Causal Discourse in a Game of Incomplete Information

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1 Causal Discourse in a Game of Incomplete Information Halbert White Department of Economics University of California, San Diego Haiqing Xu Department of Economics Penn State University Abstract Notions of cause and e ect are central to economics. Despite the immediate intuitive content of price e ects, income e ects, and the like, rigorous foundations justifying well-posed discussions of cause and e ect in the wide range of settings relevant to economics are still lacking. This paper introduces topological settable systems, an extension of White and Chalak s (2009) settable systems causal framework su cient to provide foundations justifying rigorous discussion of cause and e ect in contexts general enough to include games of incomplete information, where causal variables may be functions. We apply this framework to gain causal insight for a speci c incomplete information game, an N bidder private-value auction, posing and resolving a variety of relevant causal questions. This game not only illustrates the central issues but also suggests how topological settable systems can apply in other contexts. 1 Introduction: An N Bidder Private-Value Auction Causal discourse that is, discussion of cause and e ect lies at the heart of and permeates economics. It appears naturally and unselfconsciously throughout Adam Smith s Wealth of Nations (1790) and in all the major economic contributions of the nineteenth century (e.g., Mill, 1848; Marshall, 1890) and a good part of the twentieth (e.g., Hicks, 1939; Samuelson, 1947). As twentieth-century economists began to think carefully about systems of structural or simultaneous equations, work began on formalizing notions of causality and structure. Classical e orts in this area include the work of Haavelmo (1943, 1944), Marschak (1950), Simon (1953), Strotz and Wold (1960), and Granger (1969). Unfortunately, no clear consensus emerged from this work. Causal notions remained murky, in part due to the causal paradoxes associated with simultaneity, which nevertheless plays an indispensable role in describing economic phenomena. This lack of clarity contributed to economists tending to avoid not only formal discussion of causality, but even informal discussion, as Hoover (2004) documents. Nevertheless, causal discourse is so central to economics and so intuitive that explicit causal discussion has re-emerged in the last few decades, together with renewed and deep consideration of causal foundations (e.g., Heckman, 2005). Despite this renewal, causal discourse in economics still occupies an ambivalent status. Although it is intuitive and natural to speak about income e ects and price e ects and the like, rigorous foundations justifying well-posed discussions of cause and e ect in the wide range of 1

2 settings relevant to economics are still lacking. Intuition can only go so far. Without rm foundations, discussions of identi cation or estimation of causal e ects or attempts to draw policy conclusions or economic insight from such estimates are problematic. There remains a clear need to nd broadly applicable rigorous foundations for causal discourse in economics. Outside of economics, the Pearl causal model (PCM; e.g., Pearl, 2000) has emerged as a leading paradigm for understanding cause and e ect. Although the PCM does apply in some cases in economics, it also fails to apply in important economic contexts, as White and Chalak (2009, WC) and White, Chalak, and Lu (2010) document. To provide more broadly applicable foundations for causal discourse in economics, WC introduced the settable systems framework, which builds on the PCM and the insights of Strotz and Wold (1960) to extend the PCM to accommodate features essential to economic analysis: optimization, possibly non-unique equilibrium, and learning. In economics, the PCM has been productively applied to game theory, and, speci cally, to games of incomplete information, also known as "Bayesian" games (Harsányi, 1967). In seminal work, Koller and Milch (2001, 2003) introduced the PCM-based Multi-Agent In uence Diagrams (MAIDs) to represent and compute equilibrium in non-cooperative games. The PCM has been explicitly applied to incomplete information games by Penalva-Zuasti and Ryall (2003), Jiang and Leyton-Brown (2010), and Wellman, Hong, and Page (2011), among others. Nevertheless, these applications all inherit the limitations of the PCM: among other things, the PCM cannot support causal discourse in games with non-unique equilibrium; and by ruling out a causal role for "background" variables, the PCM does not