , a 1 , a 2 } contains 1 * , and because the set {2 * , a 1 , a 2 } contains 2 * . This suggests that if a form has two choices whose order is not exogenously determined, then any sequence that lists the two choices must also list another choice (or set of choices) that determines their order. Showing that this can be done, whenever there is no-absentmindedness, is the interesting part of the proposition's proof.
Meanwhile, the reverse direction of Proposition 3.0 shows that no-absentmindedness is necessary for injectivity. For example, consider Fig. 4, which replicates the classic example of absentmindedness in Piccione and Rubinstein (1997), Fig. 1. Here R takes both the sequence (a) and the sequence (a,a) to the set {a}. Thus, R|N is not injective. The proposition's proof shows that something similar happens whenever no-absentmindedness is violated.
Theorem 3.1 uses the forward direction of Proposition 3.0 to show that every OR form with no shared alternatives and no absentmindedness is equivalent to a choiceset form. The reverse direction of Proposition 3.0 shows that Theorem 3.1 cannot be extended to include any OR forms that have absentmindedness.

Theorem 3.1 (choice-set ← OR) Suppose (C,N ), (P, (H i ) i∈I ) is an OR form with no shared alternatives and no absentmindedness. Define N = {R(n)|n∈N }. Then (a) (C, N ) is a choice-set tree and R|N is a bijection fromN onto N . Further, define
Conversely, Theorem 3.2 shows that every choice-set form is equivalent to an OR form with no shared alternatives and no absentmindedness. The theorem's proof constructs the OR form, and is the longest proof in the paper. N ) is an OR tree and R|N is a bijection fromN onto N . Further, deriveT , (Ct )t ∈T , andX from this OR tree. Also defineP:X →I at each t ∈X by settingP(t) equal to the unique i for which Ct is a well-defined OR form with no shared alternatives and no absentmindedness. (Proof A.8.)

Definition
A KS form here is identical to a KS game (KL16, p. 89) without the players' preferences. 11 The letters K and S refer to Kuhn (1953) and Selten (1975).
Begin with a set T of nodes t, and a set E of edges e. By definition, a pair (T , E) is a directed graph if and only if E ⊆ {(t, t )∈T 2 |t =t } (Bang-Jensen and Gutin 2009, p. 2). Say that one node t immediately precedes another node t if and only if t is immediately succeeded by t if and only if (t, t ) ∈ E. Further, say that a finite walk Bang-Jensen and Gutin 2009, p. 11). Similarly, an infinite walk from t 1 is a sequence is a directed graph and r ∈ T , and [KS2] for each t ∈ T {r } there is a unique finite walk from r to t . Call r the root node of the graph-tree. Further, let X = {t∈T |(∃t ∈T )(t, t )∈E} be the set of nodes with at least one successor, and call each t ∈ X a decision node.
Next associate, with each node t ∈ T , a set C t of feasible choices c. Such a feasible set can be empty. Further, associate, with each node t ∈ T , a choice-to-successor In other words, assume each player's information-set collection H i partitions the player's set Incidentally, the collection of outcomes is W r W X r , 12 where W r is the collection of (finite and infinite) walks from r , and where W X r = { (t k ) K k=0 ∈W r | K <∞, t K ∈X } is the collection of (finite) walks from r to a decision node. The collection W r W X r 11 Several changes have been made to facilitate comparison across game specifications. First, t∈T here. Second, a "KS-tree" there becomes a "KS augmented-tree" here (this accords with note 4). Third, V D becomes X . Fourth, P and P f and P D become {(r )}∪W r and {(r )}∪W T r and {(r )}∪W X r . Fifth, both p and w∪{(r , (r ))} become w. Sixth, i ∈ N and (I i ) i∈N become i ∈ I and (H i ) i∈I . Seventh, P i becomes P −1 (i). 12 This construction fails when |T | = 1, which is a trivial case. (In this case there are no walks .) consists of [i] the infinite walks from r , and [ii] the (finite) walks from r to nondecision nodes. Later, preferences can be defined over W r W X r , or more generally if appropriate assumptions are introduced, over some space of probability distributions over W r W X r .

