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

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## Abstract

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.

## Keywords

Game tree Extensive form game## Mathematics Subject Classification

91A70## 1 Introduction

### 1.1 Introducing choice-set forms

### 1.2 Linking choice-set forms and OR forms

The second contribution of this paper is to build a formal connection between choice-set 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).

### 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 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.

### 1.4 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).

## 2 Choice-set Forms

The following game specification is new.

*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 \(n{ \ }{ \subseteq }{ \ }C\). 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 \(T = { \lbrace }{ \ }n{ \in }N{ \ }|{ \ }n{ \ }\text {is a finite set}{ \ }{ \rbrace }\). 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 \(c{ \in }t\) such that Open image in new window 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. For example, the pair \(C = { \lbrace }\mathsf {a}{ \rbrace }\) and \(N = { \lbrace }{ \lbrace }\mathsf {a}{ \rbrace }{ \rbrace }\) does not satisfy [cs2] because \(T = N\) and the node \(t = { \lbrace }\mathsf {a}{ \rbrace }\) does not have a last choice. In contrast, the pair \(C = { \lbrace }\mathsf {a}{ \rbrace }\) and \(N = { \lbrace }{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {a}{ \rbrace }{ \rbrace }\) does satisfy [cs2] because \({ \lbrace }\mathsf {a}{ \rbrace }\) is the only nonempty finite node and its last choice is \(\mathsf {a}\). For another example, the pair \(C = { \lbrace }\mathsf {a,b}{ \rbrace }\) and \(N = { \lbrace }{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {a}{ \rbrace },{ \lbrace }\mathsf {b}{ \rbrace },{ \lbrace }\mathsf {a,b}{ \rbrace }{ \rbrace }\) violates [cs2] because both \(\mathsf {a}\) and \(\mathsf {b}\) are last choices of the node \({ \lbrace }\mathsf {a,b}{ \rbrace }\). In contrast, the pair \(C = { \lbrace }\mathsf {a,b}{ \rbrace }\) and \(N = { \lbrace }{{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {b}{ \rbrace },{ \lbrace }\mathsf {a,b}{ \rbrace }}{ \rbrace }\) satisfies [cs2]. Finally, consider the set

*C*of choices and the set

*N*of nodes shown in the top left diagram of Fig. 1. This pair of sets satisfies [cs2] because each of its eight nonempty nodes has a unique last choice.

To understand [cs3], note that this equation relates the infinite nodes (that is, the members of Open image in new window ) to the finite nodes (that is, the members of *T*). By definition, a *chain* in *T* is a subcollection \(T^*{ \ }{ \subseteq }{ \ }T\) such that any two distinct nodes *t* and \(t^{ \prime }\) in \(T^*\) satisfy \(t{ \ }{ \subset }{ \ }t^{ \prime }\) or \(t{ \ }{ \supset }{ \ }t^{ \prime }\). The union of an infinite chain of finite nodes is obviously an infinite set. The \({ \supseteq }\) direction of [cs3] requires that each such union must be a node. For example, the pair \(C = { \mathbb Z}\) and \(N = { \lbrace }{{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {4}{ \rbrace },{ \lbrace }\mathsf {4,5}{ \rbrace },{ \lbrace }\mathsf {4,5,6}{ \rbrace },\ldots }{ \rbrace }\) violates [cs3] because [a] \(T = N\), [b] \(T^* = T\) is an infinite chain in *T*, and [c] \({ \cup }T^* = { \lbrace }\mathsf {4,5,6,\ldots }{ \rbrace }{ \ }{ \notin }{ \ }N\). In contrast the pair \(C = { \mathbb Z}\) and \(N = { \lbrace }{{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {4}{ \rbrace },{ \lbrace }\mathsf {4,5}{ \rbrace },{ \lbrace }\mathsf {4,5,6}{ \rbrace },\ldots }{ \rbrace }\) \({ \cup }\) \({ \lbrace }{{ \lbrace }\mathsf {4,5,6,}\ldots { \rbrace }}{ \rbrace }\) satisfies [cs3]. Meanwhile, the \({ \subseteq }\) direction of [cs3] requires that every infinite node is the union of an infinite chain of finite nodes. For example, the pair \(C = { \mathbb Z}\) and \(N = { \lbrace }{{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {4}{ \rbrace },{ \lbrace }\mathsf {4,5}{ \rbrace },{ \lbrace }\mathsf {4,5,6}{ \rbrace },\ldots }{ \rbrace }\) \({ \cup }\) \({ \lbrace }{{ \lbrace }\mathsf {4,5,6},\ldots { \rbrace },{ \lbrace }\mathsf {5,6,7},\ldots { \rbrace }}{ \rbrace }\) violates [cs3] because \({ \lbrace }\mathsf {5,6,7,\ldots }{ \rbrace }\) cannot be constructed as the union of an infinite chain of finite nodes. Incidentally, the \({ \subseteq }\) direction of [cs3] implies that every infinite node is countable. It also implies, with the help of [cs1]–[cs2], that \({ \lbrace }{ \rbrace }{ \ }{ \in }{ \ }T\).^{6} Call \({ \lbrace }{ \rbrace }\) the *root* node.

*C*,

*N*). First, define \((C_t)_{t{ \in }T}\) at each

*t*by \(C_t = { \lbrace }\,c{ \in }C\,|\,c{ \notin }t{ \ }\text {and}{ \ }t{ \cup }{ \lbrace }c{ \rbrace }{ \in }T\,{ \rbrace }\). Thus each \(C_t\) is the set of choices that are

*feasible*at the node

*t*. Second, define \(X = { \lbrace }\,t{ \in }T\,|\,C_t{ \ne }{ \varnothing }\,{ \rbrace }\). Call its members the

