Myopic equilibria, the spanning property, and sublime bundles
aa r X i v : . [ ec on . T H ] J u l Myopic equilibria, the spanning property andsubgame bundles
R. Simon, S. Spie ˙z, H. Toru´nczykJuly 28, 2020
London School of EconomicsDepartment of MathematicsHoughton StreetLondon WC2A 2AEInstitute of MathematicsPolish Academy of Sciences´Sniadeckich 8, 00–656 Warszawa bstract:
For a set-valued function F on a compact subset W of amanifold, spanning is a topological property that implies that F ( x ) = ∅ forinterior points x of W . A myopic equilibrium applies when for each actionthere is a payoff whose functional value is not necessarily affine in the strat-egy space. We show that if the payoffs satisfy the spanning property, thenthere exist a myopic equilibrium (though not necessarily a Nash equilibrium).Furthermore, given a parametrized collection of games and the spanningproperty to the structure of payoffs in that collection, the resulting myopicequilibria and their payoffs have the spanning property with respect to thatparametrization. This is a far reaching extension of the Kohberg-MertensStructure Theorem. There are at least four useful applications, when payoffsare exogenous to a finite game tree (for example a finitely repeated game fol-lowed by an infinitely repeated game), when one wants to understand a gamestrategically entirely with behaviour strategies, when one wants to extendsthe subgame concept to subsets of a game tree that are known in common,and for evolutionary game theory. The proofs involve new topological resultsasserting that spanning is preserved by relevant operations on set-valuedfunctions. Introduction
Conventionally with games payoffs are multilinear functions of the mixedstrategies, meaning that with finitely many pure strategies the payoffs canbe represented by multidimensional matrices. One can move away from amultilinear relationship between mixed strategies and payoffs and still geta Nash equilibrium. What counts is that the best reply correspondence ofeach player in mixed strategies is a convex set, allowing one to apply fixedpoint theory. If however the relationship between the mixed strategies ofa player and her payoffs is merely continuous, there may not be a Nashequilibrium. For example, if the payoff function for a player is convex in hermixed strategies, the best reply correspondence can generate disjoint sets,and the result may be no Nash equilibria.This lack of a Nash equilibrium can be rectified by expanding the conceptof what is a payoff for a player. Instead of associating a payoff to a mixedstrategy, one can associate a payoff to each pure strategy or action that theplayer can use. This broadening of what is a payoff allows one to define anew kind of equilibrium, called a myopic equilibrium. A myopic equilibriumis defined, see (Simon, Spie˙z and Toru´nczyk (2020)), to be a strategy profilesuch that each action used with positive probability by a player gives themaximum payoff possible from the use of any action of that player. In thecontext of multilinear payoffs, a myopic equilibrium is the same as a Nashequilibrium. But even when the payoff functions are concave, the myopicequilibria may be far from the Nash equilibria, as we will see in Example 1below.In the present paper, we consider parametrized relationships between themixed strategies and the payoffs, especially when that relationship satisfiesthe spanning property (Simon, Spie˙z, Toru´nczyk 2002). We show that if thepayoffs relative to a parametrization have the spanning property, then theresulting parametrized myopic equilibria also have the spanning property.This theory has multiple new applications.First, myopic equilibria are equivalent to equilibria used in evolutionarygame theory. The move away from a continuous relationship between mixedstrategies and payoffs should broaden the scope of evolutionary game theory.Second, with extensive form games, the relationship between mixed strate-gies and behaviour strategies can be seen in a new light. Behaviour strategiesare difficult to work with because they lack the multilinear relationship be-tween the strategies and payoffs. Mixed strategies are difficult to work with1ecause the dimension of the strategy space is much higher than with be-haviour strategies. The translation between behaviour strategies and mixedstrategies, accomplished by Kuhn (1953), allowed one to go between a strate-gic structure that was easier to comprehend (behaviour strateges) and onefor which a Nash equilibrium exists (mixed strategies). But with myopicequilibria and the spanning property, one can prove the existence of Nashequilibrium entirely through the behaviour strategies.Third, we can understand what is a subgame in a new way. Subgames arelinked closely to the concept of common knowledge, that the players know incommon that they have landed in the subgame. A problem arises if the subsetof starting points is not a single point; a single subgame is not defined, rathera collection of subgame parametrized by the probability distributions on thestarting points of this subset. But with the spanning property and myopicequilibria applied to this parametrization, we can understand a plurality ofstarting points as a kind of subgame.Fourth, we can prove the existence of equilibria for some new forms ofgames through induction, using the conservation of the spanning property inthe relationship between the strategies, payoffs, and equilibria.The following example demonstrates some of the uses of our theory.
