Compactification of Extensive Game Structures and Backward Dominance Procedure
CCompactification of Extensive Forms and Belief in the Opponents’ FutureRationality (cid:73)
Shuige Liu
School of Political Science and Economics, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-Ku 169-8050, Tokyo,JAPAN; EPICENTER, School of Business and Economics, Maastricht University, 6200 MD Maastricht, THENETHERLANDS
Abstract
We introduce an operation, called compactification, to reduce an extensive form to a compact onewhere each decision node in the game tree can be assigned to more than one player. Motivatedby Thompson [17]’s interchange of decision nodes, we attempt to capture the notion of a faithfulrepresentation of the chronological order of the moves in a dynamic game which plays a vital rolein fields like epistemic game theory. The compactification process preserves perfect recall and theunambiguity of the order among information sets. We specify an algorithm, called leaves-to-rootprocess, which compactifies at least as many information sets as any other compactification process.The compact extensive form provides an approach to avoid problems in dynamic game theory dueto the vague definition of the chronological order of the moves, for example, belief in the opponents’future rationality (Perea [12])’s sensitivity to the specific extensive form representation. We showthat any strategy which can rationally be chosen under common belief in future rationality in aminimal compact game if and only if it satisfies this property in every extensive form game whichis related to it via some compactification process.
Keywords: extensive forms, epistemic game theory, compactification, interchange of decisionnodes, belief in the opponents’ future rationality
1. Introduction
The extensive form has been acclaimed from the birth of game theory as a good model of thesalient features of dynamic interactive situations. Using the words of Kreps [7] (p.13), it captures“the timing of actions that players may take and the information they will have when they must takethose actions.” In many cases, there are several extensive forms to represent the same chronologicalorder of the moves. That is, even if players actually make decisions simultaneously at some stage,there are several orders to describe their moves in extensive forms. Though the orders are arbitrary,two different extensive forms representing different orders can be transformed into each other viaThompson [17]’s interchange of decision nodes. Hence some authors, for example, Thompson [17]and Elmes and Reny [6], claim that the difference among multiple representations of a chronologicalorder is inessential. (cid:73)
The author gratefullyacknowledges the support of Grant-in-Aids for Young Scientists (B) of JSPS No.17K13707and Grant for Special Research Project No. 2018K-016 of Waseda University.
Email address: [email protected] (Shuige Liu)
Preprint submitted to Elsevier May 3, 2019 a r X i v : . [ ec on . T H ] M a y et it causes severe problems sometimes, for example, in dynamic epistemic game theory.Epistemic game theory studies how a player may reason about other players before he starts toplay (see Perea [11], Dekel and Siniscalchi [5]). A primary criterion of a player’s reasoning in staticepistemic games is that he believes in the opponents’ rationality (Tan and Werlang [16]). It ismore complicated in dynamic games since it is not always possible for a player to believe thathis opponents had behaved rationally in the past (e.g., Binmore [4], Reny [13], [14]); further, hisinterpretation of the opponents’ irrational behavior determines his beliefs about their future choices(e.g., Battigalli [2], Battigalli and Siniscalchi [3], Perea [12]). Therefore, the order of informationsets matters since it specifies the meaning of past and future in a player’s reasoning. An example isthe notion of belief in the opponents’ future rationality (Perea [12]) which describes that a playeralways believes at every stage of the game that her opponents will choose rationally at the stageand in the future. The strategies optimal to the common belief in the opponents’ future rationalityis sensitive to the specific extensive form representation. In Section 6.2 of Perea [12], an exampleshows that representations with different orders of information sets, albeit representing the samedynamic situation, have different optimal strategies to the common belief in the opponents’ futurerationality.Perea [12] claims that “if we insist that the order of the information sets in the dynamic gamesfaithfully represents the actual chronological order of the moves, then there is no problem in usingcommon belief in future rationality as a concept.” Yet the meaning of a faithful representationis not specified there. Simultaneous moves are allowed and play an important role in Perea [12]’sdefinition of extensive form, in which sense the formulation can be regarded as a “compactification”of the standard one. However, there is no practical method to formulate the compactificationprocess.Perea [12] also suggests that two extensive forms which can be transformed into each otherthrough Thompson [17]’s interchange of decision nodes can be regarded as representing the sameactual chronological order of the moves. Yet if we consider the “compactified” extensive forms,it is not clear whether and how two “equivalent” (standard) extensive forms are related to acompactified form. Neither there is an effective way to connect the distinct sets of strategiesselected in “equivalent” extensive forms. Worse, interchange of decision nodes sometimes destroysthe unambiguity of the order among information sets (an example is given in Perea [10], p.141)and makes it impossible to select strategies optimal to any belief criterion which relies on thewell-defined notions of “past” and “future”.In this paper, we attempt to solve those problems. We first give a formal definition of thecompact extensive form, where each decision node in the game tree can be assigned to more thanone player. We then introduce an operation, called compactification, where an information setcontaining only the immediate successors of a node is “absorbed” into the node. It is motivatedby Thompson [17]’s interchange of decision nodes. Yet our compactification operation can beconducted only if the immediate successors form an information set, while interchange of decisionnodes allows that the absorbed information set contains other nodes. This difference mattersessentially. In Section 5 we will show that the latter causes difficulties in our context.By the compact extensive form and compactification operation, we attempt to capture theprocess of constructing a faithful representation of the chronological order of the moves in dynamicgames. We show that compactification process preserves perfect recall and the unambiguity ofthe order among information sets. A problem, however, is that the compactification process isnot order-independence. It is caused by the absorption of some information set h may hinder2ther information sets which is constituted by immediate successors of some node in h. We specifyan algorithm, called leaves-to-root process, which compactifies every information set that can beabsorbed.The compact extensive form provides an approach to alleviate the sensitivity of belief in theopponents’ future rationality (Perea [12]) to the specific extensive form representation. We showthat any strategy which can rationally be chosen under common belief in future rationality in aminimal compact extensive form game, i.e., a game whose extensive form cannot be compactifiedany more, if and only if the strategy satisfies this property in every extensive form game which canbe related to it via some compactification process.The rest of the paper is organized as follows. Section 2 defines the compact extensive form.Section 3 specifies the compactification operation. We also give some results on the properties of thecompactification process. Section 4 shows the main theorem about the minimal compact extensiveform games and the strategies optimal to common belief in the opponents’ future rationality.Section 5 provides an alternative operation which is faithful to Thompson [17]’s interchange ofdecision nodes and compare it with our compactification. Section 6 contains all the proofs.