*decision*nodes. Then assume

*information set*. Note each \({ \lbrace }t{ \in }X|c{ \in }C_t{ \rbrace }\) 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 \(C = { \lbrace }\mathsf {a,b,e,f}{ \rbrace }\) and \(N = { \lbrace }{{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {a}{ \rbrace },{ \lbrace }\mathsf {b}{ \rbrace },{ \lbrace }\mathsf {a,e}{ \rbrace },{ \lbrace }\mathsf {a,f}{ \rbrace },{ \lbrace }\mathsf {b,e}{ \rbrace }}{ \rbrace }\). Here \(X = { \lbrace }{{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {a}{ \rbrace },{ \lbrace }\mathsf {b}{ \rbrace }}{ \rbrace }\), \(C_{{ \lbrace }{ \rbrace }} = { \lbrace }\mathsf {a,b}{ \rbrace }\), \(C_{{ \lbrace }\mathsf {a}{ \rbrace }} = { \lbrace }\mathsf {e,f}{ \rbrace }\), and \(C_{{ \lbrace }\mathsf {b}{ \rbrace }} = { \lbrace }\mathsf {e}{ \rbrace }\). Thus \({ \lbrace }t{ \in }X|\mathsf {e}{ \in }C_t{ \rbrace } = { \lbrace }{{ \lbrace }\mathsf {a}{ \rbrace },{ \lbrace }\mathsf {b}{ \rbrace }}{ \rbrace }\) and \({ \lbrace }t{ \in }X|\mathsf {f}{ \in }C_t{ \rbrace } = { \lbrace }{ \lbrace }\mathsf {a}{ \rbrace }{ \rbrace }\). These two sets of decision nodes are unequal and intersecting, in violation of [cs4]. In contrast, consider Fig. 1’s choice-set tree (the figure’s dashed line is irrelevant at this point). In other words, consider \(C = { \lbrace }\mathsf {a,b,g,d,e,f}{ \rbrace }\) and \(N = { \lbrace }{{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {a}{ \rbrace },{ \lbrace }\mathsf {a,g}{ \rbrace },}\) \({ \lbrace }\mathsf {b}{ \rbrace },\) \({ \lbrace }\mathsf {a,d}{ \rbrace },\) \({ \lbrace }\mathsf {b,e}{ \rbrace },\) \({ \lbrace }\mathsf {b,f}{ \rbrace },\) \({ \lbrace }\mathsf {a,d,e}{ \rbrace },\) \({ \lbrace }\mathsf {a,d,f}{ \rbrace }{ \rbrace }\). Here \(X = { \lbrace }{{ \lbrace }{ \rbrace },{ \lbrace }\mathsf {a}{ \rbrace },{ \lbrace }\mathsf {b}{ \rbrace },{ \lbrace }\mathsf {a,d}{ \rbrace }}{ \rbrace }\), \(C_{{ \lbrace }{ \rbrace }} = { \lbrace }\mathsf {a,b}{ \rbrace }\), \(C_{{ \lbrace }\mathsf {a}{ \rbrace }} = { \lbrace }\mathsf {g,d}{ \rbrace }\), and \(C_{{ \lbrace }\mathsf {b}{ \rbrace }} = C_{{ \lbrace }\mathsf {a,d}{ \rbrace }} = { \lbrace }\mathsf {e,f}{ \rbrace }\). Thus [i] \({ \lbrace }t{ \in }X|\mathsf {a}{ \in }C_t{ \rbrace }\) \(=\) \({ \lbrace }t{ \in }X|\mathsf {b}{ \in }C_t{ \rbrace }\) \(=\) \({ \lbrace }{ \lbrace }{ \rbrace }{ \rbrace }\), and [ii] \({ \lbrace }t{ \in }X|\mathsf {g}{ \in }C_t{ \rbrace } = { \lbrace }t{ \in }X|\mathsf {d}{ \in }C_t{ \rbrace } = { \lbrace }{ \lbrace }\mathsf {a}{ \rbrace }{ \rbrace }\), and [iii] \({ \lbrace }t{ \in }X|\mathsf {e}{ \in }C_t{ \rbrace }\) \(=\) \({ \lbrace }t{ \in }X|\mathsf {f}{ \in }C_t{ \rbrace }\) \(=\) \({ \lbrace }{{ \lbrace }\mathsf {b}{ \rbrace },{ \lbrace }\mathsf {a,d}{ \rbrace }}{ \rbrace }\) (the figure’s dashed line shows this set [iii]). These three sets of decision nodes are disjoint, as [cs4] requires. They are the form’s (constructed) information sets.

Further, let *I* be a set of players *i*, and let \((C_i)_{i{ \in }I}\) assign a set \(C_i\) of choices to each player *i*. A *choice-set form* is a pair \(((C_i)_{i{ \in }I},N)\) such that \(({ \cup }_{i{ \in }I}C_i,N)\) is a choice-set tree which satisfies [cs4], and in addition, [cs5] Open image in new window and [cs6] \(({ \forall }t{ \in }X)({ \exists }i{ \in }I){ \ }C_t{ \ }{ \subseteq }{ \ }C_i\). [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 Open image in new window . The set Open image in new window consists of the infinite nodes, and the set Open image in new window consists of the finite nodes that are not decision nodes. Later, preferences can be assigned over Open image in new window , or more generally if appropriate assumptions are introduced, over some space of probability distributions over Open image in new window .

## 3 OR forms

### 3.1 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 let \(\bar{N}\) be a set of nodes \(\bar{n}\). By assumption, each node is a sequence of choices. In other words, each node is of the form \(\bar{n}= (c_k)^K_{k=1}\), where \(K{ \ }{ \in }{ \ }{ \lbrace }0,1,2,\ldots { \rbrace }{ \cup }{ \lbrace }{ \infty }{ \rbrace }\). Note that \((c_k)^0_{k=1} = { \lbrace }{ \rbrace }\) is the empty sequence. An *OR tree* is a pair \((C,\bar{N})\) such that [OR1] \(\bar{N}\) is a nonempty collection of sequences in *C*, [OR2] \(({ \forall }\,(c_k)^K_{k=1}{ \in }\bar{N},\) \(L{<}K)\) \((c_k)^L_{k=1}{ \ }{ \in }{ \ }\bar{N}\) (where both \(K = { \infty }\) and \(L = 0\) are permitted), and [OR3] \(({ \forall }\,(c_k)^{ \infty }_{k=1}{ \in }C^{ \infty })\) \((c_k)^{ \infty }_{k=1}{ \ }{ \in }{ \ }\bar{N}\) if \(({ \forall }L{<}{ \infty }){ \ }(c_k)^L_{k=1}{ \ }{ \in }{ \ }\bar{N}\) (where \(C^{ \infty }\) is the collection of infinite sequences in *C*).

This paragraph derives three entities from an OR tree. First, let Open image in new window be the collection of finite sequences \(\bar{t}\) belonging to \(\bar{N}\). Second, define \((C_{\bar{t}})_{\bar{t}{ \in }\bar{T}}\) at each \(\bar{t}\) by \(C_{\bar{t}} = { \lbrace }\,c{ \in }C\,|\,\bar{t}{ \oplus }(c){ \in }\bar{T}{ \ }{ \rbrace }\), where \(\bar{t}{ \oplus }(c)\) is the concatenation of the finite sequence \(\bar{t}\) with the one-element sequence (*c*). Thus each \(C_{\bar{t}}\) is the set of choices that are *feasible* at the node \(\bar{t}\). Third, define \(\bar{X}= { \lbrace }\,\bar{t}{ \in }\bar{T}\,|\,C_{\bar{t}}{ \ne }{ \varnothing }\,{ \rbrace }\). Call \(\bar{X}\) the set of *decision* nodes.

Next let *I* be a set of *players* *i*, and assign decision nodes to players by a *player-assignment* function \(\bar{P}\). In brief, assume [OR4] \(\bar{P}{:}\bar{X}{ \rightarrow }I\). Then associate with each player *i* a collection \(\bar{{ \mathscr {H}}}_i\) of *information sets* \(\bar{H}\). Assume [OR5] \(({ \forall }i{ \in }I){ \ }\bar{{ \mathscr {H}}}_i\) partitions \(\bar{P}^{-1}(i)\). Thus each player’s information-set collection \(\bar{{ \mathscr {H}}}_i\) partitions the player’s set \(\bar{P}^{-1}(i)\) of decision nodes. Further assume [OR6] \(({ \forall }\bar{H}{ \in }{ \cup }_{i{ \in }I}\bar{{ \mathscr {H}}}_i,\bar{t}{ \in }\bar{H},\bar{t}^{ \prime }{ \in }\bar{H})\) \(C_{\bar{t}} = C_{\bar{t}^{ \prime }}\). This is the usual assumption that two nodes in the same information set must have the same feasible choices. By definition, an *OR form* \({ \langle }(C,\bar{N}),(\bar{P},(\bar{{ \mathscr {H}}}_i)_{i{ \in }I}){ \rangle }\) is an OR tree \((C,\bar{N})\) together with a player set *I*, a player-assignment function \(\bar{P}\), and a list \((\bar{{ \mathscr {H}}}_i)_{i{ \in }I}\) of player information-set collections that satisfy [OR4]–[OR6].

Incidentally, the set of outcomes is Open image in new window . The set Open image in new window consists of the infinite sequences, and the set Open image in new window consists of the nondecision finite sequences. Later, preferences can be defined over Open image in new window , or more generally if appropriate assumptions are introduced, over some space of probability distributions over Open image in new window .