Example 1:
Consider the following game of three players. Player One chooses betweentwo states, X and Y and Players Two and Three are not informed of theoutcome of this choice. Then Players Two and Three play a simultaneousmove zero-sum game between each other (so that this game is equivalent toa simultaneous move three person game). Player Two has the actions L and R and Player Three has the actions l and r . The payoff matrices for PlayersTwo and Three are the following:State X l rL R l rL R R and l in State X and L and r in State Y . For the Rl combination inState X she gets 1 + s for some very small s > Lr combination. What are the Nash equilibria of this game and how doesit relate to myopic equilibria and a parametrized collection of subgames?2e start with the strategies for Player One. Let p ∈ [0 ,
1] be the probabil-ity for choosing the state X . Assuming that Players Two and Three knowthis probability p , (though not assumed mathematically in the definition ofan equilibrium, this can be assumed in its application), their choices are in-formed by the payoff matrix created by the mixture of p times the matrix forState X plus 1 − p times the matrix for State Y : β − βα p + 1 11 − α − p – for the sake of analysis we added α and β , β for the probability that PlayerThree will choose l and α for the probability that Player Two will choose L .If p lies in the interior (0 ,
1) there are no optimal pure strategies for eitherplayer. Assuming indifference between the two actions, we get the formulas(1 + 8 p ) β + 1 − β = β + (1 − β )(9 − p ) and(1 + 8 p ) α + 1 − α = α + (1 − α )(9 − p ),which are symmetric in β and α and solve to β = α = 1 − p . Now considerPlayer One’s payoff. Her choice for X results in a payoff of (1 + s ) β (1 − α ) =(1 + s )(1 − p ) p . Her choice for Y results in a payoff of α (1 − β ) = (1 − p ) p .As long as p is not equal to either 0 or 1, the choice for X is superior. Itfollows that there are two types of equilibria in the three player game. Inone type, Player One chooses X with certainty, Player Three chooses r withcertainty, and Player Two chooses any value for α . In the other type, PlayerOne chooses Y with certainty, Player Three chooses l with certainty, andPlayer Two chooses any value for α . In both cases Player One receives apayoff of 0. Note however that if s = 0 then all values for p can be combinedwith the appropriate above strategies for players Two and Three to defineNash equilibria of the three player game.Now lets think of this situation as a one player game in which Player Onechooses some p ∈ [0 ,
1] (representing the probability for X ) and what followsis no longer part of that game but nevertheless the payoff to Player One mustcorrespond to what the other players would do in equilibrium. The choice for X , given p , gives a payoff of (1 + s )(1 − p ) p . The choice for Y , given p , gives apayoff of (1 − p ) p . This means that the choice for p yields an expected payoffof f ( p ) := (1 + s )(1 − p ) p + (1 − p ) p . To optimize the payoff we must takethe derivative f ′ ( p ) = 1 + (2 s − p − sp and set it to zero, which solves to3 = s − √ s + s s . For s very small the solution calls for p very close to , withan expected payoff very close to . Finding the Nash equilibrium of a oneplayer game is a problem of optimization and as long as the payoff functionis continuous there will be at least one optimal strategy, and if the payofffunction is strictly concave there will be a unique optimal strategy. Indeedthe payoff function of this game is strictly concave, as the second derivative of f is 2 s − − sp , which is always negative as long as s is less than 2. Howeverthe concept of Nash equilibrium is not the right one for understanding thethree person game broken into two parts, the first part being that of PlayerOne’s choice and the second part collection of games between Players Twoand Three. It corresponds to the situation where Player One can commitherself to a mixed strategy and maximizes the payoff accordingly. But whatif Player One “takes back control” to choose the action giving the highestpayoff? By doing so she will destroy the equilibrium property of the oneplayer game and after it is re-established through the myopic equilibriumconcept she will have a worse payoff, though one relevant to the three playergame. This has similarity to the myth of Odysseus and the Sirens, whereOdysseus optimizes by binding himself.Conventionally the game played between Players Two and Three is not con-sidered a subgame because it does not involve common knowledge of a singlestate, even though the players do know in common that their choices pertainto a subset of size two. This “subgame” is really a continuum of subgames,one for each probability distribution on this subset of size 2.We generalize the concept of subgame to subsets of vertices in a game treeknown in common, and call it a subgame bundle . These subsets are closedwith respect to actions and the knowledge of the players. Some requirementof common knowledge is needed on the subsets that can define a subgamebundle, because otherwise a player could be forced to play the same in twolocations belonging to different subgames.A problem with generalizing the concept of a subgame to subsets is thatthe set of equilibria, as a function of the probability distributions on thesubset of starting positions, in general cannot be approximated by continuousfunctions. The above Example 1 used a zero-sum game played between theremaining players, whose equilibria as a function of distributions will be anupper-semi-continous and non-empty convex valued corrspondence. Howeverin the more general context of non-zero-sum games the equlibrium payoff4orrespondence will not have this nice property.We know from the Structure Theorem of Kohlberg and Mertens (1986) thatthere will be a topological structure to the equilibrium correspondence, ahomotopic relation that implies the spanning property. There are two ad-vantages to working with the spanning property over the homotopic propertyof Kohlberg and Mertens. First the spanning property is more general. Withrepeated games of incomplete information on one side the equilibrium payoffcorrespondence as a function of the probability distributions has the spanningproperty (Simon, Spie˙z, Toru´nczyk 2002), but there is no reason to believethat it has the homotopic property. Second, there is no indication yet thatthe homotopic property is robust with respect to composition, though weprove that the spanning property has such a robustness.The myopic equilibrium concept is not entirely new. The mathematical struc-ture behind a myopic equilibrium is identical to that of a “Nash equilibriaof population densities” from evolutionary game theory. Though the mathe-matical formalities are the same, the concepts are very different. With evolu-tionary game theory, there is a continuum of animals belonging to a species.Each species has a variety of types and a distribution of types is equivalentmathematically to a mixed strategy. What we call a myopic equilibrium is inthe context of evolutionary game theory a kind of Nash equilibrium becauseeach individual animal seeks to maximize a utility independent of the speciesas a whole. The distribution of types will influence the payoffs for each type,and that introduces a potentially non-affine structure to those payoffs.The rest of this paper is organized as follows. In the second section we pro-vide the necessary topological apparatus and prove new results about thespanning property, giving conditions under which it is preserved by opera-tions on correspondences like taking intersections, taking cartesian productsor taking weighted sums. In the third section we define “game bushes”,“game bundles”, “subgame bushes”, and “subgame bundles” and use resultsfrom § The spanning property of correspondences