2. Compact Extensive Forms A compact extensive form is a tuple Γ = (cid:104) T, ≥ ; I, ı ; { A } i ∈ I , α ; { H i } i ∈ I (cid:105) , where1. ( T, ≥ ) is a finite tree, that is, T is a non-empty finite set of nodes and ≥ is a partial order on T satisfies (i) there is a smallest element t (called the root or the initial node ) in T with respectto ≥ , and (ii) for each t ∈ T, ≥ is a complete order on the predecessors of t. We follow the auxiliary notations defined in Kreps and Wilson [8]. For each node x , we denoteby ( T x , ≥ ) (sometimes only T x for simplicity when no confusion is caused) the subtree of ( T, ≥ )with x as its root. We use Z to denote the set of terminal nodes in T and X the set of non-terminalnodes (called the decision nodes ), i.e., X = T \ Z. For each x ∈ X, we denote by S ( x ) the set ofimmediate successors of t, P ( x ) the set of predecessors of x , and p n ( x ) the n -th predecessor of x (we stipulate that p ( x ) = x ) . I is the finite set of players. The function ı : X → I \{ ∅ } assigns to each decision node aset of players who have to make decision there. We say that a player i is active at a decision node x ∈ X iff i ∈ ı ( x ) . For each i ∈ I, we define X i = { x ∈ X : i ∈ ı ( x ) } .
3. For each i ∈ I, A i is a non-empty set of actions of i. We define A i = A i ∪ { (cid:15) } , where (cid:15) is asymbol not belonging to any A i . By (cid:15) we mean “no actions”. The function α : T \{ t } → (cid:81) i ∈ I A i assigning to each non-initial node the last profile of actions taken to reach it. It satisfies thefollowing conditions:(3.1) α i ( t ) ∈ A i (i.e., α i ( t ) (cid:54) = (cid:15) ) if and only if i ∈ ı ( p ( t )). Here α i is the projection of α on the i -th dimension.(3.2) For each x ∈ X and y, z ∈ S ( x ) , α ( y ) (cid:54) = α ( z ) . Combined with (3.1), it means that forsome i ∈ ı ( x ) , α i ( y ) (cid:54) = α i ( z ) . (3.3) For each x ∈ X with i ∈ ı ( x ) and each y, z ∈ S ( x ) , | [ y ] α i | = | [ z ] α i | , where [ y ] α i := { y (cid:48) ∈ S ( x ) : α i ( y (cid:48) ) = α i ( y ) } . Combined with (3.1) and (3.2), it implies that for each x ∈ X, | S ( x ) | = (cid:81) i ∈ ( x ) | α i ( S ( x )) | .
4. For each i ∈ I , H i is a partition of X i . For each x ∈ X i , we use H i ( x ) to denote equivalentclass in H i containing x. We require that H i ( x ) = H i ( y ) implies α i ( S ( x )) = α i ( S ( y )). In thefollowing, we denote by α i ( h i ) the set of available actions of player i at information set h i . α assignsto each non-initial node (or, equivalently, an edge) a profile of actions where for each player activeat the immediate predecessor node there is a concrete action, and for those inactive a dummysymbol (cid:15) meaning that he does not need to act. Instead of one edge representing one action for oneplayer and different edges represent distinct actions of her in the standard extensive game, heredifferent edges may represent the same action(s) for some player(s). In (3.2) we require that twoedges radiating from one decision node should represent distinct profiles, i.e., at least one playeractive at that decision node should have different actions. Condition (3.3) describes that the edgesradiating from one decision node represent the different combinations of independent actions ofplayers active at that node. Finally, since there might be multiple players active at each node,information sets could overlap.It is easy to see that the standard extensive form is a special case of the compact one, i.e., | ı ( x ) | = 1 for every x ∈ X . In this sense, the notion of compact extensive form is a generalizationof the standard one.Like the extensive form game, a compact extensive form game is obtained by specifying eachplayer i ’s von Neumann-Morgenstern utility function u i : Z → R for terminal nodes.In Figure 1, we show how a compact extensive form (game) looks like and compare it with thestandard one. Figure 1: An extensive form game and its compactification
The graph in the left-hand side is adopted from Perea [12]’s Figure 1. The situation is, as Perea[12] put it, “[a]t the beginning of the game, ∅ , player 1 chooses between a and b, and player 2simultaneously chooses between c and d. So, ∅ is an information set that belongs to both players 1and 2. If player 1 chooses b, the game ends, and the utilities are as depicted. If he chooses a , thenthe game moves to information set h or information set h , depending on whether player 2 haschosen c or d . Player 1, however, does not know whether player 2 has chosen c or d, so player 14aces information set h after choosing a. Hence h and h are information sets that belong onlyto player 2, whereas h is an information set that belongs only to player 1.”The graph in the middle is a representation of the situation by a (standard) extensive formgame. There, since simultaneous moves are not allowed, information set ∅ is decomposed into ∅ and ∅ which correspond to player 1 and 2’s information set at the beginning of the game(to differentiate players, we use continuous-line circles to represent player 1’s information sets anddashed-line circles to represent player 2’s information sets). Also, h and h , h are displayedseparately.The graph in the right-hand side represents the situation by a compact extensive form game.Since simultaneous moves are allowed, each decision node represents a move for both players, andinformations sets overlap. Indeed, ∅ and ∅ coincide with each other, and h = h ∪ h . Notethat each edge (i.e., each non-initial node) represents a vector of the active players’ choices. It canbe seen that this form is indeed more compact than the one in the middle, with the terminal nodesunchanged.Perfect recall can be defined similarly for compact extensive form games. Formally, each player i in I has perfect recall iff the following condition is satisfied. For each x, y, y (cid:48) ∈ X with i ∈ ı ( x ) ∩ ı ( y ) ∩ ı ( y (cid:48) ) , if x < y and H i ( y ) = H i ( y (cid:48) ), then there is x (cid:48) ∈ X such that P ( y (cid:48) ) ∩ H i ( x ) = { x (cid:48) } , and if x = p n ( y ) and x (cid:48) = p m ( y (cid:48) ) , α i ( p n − ( y )) = α i ( p m − ( y (cid:48) )) .
