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 1994) with “KS forms” (named after Kuhn and Selten by Kline and Luckraz 2016). It also formally links “simple forms” with “AR forms” (both from Alós-Ferrer and Ritzberger 2016, with the former less prominent than the latter). 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.

1 Selten's "horse" in each of the five specifications. This paper [a] introduces choice-set forms, [b] links choice-set forms and OR forms, and [c] links KS forms and simple forms. (This figure shows the tree, the choices, and the information sets of each form. The figure does not show players.)

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. 1.3 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 Figure 1.1. 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 (page 145) essentially says that OR trees constitute a special case of "simple trees". This accords with Figure 1.2'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 (pages 139 and 147) say that "simple forms" 4 Peter A. Streufert 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 Figure 1.2. 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 andSelten 1975. This equivalence is repeated here as Theorems 4. 1 and 4.2, and is shown between the OR and KS columns of Figure 1.2. 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 Figure 1.2). 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 Figure 1.2).
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 Figure 1.2. 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 2-6 of this paper move from left to right across the five specification in Figure 1.2. 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 2017 and 2016.

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, 3 The distinction between tree and form can be subtle. AR16 and the present paper use "tree" to mean nodes and precedence, and use "form" to mean a tree together with choices, information sets, and players. [a] concerns trees. In particular, there is nothing in AR16 Example 6.5 that concerns the choices of the simple specification. (The choices of the OR specification do appear in AR16 Example 6.5, but only because OR nodes are specified in terms of those choices.) 4 KL16 uses the word "tree" differently than it is used in AR16, the present paper, and much of the literature. In particular, their theorems show the equivalence of "OR-trees" and "KS-trees", where a "KStree" is defined to be a tree augmented with choices. Accordingly, their "KS-tree" is about halfway from a tree to a form (as this paper and much of the literature uses those two terms).
Equivalences among Five Game Specifications ... 5 let T = { n2N | 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, [cs2] (8t2T r {{}})(9!c2C) c2t and t r {c}2T, and To understand [cs2], let a last choice of a finite node t be any choice c2t such that t r {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.
[cs2] requires that each nonempty node has a unique last choice. To understand [cs3], note that this equation relates the infinite nodes (that is, the members of N r 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 0 in T ⇤ satisfy t ⇢ t 0 or t t 0 . The union of an infinite chain of finite nodes is obviously an infinite set.
Thus each C t is the set of choices that are feasible at the node t. Second, define X = {t2T |C t 6 =? }. 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 {t2X|c2C 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. 6 Peter A. Streufert In particular, [a] the collection of information sets partitions the decision-node set X, 6 and [b] two nodes in the same information set have the same set of feasible choices. 7 For example, consider the choice-set tree defined by C = {a, b, e, f} and N = {{}, {a}, {b}, {a, e}, {a, f}, {b, e}} b, g, d, e, f} and N = {{}, {a}, {a, g}, {b}, {a, d}, {b, e}, {b, f}, {a, d, e}, {a, d, f}} , and in addition, [cs5] [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. This paper does not formally specify preferences. Rather this paragraph merely notes that the set of outcomes is N r X = (N r T )[(T r X). The set N r T consists of the infinite nodes, and the set T r X consists of the finite nodes that are not decision nodes. Later, preferences can be assigned over N r X, or more generally if appropriate assumptions are introduced, over some space of probability distributions over N r X.

Definition
An OR form here is identical to a game in Osborne and Rubinstein 1994 (page 200) without the players' preferences. This specification is repeated in KL16 (page 86). 8 6 To prove this, first note that each information set is nonempty by inspection (empty sets of the form {t2X|c2C 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 (8c2C) {t2X|c2C t } = {t2T |c2C t }). To show the reverse inclusion, take any t ⇤ 2 X. Its C t ⇤ is nonempty by the definition of X, and so, there exists c ⇤ 2 C t ⇤ such that t ⇤ 2 {t2X|c ⇤ 2C t }.
7 To prove this, suppose both t 1 and t 2 belong to the information set {t2X|c A 2C t } and yet C t 1 6 = C t 2 . Without loss of generality, the inequality implies that there exists c B 2 C t 1 r C t 2 . Thus t 1 2 {t2X|c B 2C t } and t 2 / 2 {t2X|c B 2C t }. Then {t2X|c A 2C t } and {t2T |c B 2C 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]. 8  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 = (c This paragraph derives three entities from an OR tree. First, letT =N r C • be the collection of finite sequencest belonging toN. Second, define (Ct )¯t 2T at eacht by Ct = { c2C |t (c)2T }, 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 = {t2T |Ct 6 =? }. 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] (8i2I)H i partitions P 1 (i). Thus each player's information-set collectionH i partitions the player's set This is the usual assumption that two nodes in the same information set must have the same feasible choices. By definition, an OR form h(C,N), (P, (H i ) i2I )i is an OR tree (C,N) together with a player set I, a player-assignment functionP, and a list (H i ) i2I of player information-set collections that satisfy [OR4]-[OR6].
This paper does not formally specify preferences. Rather this paragraph merely notes that the set of outcomes isN rX = (N rT )[(T rX ). The setN rT consists of the infinite sequences, and the setT rX consists of the nondecision finite sequences. Later, preferences can be defined overN rX , or more generally if appropriate assumptions are introduced, over some space of probability distributions overN rX .