Let X and Y be metrizable spaces. By a correspondence F : X → Y wemean here any compact subset of X × Y . Hence the same correspondencemay also be denoted as F ⊂ X × Y and we often switch from one notationto the other; however, the use of F : X → Y displays the asymmetric rolewhich X and Y play below. Given such an F and a subset W of X we call F ∩ ( W × Y ) the restriction of F : X → Y to W and denote it F | W . Theimage of F | W under the projection to Y is denoted by F ( W ). We also write F ( x ) := F ( { x } ) for x ∈ X and dom( F ) := { x ∈ X : F ( x ) = ∅} , the domain of F . The usual challenge we encounter is to show that a given point x ∈ X is in dom( F ), i.e, that F ( x ) = ∅ .It is convenient to note that if F : X → Y and G : Y → Z are cor-respondences, then the formula H ( x ) = G ( F ( x )) defines a correspondence,i.e., S x ∈ X { x } × H ( x ) is a compact subset of X × Z . We write G ◦ F for H .Until Theorem 1 below we let M be a compact, connected d -manifold withboundary δM (possibly empty) and W be a compact subset of M . By δW we denote the boundary of the interior of W with respect to M . Importantto us is that there exists a well-defined element of the ˇCech homology group H d ( W, δW ; Z ), denoted here by [ W ], with the following properties: Fact 1. a) If W is a compact, connected d –manifold then [ W ] is the Z –fundamental class of W , the unique non-zero element of the ˇCech homologygroup H d ( W, δW ; Z ) .b) With ∂ [ W ] ∈ H ( δW ; Z ) standing for the image of [ W ] under theboundary operator, one has that the image of ∂ [ W ] in H ( W \ { w } ; Z ) underthe inclusion–induced map is non-zero, for each point w in the interior of W . A proof with M a sphere is given in (Simon, Spie˙z, Toru´nczyk 2002);using the orientablity of M over Z a generalization causes no difficulty. Ofcourse, [ W ] depends on the ambient manifold M , e.g. it is null when W is adisk embedded in the sphere S , but non-null if this disk is being consideredas a subset of S .We say that a correspondence F : W → Y has property S if [ W ] liesin the image of the projection–induced homomorphism H ( F, F | δW ; Z ) → H ( W, δW ; Z ) of the relative ˇCech homology groups with Z –coefficients.If F is given as, say, F ⊂ W × Y × Y and we wish to consider it as acorrespondence from W to Y × Y (rather than e.g. from Y to W × Y ),6hen above we say that F has property S for W , and similarly when there aremore or two factors. Here, S is an abbreviation for ”spanning”, addressingto Fact 2a) below: Fact 2. a) If F : W → Y has property S then the domain of F contains theclosure of the interior of W relative to the ambient manifold M .b) If W ′ ⊂ W are compact subsets of M and F : W → Y has property S then F | W ′ hast it either.c) Let W and W ′ be compact subsets of M and let correspondences F : W → Y and F ′ : W ′ → Y coincide on W ∩ W ′ . If F ′ takes values insingletons only, then F ∪ F ′ has property S for W ∪ W ′ .d) Suppose a correspondence F : W → Y is such that for each neighbour-hood U of δW in M and each neighborhood V of F in W × Y there exists acompact set G ⊂ V with property S for a compact set D satisfying δD ⊂ U and D ⊂ W ∪ U . Then, F has property S for W . Claim a) above follows from Fact 1b), and for the other ones see (Simon,Spie˙z, Toru´nczyk 2002) (again, with M a sphere). Theorem 1.
Let
W, X, Y be compact manifolds and
F, G ⊂ W × X × Y be correspondences with property S for W × X and for W × Y , respectively,such that the projections F → Y and G → X are null–homotopic. Then, F ∩ G has property S for W .Proof. W, X, Y has a boundary. Let α ∈ H ( F )be a homology class mapped to [ W × X ] under the homomorphism inducedby the projection to W × X . The projection of F to Y being homotopic to aconstant (say, y o ) one has α = [ W × X × { y o } ] in H ( W × X × Y ) – by whichwe mean that the images of these two classes under the inclusion-inducedhomomorphisms coincide.Similarly, there exists a class β ∈ H ( G ) mapped to [ W × Y ] under thehomomorphism induced by the projection of G to W × Y and such that forsome x o ∈ X one has β = [ W × { x o } × Y ] in H ( W × X × Y ).Below, we’ll be using the properties of the ”intersection pairing” describedin (Dold, 1972, § VIII.13). So, there is a well-defined element α • β of H ( F ∩ G ).In H ( W × X × Y ) this element is equal to [ W × X × { y o } ] • [ W × { x o } × Y ], bywhat said earlier. The last product is however equal to [ W × { ( x o , y o ) } ], forthe manifolds X × { y o } and { x o } × Y intersect transversally at ( x o , y o ). Andsince [ W × { ( x o , y o ) } ] is mapped to [ W ] under the homomorphism inducedby the projection to W , so is α • β . Thus F ∩ G has property S for W .7). Let now δW = ∅ and δY = ∅ . We then consider Y as a subset of e Y := Y ∪ δY Y ′ , a union of two copies Y and Y ′ of Y intersecting along δY .Let h be a homeomorphism of Y onto Y ′ which restricts to identity on ∂Y ,and let G ′ := (id W × X × h )( G ) and e G := G ∪ G ′ .We first prove that e G has property S for W × e Y . To this end we takean α ∈ H ( G, G | ( W × δY )) which witnesses property S of G (i.e., is mappedto [ W × Y ] by the projection–induced homomorphism), set α ′ := h ∗ ( α ) andlet e α ∈ H ( e G, e G | ( ∂W × Y ); Z ) be the image of α ⊕ α ′ in the relative Mayer-Vietoris sequence for ( e G, e G | ( ∂W × Y )) , ( e G ′ , e G ′ | ( ∂W × Y ′ )). We readily seethat ∂ e α = 0, so e α can be considered as an element of H ( e G ). Also, theprojection-induced homomorphism maps e α to the image of [ W × Y ] ⊕ [ W × Y ′ ]under an analogous Mayer–Vietoris sequence, and this image is equal to[ W × e Y ]. This shows that e G indeed has property S for W × e Y . (We skippeda technical point dealing with why are all the pairs involved excisive.)Moreover, the projection of e G to X factors through the projection G → X and hence is null-homotopic. The analogous property of the projection F → e Y is even more obvious. Since δ e Y = ∅ , if additionally δX = ∅ then F ∩ e G has property S for W , by 1) above. However, F ∩ e G = F ∩ G and thusthe assertion is true if δW = δX = ∅ (or if δW = δY = ∅ , by symmetry).3). The assertion also holds true if both X and Y do have a boundarybut W doesn’t. This is so because then the reasoning of 2) still appliesonce the reference to 1) gets replaced by one to the conclusion of 2). (Noadditional assumption on X is being made this time.)4). Finally, if δW = ∅ then we replace W by f W := W ∪ δW W , and F and G by e F := F ◦ ( q × id X × Y ) and e G := G ◦ ( q × id X × Y ) respectively, where q : f W → W is a retraction. As in 2) above we infer that e F has property S for f W × X and e G has it for f W × Y . Hence, the correspondence e F ∩ e G hasproperty S for f W , by 1) to 3) above. Since on W it restricts to F ∩ G , thedesired conclusion follows from Fact 2b). (cid:3) Addendum to Theorem 1.