The Kline/Luckraz equivalence between OR forms and KS forms
Theorem 4.1 requires some additional notation. As in the previous paragraph, consider a KS form and let W r be the collection of walks from r . Next let N ) is an OR tree. Further, derive w:T →{(r )}∪W T r from the KS form by the previous paragraph. Also deriveT andX from the OR tree (C, N ). Also definē P:

Definition
A simple form here is virtually identical to a simple extensive form in AR16 (p. 146). 14 The difference is insignificant. 15 Let T be a set of nodes t, and let ≥ be a binary relation on T there has no equivalent here. 15 The difference is that [s4] strengthens the phrase before AR16, p. 146, Definition 6.4 (SF1), by requiring that [1] no choiceĉ ∈ ∪ i∈IĈi contains r , and [2] no choiceĉ ∈ ∪ i∈IĈi equals ∅. I argue that [1] and [2] are insignificant by considering the set p(ĉ) of nodes at which such choicesĉ would be feasible. Regarding [1], supposeĉ did contain r . Then p(ĉ) = {p(t)|t∈ĉ} would contain p(r ), which is ill-defined. Regarding Define r = max T , and call r the root node. Say that t precedes t if and only if t is succeeded by t if and only if t > t . Let X ⊆ T be the set of nodes t which have at least one successor, and call every such node t ∈ X a decision node. Define p:T {r }→X by p(t ) = min{t∈T |t>t }, and call p(t ) the (immediate) predecessor of the node t (AR16, p. 145).
By assumption, a choiceĉ will be a nonempty set of non-root nodes t ∈ T {r }. In other words, each choiceĉ satisfies ∅ =ĉ ⊆ T {r }. The set of nodes at which a choiceĉ is feasible is p(ĉ) = {p(t )|t ∈ĉ} (AR16, p. 145). Note that this equation is the standard way of defining the image of a set (such asĉ) under a function (such as p). Accordingly, p(ĉ) is the set of nodes p(t ) that immediately precede a node t in c. Further, let I be the set of players i, and let (Ĉ i ) i∈I list a collectionĈ i of choicesĉ for each player i. Incidentally, the outcomes of a simple form are the maximal chains of its simple tree (such chains can be finite or infinite). Later, preferences can be defined over the collection of maximal chains, or more generally if appropriate assumptions are introduced, over some space of probability distributions over the collection of maximal chains.

The equivalence between KS forms and simple forms
The following two theorems are new. In both theorems, part (a) is more straightforward than part (b).
The theorems use some minor conditions, all of which are discussed in Sect. 7.1. First, say that a simple form has no simultaneous decisions if and only if (∀t∈X ) J (x) is a singleton. Second, say that a KS form has no absentmindedness if and only if (∀H ∈∪ i∈I H i ,t A ∈H ,t B ∈H ) there is not a walk from t A to t B . Third, say that a KS form has no trivial decisions if and only if (∀t∈T ) |C t | = 1.
is a simple form with no simultaneous decisions, and derive its r , p, and X . t∈T at each t and eacĥ c∈C t by letting ψ t (ĉ) be the unique element of p −1 (t)∩ĉ. Also, define P:X →I at each t ∈ X by letting P(t) equal to the unique i for which (∃ĉ∈Ĉ is a well-defined KS form with no absentmindedness and no trivial decisions. (Proof A.9.)