The equivalence between choice-set forms and OR forms
[OR6] states that two nodes in the same information set have the same alternatives. 9 As usual, define CH at eachH 2 [ i2IHi by (8t2H) CH = Ct . Thus CH is the set of alternatives at the information setH. An OR form has no shared alternatives iff In other words, a form has no shared alternatives iff each of its information sets has its own 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 iff Piccione and Rubinstein 1997 page 10). In other words, a form has no absentmindedness iff 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 (page 48 Definition 2(II)), and is defended at length by AR16 (Section 4.2.3). Further, it is weaker An OR form with no absentmindedness. In accord with Proposition 3.0, R|N is injective. than perfect recall, and AR16 (page 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 theorem demonstrates. In the theorem, the function R takes any sequence (c k ) K k=1 to its range. 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 noden. Since a choice determines its information set because of the no-sharedalternatives 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 10 Imagine that Spy 1 and Spy 2 are racing to recover a document from a safe deposit box. En route one spy realizes that if she reaches the box first, she can install a bomb that will explode when the other spy reaches the box after her. But then she realizes that the other spy will be thinking the same thing, and hence, if she opens the box when she reaches it, she will find either the document or an exploding bomb. So, she considers destroying the bank without opening the box in hopes of keeping the document from the other spy.  can only be played in the order (f 2 , o 2 , o 1 ). Intuitively, this happens because the set 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 noabsentmindedness, 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 Figure 3.2, which replicates the classic example of absentmindedness in Piccione and Rubinstein 1997 Figure 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.
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. (a) there is anN such that ([ i2I C i ,N) is an OR tree and R|N is a bijection fromN onto N. Further, deriveT , (Ct )¯t 2T , andX from this OR tree. Also defineP:X!I at eacht 2X by settingP(t) equal to the unique i for which Ct i 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 page 89) without the players' preferences. 11 The letters K and S refer to Kuhn 1953 andSelten 1975. Begin with a set T of nodes t, and a set E of edges e. By definition, a pair Jensen and Gutin 2009 page 2). Say that one node t immediately precedes another node t ] iff t is immediately suc- Jensen and Gutin 2009 page 11). Similarly, an infinite walk from t 1 is a sequence (t k ) K k=1 such that 2E} be the set of nodes with at least one successor, and call each t 2 X a decision node.
Next associate, with each node t 2 T , a set C t of feasible choices c. Such a feasible set can be empty. Further, associate, with each node t 2 T , a choice-to-successor function y t : y t is a nonempty function. By definition, a KS augmented-tree h(T, E, r), (C t , y t ) t2T i is a KS graph-tree (T, E, r) together with a (C t , y t ) t2T consisting of feasible sets and choice-to-successor functions that satisfy [KS3]. Now introduce a set I of players i, and assign decision nodes to players by a player-assignment function P. Assume [KS4] P:X!I. Then associate with each player i a collection H i of information sets H. Assume [KS5] (8i2I) H i partitions P 1 (i). In other words, assume each player's information-set collection H i partitions the player's set P 1 (i) of decision nodes. Also assume [KS6] (8i2I, H2H i , t2H, t 0 2H) C t = C t 0 . Because of this, let C H denote the feasible-choice set at the information set H. By definition, a KS form h(T, E, r), This paper does not formally specify preferences. Rather, this paragraph merely suggests how other papers could add preferences to a KS form. Toward that end, take a KS form and let W r be the collection of (finite and infinite) walks from r. Then } be the collection of (finite) walks from r to a decision node. Finally, let W r r W X r be the collection of outcomes. 12 It consists of 11 Several changes have been made to facilitate comparison across game specifications. First, [i] the infinite walks from r, and [ii] the (finite) walks from r to nondecision nodes. Later, preferences can be defined over W r r W X r , or more generally if appropriate assumptions are introduced, over some space of probability distributions over W r 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.
r from the KS form by the previous paragraph. Also deriveT andX from the OR tree (C,N). Also defineP:

Definition
A simple form here is virtually identical to a simple extensive form in AR16 (page 146). 14 The difference is insignificant. 15 Let T be a set of nodes t, and let be a binary relation on More details of my adaptations, of both KL16 and AR16, are available on request. 14 Some notational changes have been made to facilitate comparison across game specifications. First, x 0 there becomes r here. Second, T = (N, ) there becomes (T, ) here. In other words, [a] N there becomes T here, and [b] T there has no equivalent here. 15 The difference is that [s4] strengthens the phrase before AR16 page 146 Definition 6.4 (SF1) by requiring that [1] no choiceĉ 2 [ i2IĈi contains r, and [2] no choiceĉ 2 [ i2IĈi 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)|t2ĉ} would contain p(r), which is ill-defined. Regarding [2], p(?) = {p(t ] )|t ] 2?} is empty, and thus ? is never feasible.
Define r = max T , and call r the root node. Say that t precedes t ] iff t is succeeded by t ] iff t > t ] . Let X ✓ T be the set of nodes t which have at least one successor, and call every such node t 2 X a decision node. Define p:T r {r}!X by p(t ] ) = min{t2T |t>t ] }, and call p(t ] ) the (immediate) predecessor of the node t ] (AR16, page 145).
By assumption, a choiceĉ will be a nonempty set of non-root nodes t ] 2 T r {r}. In other words, each choiceĉ satisfies ? 6 =ĉ ✓ T r {r}. The set of nodes at which a choiceĉ is feasible is p(ĉ) = {p(t ] )|t ] 2ĉ} (AR16 page 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 ] inĉ. Further, let I be the set of players i, and let (Ĉ i ) i2I list a collectionĈ i of choiceŝ c for each player i.
At each decision node t 2 X, let A i (t) = {ĉ2Ĉ i |t2p(ĉ) } be the set of feasible choices for player i, and let J(t) = { i2I | A i (t)6 =? } be the set of decision makers. By definition, a simple (extensive) form (AR16 page 146, and note 15 here) is a triple (T, , This paper does not formally specify preferences. Rather this paragraph merely notes that 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 Section 7.1. First, say that a simple form has no simultaneous decisions iff (8t2X) J(x) is a singleton. Second, say that a KS form has no absentmindedness iff (8H2[ i2I H i , t A 2H,t B 2H) there is not a walk from t A to t B . Third, say that a KS form has no trivial decisions iff (8t2T ) |C t | 6 = 1. Then (a) (T, E, r) is a KS graph-tree. Further, define (C t ) t2T at each t by C t = {ĉ2[ i2IĈi | t2p(ĉ) }. Also, define (y t :C t !{t ] 2T |(t,t ] )2E}) t2T at each t and eacĥ c2C t by letting y t (ĉ) be the unique element of p 1 (t)\ĉ. Also, define P:X!I at each t 2 X by letting P(t) equal to the unique i for which (9ĉ2Ĉ i ) t 2 p(ĉ). Finally, define (H i ) i2I at each i by i is a well-defined KS form with no absentmindedness and no trivial decisions. (Proof A.9.) 6 AR Forms