The conclusion of Theorem 1 holds also trueif X and Y are as before and W is any compact subset of a compact, connectedambient manifold M that is PL or is of dimension neither 4 nor 5. Proof. The interior of W is the union of an increasing sequence ( W n ) of com-pact manifolds with boundaries; see (Kirby and Siebenmann (1977), p. 108).By Theorem 1 and Fact 2b), for each n the appropriate restriction of F ∩ G has property S for W n , whence Fact 2d) applies. (cid:3) heorem 2. Let
W, X, Y be as in Theorem 1 or in the Addendum to it.Furthemore, let
Φ : W → X and Ψ : X → Y be correspondences withproperty S such that the projections Φ → X and Ψ → Y are null–homotopic.Then, Ψ ◦ Φ : W → Y has property S .Proof. Let F := Φ × Y and G := W × Ψ. Then
F, G ⊂ W × X × Y arecorrespondences with property S for W × Y and for W × X , respectively,and such that the projections F → X and G → Y are null–homotopic.We have F ∩ G = { ( w, x, y ) | w ∈ W, x ∈ Φ( w ) , y ∈ Ψ( x ) } and theprojection of F ∩ G to W is equal to the projection of F ∩ G along X toΨ ∩ Φ, followed by the projection of Ψ ∩ Φ to W . Since F ∩ G has property S for W (by Theorem 1 and the Addendum), so does Ψ ∩ Φ. (cid:3) Remark.
The assumption in Theorems 1 and 2 that the correspondingmaps be null-homotopic (i.e., homotopic to constant ones) is automaticallysatisfied if X and Y are simplices, as in the applications in § § Theorem 3.
Let W be a compact top–dimensional subset of a PL-manifoldand F i : W → Y i ( i = 1 , . . . , l ) be correspondences with property S . Then,this property is also enjoyed by the correspondence F : W → Q i Y i , definedby the formula F ( x ) = Q i F i ( x ) .Proof. We may assume all the Y i ’s to be normed linear spaces (for they canbe embedded into such). For each i let V i be an open neighborhood of F i in X × Y i . We assume this neighbourhood is triangulated consistently with thePL-structure of W × prod i Y i . By the spanning property of F i , for d = dim( W )there exists a d –chain z i in V i , with boundary in δW × Y i and such that itsclass is mapped to [ W ] by the projection–induced homomorphism.By general position we may assume that with respect to some triangula-tion T of W each d -simplex of z i projects injectively onto some simplex of T .For each tuple ( σ , . . . , σ l ) of d -simplexes σ i ∈ z i which project onto thesame simplex of T we define a ”diagonal” k -simplex in X × Q i Y i by :∆( σ , . . . , σ l ) := { ( x, y , . . . , y l ) | ( x, y i ) ∈ σ i for each i } . Then, we let z denote the chain in X × Q i V i which is the sum of ∆( σ , . . . , σ l )’sover all tuples ( σ i ) li =1 as above. One can observe that z is a d -chain in( W, ∂W ) × Q i V i and its class is mapped to [ W ] by the projection–inducedhomomorphism . Thus, F has the spanning property by Fact 2d). (cid:3) orollary 1. In Theorem 3 assume that Y i = R A for each i , where A isa finite set. Then, P i F i : W → R A defined by the formula ( P i F i )( x ) = { y + · · · + y l | y i ∈ F i ( x ) for all i } is a correspondence with property S .Proof. Let Y ⊂ ( R A ) l be a closed ball containing F ( W ), where F ( x ) = Q i F i ( x ) ⊂ ( R A ) l for x ∈ W . Obviously, P i F i = f ◦ F , where f ( y ) = P i y i for y = ( y i ) li =1 ∈ Y . Hence, it remains to apply Theorems 2 and 3. (cid:3) Given F ⊂ X × R A and a function λ : X → R we define λF ⊂ X × R A bythe formula: ( λF )( x ) = { λ ( x ) y : y ∈ F ( x ) } for x ∈ X. Corollary 2.
Let W be a compact PL–manifold and for i = 1 , , . . . , l let itbe given a continuous function λ i : W → R , a function f i : W → Y i which iscontinuous off λ − i (0) , and a correspondence G i : Y i → R A with property S .Then the correspondence P i λ i ( G i ◦ f i ) has property S .Proof. By Corollary 1 it suffices to consider the case where l = 1; we hencewrite λ, f and G for λ , f and G , respectively. We may assume that λ = 0,say sup | λ | ( W ) >
1. With K n := { x ∈ X | | λ ( x ) | > /n } it follows fromTheorem 2 and Fact 2b) that G ◦ f | K n has property S , whence H n := λG ◦ f | K n has it either. Let K and H denote the closures of S n K n and of S n H n , respectively. Using boundedness of G ( W ) we note that H is compactand H = ( λG ◦ f ) | K ; also, from Fact 2d) we infer easily that H has property S . Moreover ( λG ◦ f )( x ) = { } for x off the interior of K and so the property S of λG follows from Fact 2c). (cid:3) For the last results of this section we introduce notation which is relevantalso to the further sections. That is, if A is a finite set then we let ∆( A )denote the ( | A | − A :∆( A ) := { f : A → [0 , | X a ∈ A f ( a ) = 1 } and we consider it being embedded in the | A | –space R A of all functions from A to R , equipped with the norm k f k := ( P a | f ( a ) | ) / .Now, ( A i ) i ∈ N be a partition of A into non-empty subsets. Correspondingto it we let ∆ := Q i ∈ N ∆( A i ), considered as a subset of Q i ∈ N R A i = R A . For10 ∈ A and an element v of R A (and hence also for v ∈ ∆) we denote by v a the a -th coordinate of v . Theorem 4.