3. Compactifying Extensive Forms
We have defined the compact extensive form. In this section, we specify how to “compactify”such a form. Also, we show some important features of the compactification process.
Consider two compact extensive forms Γ = (cid:104) T, ≥ ; I, ı ; { A } i ∈ I , α ; { H i } i ∈ I (cid:105) and Γ (cid:48) = (cid:104) T (cid:48) , ≥ (cid:48) ; I, ı (cid:48) ; { A (cid:48) } i ∈ I , α (cid:48) ; { H (cid:48) i } i ∈ I (cid:105) . For simplicity, we assume that for each x ∈ X , if i ∈ ı ( x ) , then S ( x ) / ∈ H i . We say that Γ (cid:48) is a compactification of Γ, denoted by Γ → COM Γ (cid:48) , iff there exist x, y , ..., y k , z , ..., z (cid:96) , ..., z k , ..., z k(cid:96) k ∈ Z satisfying1. S ( x ) = { y , ..., y k } and S ( y r ) = { z r , ..., z r(cid:96) r } for each r = 1 , ..., k.
2. There is some i ∈ I such that i ∈ ı ( y r ) for each r = 1 , ..., k, and H i ( y ) = ... = H i ( y k ) = { y , ..., y k } . We let b = | α i ( S ( y )) | = | α i ( S ( y r )) | for each r = 1 , ..., k ). Without loss of generality,we assume that for each r = 1 , ..., k, α i ( z r ) = ... = α i ( z r,(cid:96) r /b ) = α i ( z ,(cid:96) /b ) , α i ( z r, ( (cid:96) r /b )+1 ) = ... = α i ( z r, (cid:96) /b ) = α i ( z , (cid:96) r /b ) , ..., α i ( z r,(cid:96) r − ( (cid:96) k /b )+1 ) = ... = α i ( z r,(cid:96) r ) = α i ( z ,(cid:96) ) . T (cid:48) is obtained from T by replacing T x with the following subtree (i.e., T (cid:48) − T (cid:48) x = T − T x ).(3.1) We preserve x , and let S (cid:48) ( x ) = { t , ...t b ; t , ..., t b , ..., t k , ..., t kb } where those t rs ( r =1 , ..., k, s = 1 , ..., b ) are symbols which do not belong to T. (3.2) For each r = 1 , .., k, if | ı ( y r ) | = 1 (i.e., ı ( y r ) = { i } ), then (cid:96) r = b. In this case, weregard t r , ..., t rb as the same with z r , ..., z r(cid:96) r , and each subtree T z rs can be directly “grafted”to t rs ( s = 1 , ..., b ). If | ı ( y r ) | > , then (cid:96) r > b and b is a divisor of (cid:96) r . In this case, we let S ( t r ) = { z r , ..., z r,(cid:96) r /b } , S ( t r ) = { z r, ( (cid:96) r \ b )+1 , ..., z r, (cid:96) r \ b } , ..., S ( t rb ) = { z r,(cid:96) r − ( (cid:96) k /b )+1 , ..., z r,(cid:96) r } , andthe subtrees T z rn are all preserved ( n = 1 , ..., (cid:96) r ) . This condition is satisfied when each player has perfect recall (and perhaps with some consolidation of redundantactions like coalescing of information sets in Thompson [17]).
5. For the player assignment function (cid:48) , (cid:48) ( x ) = ( x ) ∪{ i } . For each t rs ( r = 1 , ..., k , s = 1 , ..., b ),if | ( y r ) | = 1, then since we have stipulated that t rs = z rs , it follows that (cid:48) ( t rs ) = ( z rs ); if | ( y r ) | > , then (cid:48) ( t rs ) = ( y r ) \{ i } .
5. For the set of actions { A (cid:48) i } i ∈ I and action assignment function β (cid:48) , since we have assumed thatfor each x ∈ X , if i ∈ ( x ) , then there is no h i ∈ H i such that S ( y ) ⊆ h i , it follows that i / ∈ ( x ) , i.e., for each y r ( r = 1 , ..., k ), β i ( y r ) = (cid:15). Let A (cid:48) j = A j for each j ∈ I. For each r = 1 , ..., k and s = 1 , ..., b, let α (cid:48) i ( t rs ) = α i ( z ,s(cid:96) r \ b ) and α (cid:48) j ( t rs ) = α j ( y r ) for each j (cid:54) = i . Let α (cid:48) j ( t ) = α j ( t ) for allother nodes t and all j ∈ I.