permit discussion of the causal role played by structurally exogenous variables, such as Harsányi s (1967) agent types in incomplete information games. The PCM thus does not provide a satisfactory general foundation for gametheoretic causal discourse, and, in particular, for games of incomplete information. Although the settable systems framework succeeds in providing the desired causal foundations for economics in many instances, and for important games in particular (see WC, section 6), it still does not apply to certain major classes of problems, such as incomplete information games. The di culty is that optimal choices in these games can be rather general functions, but settable systems only admit optimal choices that are functions representable by a countable number of real variables, such as elements of a Hilbert space. 1 The settable systems framework does not admit optimal choices that are elements of general function spaces. Not surprisingly, the lack of formal foundations for game-theoretic causal discourse leads either to informal discussion of cause and e ect, which may be limited or misleading, or, more commonly, to the avoidance of such discussion altogether. This informal use is illustrated by the Wikipedia entry on complete information, where we nd the following: 1 In contrast, the PCM only allows real-valued causal variables. 2

3 If a game is not of complete information, then the individual players would not be able to predict the e ect that their actions would have on the others players (even if the actor presumed other players would act rationally). In what sense do players actions or strategies a ect other players, whether rational or irrational? So far, there is no adequate basis for rigorously addressing this question. In contrast and signi - cantly, there is no mention of causality in Koller and Milch s (2001, 2003) careful work. The goals and contributions of this paper, therefore, are two-fold. First, we seek an extension of WC s settable systems general enough to provide foundations justifying rigorous discussion of cause and e ect in contexts that include games of incomplete information. We achieve this goal by introducing topological settable systems, extending settable systems in a direct and natural way. Second, we seek causal insight for a speci c incomplete information game, an N bidder private-value auction, posing a variety of relevant causal questions. We focus on this game, as it is simple enough to allow straightforward analysis, yet rich enough to illustrate all of the central issues. We contribute by answering these causal questions within our new framework. The causal discourse illustrated for this particular game makes clear how analogous causal discourse can be supported in more elaborate situations, such as dynamic games of incomplete information or games with a countable in nity of types. Further, the generality of topological settable systems makes them broadly useful for application in economics, as well as in other quite di erent elds. For example, topological settable systems may apply to the study of causality in the spatial-temporal manifolds used to analyze neural activity in the brain (Valdés-Sosa, Bornot- Sánchez, et al., 2011; Valdés-Sosa, Roebroeck, et al., 2011). To pose our causal questions, we consider the rst-price private-value auction studied by Guerre, Perrigne, and Vuong (2000) and others in the empirical auction literature and treated by Krishna (2010, chapter 2). There is a single object for sale, and 1 < N < 1 potential buyers bid for the object. The highest bidder pays the amount they bid ( rst price) and gets the object. If there is a tie, then the winner can be decided randomly, e.g., by ipping a fair coin. It is assumed that there are no liquidity or budget constraints for any bidder. Thus, each bidder is both willing and able to pay the amount bid. Bidder i assigns a value of X i to the object this is the maximum amount the bidder is willing to pay. The value X i corresponds to Harsányi s (1967) notion of agent type. The X i s are random and assumed to be independently and identically distributed on an interval [ v_; v] ( v_; v 2 R + [0; 1)) according to the strictly increasing distribution function F with full support on [ v_; v], with EX i < 1. For simplicity and without essential loss of generality, we let [ v_; v] be the unit interval, I [0; 1]: F is also assumed to admit a continuous density f F 0. Bidder i observes the realization x i of X i ; but not other bidders realizations, x j ; j 6= i (private 3