Definition
An AR form here is virtually identical to a discrete extensive form in AR16 (p. 138). 16 The difference is insignificant. 17 LetṄ be a nonempty collection of nonempty setsṅ. Define Ω = ∪Ṅ . CallṄ the set of nodesṅ, and call Ω the space of outcomes ω. Notice that every nodeṅ is a subset of Ω. In other words, every nodeṅ is a set of outcomes ω. By definition, the nodeṅ ∈Ṅ precedes the nodeṅ ∈Ṅ if and only ifṅ is succeeded byṅ if and only ifṅ ⊃ṅ . Note that Ω itself can be a member ofṄ . If so,ṅ = Ω is a node which precedes all other nodesṅ ∈Ṅ {Ω}. Now introduce a set I of players i, and letĊ i denote player i's set of choicesċ. By assumption, a choiceċ is a nonempty subset of Ω. Further, let P(ċ) denote the set of nodesṅ at which the choiceċ is feasible, and define it byṖ(ċ) = {ṗ(ṫ ) |ṫ ∈Ṫ ,ċ⊇ṫ , and ( ṫ∈Ṫ )ċ⊇ṫ⊃ṫ } (AR16, p. 134, Proposition 6.2(b)). ThisṖ is not related to the player-assignment functionsP and P of OR and KS forms.
At each decision nodeṫ ∈Ẋ , let A i (ṫ) = {ċ∈Ċ i |ṫ∈P(ċ) } be the set of feasible choices for player i, and let J (ṫ) = { i∈I | A i (ṫ) =∅ } be the set of decision makers. By definition, a (discrete) AR form (p. 138 in A16, and note 17 here) is a triple (Ṅ , ⊇, (Ċ i ) i∈I ) such that (Ṅ , ⊇) is a (discrete) AR tree, [AR7] everyċ ∈ ∪ i∈IĊi is a nonempty proper subset of Ω which is both the union of a subcollection ofṄ and a superset of some member ofṪ {Ω}, does not appear in AR16, p. 47, Definition 2.4, but does appear in Alós-Ferrer and Ritzberger 2013 , p. 78, Definition 1. The latter is relevant in the present (discrete) context. See, for example, the use of rootedness in AR16, p. 144, Proposition 6.5, first sentence of part (b)'s proof. 19 [AR6] implies that each non-root node without an immediate predecessor is necessarily a nondecision node (AR16, p. 135, Proposition 6.3, second sentence of proof). In other words, (Ṅ {Ω}) Ṫ ⊆(Ṅ {Ω}) Ẋ . This is equivalent toẊ {Ω} ⊆Ṫ {Ω}. This impliesẊ ⊆Ṫ since Ω ∈Ṫ by definition. 20 This sentence fails when |Ṅ | = 1, which is a trivial case. (In such a case Ω is a nondecision node.) 21 For example,Ṅ {{ω}|ω∈Ω} in the AR16 tree that Theorem 6.2 constructs from any simple tree having infinite maximal chains. This accords with the theorem's claim that the AR16 tree is order-isomorphic to the simple tree: neither the AR16 tree nor the simple tree has (nondecision) nodes corresponding to the simple tree's infinite maximal chains. AR16 (p. 138) explains that [AR8] states the standard properties of information sets, and that [AR9] describes how choices determine successors when simultaneous decisions are allowed.
Incidentally, recall that the space Ω of outcomes is primitive (or, virtually the same, thatṄ is primitive and Ω is defined as ∪Ṅ ). Thus it is straightforward to define preferences over Ω. Recent contributions which do so include Alós-Ferrer and Ritzberger (2016bRitzberger ( , 2017b. More generally, preferences might be defined over some space of probability distributions over Ω, if appropriate assumptions are introduced. AR16,p. 144,Proposition 6.5(b). (b) Corollary of AR16,p. 139,Theorem 6.2 (DEF⇒EDP),and AR16,p. 147,Theorem 6.4(b).]

Four minor features of the five specifications
Theorems 3.1 and 3.2 restrict OR forms by no absentmindedness and no shared alternatives. 22 Theorems 5.1 and 5.2 restrict KS forms by no absentmindedness and no trivial decisions, and also restrict simple forms by no simultaneous decisions. Absentmindedness, shared alternatives, trivial decisions, and simultaneous decisions are all features of game specifications. Each of these features corresponds to a row in Table 1, and each is discussed in a paragraph below.
