Equivalences of Extensive Forms with Perfect Recall Carlos Alós-Ferrer and Klaus Ritzberger University of Cologne and Royal Holloway, University of London, 1 and VGSF 1 as of Aug. 1, 2016
1 Introduction Extensive form is the basic representation of a game... needed to verify that the rules are complete. Yet, extensive forms may differ even though the game is the same. When are details of the extensive form strategically relevant? Solutions are often defined in the normal form (as strategy profiles). I Some refinement concepts are defined purely for the extensive form, e.g., subgame perfection, perfect Bayesian, sequential equilibrium. I But these are sensitive to inessential details of the extensive form. I Therefore some have argued that strategically stable solutions should only depend on the normal form... or even only on the reduced normal form (Kohlberg & Mertens 1986, Mertens 1989, 1991, 1992,...). 2
1.1 Which Details are inessential? One approach: I Transformations of the extensive form that do not affect the semireduced normal form (Thompson 1952, Elmes & Reny 1994). Thompson: Interchange of moves, Coalescing of moves, Addition of a superfluous move, Inflation/Deflation. Elmes & Reny: Interchange of moves, Coalescing of moves, Modified addition of a superfluous move to preserve perfect recall. F F Inflation/Deflation and Addition can destroy perfect recall! I Thompson-transformations take payoffs! Thompson s Theorem: Two extensive form games have the same semireduced normal form game if and only if one emerges from the other by iterative application of the four transformations or their inverses. 3
Here are the four Thompson-transformations: 4
from this paper 5
Complementary approach: I Which information is lost/preserved in the transition from the extensive to the normal form? I But now without payoffs in a pure representation of the rules. I No transformations, but identify what is essential. I What are the structures in the extensive form that can be used for solution concepts that are as robust as normal form concepts? Related (because also without payoffs) but different is... Battigalli s Conjecture: Two extensive forms have the same reduced 2 normal form if and only if one emerges from the other by iterative application of the first two Thompson transformations (Interchange & Coalescing) or their inverses (Leonetti 2015). 2 without payoffs = semi-reduced 6
2 Definitions A game tree =( ) is a collection of nonempty subsets (the nodes) of an underlying set (of plays) partially ordered by set inclusion such that, { } for all, and (GT1) is a chain if and only if :, (GT2) every chain in the set = \{{ }} of moves has a maximum and either an infimum in the set = {{ }} of terminal nodes or a minimum. A node is finite if the set \{ } = { } has a minimum ( ), otherwise it is infinite. If is a union of nodes, define = { } and ( ) ={ : = \ } (1) as the nodes where 2 is available. 7
A discrete extensive form (DEF) with player set is a pair ( ), where =( ) isagametreewithset of plays and =( ) is a system consisting of collections (the set of s choices) of nonempty unions of nodes such that (DEF1) if ( ) ( 0 ) 6= and 6= 0,then ( ) = ( 0 ) and 0 =, for all 0 and n all, and (DEF2) 1 ( ) = ( o ( ) ) ( ) ( ) ( ) for all, where ( ) ={ ( )} are the choices available to at and ( ) ={ ( ) 6= } 6= are the decision makers at. The set of pure strategies of player is the set of all functions : = { ( )} that satisfy 1 ( ) = ( ) for all ( ) (2) and = is the set of all pure strategy combinations. 8
I For every DEF there is a surjection : that assigns to every strategy combination the play that it induces (AR 2008, Th. 4 & 6). I For let ( ( )) = ( ) denote the plays passing through information set ( ). I Assume w.l.o.g. that ( ( )) for all and all. I For a DEF no-absent-mindedness holds (AR 2005, Prop. 13), but not necessarily perfect recall. ADEF( ) satisfies perfect recall (Kuhn 1953) if and only if if 0 6= then either 0 or 0 (3) for all 0 { ( ( )) } and all (AR 2016, Th. 2). 9
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2.1 Normal Form The normal form of a DEF ( ) is the triplet ( ), where = are the strategy combinations and : assigns the induced plays. Two strategies 0 ofthesameplayerinadef( ) are strategically equivalent if ( )= ( 0 ) for all 6=. Proposition 1 Two strategies 0 are strategically equivalent if and only if, for all, ( ) 6= 0 ( ) ( ( ) ( 0 )) =. Denote the quotient space w.r.t. strategic equivalence for by and = ( ). By definition : induces a surjection :,forall [ ], by ([ ]) = ( ). ³ The triplet is the reduced normal form of the DEF ( ). 11
2.1.1 Comparing Normal Forms Let ( ) and 0 0 0 be the normal forms of two DEFs, ( ) and ( 0 0 ). They are isomorphic if there are bijections : 0, : 0, : 0,and : 0 for all such that 0 ( ( )) = ( ( )) for all (4) and ( ) ( ) = ( ) for all and all. Example 1 The following two normal forms are isomorphic. a b A 1 1 B 2 3 C 2 4 X Y Z 1 2 2 1 3 4 12
2.2 Decision System Beginwithasingleplayer,thedecision maker (DM). A (sequential) decision problem (with perfect recall) for a DM on a set is a pair ( ) where is a collection of nonempty subsets of and is a partition of such that, for all 0, if 0 and 0 6= then = 0 (5) and, for all 0 { ( ) }, if 0 6= then either 0 or 0. (6) Denote ( ) = and ( ) ={ ( ) }. Decision problems can be partially ordered. Say that decision problem ( 0 0 ) is smaller than decision problem ( ) if 0 and ( 0 ) ( ). 13
Decision problems may contain redundancies, though. E.g., = { } for and. Say that for a decision problem ( ) the pair ( ) is redundant if = ( ). Proposition 2 For every decision problem ( ) there is a unique largest decision problem that is smaller than ( ) and contains no redundant pairs. Call this the reduced decision problem of ( ). Remark 1 In epistemics an information structure is a function Π : W such that Π ( ) and 0 Π ( ) Π ( 0 ) Π ( ) for all. Such a function can be associated to a decision problem by Π ( ) =min{ ( ) ( ) } provided it is the reduced decision problem. 14