6. For the information sets, let H (cid:48) i = ( H i \ H i ( y )) ∪ , and in the following sometimes wesay that H i ( y ) is absorbed by/into x . For j (cid:54) = i, let H (cid:48) j = ( H j \ k (cid:91) r =1 H j ( y r )) ∪ k (cid:91) r =1 ( H j ( y r ) \{ y , ..., y k } ) ∪ (cid:91) s : y s ∈ H j ( y r ) { t s , ..., t sb } (1)In the above formula, we manipulate that H j ( y r ) = ∅ if j (cid:54) = ι ( y r ) for each r = 1 , ..., k. One property of the compactification operation is that it has no influence on the “grandsons”of x , i.e., the nodes z with p ( z ) = x (with respect to isomorphism). Hence it implies that a com-pactification operation does not alter the terminal nodes and, consequently, any utility functions.Here we use an example to illustrate the compactification operation. Figure 2: Compactifying an extensive form
Consider the (standard) extensive form on the left-hand side of Figure 2. There are threeplayers. Player 1 moves at first, choosing a or b . Player 2 moves without knowing player 1’s choice,and he can choose either c or d. It turns out that if player 1 chooses b, then the game terminatesafter player 2 makes a choice, while if player 1 chooses a, then after player 2’s move it is player3’s turn. Player 3 knows what player 1 has chosen yet does not know player 2’s choice. The gameterminates after player 3 makes up his choice.Here, player 3’s two decision nodes in h can be absorbed by their immediate predecessor, i.e.,the node on the left-hand side in player 2’s information set h . By doing this, the number of edges6adiating from that node increases from two to four since now each edge represents a pair of player2 and 3’s actions. Now we obtain the compact extensive form in the middle.Further, we can combine player 2’s information set h with player 1’s h . Now four edges radiatefrom the root which is occupied by both players 1 and 2. Note that h cannot be absorbed by theroot since h does not contain all immediate predecessors of the root node. In the reduced form onthe right-hand side, h now contains two nodes which represent player 1’s action a and player 2’stwo actions. This faithfully describes player 3’s knowledge at h : he knows that player 1’s actionis a, yet he has no idea on player 2’s choice.The following proposition states that perfect recall is preserved in the compactification opera-tion. Proposition 1. (Preservation of perfect recall)
Consider two compact extensive forms Γ and Γ (cid:48) with Γ → COM Γ (cid:48) . If every player has perfect recall in Γ , then so does everyone in Γ (cid:48) . Another problem concerns epistemic game theorists is the unambiguity of the order amonginformation sets. Consider a compact extensive form Γ = (cid:104) T, ≥ ; I, ı ; { A } i ∈ I , α ; { H i } i ∈ I (cid:105) and twoinformation sets h and h (cid:48) (they may belong to different players). We say that h is followed by h (cid:48) (or h (cid:48) follows h ) , denoted by h (cid:31) h (cid:48) , iff there is x ∈ h and y ∈ h (cid:48) such that x is on the uniquepath from the root to y. We say that h and h (cid:48) are simultaneous , denoted by h ∼ h (cid:48) , iff h ∩ h (cid:48) (cid:54) = ∅ .An information set h is weakly followed by h (cid:48) (or h (cid:48) weakly follows h ), denoted by h (cid:37) h (cid:48) , iff h is followed by h (cid:48) or they are simultaneous. We say the order among the information sets is unambiguous iff for every information sets h, h (cid:48) , if h (cid:45) h (cid:48) , then it does not hold that h (cid:31) h (cid:48) . The unambiguity of the order among information sets is vital in dynamic epistemic game theory.The algorithm which screens out strategies that is optimal to common belief in the opponents’future rationality, called backward dominance procedure, works if and only if the information setsare ordered unambiguously. Hence it is relevant to consider whether the compactitification processpreserves the unambiguity of the order among information sets. The following statement gives apositive answer.
Proposition 2. (Preservation of unambiguity of the order among information sets)
Consider two compact extensive forms Γ and Γ (cid:48) with Γ → COM Γ (cid:48) . If Γ has an unambiguous orderamong information sets, then so does Γ (cid:48) . Figure 2 actually shows a compactification process. Indeed, the operation can be appliedrepeatedly, and we can obtain a compactification sequence. Formally, let Γ be a compact extensiveform. A compactification process from Γ is a sequence (cid:104) Γ , Γ , ..., Γ m (cid:105) where Γ = Γ , for each r = 0 , ..., m − , Γ r → COM Γ r +1 , and Γ m can not be compactified anymore. A compact extensiveform is said to be minimal iff it cannot be compactified any more.The problem of the compactification process is that it is not order-independent. In Figure 3we give an example. Consider the extensive form on the left-hand side. The symbols A, ..., N represent the terminal nodes. We give two compactification processes from it. In the process onthe top, first the four nodes in h (cid:48) are absorbed by the node on the left-hand side of h (as wellas h ). Then the three nodes in h are absorbed by the root. In the process on the button, incontrast, the absorption of h is conducted first. However, this makes the compactification processinable to continue, since now no node has its immediate successors in one information set. It isclear that the two processes have distinct terminal terms.7 igure 3: Order-dependence of the compactification process Though order-independence does not hold in general, by peering closely at Figure 3, one maynotice that the process on the top makes the original extensive form more compact, i.e., one moreinformation set is absorbed. Here we define a compactification process which absorbs at least asmany informations sets as any other process. We start from the nodes next to the terminal nodes,i.e., { y ∈ X : S ( y ) ⊆ Z } and check whether some of them can be absorbed by their immdiatepredecessor, if so, then conduct the compactification operation on them. When all compactificationis done for those nodes, we go to the nodes next to them, etc. We call it a leaves-to-root (LTR)compactification process. We have the following statement. Proposition 3. (LTR process absorbs more information sets)
Consider the LTR compact-ification process and an arbitrary compactification processes from a compact extensive form Γ . Ifan information set in Γ is absorbed in the latter, then so it is in the former. For each compact extensive form Γ, we use
COM (Γ) to denote the set of all compact extensiveforms (not necessarily minimal) which can be reached through a compactification process from Γ.Let Γ ∗ be a minimal compact extensive form. We define [Γ ∗ ] = (cid:83) { COM (Γ) : Γ ∗ is the terminalterm of the LTR compactification process from Γ } . Each Γ (cid:48) ∈ [Γ ∗ ] can be said as related to Γ ∗ via some compactification process; either it is a compact extensive form which has Γ ∗ as the finalterm of the LTR process, like the one on the left-hand side of Figure 3, or it can be reached via acompactification process from another compact extensive form which reaches Γ ∗ through the LTRprocess (thought itself may not be able to reach Γ ∗ ), like the terminal term on the bottom processin Figure 3.As we mentioned in Section 3.1, compactification operations have no influence on the terminalnodes. Hence any utility function applied to one Γ (cid:48) ∈ [Γ ∗ ] can be applied to any compact extensiveform in [Γ ∗ ] without any alteration.