Absentmindedness is listed first because it is the only feature whose absence limits the range of social interactions that can be modeled. At the same time, this limitation is very unimportant (Sect. 3.2, second paragraph). Absentmindedness is currently OR forms and KS forms allow information sets to share alternatives (i.e., choices), and this can be convenient when defining a repeated game. Yet, disallowing shared alternatives is an innocuous assumption in the sense that one can always introduce enough alternatives so that each information set has its own alternatives. Choice-set forms disallow shared alternatives because {t∈X |c∈C t } (if nonempty) is the unique information set associated with the choice c (recall [cs4]).
Trivial decisions can be convenient for expanding game trees. Trivial decisions are currently allowed in choice-set forms, OR forms, and KS forms. It seems they might later be allowed in simple forms by altering [s3] and [s8], and in AR forms by pursuing AR16, pp. 64-65.
Simultaneous decisions are more convenient than cascading information sets in the sense of AR16, pp. 140-142. Simultaneous decisions are already built into simple forms and AR forms. A similar construction seems possible for choice-set forms, OR forms, and KS forms (as in Osborne and Rubinstein 1994, p. 102).

General discussion
Although all four features are of minor importance, Table 1 and the preceding paragraph argue that OR forms and KS forms have more of the features than choice-set forms and AR forms. Further, simple forms seem able to gain absentmindedness by removing [s7] and to gain trivial decisions by altering [s3] and [s8]. In this sense, the three middle specifications appear to be slightly more general than the two specifications on the ends. Now consider these three middle specifications in the context of Fig. 2. The leftright spectrum there is identical to the left-right spectrum in Table 1. KS forms are special because both their nodes and their choices are abstract (see the top two rows of Fig. 2). This allows one to specify both nodes and choices flexibly, as desired. OR forms are less flexible but more efficient notationally since they express nodes in terms of choices. Symmetrically, simple forms are less flexible but more efficient notationally since they express choices in terms of nodes.
At the two ends of the spectrum, choice-set forms and AR forms sacrifice small amounts of generality for even more notational efficiency. In both cases, the extra efficiency is gained by using more set theory. For example, precedence becomes set inclusion: On the left, a choice-set form has t preceding t if and only if t ⊂ t , while on the right, an AR form hasṫ precedingṫ if and only ifṫ ⊃ṫ .
As the last sentence suggests, the two spectrum ends are opposites in some sense. On the spectrum's left, nodes are expressed in terms of choices (see Fig. 2's top row). Since that is done in terms of past choices, the notation looks backward more efficiently. For example, it can be relatively easy [a] to find the product of the probabilities of past choices, [b] to sum the rewards and costs from past choices, or [c] to sum the infinite relative likelihoods of past choices (Streufert 2015a, Sects. 3.3, 4.1). In contrast, on the spectrum's right, nodes and choices are expressed in terms of outcomes (see Fig. 2's top rows). Since outcomes are in the future, this notation looks forward more efficiently. For example, it can be relatively easy [a] to abstractly analyze preferences over outcomes without even referring to the time horizon (Alós-Ferrer and Ritzberger 2016b), or [b] to connect game forms with the standard statistical foundations of stochastic processes.
Although it is natural to have a favourite game specification, there appears to be no objective sense in which one game specification is best for all purposes. Thus it may be advantageous to be fluent in several specifications, so as to be able to freely choose the specification that best fits the purpose at hand.
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Lemma A.1 Suppose T is a collection of finite subsets t of C. Then, for any s ⊆ C, (a) there is an infinite chain T * in T such that
The reverse direction is proved by setting T * = {t m |m≥1}.