### 3.2 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 \(C_{\bar{H}}\) at each \(\bar{H}{ \ }{ \in }{ \ }{ \cup }_{i{ \in }I}\bar{{ \mathscr {H}}}_i\) by \(({ \forall }\bar{t}{ \in }\bar{H}){ \ }C_{\bar{H}} = C_{\bar{t}}\). Thus \(C_{\bar{H}}\) is the set of alternatives at the information set \(\bar{H}\). Say that an OR form has *no shared alternatives* if and only if \(({ \forall }\bar{H}{ \in }{ \cup }_{i{ \in }I}{ \mathscr {H}}_i,\bar{H}^{ \prime }{ \in }{ \cup }_{i{ \in }I}\bar{{ \mathscr {H}}}_i){ \ }\bar{H}{ \ }{ \ne }{ \ }\bar{H}^{ \prime }\) implies \(C_{\bar{H}}{ \cap }C_{\bar{H}^{ \prime }} = { \varnothing }\). Hence a form has no shared alternatives if and only if 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* if and only if Open image in new window (Piccione 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, *R* takes \((c_k)^K_{k=1} = (c_1,c_2,\ldots \,c_K)\) to \({ \lbrace }c_1,c_2,\ldots \,c_K{ \rbrace }\). Similarly, if *K* is infinite, *R* takes \((c_k)^K_{k=1} = (c_1,c_2,\ldots \,)\) to \({ \lbrace }c_1,c_2,\ldots \,{ \rbrace }\).

### Proposition 3.0

Consider an OR form with no shared alternatives, and let \(\bar{N}\) be its collection of nodes. Then the form has no absentmindedness iff \(R|_{\bar{N}}\) is injective. (Proof: Lemma A.5(a\({ \Leftrightarrow }\)c).)

But the forward direction of Proposition 3.0 goes further. It shows that \(R|_{\bar{N}}\) 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 \(\mathsf {a}^\mathsf {1}\) and \(\mathsf {a}^\mathsf {2}\) in Fig. 3 have been played in different orders at the nodes \((\mathsf {1}^\mathsf {*},\mathsf {a}^\mathsf {1},\mathsf {a}^\mathsf {2})\) and \((\mathsf {2}^\mathsf {*},\mathsf {a}^\mathsf {2},\mathsf {a}^\mathsf {1})\). However, the choices in \(R((\mathsf {1}^\mathsf {*},\mathsf {a}^\mathsf {1},\mathsf {a}^\mathsf {2})) = { \lbrace }\mathsf {1}^\mathsf {*},\mathsf {a}^\mathsf {1},\mathsf {a}^\mathsf {2}{ \rbrace }\) can only be played in the order \((\mathsf {1}^\mathsf {*},\mathsf {a}^\mathsf {1},\mathsf {a}^\mathsf {2})\), and the choices in \(R((\mathsf {2}^\mathsf {*},\mathsf {a}^\mathsf {2},\mathsf {a}^\mathsf {1})) = { \lbrace }\mathsf {2}^\mathsf {*},\mathsf {a}^\mathsf {1},\mathsf {a}^\mathsf {2}{ \rbrace }\) can only be played in the order \((\mathsf {2}^\mathsf {*},\mathsf {a}^\mathsf {2},\mathsf {a}^\mathsf {1})\). Intuitively, this happens because the set \({ \lbrace }\mathsf {1}^\mathsf {*},\mathsf {a}^\mathsf {1},\mathsf {a}^\mathsf {2}{ \rbrace }\) contains \(\mathsf {1}^\mathsf {*}\), and because the set \({ \lbrace }\mathsf {2}^\mathsf {*},\mathsf {a}^\mathsf {1},\mathsf {a}^\mathsf {2}{ \rbrace }\) contains \(\mathsf {2}^\mathsf {*}\). 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.

*R*takes both the sequence \((\mathsf {a})\) and the sequence \((\mathsf {a,a})\) to the set \({ \lbrace }\mathsf {a}{ \rbrace }\). Thus, \(R|_{\bar{N}}\) 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 choice-set 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** \({ \leftarrow }\) **OR**) Suppose \({ \langle }(C,\bar{N}),(\bar{P},(\bar{{ \mathscr {H}}}_i)_{i{ \in }I}){ \rangle }\) is an OR form with no shared alternatives and no absentmindedness. Define \(N = { \lbrace }R(\bar{n})|\bar{n}{ \in }\bar{N}{ \rbrace }\). Then (a) (*C*, *N*) is a choice-set tree and \(R|_{\bar{N}}\) is a bijection from \(\bar{N}\) onto *N*. Further, define \((C_i)_{i{ \in }I}\) at each *i* by \(C_i = { \cup }_{\bar{H}{ \in }\bar{{ \mathscr {H}}}_i}C_{\bar{H}}\) , where \((C_{\bar{H}})_{\bar{H}{ \in }\bar{{ \mathscr {H}}}_i}\) is derived from the OR form. Then (b) \(((C_i)_{i{ \in }I},N)\) is a choice-set form. (Proof A.7.)

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.

### Theorem 3.2

(**choice-set** \({ \rightarrow }\) **OR**) Suppose \(((C_i)_{i{ \in }I},N)\) is a choice-set form. Then (a) there is an \(\bar{N}\) such that \(({ \cup }_{i{ \in }I}C_i,\bar{N})\) is an OR tree and \(R|_{\bar{N}}\) is a bijection from \(\bar{N}\) onto *N*. Further, derive \(\bar{T}\), \((C_{\bar{t}})_{\bar{t}{ \in }\bar{T}}\), and \(\bar{X}\) from this OR tree. Also define \(\bar{P}{:}\bar{X}{ \rightarrow }I\) at each \(\bar{t}{ \ }{ \in }{ \ }\bar{X}\) by setting \(\bar{P}(\bar{t})\) equal to the unique *i* for which \(C_{\bar{t}}{ \ }{ \subseteq }{ \ }C_i\). Also define \((\bar{{ \mathscr {H}}}_i)_{i{ \in }I}\) at each *i* by \(\bar{\mathscr {H}}_{i}={ \lbrace }\,{ \lbrace }\bar{t}{ \in }\bar{X}|c{ \in }C_{\bar{t}}{ \rbrace }{ \ne }{ \varnothing }\,|\,c{ \in }C_i\,{ \rbrace }\). Then (b) \({ \langle }({ \cup }_{i{ \in }I}C_i,\bar{N}),\) \((\bar{P},(\bar{{ \mathscr {H}}}_i)_{i{ \in }I}){ \rangle }\) is a well-defined OR form with no shared alternatives and no absentmindedness. (Proof A.8.)

## 4 KS forms

### 4.1 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 \(E{ \ }{ \subseteq }{ \ }{ \lbrace }(t,t^{\sharp }){ \in }T^2|t{ \ne }t^{\sharp }{ \rbrace }\) (Bang-Jensen and Gutin 2009, p. 2). Say that one node *t* *immediately precedes* another node \(t^{\sharp }\) if and only if *t* is *immediately succeeded* by \(t^{\sharp }\) if and only if \((t,t^{\sharp }){ \ }{ \in }{ \ }E\). Further, say that a *finite walk from* \(t^1\) *to* \(t^K\) is a sequence \((t^k)^K_{k=1}\) such that \(K{ \in }{ \lbrace }2,3,\ldots { \rbrace }\) and \(({ \forall }k{<}K){ \ }(t^k,t^{k+1}){ \ }{ \in }{ \ }E\) (Bang-Jensen and Gutin 2009, p. 11). Similarly, an *infinite walk from* \(t^1\) is a sequence \((t^k)^K_{k=1}\) such that \(K = { \infty }\) and \(({ \forall }k{<}K){ \ }(t^k,t^{k+1}){ \ }{ \in }{ \ }E\). By definition, a *KS graph-tree* is a triple (*T*, *E*, *r*) such that [KS1] (*T*, *E*) is a directed graph and \(r{ \ }{ \in }{ \ }T\), and [KS2] for each Open image in new window there is a unique finite walk from *r* to \(t^{\sharp }\). Call *r* the *root* node of the graph-tree. Further, let \(X = { \lbrace }t{ \in }T|({ \exists }t^{\sharp }{ \in }T)(t,t^{\sharp }){ \in }E{ \rbrace }\) be the set of nodes with at least one successor, and call each \(t{ \ }{ \in }{ \ }X\) a *decision* node.