Let F : W × ∆ → R A be a correspondence with property S ,where W is compact manifold or is as in the Addendum to Theorem 1. Letfurther D ⊂ ∆ × R A denote the set of all pairs ( σ, v ) such thatfor all i ∈ N and a ∈ A i , if σ a > then v a = max { v b : b ∈ A i } . (1) Then, F ∩ ( W × D ) has property S for W . Corollary 3.
With W and F as above define E : W → R N by ( w, r ) ∈ E ifand only if for some ( w, σ, v ) ∈ F one has both (1) above and r i = max a ∈ A i v a for each i ∈ N . Then, E has property S .Proof. Theorems 2 and 4 do the job, because E = f ◦ ( F ∩ ( W × D )) where f : R A → R N is defined by the formula f ( v ) = (max { v a | a ∈ A i } ) i ∈ N . (cid:3) Proof of Theorem 4.
Let G denote the closure of D in ∆ × S , where S isthe sphere R A ∪ {∞} . By Theorem 1 and its Addendum, combined with thecontractibility of ∆ and of R A , it suffices to prove that G has property S for S .To this end let us recall that there exists a retraction r : R A → ∆ suchthat, for σ ∈ ∆ and v ∈ R A , condition (1) holds true if and only if r ( σ + v ) = σ . (See (Simon, Spie˙z, Toru´nczyk (2020), Lemma 1). This r is the product ofnearest–point retractions r i : R A i → ∆( A i ) , i ∈ N .) The formulas φ ( σ, v ) = σ + v and ψ ( v ) = ( r ( v ) , v − r ( v )) hence define mutually inverse mappings ψ : R A → D and φ : D → R A . It follows that φ : D → R A is a homeomorphismwhich is at finite distance from the projection ( σ, v ) v . The straight-linehomotopy between these two mappings is thus proper, and accordingly theprojection p : D ∪ {∞} → S is homotopic to the homeomorphism φ ∪ id {∞} .However, the projection of G to S is equal to the composition p ◦ q , where q : G → D ∪ {∞} is a map that squeezes the set G ∞ := G ∩ (∆ × {∞} )to {∞} and is the identity elsewhere. It is readily seen that G ∞ = ∆ × {∞} ,whence q induces a homology–isomorphism by Vietoris’ theorem. Since sodid p , also p ◦ q induces such an isomorphism. (cid:3) We have to modify the concept of a finite game tree so that the terminalpoints of the game contain variable payoffs and also there could be a variety11f locations where the game starts. Without first stipulating the payoffs,we call this modification a game bush . Some may prefer the term “gameforest”. But our structure involves an informational interaction betweenthe different starting points (roots) and terminal points (leaves); it is notmerely an independent collection of games. After we add a payoff structure,it becomes a “game bundle”. In Simon, Spie˙z, and Toru´nczyk (2020) theconcept of a “truncated game tree” was introduced, a game tree with a uniquestarting point and an informational structure to the terminal points. Howeverwe prefer a “game bush” to a “truncated game bundle” and prefer a “subgamebush” to a “truncated subgame bundle”. Except for the terminology, theconcepts follow in parallel those of that article.Let there be a finite set N of players. There is a finite directed graph ( V, V )(arrows between vertices) such that the directed graph is acyclic without theorientation of the edges (meaning that for every vertex there is only one pathleading to that vertex from the roots. The subset R of initial vertices is calledthe roots. T is the set of terminal points and every path of arrows starts ata root and ends at a terminal point, with each terminal point determininga unique such path of arrows. We allow T and R to have a non-emptyintersection, meaning that in part of the game bush the game could startand end simultaneously. The set D of nodes is the subset V \ T and theseare the vertices (except for the roots) to which comes exactly one arrow andfrom which, without loss of generality, come at least two arrows. Toward aroot, there is no arrow, and from a root comes at least two arrows.For each player n ∈ N there is a subset D n ⊆ D such that ∀ i = n D i ∩ D n = ∅ .Define D to be the set D \ ( ∪ n ∈ N D n ). Also for every n ∈ N there is apartition P n of D n . For every W ∈ P n there is a set of actions A nW such thatthere is a bijective relationship between A nW and the arrows leaving every v ∈ W . For every v ∈ D there is a probability distribution p v on the arrowsleaving the node v , and therefore also on the nodes following directly after v in the tree. We assume without loss of generality that p v gives positiveprobability to every arrow leaving v . The subset D is where Nature makesa choice. Nature is not a player because it has no payoffs and its actions areinvoluntary and randomized rather than choices in the usual sense.At any node v ∈ W ∈ P n only the player n is making any decision, andthis decision determines completely which vertex follows v . At the nodes v in V nature is making a random choice, according to p v , concerning which12ertex follows v . If the game is at the node v ∈ D n and v ∈ W ∈ P n thenPlayer n is informed that the node is in the set W and that player has noadditional information, so that inside W player n cannot distinguish betweennodes within W . A set W ∈ P n is called an information set of Player n .The inspiration behind its definition is that a game bush could be part of alarger game, either at the start, the end, or in the middle. With conventionalgame trees, there is only one root. The introduction of multiple roots createsnew problems. As a game bush could represent a kind of subgame, informa-tion may be inherited from previous play. Therefore for each player n ∈ N we require that there is a partition R n of the roots R representing what eachplayer knows at the start of the game bush. And then to define the payoffsand potentially to connect the game bush to further play in subsequent gamebushes, we require for each player n ∈ N that there is a partition Q n of theterminal points.Now we extend the definition of a game bush to that of a game