4 value). All elements of the auction other than the realized values are assumed to be commonly known to all bidders. In particular, F is common knowledge, as is N: Thus, bidder i knows that other bidders values are independently distributed according to F. A strategy for bidder i is a non-decreasing (monotone) function i that maps types to actions (bids), 2 determining his or her bid for every value of x i. Bidders are assumed risk neutral; they seek to maximize the expected value of their surplus or pro t, X i i (X i ): Because negative pro t can always be avoided, we further require that i (X i ) X i ; implying that i maps I to I. Under individual rational behavior, i.e., one-person partial equilibrium, bidder i simply takes other bidders strategies as given and best responds. Bidder i s best response function describes how his or her optimal strategy varies with the other bidders strategy pro le, denoted i : This (N 1) vector contains all strategies j ; j 6= i. Thus, the bidder s partial equilibrium best response function is de ned by i ( i ) arg max E P i (X i ) max j(x j ) j X i (X i i (X i )) ; (1) i 2H j6=i where H is the collection of all non decreasing functions mapping I to I. The given conditions ensure that i exists; this is veri ed in the Appendix. We further assume that a unique solution exists or can be selected. Note the distinction between the best response function i : i6=j H! H and the strategies j : I! I; i 6= j; which need not be optimal. Thus, i describes player i s rational behavior, whereas i is a vector of other players possibly suboptimal strategies, which bidder i knows (i.e., correctly believes). In this game, the individual s partial equilibrium best response is a function depending on other bidders strategies, rather than a real variable determined in response to other bids. To express the optimal bid, we observe that the best response function i is equivalent to the following: for x i 2 I almost surely (a:s:), b i ( i ; x i ) e( i ( i ); x i ) = arg max P b i max j(x j ) (x i b i ) ; b i 2I j6=i where e is the evaluation functional, e(; x) = (x): This represents bidder i s optimal bid once she knows that her private value is actually x i, and given other bidders strategies i. Although the other bidder s strategies may or may not represent their best responses in partial equilibrium, in Bayes-Nash full equilibrium, each bidder s best response must be to the other bidders best responses. This full equilibrium condition is represented by the xed point requirement, y i = i ( y i ); i = 1; :::; N; (2) 2 This is in fact a pure strategy; mixed strategies involve random choice among pure strategies. For now, all strategies are pure strategies. 4

5 where y i : I! I denotes the full equilibrium best response function for bidder i: In Bayes- Nash equilibrium, each bidder correctly believes that the other players will best respond. Thus, Bayes-Nash equilibrium is a rational belief equilibrium. For simplicity, assume for now that a unique solution to this system of equations exists or can be selected. In Bayes-Nash full equilibrium, the optimal responses are mappings that can depend only on the bidder s own realized value; these yield full equilibrium optimal bids b y i = y i (x i): With this background, we can pose a number of central questions about causality in this auction game: In what sense is a bidder s strategy or action causally a ected by other bidders strategies or actions? How do rationality in behavior and belief matter for causal discourse? What is the causal role of types, X i ; if any? Do N and F have e ects? If so, how? If not, why not? What are the structural equations here? Speci cally, are the simultaneous equations (2) structural? How about the full equilibrium "reduced form"? Is it structural? In particular, do y i and b y i have structural meaning and/or causal content? The plan of the paper is as follows. In Section 2, we introduce basic elements of topological settable systems and relate these to one-person partial equilibrium in the N bidder rst-price private-value auction, resolving several of these causal questions. In Section 3, we introduce further elements of topological settable systems and relate these to full equilibrium, resolving the rest. Section 4 contains further discussion, and Section 5 summarizes and concludes. 