Now return to multi-player games. The decision system associated with a DEF ( ) is the collection ( ) of decision problems where, for each, the partition of is induced by, for all 0, 0 ( ) = ( 0 ). Each is a choice and each is an information set. If in a decision system the decision problems ( ) arereplacedbytheir reduced decision problems, the resulting collection is called the reduced decision system of the DEF. With the decision problem ( ) of each player comes the space of s choice functions. Θ = { : ( ), } (7) 15
Choice functions and strategies are bijective and preserve. Lemma 1 For each there is a bijection : Θ such that, with = ( ) Θ = Θ and = ( ) : Θ, { ( ) ( ) } = ( ( )). 2.2.1 Comparing Decision Systems The decision system ( ) of a DEF ( ) and the decision system ( 0 0 ) 0 of a DEF ( 0 0 ) are isomorphic if there are bijections : 0 and : 0 such that 0 ( ) = { ( ) } and 0 ( ) = {{ ( ) } } for all, where ( ) ={ ( ) } for all 2. 16
Example: Let = {1 2 3 4} be the plays and for two players, = {1 2}, 1 = {{1} {2 3 4} {3} {4}} with 1 = {{{1} {2 3 4}} {{3} {4}}} and 2 = {{2} {3 4}} with 2 = {{{2} {3 4}}}. 17
3 First Results I For what follows let ( ) and ( 0 0 ) be two DEFs with normal forms ( ) and ³ 0 0 0 respectively, 0 reduced normal forms and ³ 0 0, decision systems ( ) and ( 0 0 ) respectively, and 0 reduced decision systems and 0 0 I All DEFs involved satisfy perfect recall. Theorem 1 Two DEFs have isomorphic normal forms if and only if their decision systems are also isomorphic. Conjecture: This corresponds to Thompson s Interchange of moves." Choices and information sets stay constant, only the tree changes. 0. 18
Idea of proof for Theorem 1: if: From the isomorphism between decision systems construct bijections between the choice functions. Use Lemma 1 to construct an isomorphism between normal forms. only if: Exploits perfect recall on every path to an information set of player this player has to take the same choices. Now take the chain of image choices (under the isomorphism of normal forms), intersect them, and obtain the images of choices. Show that this results in an isomorphism between decision systems. (This takes DEF1 and DEF2.) I Important: No reduction step needed! And multiple strategically equivalent strategies still show up. I Hence, the only difference between the two DEFs concerns the trees but not choices and information sets. 19
Theorem 2 Two DEFs have isomorphic reduced normal forms if and only if their reduced decision systems are also isomorphic. Conjecture: This corresponds to the first two Thompson-transformations, Interchange of moves and Coalescing of moves. Idea of proof for Theorem 2: Similar to proof of Theorem 1, but now reduction step is involved. DEF 7 decision system 7 reduced decision system 7 new DEF Involves an algorithm that reconstructs a DEF from a (possibly reduced) decision system. Let 0 = 0 = { }. For each =1 2 and each 1 1 \{{ } } let ( ) ={ : ( )} and 1 ( ) = ( ) 6= ( ) ( ) ª and set = 1 1 ( ) and = \{{ } }. 20
4 What s that Saying? I If the conjectures are correct, then Battigalli s conjecture is correct, but canberefined: Two extensive forms have the same normal form if and only if one emerges from the other by iterative application of the first Thompson transformation (Interchange) or its inverse. Two extensive forms have the same reduced normal form if and only if one emerges from the other by iterative application of the first two Thompson transformations (Interchange & Coalescing) or their inverses. F Beware: This applies to forms not to games. E.g., the following is not the normal form of a DEF with = { }. 1 2 2 2 1 1 2 1 21
4.1 Implications for Solutions I Robust solutions for extensive form games cannot depend on the tree! For instance, Myerson & Reny s attempts to generalize sequential equilibrium to large games will have to work without the notion of beliefs. For, beliefs are definedonthe nodes of the tree and the tree changes both under Theorem 1 and 2. I Will robust solutions have to work without a sequential structure? No! The decision system captures all the relevant sequential structure, because (for a given player) choices together with information sets are like a tree. But the appropriate domain for probabilistic assessments is the set of plays not the nodes. I Why not work with the (reduced) normal form right away? The normal form does not identify subgames and information sets (recall, though, Mailath, Samuelson, and Swinkels 1993, 1994). Hence, backwards induction intuition can only be captured indirectly, e.g., by proper equilibrium. But (proper) strategy perturbations in large games are bound to depend on the topology and for large games there is no natural topology on strategy space. 22
4.2 ABitofSpeculation I The existing results show that the decision system captures precisely the same information as the normal form, and the reduced decision system the same as the reduced normal form. I If the normal form (resp. reduced normal form) contains all strategically relevant information, then so does the decision system (resp. reduced decision system). This raises the following question: Is it possible to represent a game purely by its decision system a game in decision form, as it were? One would need to find conditions on a decision system such that the algorithm produces a DEF that has the original decision system (preferably also for the reduced version). 23
5 Conclusions Working with extensive forms and normal forms without payoffs, but with perfect recall, we characterize equivalence classes of DEFs that have isomorphic normal forms that have isomorphic reduced normal forms by the property that they have isomorphic decision systems they have isomorphic reduced decision systems. This is most likely related to the first two Thompson transformations (Interchange & Coalescing) and to Battigalli s conjecture. 24
Thank you for your attention! 25