bundle.Define Q to be the meet partition ∧ n ∈ N Q n , the unique finest partition suchthat for every n ∈ N every member of Q n is contained in some member of Q . For any C ∈ Q we assume there is a correspondence F C ⊆ ∆( C ) × R C × N of continuation payoffs. The correspondences F C for all C ∈ Q together withthe game bush define a game bundle. The partition Q represents commonknowledge in the end ponts. We believe that the payoffs must be definedby such a partition. A non-empty overlap of different subsets could createan unresolvable ambiguity concerning what are the payoffs. And an overlapbetween an information set and a set defining the subgame would createproblems for the subgame concept.A game bundle is not a single game, rather a collection of games parametrizedby the distributions on the set R of roots. Furthermore, because the payoffsare defined by correspondences, there could be different payoff consequencesfor the same set of actions and ending at the same terminal point (or indeednone where the correspondence may be empty).A game tree, the conventional game in extensive form, is a game bundlesuch that the set R (of roots) is a singleton, Q is the discrete partition on T (meaning the collection {{ t }| t ∈ T } ) and the correspondence F t for every t ∈ T is defined by a singleton { ( δ t , r ) } for some payoff r ∈ R N (where δ r isthe Kroniker delta). 13or every player n ∈ N let S n be the finite set of pure decision strategies ofthe players in the game bush, by which we mean a function that decides, atevery set W in P n deterministically which member of A nW should be chosen.If each such A nW has cardinality l and there are k such sets in P n then thecardinality of S n is l k .For our purposes, the set ∆ := Q n ∈ N ∆( S n ) will be the set of strategies, whatare called the mixed strategies. A behaviour strategy is a choice of a pointin ∆( A nW ) for each choice of n ∈ N and W ∈ P n . Later we will establishnew relations between these different types of strategies through subgamebundles and myopic equilibria.For every q ∈ ∆( R ), σ ∈ ∆ := Q n ∈ N ∆( S n ), and C ∈ Q let p q,σ ( C ) be theprobability of reaching C through σ and q ∈ ∆( R ). If this probability ispositive, define P q,σ ( ·| C ) to be the conditional probability on C induced by q and σ .Given a game bundle with q ∈ ∆( R ) and σ ∈ ∆ define a plan φ for ( q, σ )to be a choice of y ∈ R N × T such that y N × C ∈ F C ( p q,σ ( ·| C )) if p q,σ ( C ) > y N × C ∈ F C ( w ) for some w ∈ ∆( C ). As a priori the payoffcorrespondence may fail to be a non-empty, there may be no plan.Given a plan φ for ( q, σ ), a player n ∈ N and a choice s ∈ S n define f nφ ( s ) to be the expected value of y n,e as determined by the q ∈ ∆( R ) andtransitions determined by the s and the ( σ j | j = n ), whereby the conditionalprobabilities p q,σ are still determined by the entire σ = ( σ j | j ∈ N ). A plan φ for ( q, σ ) is an m-equilibrium if for all s ∈ S n with σ n ( s ) > f nφ ( s ) = max t ∈ S n f nφ ( t ). The payoff of a plan φ is a vector in R N suchthat the n coordinate is the expected value of of the f nφ ( s ), which for anm-equilibrium is the common value for the f nφ ( s ) for all those s with positiveprobability.Next we show that the correspondence of m-equilibria has a certain specialproperty as long as the correspondences defining the continuation payoffshave that same special property.The next theorem applies to the context of a game bundle defined by apartition Q and correspondences F C for every C ∈ Q . Theorem 5.
If for every C ∈ Q the correspondence F C ⊆ ∆( C ) × R C × N has the spanning property then the correspondence defined by the payoffs ofm-equilibria has the spanning property with respect to ∆( R ) . emark: We follow approximately a previous argument of Simon, Spie˙z,and Toru´nczyk (2020).
Proof:
Let ǫ > B be a positive quantity larger than anypayoff from the correspondences F C . If p q,σ ( C ), the probability of reaching C with q and σ , is at least 2 ǫ then define λ q,σ ( C ) = 1. If p q,σ ( C ) ≤ ǫ then λ q,σ ( C ) = 0. And if ǫ < p q,σ ( C ) < ǫ then let λ q,σ ( C ) = p q,σ ( C ) − ǫǫ .For every choice of ǫ > C ∈ Q define the correspondence F ǫ ⊆ ∆( R ) × ∆ × R N × T by F ǫ ( q, σ ) N × C = λ q,σ F C ( q, p q,σ ( ·| C ) + (1 − λ q,σ ( C )) { r B } , where r B is the point whose entry is B for all choices. Due to Corollary 2 thecorrespondence F ǫ has the spanning property. Due to Corollary 3 (with∆( R ) taking the role of W in Corollary 3), the correspondence G ǫ defined bythe myopic equilibria from F ǫ plans has the spanning property. And becausethe spanning property is defined using ˇCech homology, the cluster limit G ofthe G ǫ for any sequence of ǫ going to zero also has the spanning property.Now consider any payoff obtained from G ; we need to show that it comes froman m-equilibrium. Suppose for a given ( q, σ ) that y N × A is from a sequenceof the F ǫ . As there are fixed values for the minimal non-zero probability forreaching any terminal point e ∈ T , and the sequence of ǫ went to zero, itfollows for any C ∈ Q given positive probability by q and σ that λ < G ǫ for some sufficiently small ǫ .And if C ∈ Q is given zero probability by the q and σ , due to the largepositive B , any cluster of vectors from the correspondence F C can be usedfor the event that some terminal point in C is reached (and if there is nosuch cluster, meaning that eventually only B was used in the definition of the G ǫ , then any vecctor from F C would do). Given the inequalities defining themyopic equilibria of G , after removing the use of B those inequalities remainto show that G is a subset of the myopic equilibria of the game bundle. (cid:3) We extend the definition of a subgame from that defined on a single state toone defined on a subset of possible states.