To prove the forward direction, take any s and assume T * is an infinite chain in T such that ∪T * = s. Define (t m ) m≥1 recursively by t 1 = min T * and (∀m≥2) t m = min T * {t 1 , t 2 , . . . t m−1 }. Every step in this infinite recursion is well-defined because [a] T * is infinite by assumption and [b] every nonempty subcollection of T * has a minimum because T * is a chain of finite sets by assumption. By inspection, (∀m≥1) t m ⊂ t m+1 . Thus it remains to show ∪ m≥1 t m = s. Note that ∪ m≥1 t m ⊆ ∪T * = s, where the set inclusion holds by (∀m≥1) t m ∈ T * , and where the equality holds by assumption. Conversely, the next two paragraphs show s ⊆ ∪ m≥1 t m . This paragraph shows by induction that (∀m≥0) m ≤ |t m+1 |. The initial step (m = 0) is 0 ≤ |t 1 |, which holds trivially. The inductive step (m ≥ 1) is m = (m−1)+1 ≤ |t m |+1 ≤ |t m+1 |, where the first inequality holds by the inductive hypothesis and the second inequality holds by t m ⊂ t m+1 .
{t∈T |n k ∈Ct }. These are information sets by Lemma A.3. Since these two information sets intersect, [OR4] and [OR5] imply that the two are equal. Note that 1nk−1 belongs to the latter. Thus it belongs to the former. In other words,n ∈ C 1nk−1 . Thust * = 1nk−1 ⊕(n ) is well-defined. Since < k,t * is well-defined and equalsn . Since < k and botht * andt * k equaln , and [c]t =t k , both 1t −1 and 1tk−1 belong to {t∈T |t k ∈Ct }. By Lemma A.3, this is an information set. Since < k, the last two sentences imply that this information set contains both 1t −1 and its successor 1tk−1 .

Proof Not (a) ⇒ not (b). Suppose there is absentmindedness. Then Lemma A.4 (a⇔c) implies there is a sequencet such that
. This is obvious sinceT ⊆N . Not (c) ⇒ not (a). Assume thatn 1 andn 2 are distinct elements ofN such that R(n 1 ) = R(n 2 ).

Lemma A.6 Consider an OR form with no shared alternatives and no absentminded-
where the equalities hold by two applications of Lemma A.4(a⇔c), and where the inequality holds because R(t ) ⊆ R(t). This is the first of the lemma's two conclusions. For the second conclusion, it suffices to show that 1t K (t ) = 1tK (t ) . For this, it suffices that the next two paragraphs show, by induction on k ∈ {1, 2, . . . K (t )}, that (∀k≤K (t )) 1t k = 1tk .
. Because of Claim 4, it suffices to show the reverse direction. Toward that end,  is ∈t and t ∪{c}=t by manipulation; which is [5] equivalent to (∃t ∈T ) c∈t and t {c}=t by manipulation; which is [6] equivalent to c∈t and t {c}∈T . Claim 7: [cs2] holds. Take t = {}. It must be shown that t has a unique last choice.