2 Topological Settable Systems and Individual Rationality We extend the settable systems framework of WC to topological settable systems, which support causal discourse for phenomena involving random variables taking values in general topological spaces, rather than just the real line. In particular, this makes it possible to talk sensibly about the causal e ects of one agent s strategy on other agents best responses. Some of the material below is unavoidably abstract, but we promptly follow each general concept or de nition with discussion relating these to the auction of Section 1. We draw on and refer the interested reader to Corbae, Stinchcombe, and Zeman (2009, ch.10) for topological background. Recall that a topological space is a pair (X; ); where X is a non-empty set and is a collection of subsets of X containing X and?; closed under arbitrary unions and nite intersections. The elements of are open sets. Topological spaces may or may not have associated norms or metrics. For example, consider the set of strategies H; i.e., the collection of non decreasing functions mapping I to I. This set has a topology (described next), say H : The topological space (H; H ) is known as Helly space, named after mathematician Eduard Helly (see, e.g., Kelley, 1975, p.164). H is a closed subset of the set I I ; the set of all mappings from I to I. That set has an associated topology, say I. We can take H to be the relative topology, de ned as the collection of all sets 5

6 of the form H \ A; where A belongs to I. Helly space is not a metric space. 3 To de ne random elements of topological spaces, we let B(X; ) denote the Borel generated by the topology; this is the smallest collection of subsets of X that includes all elements of, the complement of any set in B(X; ); and the countable union of any collection of sets in B(X; ): Let (; F) be a measurable space. Random elements of (X; ) are mappings, say Z :! X; that are measurable F=B(X; ), i.e., for all B 2 B(X; ); Z 1 (B) f! : Z(!) 2 Bg 2 F: Random scalars and vectors are simple examples of this concept. eld For example, for types, X i :! I; take X = I and let I be the usual open sets of I. This also permits us to de ne random strategies. A random strategy B is a mapping from to H (i.e., = B(!) belongs to H for each! in ) with the property that for all "strategy events" H 2 B(H; H ); 4 we have that B 1 (H) f! : B(!) 2 Hg belongs to F. If P is a probability measure on (; F); H is a strategy event, and B is a random strategy, then we can de ne the probability of H under B as P B [H] P [B 2 H] = P f! : B(!) 2 Hg: For example, a measurable selection from a strategy event H; say s H :! H; gives a random strategy B such that B(!) = s H (!) 2 H for all! 2 : Thus, even when there are multiple oneperson or full equilibrium solutions, we can apply a suitable measurable selection to the solution set to obtain a single solution, a randomized strategy. We also use the concept of the product topology. Let N + denote the positive integers, and de ne N + N + [ f1g: When n = 1; we interpret i = 1; :::; n as i = 1; 2; :::. Let n 2 N + ; and for i = 1; :::; n; let (X i ; i ) be topological spaces: De ne the Cartesian product X n n i=1 X i; the projection mapping i : X n! X i is the mapping such that i (x n ) = x i for each x n = (x 1 ; :::; x n ) 2 X n : The product topology, say n = n i=1 i; is the smallest topology that makes each i ; i = 1; :::; n; continuous, i.e., i 1 ( i ) n : When n is countable in nity, B(X 1 ; 1 ) denotes the Borel eld generated by the measurable nite-dimensional product cylinders, i.e. the smallest collection of subsets of X 1 that includes: (i) sets of the form 1 i=1 B i; where B i 2 B(X i ; i ); i = 1; 2; :::; and B i = X i except for nitely many i; (ii) the complement of any set in B(X 1 ; 1 ); and (iii) the countable union of any collection of sets in B(X 1 ; 1 ): For example, consider the individually rational best response functions, i : i6=j H! H: We require the domain (here, H N 1 i6=j H) to be a Borel set of the product space (H N 1 ; N 1 ); say, which it obviously is. For the evaluation functional, e : H I! I; we require the domain H I to be a Borel set of the product space (H I; H I ); which again it obviously is. 3 It is also known that Helly space is compact convex Hausdor, rst (but not second) countable, and separable (Kelley, 1975, p.164). 4 We use the term "event" to denote a Borel measurable set. A "strategy set" could be any subset of H. 6