Definition:
A subset S of the vertices V of a game bush defines a subgamebush on the vertices S if and only if(1) one cannot leave S through any choice of action of a player or a choice15f nature (meaning closure by arrows),(2) for every player n ∈ N and every W ∈ P n or W ∈ Q n either W iscontained in S or is disjoint from S .At first the condition with Q n may seem unnecessary. But if we dropped thiscondition then every subset of T would define a subgame bush. That wouldcreate a problem if the game bush is followed by further play.If the vertices S define a subgame bush, let R ′ be the roots of the subgamebush (the vertices in S that are in R or whose predecessor is not in S ) and T ′ is the subset of terminal vertices T ′ = T ∩ S . The partitions Q ′ n of T ′ aredefined by Q ′ n = { F ∩ S | F ∈ Q n } which by the definition of a subgamebush are equal to { F | F ∈ Q n , F ⊆ T ′ } . Likewise define Q ′ to be the meetpartition ∧ n ∈ N Q ′ n . The partition R ′ n of R ′ is defined by u, v belonging to thesame member of R ′ n if and only if u and v share the same last member of P n in the paths to u and v , and otherwise, if no such member of P n exists, theyshare the same member of R n in the paths to u and v .Given that additionally a game bundle Γ is defined, meaning with correspon-dences F C for every C ∈ Q , another game bundle is defined by the subgamebush defined by S and the same correspondences F C for the sets C ∈ Q ,since Q ′ is just a subset of Q . The resulting subgame game bundle we callΓ | S , and we see that it is also a game bundle.Notice that it is possible for a set S defining a subgame bush to have a non-empty intersection with the roots R . When this happens, the roots R ′ of thesubgame bush have a non-empty intersection with R . Also it is possible for R ′ to have a non-empty intersection with T (and therefore also with T ′ ).For every S that defines a subgame bush, we define also a factor game bush.Its roots R ′′ are the same as R and its partitions R ′′ n are the same as R n .Its terminal points T ′′ are defined to be T ′′ := ( T \ S ) ∪ R ′ , where R ′ are theroots of the subgame bush defined by S . The partition Q ′′ n of T ′′ are definedby { A | A ∈ Q n , A ∩ S = ∅} ∪ R ′ n , (where R ′ n is the partition on the roots ofthe subgame bush). Because S is containing or disjoint from every memberof Q n , the definition of Q ′′ n is straightforward. We define Q ′′ to be ∧ n ∈ N Q ′′ n . Definition:
A game bush has perfect recall for every player n if for every v ∈ W ∈ P n the member of R n followed by the information sets in P n and actions taken by player n leading to v ∈ W has no repetitions and thissequence is the same for every v ∈ W . The game bush has perfect recall if it16s has perfect recall for every player.Next we would like to define a factor game bundle. If C ∈ Q ′′ ∩ Q is disjointfrom S , we don’t change the payoff correspondence from the previous defini-tion of F C . But we need to know what should be correspondence F C when C ∈ Q ′′ is a subset of R ′ , the roots of the subgame bundle. We would liketo say that C also defines a subgame bush, but unfortunately we cannot dothat without the assumption of perfect recall. It makes sense to define F C tobe some subset of the m-equilibria of the subgame bundle restricted to ∆( C )as a subset of ∆( R ′ ).If a set S of vertices defines a subgame bundle then an m-equilibrium forsome q ∈ ∆( R ) is S -perfect if restricted to the subgame bundle defined by S it defines an m-equilibrium for some distribution on the roots R ′ of S such that when there is positive probability of reaching S that distributionis the conditional probability as implied by the strategies and the initialdistribution q .An m-equilibrium is subgame bundle perfect if it is S -perfect for all sub-sets S of vertices that define subgame bundles.Unless otherwise stated, for the factor game bundle Γ /S we define thecorrespondence F C for every C ∈ Q ′′ with C ⊆ R ′ to be the subgame bundleperfect m-equilibria of the subgame Γ | S as restricted to ∆( C ). If there issome ambiguity concerning what should be F C , we can denote it by Γ F /S ,where F stands for alternative corespondences. Lemma 1.
If the sets S and T define subgame bushes then their union S ∪ T and their intersection S ∩ T also define subgame bushes. Proof:
It follows directly from the definition of subgame bushes as sets thatare closed with respect to information sets, partitions, and consequences ofactions. (cid:3)
In general, it may be impossible for a player to combine her strategies fromΓ /S followed by Γ S in the way desired, as this player may not possess therequired memory to do this. If we assumed that these were two distinctplayers, one playing in the factor game and a different one playing in thesubgame bundle, such a combination would not be problematic. But a playerperforming in both the factor game and the subgame may fail to be capableof performing the necessary mixed strategy in the subgame due to a lack of17emory of what happened in the factor game. To go further, we need theproperty of perfect recall. Lemma 2:
Let Γ be a game bundle, S a set defining the subgame bundle Γ | S of G , with R ′ the roots of the subgame defined by S . Assume that the gamebush of Γ has perfect recall. Let φ be an m-equilibrium of the factor game Γ /S and q be its induced conditional probability on R ′ , the roots of S . Theplan φ combined with any m-equilibria of Γ | S corresponding to the conditionalprobability q of R ′ (with q equal to anything if the probability of reaching S is zero) defines an m-equilibrium of Γ . Proof:
Let S n and S n be the decision functions for Γ /S and Γ S respectively.Let s be a member of S n and P an information set of Player n in the subgamebundle Γ S that is reached with positive probability. If r is the expected payofffor Player n from s conditioned on reaching P in the subgame bundle fromthe strategies used in the factor game Γ /S , we need to know that r is also theexpected payoff from the combination of s with s conditioned on reaching P . This follows from the property of perfect recall, because the distributionon P from using s is not any different from the distribution on P used tocalculate the expected payoff r in the subgame bundle. Theorem 6:
If there is perfect recall, the correspondence of subgame bundleperfect m-equilibria has the spanning property, meaning that for every intialprobability distribution on the roots there is a subgame bundle perfect m-equilibrium.