4. Compactification and Belief in the Opponents’ Future Rationality
We have developed the compact extensive form as a faithful representation of the actual chrono-logical order of the moves in Perea [12], a notion essential for the application of backward dominance8rocedure which screens out strategies optimal to types with common belief in future rationality.This section discuss this relationship between compactification process and belief in the opponents’future rationality in detail.Consider a compact extensive form game G = (Γ , { u i } i ∈ I ) , where Γ = (cid:104) T, ≥ ; I, ; { A } i ∈ I , β ; { H i } i ∈ I (cid:105) is a compact extensive form and for each i ∈ I , u i is player i ’s von Neumann-Morgenstern utility function for terminal nodes. An information set h i ∈ H i is called initial forplayer i iff for any x ∈ h i and any y < x, there is no h (cid:48) i ∈ H i such that y ∈ h (cid:48) i . A strategy for player i is a function s i : (cid:98) H i → (cid:83) h i ∈ (cid:98) H i β i ( h i ) which assigns each h i ∈ (cid:98) H i ⊆ H i a choice s i ( h i ) ∈ β i ( h i )satisfying(1) Every initial information set of i belongs to (cid:98) H i , and(2) (cid:98) H i = H i ( s i ) , where H i ( s i ) is the information sets that strategy s i allows for.Consider an information set h. We adopt the following symbols from Perea [12]: S ( h ) := { ( s i ) i ∈ I ∈ (cid:81) i ∈ I S i : ( s i ) i ∈ I reaches some node in h } ,S i ( h ) := { s i ∈ S i : ( s i , s − i ) ∈ S ( h ) for some s − i ∈ S − i } ,S − i ( h ) := { s − i ∈ S − i : ( s i , s − i ) ∈ S ( h ) for some s i ∈ S i } . Consider a compact extensive form game G = (Γ , { u i } i ∈ I ) . An epistemic model for G is a tuple M = ( T i , b i ) i ∈ I where for each i ∈ I, (1) T i is a finite set of types for player i, (2) b i is a function that assigns to every type t i ∈ T i , and every information set h i ∈ H i , aprobability distribution b i ( t i , h i ) ∈ ∆( S − i ( h i ) × T − i ) . Consider a type t i , a strategy s i , and an information set h i ∈ H i ( s i ) . By u i ( s i , t i | h i ) we denotethe expected utility from choosing s i under the conditional belief that t i holds at h i . Strategy s i is optimal for type t i iff u i ( s i , t i | h i ) ≥ u i ( s (cid:48) i , t i | h i ) for all s (cid:48) i ∈ S i ( h i ) . Strategy s i is rational for a type t i iff s i is optimal for t i at every h i ∈ H i ( s i ) . Consider a type t i , an information set h i ∈ H i , and an opponent j (cid:54) = i. Type i believes at h i in j ’ s future rationalit y iff b i ( t i , h i ) only assigns positive probability to j ’s strategy-type pairs ( s j , t j )where s j is optimal for t j at every h j ∈ H j ( s j ) that weakly follows h i . Type t i believes in theopponents’ future rationality iff at every h i ∈ H i , type t i believes in every opponents’ rationality.Consider a compact extensive form game G and an epistemic model M = ( T i , b i ) i ∈ I for G. Initial step . Define for every player i the set of types T i = { t i ∈ T i : t i believes in the opponents’ future rationality } Inductive step . Let k ≥
2, and suppose that T k − i has been defined for all player i . Then,we define T ki = { t i ∈ T k − i : b i ( t i , h i )( S − i × T k − − i ) = 1 for all h i ∈ H i } . A type t i expresses common belief in future rationality iff t i ∈ T ki for every k. A strategy can rationally be chosen under common belief in future rationality iff there is someepistemic model M = ( T j , b j ) j ∈ I and some type t i ∈ T i such that t i expresses common belief infuture rationality and s i is rational for t i .We have the following statement Theorem 4. (Minimal form and the smallest set of rational strategies ) Let Γ ∗ be a min-imal compact extensive form and ( u i ) i ∈ I a vector of utility functions for each players on the ter-minal nodes of Γ ∗ . A strategy can rationally be chosen under common belief in future rationalityin (Γ ∗ , ( u i ) i ∈ I ) if and only if for each Γ ∈ [Γ ∗ ] , it can rationally be chosen under common belief infuture rationality in (Γ , ( u i ) i ∈ I ) . . Concluding Remarks: An Alternative Compactification In the definition of compactification in Section 3.1, we require that the immediate successorsof x form an information set for some player i , i.e., H i ( y ) = ... = H i ( y k ) = { y , ..., y k } . It mightbe wondered why we did not follow Thompson [17]’s interchange of decision nodes faithfully andrequire only H i ( y ) = ... = H i ( y k ) . Technically, it is possible, and the definition of compactificationcan be preserved except that, in term 6, we have to define H (cid:48) i = ( H i \ H i ( y )) ∪ { ( H i ( y ) \{ y , ..., y k } ) ∪ { x }} . (2)We call it a Thompson compactification ( T-compactification ), denoted by Γ → T COM Γ (cid:48) . Thisdefinition allows that only some nodes in an information set, instead of the whole set, are absorbed.The advantage of T-compactification is that the process is order-independence. Formally, we havethe following statement.
Proposition 5. (Order-independence of T-compactification process)
Consider a compactextensive form Γ and two T-compactification processes (cid:104) Γ , Γ , ..., Γ m (cid:105) and (cid:104) Γ (cid:48) , Γ (cid:48) , ..., Γ (cid:48) n (cid:105) from Γ .Then Γ m = Γ (cid:48) n . However, on the other hand, T-compactification has a severe disadvantage: the partial absorp-tion may reverse the order of two information sets, sometimes even destroy the unambiguity of theorder. Figure 4 gives an example.