Next associate, with each node \(t{ \ }{ \in }{ \ }T\), a set \(C_{t}\) of *feasible choices* *c*. Such a feasible set can be empty. Further, associate, with each node \(t{ \ }{ \in }{ \ }T\), a *choice-to-successor function* \({ \psi }_{t}{:}C_{t}{ \rightarrow }{ \lbrace }t^{\sharp }{ \in }T|(t,t^{\sharp }){ \in }E{ \rbrace }\), which is a bijection from *t*’s feasible set \(C_{t}\) of choices *c* onto the set \({ \lbrace }t^{\sharp }{ \in }T|(t,t^{\sharp }){ \in }E{ \rbrace }\) of nodes \(t^{\sharp }\) that immediately succeed *t*. For future reference, call this bijectivity assumption [KS3]. [KS3] implies [a] \(t{ \ }{ \in }{ \ }X\) if and only if [b] \(C_t{ \ }{ \ne }{ \ }{ \varnothing }\) if and only if [c] \({ \psi }_t\) is a nonempty function. By definition, a *KS augmented-tree* \({ \langle }(T,E,r),(C_{t},{ \psi }_{t})_{t{ \in }T}{ \rangle }\) is a KS graph-tree (*T*, *E*, *r*) together with a \((C_t,{ \psi }_t)_{t{ \in }T}\) 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] \(P{:}X{ \rightarrow }I\). Then associate with each player *i* a collection \({ \mathscr {H}}_i\) of *information sets* *H*. Assume [KS5] \(({ \forall }i{ \in }I){ \ }{ \mathscr {H}}_i\) partitions \(P^{-1}(i)\). In other words, assume each player’s information-set collection \({ \mathscr {H}}_i\) partitions the player’s set \(P^{-1}(i)\) of decision nodes. Also assume [KS6] \(({ \forall }i{ \in }I,\) \(H{ \in }{ \mathscr {H}}_i,\) \(t{ \in }H,\) \(t^{ \prime }{ \in }H)\) \(C_t = C_{t^{ \prime }}\). Because of this, let \(C_{H}\) denote the feasible-choice set at the information set *H*. By definition, a *KS form* \({ \langle }(T,E,r),(C_t,{ \psi }_t)_{t{ \in }T},(P,({ \mathscr {H}}_i)_{i{ \in }I}){ \rangle }\) is a KS augmented-tree \({ \langle }(T,E,r), (C_{t},{ \psi }_{t})_{t}{ \rangle }\) together with a player set *I*, a player-assignment function *P*, and a list \(({ \mathscr {H}}_i)_{i{ \in }I}\) of player information-set collections that satisfy [KS4]–[KS6].

Incidentally, the collection of outcomes is Open image in new window ,^{12} where \({ \mathscr {W}}_r\) is the collection of (finite and infinite) walks from *r*, and where \({ \mathscr {W}}_r^X = { \lbrace }\,(t^k)^K_{k=0}{ \in }{ \mathscr {W}}_r\,|\,K{<}{ \infty },\,t^K{ \in }X\,{ \rbrace }\) is the collection of (finite) walks from *r* to a decision node. The collection Open image in new window consists of [i] the infinite walks from *r*, and [ii] the (finite) walks from *r* to nondecision nodes. Later, preferences can be defined over Open image in new window , or more generally if appropriate assumptions are introduced, over some space of probability distributions over Open image in new window .

### 4.2 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 \({ \mathscr {W}}_r\) be the collection of walks from *r*. Next let \({ \mathscr {W}}_r^T\) \(=\) \({ \lbrace }\,(t^k)^K_{k=0}{ \in }{ \mathscr {W}}_r\,|\,K{<}{ \infty }\,{ \rbrace }\) be the collection of finite walks from *r*. Finally, let the *node-to-walk function* \(w{:}T{ \rightarrow }{ \lbrace }(r){ \rbrace }{ \cup }{ \mathscr {W}}_r^T\) be the bijection mapping [i] \(r{ \ }{ \in }{ \ }T\) to (*r*) and [ii] each Open image in new window to [KS2]’s walk from *r* to \(t^{\sharp }\).

### Theorem 4.1

(**OR** \({ \leftarrow }\) **KS**) Suppose \({ \langle }(T,E,r),(C_{t},{ \psi }_{t})_{t{ \in }T},(P,({ \mathscr {H}}_i)_{i{ \in }I}){ \rangle }\) is a KS form, and derive its \({ \mathscr {W}}_r\). Define \(C = { \cup }_{t{ \in }T}C_{t}\). Also define \({ \alpha }\) and \(\bar{N}\) by letting \({ \alpha }\) be the surjective function, from \({ \lbrace }(r){ \rbrace }{ \cup }{ \mathscr {W}}_r\) onto \(\bar{N}{ \ }{ \subseteq }{ \ }C^{ \infty }\), such that \({ \alpha }((r)) = { \lbrace }{ \rbrace }\) and \(({ \forall }(t^k)^K_{k=0}{ \in }{ \mathscr {W}}_r)\) \({ \alpha }((t^k)^K_{k=0}) = ({ \psi }^{-1}_{t^{k-1}}(t^k))^K_{k=1}\). Then (a) \({ \alpha }\) is a well-defined bijection and \((C,\bar{N})\) is an OR tree. Further, derive \(w{:}T{ \rightarrow }{ \lbrace }(r){ \rbrace }{ \cup }{ \mathscr {W}}_r^T\) from the KS form by the previous paragraph. Also derive \(\bar{T}\) and \(\bar{X}\) from the OR tree \((C,\bar{N})\). Also define \(\bar{P}{:}\bar{X}{ \rightarrow }I\) at each \(\bar{t}{ \ }{ \in }{ \ }\bar{X}\) by \(\bar{P}(\bar{t}) = P{ \circ }w^{-1}{ \circ }{ \alpha }^{-1}(\bar{t})\). Also define \((\bar{{ \mathscr {H}}}_i)_{i{ \in }I}\) at each *i* by \(\bar{{ \mathscr {H}}}_i = { \lbrace }{ \ }{ \lbrace }{ \alpha }{ \circ }w(t)|t{ \in }H{ \rbrace }{ \ }|\) \(H{ \in }{ \mathscr {H}}_i{ \ }{ \rbrace }\). Then (b) \({ \langle }(C,\bar{N}),(\bar{P},(\bar{{ \mathscr {H}}}_i)_{i{ \in }I}){ \rangle }\) is an OR form. (Corollary of KL16 Lemma 2 and KL16 Theorem 1.)^{13}

### Theorem 4.2

(**OR** \({ \rightarrow }\) **KS**) Suppose \({ \langle }(C,\bar{N}),(\bar{P},(\bar{{ \mathscr {H}}}_i)_{i{ \in }I}){ \rangle }\) is an OR form, and derive its \(\bar{T}\) and \((C_{\bar{t}})_{\bar{t}{ \in }\bar{T}}\). Define \(E = { \lbrace }{ \ }(\bar{t},\bar{t}^{\,\sharp }){ \in }\bar{T}^2{ \ }|{ \ }({ \exists }c{ \in }C){ \ }\bar{t}{ \oplus }(c){=}\bar{t}^{\,\sharp }{ \ }{ \rbrace }\). Then (a) \((\bar{T},E,{ \lbrace }{ \rbrace })\) is a KS graph-tree. Further, define \(({ \psi }_{\bar{t}}{:}C_{\bar{t}}{ \rightarrow }\bar{T})_{\bar{t}{ \in }\bar{T}}\) at each \(\bar{t}\) by \(({ \forall }c{ \in }C_{\bar{t}}){ \ }{ \psi }_{\bar{t}}(c) = \bar{t}{ \oplus }(c)\). Then (b) \({ \langle }(\bar{T},E,{ \lbrace }{ \rbrace }),(C_{\bar{t}},{ \psi }_{\bar{t}})_{\bar{t}{ \in }\bar{T}},(\bar{P},(\bar{{ \mathscr {H}}}_i)_{i{ \in }I}){ \rangle }\) is a KS form. (Corollary of KL16 Theorem 2.)