7 2.1 Topological Settable Systems: Formal De nition We now have su cient background to formally de ne topological settable systems. We let N f0; 1; :::g, and de ne N N [ f1g: When m = 0; we ignore references to i = 1; :::; m. De nition 2.1 (Partitioned Topological Settable System) Let A be a set and let attributes a 2 A be given. Let (; F; a ) be a complete measure space such that contains at least two elements and a is nite. Let m 2 N; let (X 0;i ; 0;i ) be topological spaces, i = 1; :::; m; let S 0 be a multi-element Borel subset of the product topological space (X m 0 ; m 0 ); and let the fundamental settings Z 0 :! S 0 be measurable ( F=B(X m 0 ; m 0 )). Let n 2 N + ; and for units i = 1; 2; :::; n, let (X i ; i ) be a topological space and let S i be a multi-element subset of B(X i ; i ): Let = f b g be a partition of f1; :::; ng; with cardinality B 2 N + : For i = 1; 2; :::; n; let settings Z i :! S i be measurable functions, and for blocks b = 1; :::; B; let Z (b) be the vector containing Z 0 and Z i ; i =2 b; and taking values in the non-empty Borel set S (b) S 0 i=2b S i : For b = 1; :::; B; let S [b] i2 b S i be a non-empty Borel set, and suppose there exists a measurable joint response function r [b] ( ; a) : S (b)! S [b] such that responses Y[b] () (Y i (); i 2 b ); are jointly determined as De ne the settable variables X i Y [b] = r [b] (Z (b) ; a): : f0; 1g! S i as X i (0;!) Y i (!) and X i (1;!) Z i (!)! 2 ; i = 1; :::; n; and X 0 (0;!) Y 0 (!) X 0 (1;!) Z 0(!);! 2 : Put X fx 0 ; X 1 ; X 2 ; :::g: The triple S f(a; a); (; F; a ); (; X )g is a partitioned topological settable system. This de nition includes the settable systems originally de ned by WC as the special case where all the topological spaces are (R; R ); where R is the usual collection of open sets of R: Because the auction game of Section 1 involves other topological spaces, WC s original settable systems do not apply. But as we now show, our auction game maps directly to topological settable systems. In general, the attributes a implicitly contain all items referenced within the de nition except those that explicitly depend on a: For games, attributes can exhaustively specify the rules, including the number of players, their utility functions, the payo s, the form of equilibrium, and any equilibrium selection mechanisms, among other things. In our auction, the given number of players, N; and the common distribution of types, F; are elements of a: It is often convenient to take the measure a to be a probability measure, in which case we write a as P a : In our auction game, the probability measure P a (= P) is determined by F and 7

8 N and by the assumed independence of types. The independence assumption is also a component of a; expressed, for example, as a speci c choice of copula (Sklar, 1959; Nelsen, 1999). The vector of types, X (X 1 ; :::; X N ); corresponds to the fundamental settings, Z 0 ; as these are determined outside the system. Thus, m = N: For the agent types, the spaces (X 0;i ; 0;i ) = (I; I ) are implicitly components of a; the product topological space for the types is (X m 0 ; m 0 ) = (I N ; N I ): The admissible values for the fundamental settings are S 0 = I N. We designate two units for each player, a "strategy" unit and a "bid" unit. Thus, n = 2N: For strategy units (i = 1; :::; N), the topological space is Helly space, (X i ; i ) = (H; H ): For bid units (i = N + 1; :::; 2N), the topological space is (X i ; i ) = (I; I ): These spaces are also implicitly components of a. Here, the admissible values are S i = H for strategies and S i = I for bids, implicitly components of a and obviously elements of B(H; H ) and B(I; I ); respectively. The partition acts to specify blocks of units that jointly respond to settings of system units outside their block. The elementary partition is e = ff1g; :::; fngg. The elementary settable system S e has this partition and represents the response of each individual unit to every other unit of the system. To examine the individually rational case, we can use the agent partition, a = f a b ; b = 1; :::; Ng; where block b = i has a b = fi; i + Ng: This partition groups together all responses governed by each agent. The corresponding settable system describes how each agent s optimal strategy and optimal bid jointly depend on settings for all other agents. Other partitions, discussed below, correspond to jointly rational full equilibrium. The joint response function r [b] ( ; a) speci es how the settings outside block b; Z (b) ; determine the joint responses inside block b; Y[b] : The distinction between the structurally determined responses (left-hand-side variables, Y[b] ) and the settings (right-hand-side variables, Z (b)), which can take any admissible value, enforces the Strotz-Wold (1960) dichotomy between left-hand side and right-hand side variables in structural equations: the same variables never appear on both the right- and left-hand sides of the system s equations. This rules out simultaneity. Settings, response functions, and responses are partition-speci c. For the agent partition, a, block b references bidder i; and r a [i] = ra [i] = (ri a; ra i+n ); say, where strategy units have responses r a i (z a (i) ; a) = i ( i ); b = i 2 f1; :::; Ng; and bid units have responses r a i+n(z a (i) ; a) = b i ( i ; x i ); b = i 2 f1; :::; Ng: By de nition, z(i) a = za (i) contains setting values outside block i; that is, setting values for other agents arbitrary strategies and bids and setting values for types: z(i) a = ( i; b i ; x): Observe that r[i] a ( ; a) makes explicit the dependence of the response on a, whereas the dependence of i and b i on N and F is implicit. Also, observe that whereas r[i] a (za (i); a) allows 8