Proof:
The proof is by induction on the partially ordered structure of sub-game bundles. If there are no subgame bundles, it is true by Theorem 1.Otherwise let S be any subset of vertices defining a subgame. By inductionwe assume that the correspondence of subgame bundle perfect m-equilibriaof both G | S and G/S have the spanning property. Let U be a set of verticesdefining a subgame. If U is contained in S , we have already assumed theconclusion. If U is not contained in S , then by Lemma 1 G | U is a combina-tion of G | U /S and G | U ∩ S . By Lemma 2 and Theorem 2 the subgame perfectm-equilibria of G | U has the spanning property. (cid:3) Further problems
The main advantage of mixed strategies over behaviour strategies is that onecan define the relationship between strategies and payoffs in a straightforwardway, through multi-dimensional matrices. However the dimensions of thestrategy spaces can become very large, and we would like to reduce thesedimensions.
Definition:
An extensive form game is solvable if there is a sequence ∅ = S ⊆ · · · ⊆ S n = V such that for each game G | S i /S i − no player is called uponto make a choice of action twice (meaning that there are no two P , P ∈ P n such that P follows from P in some path).In a solvable game, in each factor games G | S i /S i − there is no distinctionbetween behaviour strategies and pure strategies. Therefore one can under-stand equilibria of the game entirely through behaviour strategies using theabove theorems. The question however is whether there is any computationaladvantage won through this perspective. So far, the only clear advantage per-tains to standard games of incomplete information on one side where a finitegame is followed by an infinitely repeated one, as is done in Simon, Spie˙z,and Toru´nczyk (2020). There we proved that there exists an equilibriumand if the payoffs from both games come from the same matrices and thepayoff is a combination of the discounted and undiscounted, we showed that ǫ -equilibria exist for every positive ǫ . With the above Theorem 5 one couldhope to improve this result to the existence of an equilibrium.The following examples demonstrate some of the complexity of what is asubgame and what constitutes subgame perfection. Example 2:
There are two players. Player One has the choice between A for aggressionand P for passivity. If P is chosen the game moves to State X and the payoffsare 1 for Player One and 2 for Player Two. If A was chosen the game movesto State Y and Player Two has the choice between a for aggression and p forpassivity. If a is chosen the payoff is 0 for both players and if p is chosen thepayoff is 2 for Player One and 1 for Player Two.There is an equilibrium where Player Two chooses a with certainty and PlayerOne choose P with certainty. Player One is afraid of getting 0, hence willingto choose P . And because Player One chooses P with certainty, the “foolish-19ess” of Player Two choosing a never upsets the equilibrium property. Thisis the model of “mutually assured destruction”, that a player promises toharm all players in the game and that threat works to keep that player fromever confronting the situation where that threat must be carried out. On theother hand the state Y defines a subgame whose only equilibrium involvesPlayer Two choosing p , hence also Player One choosing A , defining a secondand very different equilibrium. We recongnize the first equilibrium does notinduce an equilibrium of the subgame defined by the state Y , but the secondequilibrium does. The first equilibrium is subgame perfect, the second is not.But before we discared the second equilibrium as irrelevant, notice that itgives a superior payoff to Player One, the player who is performing the inte-rior action in the subgame. Example 3:
Now we alter this game slightly so that Player Two cannotdistinguish between State X and State Y and has the actions a and p also inState X and the payoff consequences of both a and p from State X are thesame as before, a payoff of 2 to Player One and 1 to Player Two. Conven-tionally, this game has no subgames, so every equilibrium is subgame perfect.In some way it is strategically equivalent to the previous game, as it can berepresented in normal form by the same matrix p aP (1 ,
2) (1 , A (2 ,
1) (0 , . We perceive X and Y together as a subset defining a subgame bundle, witha certainty that the play will reach this subgame bundle. Both equilibria dis-covered in the previous example, A with p and P with a , induce equilibria ofthe subgame bundle defined by the set { X, Y } . We introduced a new collec-tion of subgames, and yet the A and p combination cannot be eliminated asa subgame bundle perfect equilibrium. This shows that the subgame bundleperfect concept is not determined exclusively by the normal, or matrix, formof the game.One would like a concept of subgame bundle perfection that incorporatesthe most likely probability distribution on a subset C ∈ Q given that theprobability for C is zero. With probability theory, this question motivatesthe concept of a random variable representing conditional probability withthe conditioned set has zero probability. However with games, especially if20he game tree is finite, this question cannot be divorced from the relatedquestion of who deviated to bring the play to this forbidden subset and why.Unfortunately hitherto there is no good general answer to this question. Dold, A. (1972),
Lectures on Algebraic Topology , Springer Verlag.Kohlberg, E. and Mertens, J.-F. (1986), On the Strategic Stability of Equi-libria,
Econometrica ,
54 (5) , pp. 1003 –1037.Kirby, R. C. and Siebenmann, L. C. (1977),
Foundational essays on topolog-ical manifolds, smoothings, and triangulations , Annals of MathematicsStudies 88 (Princeton University Press)Kuhn, H. (1953), Extensive Games and the Problem of Information, in
Contributions to the Theory of Games I , Princeton University Press,eds. Kuhn and Tucker, pp. 193-216.Simon, R.S., Spie˙z, S., Toru´nczyk, H. (2002), Equilibrium Existence andTopology in Games of Incomplete Information on One Side,
Transac-tions of the American Mathematical Society , Vol. 354, No. 12, pp.5005-5026.Simon, R.S., Spie˙z, S., Toru´nczyk, H. (2020), Games of Incomplete Informa-tion and Myopic Equilibria,