## 5 Simple forms

### 5.1 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 \({ \ge }\) be a binary relation on *T*. A *simple tree* (AR16, p. 143) is a pair \((T,{ \ge })\) such that [s1] \((T,{ \ge })\) is a partial ordering (AR16, p. 20) with a maximum, [s2] \(({ \forall }s{ \in }T)\) \({ \lbrace }t{ \in }T|t{ \ge }s{ \rbrace }\) is a finite chain, and [s3] \(({ \forall }t{ \in }T,t^A{ \in }T){ \ }t > t^A\) implies \(({ \exists }t^B{ \in }T){ \ }t > t^B\) and neither \(t^A{ \ }{ \ge }{ \ }t^B\) nor \(t^B{ \ }{ \ge }{ \ }t^A\). Define *r* = max *T*, and call *r* the *root* node. Say that *t* *precedes* \(t^{\sharp }\) if and only if *t* is *succeeded* by \(t^{\sharp }\) if and only if \(t > t^{\sharp }\). Let \(X{ \ }{ \subseteq }{ \ }T\) be the set of nodes *t* which have at least one successor, and call every such node \(t{ \ }{ \in }{ \ }X\) a *decision* node. Define Open image in new window by \(p(t^{\sharp })\) = min\({ \lbrace }t{ \in }T|t{>}t^{\sharp }{ \rbrace }\), and call \(p(t^{\sharp })\) the *(immediate) predecessor* of the node \(t^{\sharp }\) (AR16, p. 145).

By assumption, a choice \(\hat{c}\) will be a nonempty set of non-root nodes Open image in new window . In other words, each choice \(\hat{c}\) satisfies Open image in new window . The set of nodes at which a choice \(\hat{c}\) is *feasible* is \(p(\hat{c}) = { \lbrace }p(t^{\sharp })|t^{\sharp }{ \in }\hat{c}{ \rbrace }\) (AR16, p. 145). Note that this equation is the standard way of defining the image of a set (such as \(\hat{c}\)) under a function (such as *p*). Accordingly, \(p(\hat{c})\) is the set of nodes \(p(t^{\sharp })\) that immediately precede a node \(t^{\sharp }\) in \(\hat{c}\). Further, let *I* be the set of players *i*, and let \((\hat{C}_i)_{i{ \in }I}\) list a collection \(\hat{C}_i\) of choices \(\hat{c}\) for each player *i*.

*i*, and let \(J(t) = { \lbrace }\,i{ \in }I\,|\,A_i(t){ \ne }{ \varnothing }\,{ \rbrace }\) be the set of decision makers. By definition, a

*simple (extensive) form*(p. 146 in AR16, and note 15 here) is a triple \((T,{ \ge },(\hat{C}_i)_{i{ \in }I})\) such that \((T,{ \ge })\) is a simple tree, [s4] each \(\hat{C}_i\) is a collection of nonempty subsets \(\hat{c}\) of Open image in new window ,

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.

### 5.2 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 \(({ \forall }t{ \in }X)\) *J*(*x*) is a singleton. Second, say that a KS form has *no absentmindedness* if and only if \(({ \forall }H{ \in }{ \cup }_{i{ \in }I}{ \mathscr {H}}_i,\!\) \(t^A{ \in }H,\!\) \(t^B{ \in }H)\) 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 \(({ \forall }t{ \in }T){ \ }|C_{t}|{ \ }{ \ne }{ \ }1\).

### Theorem 5.1

(**KS** \({ \leftarrow }\) **simple**) Suppose \((T,{ \ge },(\hat{C}_i)_{i{ \in }I})\) is a simple form with no simultaneous decisions, and derive its *r*, *p*, and *X*. Define \(E = { \lbrace }(t,t^{\sharp }){ \in }T^2|t{=}p(t^{\sharp }){ \rbrace }\). Then (a) (*T*, *E*, *r*) is a KS graph-tree. Further, define \((C_t)_{t{ \in }T}\) at each *t* by \(C_t = { \lbrace }{ \ }\hat{c}{ \in }{ \cup }_{i{ \in }I}\hat{C}_i{ \ }|{ \ }t{ \in }p(\hat{c}){ \ }{ \rbrace }\). Also, define \(({ \psi }_t{:}C_t{ \rightarrow }{ \lbrace }t^{\sharp }{ \in }T|(t,t^{\sharp }){ \in }E{ \rbrace })_{t{ \in }T}\) at each *t* and each \(\hat{c}{ \in }C_t\) by letting \({ \psi }_t(\hat{c})\) be the unique element of \(p^{-1}(t){ \cap }\hat{c}\). Also, define \(P{:}X{ \rightarrow }I\) at each \(t{ \ }{ \in }{ \ }X\) by letting *P*(*t*) equal to the unique *i* for which \(({ \exists }\hat{c}{ \in }\hat{C}_i)\) \(t{ \ }{ \in }{ \ }p(\hat{c})\). Finally, define \(({ \mathscr {H}}_i)_{i{ \in }I}\) at each *i* by \({ \mathscr {H}}_i = { \lbrace }p(\hat{c})|\hat{c}{ \in }\hat{C}_i{ \rbrace }\). Then (b) \({ \langle }(T,E,r),\) \((C_t,{ \psi }_t)_{t{ \in }T},\) \((P,({ \mathscr {H}}_i)_{i{ \in }I}){ \rangle }\) is a well-defined KS form with no absentmindedness and no trivial decisions. (Proof A.9.)

### Theorem 5.2

(**KS** \({ \rightarrow }\) **simple**) Suppose \({ \langle }(T,E,r),\!\) \((C_t,{ \psi }_t)_{t{ \in }T},\!\) \((P,({ \mathscr {H}}_i)_{i{ \in }I}){ \rangle }\) is a KS form with no absentmindedness and no trivial decisions. Let > be \({ \lbrace }\,(t,t^{\sharp }){ \in }T^2\,|\!\) there is a walk from *t* to \(t^{\sharp }{ \rbrace }\), and let \({ \ge }\) be \({ \lbrace }\,(t,t^{\sharp }){ \in }T^2\,|\!\) \(t{=}t^{\sharp }\) or \(t{>}t^{\sharp }\,{ \rbrace }\). Then (a) \((T,{ \ge })\) is a simple tree. Further, define \((\hat{C}_i)_{i{ \in }I}\) at each *i* by \(\hat{C}_i = { \lbrace }{ \ }{ \lbrace }{ \psi }_t(c)|t{ \in }H{ \rbrace }{ \ }|{ \ }c{ \in }C_{H},{ \ }H{ \in }{ \mathscr {H}}_i{ \ }{ \rbrace }\) where \((C_H)_{H{ \in }{ \mathscr {H}}_i}\) is derived from the KS form. Then (b) \((T,{ \ge },(\hat{C}_i)_{i{ \in }I})\) is a simple form with no simultaneous decisions. (Proof A.10.)