9 the response to potentially depend on settings for every unit except those in a i ; the economics of the auction game dictate that the best response for strategy unit i depends only on the settings of strategies for the other strategy units, i ; and not on any of the agent values (types) or other agents bid settings. Similarly, the best response for bid unit i + N depends only on i and the own type, x i. The economics of the game thus impose variable exclusion restrictions. For block b; the sets S (b) and S [b] can impose partition-speci c restrictions on the admissible values for Z(b) and Y [b] ; respectively. Under the agent partition, Sa (i) = IN H N 1 I N 1 and = H I. For this partition, there are no restrictions imposed on the admissible setting and S a [i] response values. Thus, each aspect of one-person partial equilibrium in our N bidder rst-price private-value auction maps to the agent partition of the settable systems framework, permitting us to de ne agent-partition-speci c settable variables X a fx 0 ; X1 a; :::; X 2N a g for this game.5 Because of the structure imposed by the settable system, we refer to any of its elements or aspects as structural. In particular, the response equations Y[b] = r [b] (Z (b); a) are structural equations. The fundamental settings, Z 0 ; are determined outside the system; they are thus structurally exogenous. 6 The responses Y[b] are determined within the system; they are therefore structurally endogenous. It follows that in our auction game, the structural equations for the individually rational one-person partial equilibrium are a i = i ( i ) b a i = b i ( i ; x i ); i = 1; :::; N: (3) The types X are structurally exogenous. The strategies and bids are structurally endogenous. 2.2 Cause and E ect in Topological Settable Systems In settable systems, de nitions of cause and e ect are partition-speci c. De nition 2.2 (Intervention, Direct Cause, Direct E ect) Let S f(a; a); (; F; a ); (; X )g be a partitioned topological settable system. For b 2 f1; :::; Bg; let z (b) and ~z (b);j be distinct and admissible, i.e., z (b) ; ~z (b);j 2 S (b) with z (b) and ~z (b);j identical, except that z (b) has a component z j ; whereas ~z (b);j has ~z j 6= z j instead. Then the ordered pair (z (b) ; ~z (b);j ); denoted z (b)! ~z (b);j ; is an intervention to Z (b) : 5 It would be formally correct and more explicit to write X a a fx a;0; X a a;1; :::; X a a;2n g instead of X a, making clear the dependence of the settable variables on the attributes. We leave this implicit to keep the notation simpler. 6 This notion of structural exogeneity has no necessary relation to econometric notions of exogeneity, which involve various forms of stochastic orthogonality (e.g., independence or non-correlation) between observed regressors and unobserved "errors". 9

10 Let i 2 b ; j 62 b : Then X j directly causes X i in S if there exists z (b)! ~z (b);j such that r i (~z (b);j ; a) 6= r i (z (b) ; a); where r [b] (~z (b);j; a) and r [b] (z (b); a) are both admissible, i.e., r [b] (~z (b);j; a), r [b] (z (b); a) 2 S [b] : Otherwise, we say X j does not directly cause X i in S: The ordered pair (y i ; ~y i;j ) = (r i (z (b); a); r i (~z (b);j; a)) is the direct e ect of X j of z (b)! ~z (b);j. on X i According to this de nition, causality is just structural functional dependence, given a suitable context. This now enables us to answer several of the questions posed in the Introduction. Speci cally, agent types can play a causal role in our auction game, as fundamental variables can indeed act as direct causes under this de nition. As we see from structural equations (3), however, in one-person partial equilibrium, other agent types do not directly cause either strategy or bid best responses for a given agent, as the agent-partition joint response functions r[i] a do not depend on x i : Nor does an agent s own type directly cause her strategy best response, as ri a does not depend on x i : But an agent s own type does directly cause her bid best response. 