Definition
An AR form here is virtually identical to a discrete extensive form in AR16 (page 138). 16 The difference is insignificant. 17 LetṄ be a nonempty collection of nonempty setsṅ. Define W = [Ṅ. CallṄ the set of nodesṅ, and call W the space of outcomes w. Notice that every nodeṅ is a subset of W . In other words, every nodeṅ is a set of outcomes w. By definition, the nodeṅ 2Ṅ precedes the nodeṅ ] 2Ṅ iffṅ is succeeded byṅ ] iffṅ ṅ ] . Note that W itself can be a member ofṄ. If so,ṅ = W is a node which precedes all other nodeṡ n ] 2Ṅ r {W }.
At each decision nodeṫ 2Ẋ, let A i (ṫ) = {ċ2Ċ i |ṫ2P(ċ) } be the set of feasible choices for player i, and let J(ṫ) = { i2I | A i (ṫ)6 =? } be the set of decision makers. By definition, a (discrete) AR form (AR16 page 138, and note 17 here) is a triple AR16 (page 138) explains that [AR8] states the standard properties of information sets, and that [AR9] describes how choices determine successors when simultaneous decisions are allowed. This paper does not formally specify preferences. Rather this paragraph merely recalls that the space W of outcomes is primitive (or, virtually the same, thatṄ is primitive and W is defined as [Ṅ). Thus it is straightforward to define preferences over W . Recent contributions which do so include Alós-Ferrer and Ritzberger 2016band Ritzberger , 2017band Ritzberger , and 2017c. More generally, preferences might be defined over some space of probability distributions over W , if appropriate assumptions are introduced. 19 [AR6] implies that each non-root node without an immediate predecessor is necessarily a nondecision node (AR16 page 135 Proposition 6.3, second sentence of proof). In other words, (Ṅ r {W }) rṪ ✓ (Ṅ r {W }) rẊ . This is equivalent toẊ r {W } ✓Ṫ r {W }. This impliesẊ ✓Ṫ since W 2Ṫ by definition,. 20 This sentence fails when |Ṅ| = 1, which is a trivial case. (In such a case W is a nondecision node.) 21 For example,Ṅ 6 ◆ {{w}|w2W } 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 orderisomorphic to the simple tree: neither the AR16 tree nor the simple tree has (nondecision) nodes corresponding to the simple tree's infinite maximal chains.

The Alós-Ferrer/Ritzberger equivalence between simple forms and AR forms
Theorem 6.1 (simple AR) Suppose (Ṅ, ◆, (Ċ i ) i2I ) is a (discrete) AR form, and derive its W ,ṗ, andṪ . Then (a) Corollary of AR16 page 144 Proposition 6.5 (b). (b) Corollary of AR16 page 139 Theorem 6.2 (DEF)EDP) and AR16 page 147 Theorem 6.4 (b).) The order isomorphism in Theorem 6.2 means that there is a bijection j:T !Ṅ such that (8t2T,t ] 2T ) t t ] iff j(t) ◆ j(t ] ) (AR16 page 20). In this case, the bijection is T 3 t 7 ! {w2W |t2w} 2Ṅ (AR16 page 144 note 7).   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 (Section 3.2, second paragraph). Absentmindedness is currently allowed in OR forms and KS forms, and might later be allowed in simple forms by removing [s7]. Absentmindedness is incompatible with choice-set forms (Proposition 3.0) and AR forms (AR16 Section 4.2.3).
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 {t2X|c2C t } (if nonempty) is the unique information set associated with the choice c (recall [s4]).
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 pages 64-65.
Simultaneous decisions are more convenient than cascading information sets in the sense of AR16 pages 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 (see for example Osborne and Rubinstein 1994 page 102).

General discussion
Although none of the four features is that important, Table 7.1 and the preceding paragraph argue that OR forms and KS forms have more 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 Figure 1.2. The left-right spectrum there is identical to the left-right spectrum in Table 7.1. KS forms are special because both their nodes and their choices are abstract (see the top two rows of Figure 1.2). 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 ] iff t ⇢ t ] , while on the right, an AR form hasṫ precedingṫ ] iffṫ 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 Figure 1.2'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 Sections 3.3 and 4.1). In contrast, on the spectrum's right, nodes and choices are expressed in terms of outcomes (see Figure 1.2'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.