## 6 AR forms

### 6.1 Definition

An AR form here is virtually identical to a discrete extensive form in AR16 (p. 138).^{16} The difference is insignificant.^{17}

Let \(\dot{N}\) be a nonempty collection of nonempty sets \(\dot{n}\). Define \({ \varOmega } = { \cup }\dot{N}\). Call \(\dot{N}\) the set of *nodes* \(\dot{n}\), and call \({ \varOmega }\) the space of *outcomes* \({ \omega }\). Notice that every node \(\dot{n}\) is a subset of \({ \varOmega }\). In other words, every node \(\dot{n}\) is a set of outcomes \({ \omega }\). By definition, the node \(\dot{n}{ \ }{ \in }{ \ }\dot{N}\) *precedes* the node \(\dot{n}^{\sharp }{ \ }{ \in }{ \ }\dot{N}\) if and only if \(\dot{n}\) is *succeeded* by \(\dot{n}^{\sharp }\) if and only if \(\dot{n}{ \ }{ \supset }{ \ }\dot{n}^{\sharp }\). Note that \({ \varOmega }\) itself can be a member of \(\dot{N}\). If so, \(\dot{n}= { \varOmega }\) is a node which precedes all other nodes Open image in new window .

By definition, a (discrete) *AR tree* (AR16, p. 47, Definition 2.4; AR16, p. 112, Definition 5.1; and AR16, p. 135, Definition 6.1) is a pair \((\dot{N},{ \supseteq })\) which satisfies two sets of conditions. First, it satisfies [AR1] \((\dot{N},{ \supseteq })\) is a partially ordered set, [AR2] \({ \varOmega }{ \ }{ \in }{ \ }\dot{N}\), where \({ \varOmega }\) is defined to be \({ \cup }\dot{N}\),^{18} [AR3] \(({ \forall }\dot{N}^*{ \subseteq }\dot{N})\) \(\dot{N}^*\) is a chain if and only if \(({ \exists }{ \omega }{ \in }{ \varOmega })({ \forall }\dot{n}^*{ \in }\dot{N}^*){ \ }{ \omega }{ \ }{ \in }{ \ }\dot{n}^*\), and [AR4] \(({ \forall }{ \omega }{ \in }{ \varOmega },{ \omega }^{ \prime }{ \in }{ \varOmega })\) \({ \omega }{ \ }{ \ne }{ \ }{ \omega }^{ \prime }\) implies Open image in new window and Open image in new window . Call \({ \varOmega }\) the *root* node, and let \(\dot{X}= { \lbrace }\,\dot{n}{ \in }\dot{N}\,|\,({ \exists }\dot{n}^{\sharp }{ \in }\dot{N})\)\(\dot{n}{ \supset }\dot{n}^{\sharp }\,{ \rbrace }\) be the set of *decision* nodes (AR16, p. 69). Second, it satisfies [AR5] each nonempty chain in \(\dot{N}\) has a maximum, and [AR6] \(({ \forall }\dot{n}{ \ne }{ \varOmega })\) \({ \lbrace }\dot{n}^{\flat }{ \in }\dot{N}|\dot{n}^{\flat }{ \supset }\dot{n}{ \rbrace }\) has an infimum in Open image in new window . Let \(\dot{p}(\dot{n}) = \text {min}{ \lbrace }\dot{n}^{\flat }{ \in }\dot{N}|\) \(\dot{n}^{\flat }{ \supset }\dot{n}{ \rbrace }\) be the *(immediate) predecessor* of \(\dot{n}\) (AR16, p. 133), and let \(\dot{T}= { \lbrace }{ \varOmega }{ \rbrace }{ \cup }\) \({ \lbrace }\dot{n}{ \in }\dot{N}|\dot{p}(\dot{n}){ \ }\text {exists}{ \rbrace }\). Among other things, [AR6] implies \(\dot{X}{ \ }{ \subseteq }{ \ }\dot{T}\).^{19} Thus \(\dot{N}\) is partitioned by Open image in new window . Open image in new window is the set of nondecision nodes without immediate predecessors, Open image in new window is the set of nondecision nodes with immediate predecessors, and \(\dot{X}\) is the set of decision nodes.^{20}

Nondecision (i.e., “terminal”) nodes are closely related to outcomes. In particular, nondecision nodes are singleton nodes in the sense that \(({ \forall }\dot{n}{ \in }\dot{N})\) \(\dot{n}{ \ }{ \notin }{ \ }\dot{X}\) if and only if \(({ \exists }{ \omega }{ \in }{ \varOmega })\) \(\dot{n}={ \lbrace }{ \omega }{ \rbrace }\) (AR16, p. 86, Lemma 4.1(b)). This does not imply \(\dot{N}{ \ }{ \supseteq }{ \ }{ \lbrace }{ \lbrace }{ \omega }{ \rbrace }|{ \omega }{ \in }{ \varOmega }{ \rbrace }\).^{21} However, \(\dot{N}{ \ }{ \supseteq }{ \ }{ \lbrace }{ \lbrace }{ \omega }{ \rbrace }|{ \omega }{ \in }{ \varOmega }{ \rbrace }\) does hold when \(\dot{N}\) is finite. Further, there is a compelling sense in which any \(\dot{N}\) can be expanded to \(\dot{N}{ \ }{ \cup }{ \ }{ \lbrace }{ \lbrace }{ \omega }{ \rbrace }|{ \omega }{ \in }{ \varOmega }{ \rbrace }\) without changing its meaning (AR16, p. 50, Proposition 2.11).

Now introduce a set *I* of *players* *i*, and let \(\dot{C}_i\) denote player *i*’s set of *choices* \(\dot{c}\). By assumption, a choice \(\dot{c}\) is a nonempty subset of \({ \varOmega }\). Further, let \(P(\dot{c})\) denote the set of nodes \(\dot{n}\) at which the choice \(\dot{c}\) is *feasible*, and define it by \(\dot{P}(\dot{c}) = { \lbrace }{ \ }\dot{p}(\dot{t}^{\sharp }){ \ }|{ \ }\dot{t}^{\sharp }{ \in }\dot{T},{ \ }\dot{c}{ \supseteq }\dot{t}^{\sharp },\) \(\text {and}\) \((\not \exists \dot{t}{ \in }\dot{T}){ \ }\dot{c}{ \supseteq }\dot{t}{ \supset }\dot{t}^{\sharp }{ \ }{ \rbrace }\) (AR16, p. 134, Proposition 6.2(b)). This \(\dot{P}\) is not related to the player-assignment functions \(\bar{P}\) and *P* of OR and KS forms.

*i*, and let \(J(\dot{t}) = { \lbrace }\,i{ \in }I\,|\,A_i(\dot{t}){ \ne }{ \varnothing }\,{ \rbrace }\) be the set of decision makers. By definition, a (discrete)

*AR form*(p. 138 in A16, and note 17 here) is a triple \((\dot{N},{ \supseteq },(\dot{C}_i)_{i{ \in }I})\) such that \((\dot{N},{ \supseteq })\) is a (discrete) AR tree, [AR7] every \(\dot{c}{ \ }{ \in }{ \ }{ \cup }_{i{ \in }I}\dot{C}_i\) is a nonempty proper subset of \({ \varOmega }\) which is both the union of a subcollection of \(\dot{N}\) and a superset of some member of Open image in new window ,

Incidentally, recall that the space \({ \varOmega }\) of outcomes is primitive (or, virtually the same, that \(\dot{N}\) is primitive and \({ \varOmega }\) is defined as \({ \cup }\dot{N}\)). Thus it is straightforward to define preferences over \({ \varOmega }\). Recent contributions which do so include Alós-Ferrer and Ritzberger (2016b, 2017b, c). More generally, preferences might be defined over some space of probability distributions over \({ \varOmega }\), if appropriate assumptions are introduced.