Figure 4: Change of the unambiguity of the order between information sets
On the left-hand side of Figure 4 is a fragment of some game. The symbols
A, ..., M representthe continuation of the game. Only player 1’s information set h and player 2’s h are depictedwith h unambiguously follows h , and all other nodes belong to players other than 1 and 2. It canbe seen that four nodes in h can be absorbed by their immediate predecessors respectively, and weobtain the fragment in the middle. Now, h is simultaneous with h and follows h . Further, the twonodes on the left-hand side in h in the middle can be absorbed by their immediate predecessor, andwe obtained the fragment on the right-hand side. Here, the unambiguity of the order between h and h is destroyed; indeed, h follows h and is also followed by h , which obfuscates the meaning10f “future” for moves in h and makes it impossible to apply Perea [12]’s backward dominanceprocedure.The requirement of absorption of the whole information set can avoid this difficulty. Yet itis difficult to deny the validity of combining only some nodes in an information set since thestrategic features of the original game seems invariant in this process. Perhaps we can considersome restrictions on the original game which guarantee the preservation of the unambiguity ofthe order of the information sets, for example, every two nodes in an information set should haveexperienced the same history in the sense of information sets, i.e., if the path to one node passesan information set h, so does the other. More research is expected in this direction.
6. Proofs
In this subsection, we will prove Proposition 1 in Section 3.1. First, it can be seen that acompactification operation can be characterized by a pair ( x, i ) where x is the node which absorbsan information set constitute by its immediate successors and i is the player who possesses thatinformation set. Hence for each compact form Γ = (cid:104) T, ≥ ; I, ı ; { A } i ∈ I , α ; { H i } i ∈ I (cid:105) , each x ∈ X , andeach i ∈ I with S ( x ) ∈ H i , we can define ψ (Γ; ( x, i )) to be the compact extensive form obtainedfrom Γ via the compactification operation where player i ’s information set S ( x ) ∈ H i is absorbedby x . Conversely, for each Γ , Γ (cid:48) with Γ → COM Γ (cid:48) , there exist uniquely x ∈ X and i ∈ I such thatΓ (cid:48) = ψ (Γ; ( x, i )) . Consider a compact extensive form Γ which satisfies perfect recall, and Γ (cid:48) = ψ (Γ; ( x, i )) . Let y, z, z (cid:48) ∈ X with j ∈ ı ( y ) ∩ ı ( z ) ∩ ı ( z (cid:48) ), y < z, and H j ( z ) = H j ( z (cid:48) ) . It can be seen straightforwardlythat condition for perfect recall is satisfied in Γ (cid:48) if one of the following conditions is satisfied: (i) j (cid:54) = i ; (ii) y, z, z (cid:48) and x are on different subtrees; (iii) H i ( z ) > x, (iv) p n ( y ) = x with n ≥ . Thereason is, in each case, the compactification has no essential effect on the configuration of the treebetween y and z , z (cid:48) .Hence we only need to consider the case where j = i, y, z, z (cid:48) and x are on the same sub-tree,and x = p ( y ) or y < x. Suppose that x = p ( y ). Since the original Γ satisfies perfect recall, thereis some y (cid:48) ∈ H i ( y ) leading to z (cid:48) via the same action as y for z. After the compactification, y (cid:48) and y are both absorbed into x , it can be seen that in Γ (cid:48) , now z and z (cid:48) are reached from x (now H i ( y ) isreplaced by { x } ) through two edges sharing the same action of i. For the case of y < x, it is easy tosee that the existence of y (cid:48) and the action from it leading to z (cid:48) is invariant in the compactification.Here we have shown that perfect recall is satisfied in Γ (cid:48) . In this subsection, we will show Proposition 2 in Section 3.1. Consider two compact ex-tensive form Γ and Γ (cid:48) with Γ (cid:48) = ψ (Γ; ( x, i )) . It can be seen that for each j ∈ I, there is abijection (cid:37) j from H j to H (cid:48) j . Indeed, using the symbols in the definition of the compactifica-tion in Section 3.1, (cid:37) i ( H i ( y )) = { x } , and for each j (cid:54) = i with j ∈ ı ( y r ) for some y r ∈ S ( x ) ,(cid:37) j ( H j ( y r )) = ( H j ( y r ) \{ y , ..., y k } ) ∪ (cid:83) s : y s ∈ H j ( y r ) { t s , ..., t sb } , and for every other information set,since it is preserved in the compactification, (cid:37) i maps it to itself. Here the case y = x is impossible since it goes against our assumption that if i ∈ ı ( x ) , then S ( x ) ∈ H i . Also,the relation between H i ( z ) and x is not essential here. (cid:48) , we only need to consider those information sets h j with (cid:37) j ( h j ) (cid:54) = h j . It iseasy to see that H i ( y ) does not cause any trouble, because in Γ (cid:48) it is degenerated into a singleton.For every j (cid:54) = i with j ∈ ı ( y r ) for some y r ∈ S ( x ), since the only difference between H j ( y r ) and (cid:37) j ( H j ( y r )) is that every y s ∈ H j ( y r ) with y s ∈ S ( x ) is now replaced by a group of doppelgangers t s , ..., t sb , the order between H j ( y r ) and any information set other than H i ( y ) in Γ is preserved inΓ (cid:48) (we only need to replace H j ( y r ) by (cid:37) j ( H j ( y r ))). Also, it is clear that { x } (cid:31) (cid:37) j ( H j ( y r )). Here,we have shown that the order of information sets in Γ (cid:48) are ordered unambiguously.Through the proof above, it can be seen that the order of information sets are weakly preservedin the compactification operation. Formally, we have the following statement. Corollary 6. (Weak preservation of the order of information sets)