7 We also see that in one-player partial equilibrium, other bidders (suboptimal) strategies but not their actions (bids) directly cause agent i s rational behavior, as both the strategy and bid best responses depend on i but not b i : Indeed, each bidder has enough information to determine precisely how her own strategy and action will causally a ect every other bidders rational responses in one-player partial equilibrium, despite the incomplete information and contrary to the Wikipedia entry quoted in the Introduction. Only when the other players response functions are unknown is a given bidder unable to determine the e ects of her behavior. Rationality in behavior is a su cient but not necessary condition for other bidders response functions to be known to every bidder. This de nition of partition-speci c direct cause and e ect admits the possibility of mutual causality, or what Strotz and Wold (1960) call mutual causation: we can have both Xi directly causing Xj and Xj directly causing Xi : Because of the distinction between settings and responses, however, there is no simultaneity or instantaneous causality. Framing causal statements in terms of settable variables enforces this distinction, making it explicit that causal relations do not simply hold between random variables or events (elements of F); rather, they hold between more structured objects, i.e., settable variables. In one-person partial equilibrium for our auction game, we immediately see that mutual causality is present, as each agent s strategy directly causes every other agent s best response. There is, however, no simultaneity. 7 Strictly speaking, the presence of x i in agent i s bid best response function implies only that the associated settable variable is a potential direct cause. In our auction game, however, this potential direct cause is generally an actual direct cause, as the bid best response function generally is not constant in x b. in S 10

11 By construction, the e ects just de ned are ceteris paribus: z (b) and ~z (b);j di er in only one component. This is just for simplicity and concreteness. Any pair (z (b) ; ~z (b) ) of distinct admissible values is an admissible intervention, and the joint direct e ect of this intervention is (y [b] ; ~y [b] ) = (r [b] (z (b); a); r [b] (~z (b); a)): Although the term "intervention" suggests some mechanical process, there is no need for physical manipulation. As the de nition makes clear, in settable systems, interventions are simply pairs of admissible values for the settings. CW de ne the e ect of the intervention z (b)! ~z (b) as the di erence ~y [b] y [b] : General topological spaces need not be vector spaces, so di erences need not be de ned for (X i ; i ), motivating our de nition of e ects as pairs of response values. Nevertheless, when X i is a subset of a vector space, di erences taking values in that vector space are de ned and can be interpreted as e ects, analogous to CW. For example, consider the strategy intervention i! ~ i;j ; where ~ i;j di ers from i with respect to the strategy of bidder j: One e ect of this intervention is ( i ( i ); i ( ~ i;j )): The di erence i ( ~ i;j ) i ( i ) is generally not an element of H, as it takes values in [ 1; 1] and need not be monotonic. Nevertheless, H is a subset of the set of bounded Borel-measurable functions from I to R; a vector space. The di erence i;j i ( ~ i;j ) i ( i ) is thus an element of this set. The usual L p norms are de ned for these di erences, so the magnitude of the e ect can be de ned as the norm jj i;j jj p = [ R 1 0 j i;j(x)j p dq(x)] 1=p ; where Q is a speci ed probability measure on the measurable space (I; B(I; I )): Observe that only variables can have e ects. Constants, like N and F; cannot take more than one value, so interventions are not possible (cf. Holland, 1986). Thus, N and F do not have e ects, resolving several more questions posed in the Introduction. The same is true for all system attributes, a: If interest attaches to e ects of F; say, then F must be removed from the system attributes and assigned status as a variable taking values in a suitable topological space. For example, F can be a fundamental variable belonging to a collection of distributions de ning measures absolutely continuous with respect to a given nite measure a : These distributions can be indexed by a nite- or in nite-dimensional parameter. Cause and e ect are therefore not only partition-speci c, but are relative to the attributes chosen to describe a given phenomenon. 3 Cause and E ect in Full Equilibrium 3.1 Comparable and Compatible Settable Systems To relate individual best responses (one-person partial equilibrium) to jointly rational responses (Bayes-Nash full equilibrium), we use the concepts of comparable and compatible settable systems. 11