### 6.2 The Alós–Ferrer/Ritzberger equivalence between simple forms and AR forms

### Theorem 6.1

(**simple** \({ \leftarrow }\) **AR**) Suppose \((\dot{N},{ \supseteq },(\dot{C}_i)_{i{ \in }I})\) is a (discrete) AR form, and derive its \({ \varOmega }\), \(\dot{p}\), and \(\dot{T}\). Then (a) \((\dot{T},{ \supseteq })\) is a simple tree. Further, define \((\hat{C}_i)_{i{ \in }I}\) at each *i* by Open image in new window . Then (b) \((\dot{T},{ \supseteq },(\hat{C}_i)_{i{ \in }I})\) is a simple form [(a) Corollary of AR16, p. 144, Proposition 6.5(b). (b) Corollary of AR16, p. 139, Theorem 6.2 (DEF\({ \Rightarrow }\)EDP), and AR16, p. 147, Theorem 6.4(b).]

The order isomorphism in Theorem 6.2 means that there is a bijection \({ \varphi }{:}T{ \rightarrow }\dot{N}\) such that \(({ \forall }t{ \in }T,t^{\sharp }{ \in }T)\) \(t{ \ }{ \ge }{ \ }t^{\sharp }\) if and only if \({ \varphi }(t){ \ }{ \supseteq }{ \ }{ \varphi }(t^{\sharp })\) (AR16, p. 20). In this case, the bijection is \(T{ \ }{ \ni }{ \ }t \mapsto { \lbrace }{ \omega }{ \in }{ \varOmega }|t{ \in }{ \omega }{ \rbrace }{ \ }{ \in }{ \ }\dot{N}\) (AR16, p. 144, note 7).

### Theorem 6.2

(**simple** \({ \rightarrow }\) **AR**) Suppose \((T,{ \ge },(\hat{C}_i)_{i{ \in }I})\) is a simple form. Let \({ \varOmega }\) be \((T,{ \ge })\)’s collection of maximal chains, and let \(\dot{N}= { \lbrace }{ \ }{ \lbrace }{ \omega }{ \in }{ \varOmega }|t{ \in }{ \omega }{ \rbrace }{ \ }|{ \ }t{ \in }T{ \ }{ \rbrace }\). Then (a) \((\dot{N},{ \supseteq })\) is a (discrete) AR tree which is order-isomorphic to \((T,{ \ge })\). Further, define \((\dot{C}_i)_{i{ \in }I}\) at each *i* by \(\dot{C}_i = { \lbrace }{ \ }{ \cup }_{t^{\sharp }{ \in }\hat{c}}{ \lbrace }{ \omega }{ \in }{ \varOmega }|t^{\sharp }{ \in }{ \omega }{ \rbrace }{ \ }|{ \ }\hat{c}{ \in }\hat{C}_i{ \ }{ \rbrace }\). Then (b) \((\dot{N},{ \supseteq },(\dot{C})_{i{ \in }I})\) is a (discrete) AR form. [(a) Corollary of AR16, p. 144, Proposition 6.5(a). (b) Corollary of AR16, p. 147, Theorem 6.4(a), and AR16, p. 139, Theorem 6.2 (EDP\({ \Rightarrow }\)DEF).]

## 7 Advantages and disadvantages

### 7.1 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.

Four minor features of the five specifications. The inequalities slightly complicate this paper’s new equivalence theorems. (Distinctions between “later” and “never” are conjectural.)

Feature | Choice-set | OR | KS | Simple | AR | ||
---|---|---|---|---|---|---|---|

Absentmindedness | Never | < | Yes | Yes | > | Later | Never |

Shared alternatives | Never | < | Yes | Yes | –\(^{\mathrm{a}}\) | –\(^{\mathrm{a}}\) | |

Trivial decisions | Yes | Yes | Yes | > | Later | Later | |

Simultaneous decisions | Later | Later | Later | < | Yes | Yes |

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 \({ \lbrace }t{ \in }X|c{ \in }C_t{ \rbrace }\) (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).

### 7.2 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 left-right 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 \(t^{\sharp }\) if and only if \(t{ \ }{ \subset }{ \ }t^{\sharp }\), while on the right, an AR form has \(\dot{t}\) preceding \(\dot{t}^{\sharp }\) if and only if \(\dot{t}{ \ }{ \supset }{ \ }\dot{t}^{\sharp }\).

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.

## Footnotes

- 1.
This reference to the literature excludes an earlier multi-paper version of the present paper. Streufert (2015b) links OR forms and choice-set forms, and Streufert (2015c) links choice-set forms and AR forms. That connection between OR forms and AR forms is less straightforward than the one in this paper.

- 2.
- 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 “KS-tree” 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).

- 5.
\({ \lbrace }{ \rbrace }\) and \({ \varnothing }\) are alternative notations for the empty set. I use \({ \lbrace }{ \rbrace }\) for a root node, and use \({ \varnothing }\) for all other purposes.

- 6.
To prove this, note

*N*is nonempty by [cs1]. Thus, either*T*is nonempty, or Open image in new window is nonempty. The latter also implies that*T*is nonempty by the \({ \subseteq }\) direction of [cs3]. Hence there is a \(t{ \ }{ \in }{ \ }T\). Hence |*t*| applications of [cs2] imply \({ \lbrace }{ \rbrace }{ \ }{ \in }{ \ }T\). - 7.
To prove this, first note that each information set is nonempty by inspection (empty sets of the form \({ \lbrace }t{ \in }X|c{ \in }C_t{ \rbrace }\) 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 \(({ \forall }c{ \in }C)\) \({ \lbrace }t{ \in }X|c{ \in }C_t{ \rbrace } = { \lbrace }t{ \in }T|c{ \in }C_t{ \rbrace }\)). To show the reverse inclusion, take any \(t^*{ \ }{ \in }{ \ }X\). Its \(C_{t^*}\) is nonempty by the definition of*X*, and so, there exists \(c^*{ \ }{ \in }{ \ }C_{t^*}\) such that \(t^*{ \ }{ \in }{ \ }{ \lbrace }t{ \in }X|c^*{ \in }C_t{ \rbrace }\). - 8.
To prove this, suppose both \(t^1\) and \(t^2\) belong to the information set \({ \lbrace }t{ \in }X|c^A{ \in }C_t{ \rbrace }\) and yet \(C_{t^1}{ \ }{ \ne }{ \ }C_{t^2}\). Without loss of generality, the inequality implies that there exists Open image in new window . Thus \(t^1{ \ }{ \in }{ \ }{ \lbrace }t{ \in }X|c^B{ \in }C_t{ \rbrace }\) and \(t^2{ \ }{ \notin }{ \ }{ \lbrace }t{ \in }X|c^B{ \in }C_t{ \rbrace }\). Then \({ \lbrace }t{ \in }X|c^A{ \in }C_t{ \rbrace }\) and \({ \lbrace }t{ \in }T|c^B{ \in }C_t{ \rbrace }\) 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].

- 9.
Several changes have been made to facilitate comparison across game specifications. First, \(a\,{ \in }\,{ \mathscr {A}}\) and \((a^k)^K_{k{ \in }1}\,{=}\,h\,{ \in }\,{ \mathscr {H}}\) in KL16 become \(c\,{ \in }\,C\) and \((c_k)^K_{k=1}\,{=}\,\bar{n}\,{ \in }\,\bar{N}\) here. Second, OR1 and OR2 there become [OR2] here. Third, \({ \mathscr {H}}_f\) and \({ \mathscr {H}}^D\) become \(\bar{T}\) and \(\bar{X}\). Fourth, \(({ \mathscr {A}}_{h})_{h{ \in }{ \mathscr {H}}_f}\) becomes \((C_{\bar{t}})_{\bar{t}{ \in }\bar{T}}\). Fifth, \(i\,{ \in }\,N\) and \((I_i)_{i{ \in }N}\) become \(i\,{ \in }\,I\) and \((\bar{{ \mathscr {H}}}_i)_{i{ \in }I}\). Sixth,

*P*and \(P_i\) become \(\bar{P}\) and \(\bar{P}^{-1}(i)\). - 10.
The terms “alternative”, “action”, and “choice” are fundamentally synonymous (see note 22).