Consider two compactextensive forms Γ , Γ (cid:48) with Γ → COM Γ (cid:48) and two information sets h i , h (cid:48) j in Γ . If h i ≺ h (cid:48) j , then (cid:37) i ( h i ) (cid:45) (cid:37) j ( h (cid:48) j ) . In this subsection we show Proposition 3 in Section 3. Suppose that some information set h is absorbed in some compactification process but not in the LTR process. The reason can onlybe that some information set smaller than h (i.e., nearer to the root) is absorbed before. Yet itis contradictory to the definition of “leaf to root”. Hence LTR process absorbs at least as manyinformation sets as any other compactification process.Also, it will be shown in Section 6.5 that, even for those processes with which LTR shares thesame terminal term, LTR is at least as fast as any of them. In summary, we can say that LTR isthe most efficient compactification process. In this subsection we show Theorem 4 in Section 4. We need an algorithm called backwarddominance procedure defined in Perea [12].For a given information set h , a subset Λ( h ) ⊆ S ( h ) is called a decision problem at h iff forevery active player i at h there are some D i ⊆ S i ( h ) and D − i ⊆ S − i ( h ) such that Λ( h ) = D i × D − i .Consider an information set h , a player i who is active at h , and a decision problem Λ( h ) = D i × D − i at h. We say that a strategy s i ∈ D i is strictly dominated within the decision problem Λ( h ) iffthere is some randomized strategy µ i ∈ ∆( D i ) such that u i ( µ i, s − i ) > u i ( s i , s − i ) for all s − i ∈ S − i . By sd i (Λ( h )) we denote the set of strategies in D i that are strictly dominated within Λ( h ) for theactive player i. Similarly, we define sd (Λ( h )) = { ( s j ) j ∈ I ∈ Λ( h ) : s i ∈ sd i (Λ( h )) for some i that is active at h } (3)The backward dominance procedure is defined as follows: Initial step . For every information set h, let Λ ( h ) = S ( h ); Inductive step . Let k ≥
1, and suppose that Λ k − ( h ) has been defined for every informationset h. Then, at every information set h we defineΛ k ( h ) = Λ k − ( h ) \ (cid:91) h (cid:48) ≥ h sd (Λ k − ( h (cid:48) )) (4)12 strategy s i survives the backward dominance procedure iff there is some s − i ∈ S − i such that( s i , s − i ) ∈ Λ k ( h o ) for each k ∈ N. Here h o is the information set which contains (only) the root.Perea [12] showed the following result (also see Perea [11]). Lemma 7. (Strategies surviving the backward dominance procedure)
Player i can ratio-nally choose s i under common belief in future rationality if and only if s i survives the backwarddominance procedure. Based on Corollary 6, we can show the following statetement.
Lemma 8. (Decrease of strategies surviving the backward dominance procedure in acompactification)
Consider compact extensive forms Γ , Γ (cid:48) with Γ → COM Γ (cid:48) , and a vector ofutility functions ( u i ) i ∈ I . If s i survives the backward dominance procedure in (Γ (cid:48) , ( u i ) i ∈ I ) , so it doesin (Γ , ( u i ) i ∈ I ) . Indeed, through a compactification operation, an information set h which used to follow anotherinformation set h (cid:48) is now simultaneous with h (cid:48) , which may lead to that more strategies in some S i ( h ) as well as strategies reaching some information sets before h and h (cid:48) to be eliminated. Basedon this lemma, it is easy to see by mathematical induction that, if Γ is the final term of some LTRprocess from Γ (cid:48) , then every s i survives the backward dominance procedure in Γ , so it does in Γ (cid:48) . In other words, if a strategy s i does not survive the procedure for some compact extensive form,then s i does not survive in the terminal of the LTR process from it.Consider a minimal compact extensive form Γ ∗ , some Γ from which the LTR process leads toΓ ∗ , and some Γ (cid:48) which is the terminal term of a compactification process from Γ. Remember thatwe have shown in Proposition 3 that for every information set, if it is absorbed in Γ (cid:48) , then so it doesin Γ ∗ . For simplicity, we assume that at each node, each player has more than two actions. It can beseen that (1) every information set in Γ can be absorbed for at most one time, and it is determinedthat to where they are absorbed. (2) If Γ (cid:48) (cid:54) = Γ ∗ , it is because in the compactification process fromΓ to Γ (cid:48) , some information set h is absorbed while some other information set h (cid:48) following it is stillnot absorbed and is not able to absorbed due to the absorption of h. Therefore, Γ (cid:48) can be informallysaid as a “incomplete” version of Γ ∗ . Hence, similarly to the argument in the previous paragraph,it can be seen that every strategy which does not survive the backward dominance procedure inΓ (cid:48) does not survive the procedure in Γ ∗ . Combined with Lemma 8, it follows that the statementholds for any middle term in any compactification process from Γ . Hence, we have shown the onlyif part of Theorem 4.The if part of Theorem 4 holds straightforwardly since Γ ∗ itself belongs to [Γ ∗ ] . Here we haveshown Theorem 4.