Claim 6 establishes that the last elements of the sequence (R|T ) −1 (t) are identical to the last choices (Sect. 2) of the set t. Since the sequence (R|T ) −1 (t) is nonempty because the set t is nonempty, the sequence has a unique last element. By the previous two sentences, the set t has a unique last choice.   Claim 14:[cs4] holds. Suppose there were c A ∈ ∪ i C i and c B ∈ ∪ i∈I C i such that {t∈X |c A ∈C t } and {t∈X |c B ∈C t } were unequal but intersecting. Then, without loss of generality, assume that t 2 is in both sets and that t 1 is in the former but not the latter. Then t 1 ∈ X and t 2 ∈ X are such that {c A , c B } ⊆ C t 2 , c A ∈ C t 1 , and c B / ∈ C t 1 . By Claim 13 andX ⊆T , there existt 1 andt 2 inT such that R(t 1 ) = t 1 and R(t 2 ) = t 2 . Thus by the second-previous sentence and Claim 12, {c A , c B } ⊆ Ct2 , c A ∈ Ct1, and c B / ∈ Ct1. Thust 2 is in both {t∈T |c A ∈Ct } and {t∈T |c B ∈Ct }, whilet 1 is in the former but not the latter. Therefore, since these two sets belong to ∪ i∈IHi by Lemma A.3, ∪ i∈IHi is not pairwise disjoint. This contradicts [OR4] and [OR5]. Claim 15:[cs5] holds. Suppose there were i ∈ I , j ∈ I {i}, and c ∈ C i ∩C j . By the theorem's definition of takes each nonempty t ∈ T to its unique last choice c * (t). Third, define (T k ) k≥0 by T k = { t∈T | |t|=k }. Note T = ∪ k≥0 T k . Also note T 0 = {{}} by note 6. Definition of (Q k ) k≥0 . This paragraph recursively defines a sequence (Q k ) k≥0 of surjective functions which map choice sets to choice sequences. More precisely, each Q k will map each t ∈ T k to some finite sequence in C. To begin, define the oneelement function Q 0 :T 0 →Q 0 (T 0 ) by Q 0 ({}) = {}. Note that the codomain of Q 0 has been set equal to its range Q 0 (T 0 ) = {{}}. Then, for any k ≥ 1, use Q k−1 to define T k−1 is the domain of the function Q k−1 which was defined in the last step of the recursion. Also note that the codomain of Q k has been set equal to its range Q k (T k ).
Thus the ⊇ half of the definition ofN impliess ∈N . Claim 3:T = ∪ k≥0 Q k (T k ). Section 3.1 definesT to be the collection of finite sequences inN . Thus the claim holds by inspecting the definition ofN . Claim 4: (∀k≥0, t∈T k ) K (Q k (t)) = k. This can be shown by induction. The initial step (k = 0) holds because T 0 = {{}} by the definition of T 0 , and because K (Q 0 ({})) = K ({}) = 0 by the definition of Q 0 . The inductive step (k ≥ 1) holds because for any t ∈ T k , K ( Q k (t) ) equals ) ) by the definition of Q k , which equals K ( Q k−1 (t {c * (t)}) ) + 1 by inspection. This equals (k−1) + 1 by [a] the inductive hypothesis and [b] the fact that t {c * (t)} ∈ T k−1 by t ∈ T k and [cs2]. Finally, (k−1) Claim 5. Thus there is t ∈ T K (t) such thatt = Q K (t) (t). Sincet = {} by assumption, K (t) ≥ 1. The last two sentences and the definition of [OR2] holds for K < ∞. Toward that end, take anyn ∈N and any ≥ 0 such that < K (n) < ∞. By K (n) < ∞ and the definition ofT ,n ∈T . Thus 1n belongs toT by K (t)− applications of Claim 6; which is a subset ofN by the definition ofT .
Take t ∈ T . The definition of (T k ) k≥0 implies [1] by the inductive hypothesis, which [d] equals t. Claim 10:R|T :T →T is the inverse of ∪ k≥0 Q k . Claim 9 implies that (∀k≥0) R| Q k (T k ) = Q −1 k and that it maps from Q k (T k ) onto T k . Claim 5 implies that the members of {Q k (T k )|k≥0} are disjoint. The definition of (T k ) k≥0 implies that the members of {T k |k≥0} are disjoint. The last three sentences imply that R| ∪ k Q k (T k ) = (∪ k≥0 Q k ) −1 and that it maps from ∪ k≥0 Q k (T k ) onto ∪ k≥0 T k . This is equivalent to the claim because [a] ∪ k≥0 Q k (T k ) =T by Claim 3 and because [b] ∪ k≥0 T k = T by the definition of (T k ) k≥0 . Claim 11:(∀k≥1, t ∈T k−1 , c∈C, t∈T k ) Q k−1 (t )⊕(c) = Q k (t) iff both c / ∈ t and t ∪{c} = t. Take any such k, t , c, and t. }) and c=c * (t) by rearrangement; which [c] is equivalent to t =t {c * (t)} and c=c * (t) by applying R to both sides of the first equality and then simplifying it via Claim 10; which [d] is equivalent to t =t {c}, c∈t, and c=c * (t); which [e] is equivalent to t =t {c} and c∈t since t ∈T by assumption; which [f] is equivalent to c / ∈t and t ∪{c}=t by rearrangement.