- 11.
Several changes have been made to facilitate comparison across game specifications. First, \({ \langle }(V,E,r),(A_v,{ \psi }_v)_{v{ \in }V}){ \rangle }\) in KL16 becomes \({ \langle }(T,E,r),(C_t,{ \psi }_t)_{t{ \in }T}{ \rangle }\) here. Second, a “KS-tree” there becomes a “KS augmented-tree” here (this accords with note 4). Third, \(V^D\) becomes

*X*. Fourth, \({ \mathscr {P}}\) and \({ \mathscr {P}}_f\) and \({ \mathscr {P}}^D\) become \({ \lbrace }(r){ \rbrace }{ \cup }{ \mathscr {W}}_r\) and \({ \lbrace }(r){ \rbrace }{ \cup }{ \mathscr {W}}_r^T\) and \({ \lbrace }(r){ \rbrace }{ \cup }{ \mathscr {W}}_r^X\). Fifth, both*p*and \(w{ \cup }{ \lbrace }(r,(r)){ \rbrace }\) become*w*. Sixth, \(i\,{ \in }\,N\) and \((I_i)_{i{ \in }N}\) become \(i\,{ \in }\,I\) and \(({ \mathscr {H}}_i)_{i{ \in }I}\). 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.)

- 13.
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, \(T = (N,{ \ge })\) there becomes \((T,{ \ge })\) 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, p. 146, Definition 6.4 (SF1), by requiring that [1] no choice \(\hat{c}{ \ }{ \in }{ \ }{ \cup }_{i{ \in }I}\hat{C}_i\) contains

*r*, and [2] no choice \(\hat{c}{ \ }{ \in }{ \ }{ \cup }_{i{ \in }I}\hat{C}_i\) equals \({ \varnothing }\). I argue that [1] and [2] are insignificant by considering the set \(p(\hat{c})\) of nodes at which such choices \(\hat{c}\) would be feasible. Regarding [1], suppose \(\hat{c}\) did contain*r*. Then \(p(\hat{c}) = { \lbrace }p(t)|t{ \in }\hat{c}{ \rbrace }\) would contain*p*(*r*), which is ill-defined. Regarding [2], \(p({ \varnothing }) = { \lbrace }p(t^{\sharp })|t^{\sharp }{ \in }{ \varnothing }{ \rbrace }\) is empty, and thus \({ \varnothing }\) is never feasible. - 16.
Some notational changes have been made to facilitate comparison across game specifications. First, the set

*W*of plays*w*in AR16 becomes the set \({ \varOmega }\) of outcomes \({ \omega }\) here. Second, Open image in new window there becomes Open image in new window here. Third, a move \(x{ \ }{ \in }{ \ }X\) there becomes a decision node \(\dot{t}{ \ }{ \in }{ \ }\dot{X}\) here. Fourth, \(T = (N,{ \supseteq })\) there becomes \((\dot{N},{ \supseteq })\) here. In other words, [a]*N*there becomes \(\dot{N}\) here, and [2]*T*there has no equivalent here ([a] merits emphasis because note 14 in the*previous*section changed*N*to*T*). - 17.
The difference is that [AR7] strengthens the phrase before AR16, p. 138, Definition 6.2 (DEF1), by requiring that [1] no choice \(\dot{c}{ \ }{ \in }{ \ }{ \cup }_{i{ \in }I}\dot{C}_i\) is equal to \({ \varOmega }\), and [2] every choice \(\dot{c}{ \ }{ \in }{ \ }{ \cup }_{i{ \in }I}\dot{C}_i\) is a superset of some Open image in new window . I argue that [1] and [2] are insignificant by considering the set \(\dot{P}(\dot{c})\) of nodes at which such choices \(\dot{c}\) would be feasible. Regarding [1], \(\dot{P}({ \varOmega }) = { \lbrace }\dot{p}(\dot{t}^{\sharp })|\dot{t}^{\sharp }{ \in }\dot{T},{ \varOmega }{ \supseteq }\dot{t}^{\sharp },(\not \exists \dot{t}{ \in }\dot{T}){ \varOmega }{ \supseteq }\dot{t}{ \supset }\dot{t}^{\sharp }{ \rbrace }\) equals \({ \lbrace }\dot{p}({ \varOmega }){ \rbrace }\), and \(\dot{p}({ \varOmega })\) is not naturally defined. Regarding [2], suppose that \(\dot{c}{ \ }{ \ne }{ \ }{ \varOmega }\) (by [1]) and that \(\dot{c}\) does not contain any Open image in new window . Then \(\dot{P}(\dot{c}) = { \lbrace }\dot{p}(\dot{t}^{\sharp })|\dot{t}^{\sharp }{ \in }\dot{T},\dot{c}{ \supseteq }\dot{t}^{\sharp },(\not \exists \dot{t}{ \in }\dot{T})\dot{c}{ \supseteq }\dot{t}{ \supset }\dot{t}^{\sharp }{ \rbrace }\) is empty, and thus there is no node at which \(\dot{c}\) is feasible.

- 18.
[AR2] 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, Open image in new window . This is equivalent to Open image in new window . This implies \(\dot{X}{ \ }{ \subseteq }{ \ }\dot{T}\) since \({ \varOmega }{ \ }{ \in }{ \ }\dot{T}\) by definition.

- 20.
This sentence fails when \(|\dot{N}| = 1\), which is a trivial case. (In such a case \({ \varOmega }\) is a nondecision node.)

- 21.
For example, \(\dot{N}\not \supseteq { \lbrace }{ \lbrace }{ \omega }{ \rbrace }|{ \omega }{ \in }{ \varOmega }{ \rbrace }\) 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.

- 22.
The terms “alternative”, “action”, and “choice” are fundamentally synonymous. The only distinction is that “action” tends to be used when alternatives can be shared (e.g., Osborne and Rubinstein 1994; KL16), and “choice” tends to be used when they cannot (e.g., von Neumann and Morgenstern 1944 Sect. 9–10; Ritzberger 2002, Sect. 3.2; AR16). This paper uses “choice” because the new concept of choice-set forms implicitly assumes that alternatives cannot be shared, as shown two paragraphs hence.

- 23.
In the text, a sequence is denoted by \(\bar{n}= (c_k)^K_{k=1}\) for some \(K{ \ }{ \in }{ \ }{ \lbrace }0,1,2,\ldots { \rbrace }{ \cup }{ \lbrace }{ \infty }{ \rbrace }\), and an initial segment is denoted by \((c_k)^L_{k=1}\) for some \(L < K\) [this resembles notation from Osborne and Rubinstein (1994)]. In this appendix, the elements of a sequence \(\bar{n}\) are denoted \(\bar{n}_k\) rather than \(c_k\), the length of a sequence is denoted \(K(\bar{n}){ \ }{ \in }{ \ }{ \lbrace }0,1,2,\ldots { \rbrace }{ \cup }{ \lbrace }{ \infty }{ \rbrace }\), and an initial segment is denoted \(_1\bar{n}_{ \ell } = (\bar{n}_1,\bar{n}_2,\ldots \,\bar{n}_{ \ell })\) for some \({ \ell } < K(\bar{n})\).

## Notes

### Acknowledgements

I am very indebted to two anonymous referees, who generously and insightfully addressed a very long preliminary version of this paper.

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