In this subsection we show Proposition 5 in Section 5, i.e., the order independence of the T-compactification process. We use a lemma proved in Newman [9] (also see Apt [1]) which gives asufficient condition for order-independence in an abstract reduction system.An abstract reduction system is a pair (Θ , → ) , where Θ is a non-empty set and → is a binaryrelation on Θ . An element θ ∈ Θ is called an endpoint in (Θ , → ) if there is no θ (cid:48) ∈ Θ such that θ → θ (cid:48) . We say that { θ n : n = 0 , , ... } (it can be finite or infinite) is a → -sequence in (Θ , → ) iff θ n ∈ Θ for each n and θ n → θ n +1 (as far as θ n +1 is defined). We use → ∗ to denote the reflexive andtransitive closure of → . We say that ( X, → ) is weakly confluent iff for each θ, η, ζ ∈ Θ , if θ → η and θ → ζ, then there is some θ (cid:48) ∈ Θ such that η → ∗ θ (cid:48) and ζ → ∗ θ (cid:48) . emma 9. (Newman’s lemma) If an abstract reduction system ( X, → ) satisfies the followingtwo conditions: (N1) each → -sequence is finite, and (N2) ( X, → ) is weakly confluent, then foreach θ ∈ Θ there is a unique endpoint θ (cid:48) ∈ Θ such that θ → ∗ θ (cid:48) . The set of all compact extensive forms and → T COM form an abstract reduction system. Sincea T-compactification operation does not generate new information set which can be absorbed byits immediate predecessor, is straightforward to see that each → T COM -sequence is finite. Hence,by Newman’s lemma, to show that the T-compactification process is order-independent, we onlyneed to show that weak confluence is satisfied.First, it can be seen that a T-ompactification operation can still be characterized by a pair( x, i ) where x is the node which absorbs an information set constitute by its immediate successorsand i is the player who possesses that information set. So here we abuse the symbol and stilluse ψ (Γ; ( x, i )) to show what is changed in a T-compactification on Γ . Consider three compactextensive forms Γ , Γ (cid:48) , Γ (cid:48)(cid:48) with Γ (cid:48) = ψ (Γ; ( x, i )) and Γ (cid:48)(cid:48) = ψ (Γ; ( y, j )) . Intuitively, it can be seeneasily that if x and y are “far” from each other, then for each form we just repeat the operationthat leads to the other and we can obtain the same outcome. Formally, if x and y are on differentsubtrees (i.e., neither x ≤ y nor y ≤ x ) or x = p k ( y ) with k ≥ y = p k ( x ) with k ≥ x , S ( x ) and y, S ( y ) are preserved in Γ (cid:48)(cid:48) , Γ (cid:48) respectively, we have ψ (Γ (cid:48) ; ( y, j )) = ψ (Γ (cid:48)(cid:48) ; ( x, i )) . Suppose that x = y (i.e., x = p ( y )). If i = j, then Γ (cid:48) = Γ (cid:48)(cid:48) . Consider the case that i (cid:54) = j . Itmeans that S ( x ) ∈ H i and S ( x ) ∈ H j . By the equation (1) in the definition of T-compactification inSection 5, x is preserved in Γ (cid:48) and Γ (cid:48)(cid:48) , and S (cid:48) ( x ) ∈ H j and S (cid:48)(cid:48) ( x ) ∈ H i . Hence we can further com-bine S (cid:48) ( x ) with x in Γ (cid:48) and S (cid:48)(cid:48) ( x ) with x in Γ (cid:48)(cid:48) , and it can be seen that ψ (Γ (cid:48) ; ( y, j )) = ψ (Γ (cid:48)(cid:48) ; ( x, i )) . Finally, suppose that x = p ( y ). By our assumption, in this case i (cid:54) = j . When | ı ( y ) | = 1(i.e, ı ( y ) = { i } ) , it can be seen that ψ (Γ (cid:48)(cid:48) ; ( x, i )) = Γ (cid:48) . Suppose that | ı ( y ) | > . We still use thesymbols used in Section 3 and let b = | α i ( S ( y )) | . In Γ (cid:48) , now y is “replaced” by dummies t , ..., t b , and (cid:83) br =1 S (cid:48) ( t r ) ⊆ h j for some h j in H j . Hence we can make each t r to absorb S ( t r ). On the other hand,in Γ (cid:48)(cid:48) , x still can absorb S (cid:48)(cid:48) ( x ) (= S ( x )). It can be seen that ψ ( ...ψ ( ψ (Γ (cid:48) ; ( t , j )); ( t , j )) ... ; ( t b , j )) = ψ (Γ (cid:48)(cid:48) ; ( x, i )). Here we have shown that condition (N2) in Newman’s lemma is satisfied, that is,we have shown that T-composition process is order-independent.Though the terminal term is invariant among different T-compactification processes, it canbe seen from the above proof that the LTR process is the fastest one. Indeed, the outcomethat LTR process can achieve in two steps if we start from the node which is “nearer” to theterminal nodes (i.e., y ) . If we start from the farther one (i.e., x ), then it costs b times of operation.Therefore, LTR process still has its distinct value from the viewpoint of algorithmic complexity forT-compactification. References [1] Apt, K.R. 2011. Direct proofs of order independence.
Economics Bulletin : 106-115.[2] Battigalli, P. 1997. On rationalizability in extensive games. Journal of Economic Theory : 40-61.[3] Battigalli, P., Siniscalchi, M. 2002. Strong belief and forward induction reasoning. Journal of Economic Theory : 356-391.[4] Binmore, K. 1987. Modelling rational players, Part I.
Economics and Philosophy : 179-214. It should be noted that in this proof, all the equality holds in the sense of isomorphism, i.e., strictly speakingthere may need some re-naming of the nodes.
5] Dekel, E., Siniscalchi, M. 2015. Epistemic game theory. In: Young, H.P., Zamir, S. (Eds.), Handbook of GameTheory, vol. 4, Elsevier: 619-702.[6] Elme. S., Reny, P.J. 1994. On the strategic equivalence of extensive form games.
Journal of Economic Theory : 1-23.[7] Kreps, D.M. 1990. Game Theory and Economic Modelling . Clarendon Press.[8] Kreps, D.M., Wilson, R. 1982. Sequential equilibria.
Econometrica : 863-894.[9] Newman, M.H.A. 1942. On theories with a combinatorial definitions of equivalence. Annals of Mathematics :223-243.[10] Perea, A. 2001. Rationality in Extensive Form Games . Kluwer Academic Publishers.[11] Perea, A. 2012.
Epistemic Game Theory . Cambridge University Press.[12] Perea, A. 2014. Belief in the opponents’ future rationality.
Games and Economic Behavior : 231-254.[13] Reny, P.J. 1992. Rationality in extensive-form games. Journal of Economic Perspectives : 103-118.[14] Reny, P.J. 1993. Common belief and the theory of games with perfect information. Journal of Economic Theory : 257-274.[15] Selten, R. 1975. Re-examination of the perfectness concept for equilibrium points in extensive games. Interna-tional Journal of Game Theory : 25-55.[16] Tan, T., Werlang, S.R.C. 1988. The Bayesian foundations of solution concepts of games. Journal of EconomicTheory : 370-391.[17] Thompson, F.B. 1952. Equivalence of games in extensive form. The Rand Corporation No. 759.: 370-391.[17] Thompson, F.B. 1952. Equivalence of games in extensive form. The Rand Corporation No. 759.