A Proofs
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 [T ⇤ = s iff (b) there is (t m ) m 1 2 T • such that (8m 1) t m ⇢ t m+1 and [ m 1 t m = s.
Proof 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 (8m 2) t m = min T ⇤ r {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, ( 8m 1 This paragraph shows by induction that (8m 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 .
On the other hand, suppose there exists a k 1 such thatn 1 k 6 =n 2 k . Then let`be the smallest such k. Then [a]  This and`⇤ 6 =`imply |{k 0 1|n 1 k 0 2Ct }| 2 fort = 1n 1 1 . This implies absentmindedness by Lemma A.4 (a,b). Lemma A.6 Consider an OR form with no shared alternatives and no absentmind- 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 For this, it suffices that the next two paragraphs show, by induction on k 2 {1, 2, ...
. Because of Claim 4, it suffices to show the reverse direction. Toward that end, assume (a,c), by [i] and [ii], and by Lemma A.4 (a,c) again. So, trivially, The last two sentences yield [a]t [ = 1tK(t) 1 .
Claim 7: [cs2] holds. Take t 6 = {}. 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 (Section 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 8: N r T ✓ { [T ⇤ | T ⇤ is an infinite chain in T }. Take n 2 N r T . By the definition of N, there isn 2N such that n = R(n). Thus, since n / 2 T , R(n) is infinite. This andn 2N implyn 2N rT . Hence, by [OR2], we may define T ⇤ = {R( 1n`) |` 1}. As required, T ⇤ ✓ T because [a] R takesN to N and [b] finite sequences have finite ranges. Further, T ⇤ is an infinite chain because [i] it is a chain by inspection and [ii] (8` 1) |R( 1n`) | =`by Lemma A.4 (a,c) [e] and [f]  2C 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 2 X and t 2 2 X are such that {c A , c B } ✓ C t 2 , c A 2 C t 1 , and c B / 2 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 } ✓ C¯t2, c A 2 C¯t1, and c B / 2 C¯t1. Thust 2 is in both {t2T |c A 2Ct } and {t2T |c B 2Ct }, whilet 1 is in the former but not the latter. Therefore, since these two sets belong to [ i2IHi by Lemma A.3, [ i2IHi is not pairwise disjoint. This contradicts [OR4] and [OR5].
Claim 15: [cs5] holds. Suppose there were i 2 I, j 2 I r {i}, and c 2 C i \C j . By the theorem's definition of  Second,[cs2] implies the existence of a function c ⇤ :T r {{}}!C that takes each nonempty t 2 T to its unique last choice c ⇤ (t). Third, define (T k ) k 0 by T k = {t2T | |t|=k }. Note T = [ k 0 T k . Also note T 0 = {{}} by note 5.
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 2 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 Q k :T k !Q k (T k ) by Q k (t) = Q k 1 (t r {c ⇤ (t)}) (c ⇤ (t)). Note that Q k is well-defined at each t 2 T k because [a] t r {c ⇤ (t)} 2 T k 1 by [cs2] and [b] 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 implies (8` 0) 1s`2 [ k 0 Q k (T k ). This implies (8` 0) 1s`2N since [ k 0 Q k (T k ) ✓N by the ◆ half of the definition ofN.
Claim 4: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 8: ([ i2I C i ,N) is an OR tree. Because of Claims 1-3, it suffices to show that [OR2] holds for K < •. Toward that end, take anyn 2N and any` 0 such that < K(n) < •. By K(n) < • and the definition ofT ,n 2T . Thus 1n`b elongs toT by K(t) `applications of Claim 7; which is a subset ofN by the definition ofT .
Claim 10: (8k 0,t2T k ) R(Q k (t)) = t. This can be shown by induction. The initial step (k = 0) holds because T 0 = {{}} by the definition of T 0 and because R(Q 0 ({})) = R({}) = {} by the definition of Q 0 . To see the inductive step, take any k 1 and any Claim 11: R|T :T !T is the inverse of [ k 0 Q k . Claim 10 implies that (8k 0) R| Q k (T k ) = Q 1 k and that it maps from Q k (T k ) onto T k . Claim 6 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 4 and because [b] [ k 0 T k = T by the definition of (T k ) k 0 .
[e] and [f]    Claim 1: (a) (T, E, r) is a KS graph-tree, (b) (8t2T,t ] 2T ) t > t ] iff there is a walk from t to t ] , and (c) the decision-node set derived from (T, ) equals the decision node set derived from (T, E, r). ((c) will be used implicitly to ensure that the symbol X is unambiguous). A simple tree is specified via order theory, while a KS graph-tree is specified via graph theory. The conversion from the former to the latter is relatively straightforward. Details are available on request.
Claim 8: P is well-defined, and [KS4] holds. Take any t 2 X. By the assumption of no simultaneous decisions, J(t) is a singleton. Thus, by the definition of J, there is a unique i for which (9ĉ2Ĉ i ) t2p(ĉ).
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 2 [ i2I H i such that {t 1 ,t K } ✓ H. Since (t 1 ,t 2 ) 2 E, Claim 6 implies there isĉ 2 C t 1 such that [a] y t 1 (ĉ) = t 2 . Further, since {t 1 ,t K } ✓ H,