∈ t and t ∪{c} = t. Take t , c, and t. By Claim 8 ∈ t and t ∪{c} = t together. For the forward direction, assume [a]. Two applications of Claim 5 imply K (Q |t | (t )) = |t | and K (Q |t| (t)) = |t|. This and [a] imply |t| ≥ 1 and |t | = |t|−1. Claim 13:(∀t ∈T , c∈C,t∈T )t ⊕(c) =t iff both c / ∈ R(t ) and R(t )∪{c} = R(t). Take anyt , c, andt. By Claim 10,t ⊕(c) =t is equivalent to This, in turn, is equivalent to the combination of c / ∈ R(t ) and R(t )∪{c} = R(t) by Claim 12 applied at t = R(t ) and t = R(t Conversely, suppose t ∈ X . By Claim 10, there existst ∈T such that R(t) = t. Thus it suffices to showt ∈X . Since R(t) = t and t ∈ X , C R(t) = ∅. Thus by Claim 14, Ct = ∅. Hencet ∈X . Claim 16:P is well-defined and [OR4] holds. It suffices to showP:X →I is welldefined. This is equivalent to (∀t∈X )(∃!i∈I ) Ct ⊆ C i . By Claim 14, this is equivalent to (∀t∈X )(∃!i∈I ) C R(t) ⊆ C i . By Claim 15, this is equivalent to (∀t∈X )(∃!i∈I ) C t ⊆ C i . To show this, take t ∈ X .
[cs6] implies there is an i ∈ I such that C t ⊆ C i . [cs5] implies that this i is unique since [i] C t = ∅ since [ii] t ∈ X . Claim 17:(∀c∈∪ i∈I C i ) { R(t) |t∈X , c∈Ct } = { t∈X | c∈C t }. Take any c. For the ⊆ direction, take anyt ∈X such that c ∈ Ct . Notet ∈X and Claim 15 imply R(t) ∈ X . Further, c ∈ Ct and Claim 14 imply c ∈ C R(t) . The previous two sentences imply R(t) ∈ { t∈X | c∈C t }. Conversely, take any t ∈ X such that c ∈ C t . Note t ∈ X and Claim 15 imply the existence of at ∈X such that t = R(t  Claim 12:The KS form has no absentmindedness. Suppose there were a walk (t k ) K k=1 from t 1 to t K and an H ∈ ∪ i∈I H i such that {t 1 , t K } ⊆ H . Since (t 1 , t 2 ) ∈ E, Claim 6 implies there isĉ ∈ C t 1 such that [a] ψ t 1 (ĉ) = t 2 . Further, since {t 1 , t K } ⊆ H , Claim 10 and [KS6] implyĉ ∈ C t K . Thus we may define [b] t K +1 = ψ t K (ĉ). By the definition of (ψ t ) t∈T Further, [b] and Claim 5 imply (t K , t K +1 ) ∈ E; which implies (t k ) K +1 k=2 is a walk; which implies (∀k≤K ) (t k , t k+1 ) ∈ E; which by the definition of E implies (∀k≤K ) t k = p(t k+1 ); which by the definition of p and the transitivity of > implies t 2 > t K +1 . This and [e] contradict [s7].  (d) will be used implicitly to ensure that the symbols r and X are unambiguous.] A KS graph-tree is specified via graph theory while a simple tree is specified via order theory. The conversion from the former to the latter is relatively straightforward. Details are available on request.