12 3.1.1 Comparability De nition 3.1 (Comparability, Nesting, Finer and Coarser Partitions) Suppose topological settable systems S f and S c have identical (A; a): Then they are comparable. Let comparable topological settable systems S f and S c have partitions f = f f 1 ; :::f B f g and c = f c 1 ; :::; c B cg; respectively. If each element of c is a union of elements of f ; then c nests f, c is coarser than f ; f is ner than c ; and we write f - c : When f - c and c - f ; then f and c are the same, f = c : 8 When there are just two partitions f and c such that f 6= c and f is ner than c ; we write f c ; and we call f the ne partition and c the coarse partition. The coarsest partition is the global partition, g ff1; :::; ngg: This nests every partition; in particular, it nests the nest partition, which is the elementary partition, e ff1g; :::; fngg: We use the same nesting terminology for comparable S f and S c. For example, if c nests f ; then we also say S c nests S f or that S f is ner than S c ; and we write S f - S c : In our incomplete information game, the elementary partition e and the agent partition a are associated with comparable topological settable systems S e and S a ; say, with the coarser system S a nesting the ner system S e : This follows because a = f a i g; with a b = fb; b + Ng = fbg [ fb + Ng; each element of the agent partition is the union of elementary partition elements Compatibility To motivate the de nition of compatibility, rst recall that for c b 2 c ; we have Y[b] c = rc [b] (Zc (b); a): (4) Next, de ne r f [b] (r f i ; i 2 c b ): This is the vector of response functions for the ne partition corresponding to the coarse partition element c b. denote the set of coarse partition Let Sc (b) admissible values for the settings Z(b) c ; and suppose that for each zc (b) 2 Sc (b) there is an admissible y[b] c (2 Sc [b]) such that9 y [b] c = rf [b] (yc [b] ; zc (b); a). (5) When (5) holds, we see that y[b] c is a xed point of the ne partition c b system (5). Signi - cantly, and unlike the PCM, this xed point need not be unique. As noted above, a can embody 8 Let denote the collection of all partitions of f1; :::; ng: Then it is easily checked that (; -) is a partially ordered set. As every partition has a greatest lower bound (inf), (; -) is a complete lattice (e.g., Burris and Sankappanavar, 1981, theorem 4.2, p.17). 9 We write r f [b] in this form for notational convenience. Although rf [b] as a whole depends on all elements of yc [b]; for given i 2 c b; r f i block containing i: depends only on the subvector of yc [b] that omits the variables determined within the ne partion 12

13 equilibrium selection mechanisms, so it can specify which of possibly many xed points for (5) is represented by (4). We refer to (5) as a mutual consistency condition, as it ensures that the coarse partition is consistent with the ne partition in this speci c way. Reinforcing the message of Strotz and Wold (1960), we emphasize that although these equations are precisely a system of simultaneous equations, they are not structural equations carrying causal meaning. 10 That meaning is carried by the structural equations of S f and S c : Causal discourse is valid in both S f and S c regardless of whether (5) holds. 11 Instead, the role of the simultaneous equations system (5) is to provide functional links between ner and coarser systems. These connections make causal discourse and economic explanations coherent between partitions in a precise sense. To relate these ideas to our auction game, let the ne partition be the agent partition, a ; and let the coarse partition be the global partition, g : Clearly, g nests a : The settable system S g corresponding to g describes how all bidders strategies and bids jointly respond to all bidders types. The xed point condition (5) has the form g i = ri a ( g i ; bg i ; x; a) = i ( g i ) b g i = ri+n( a g i ; bg i ; x; a) = b i ( g i ; x i); i = 1; :::; N; where we have modi ed the arguments of ri a and ri+n a to make the dependencies clear. These simultaneous equations correspond to the Bayes-Nash full equilibrium conditions (2). Drawing on the discussion above, we can now resolve another of the questions posed at the outset: the Bayes- Nash full equilibrium conditions (2) are mutual consistency conditions, not structural equations. Suppose the strategy equations have an admissible xed point, y ; as assumed above, and suppose further that a speci es that responses for the global partition are governed by Bayes- Nash equilibrium, so that g = y : Then 12 g i = r g i (x; a) = y i b g i = r g i+n (x; a) = y i (x i); i = 1; :::; N: (6) The responses of the agent partition and those of the global partition are then mutually consistent. We formalize these ideas with the following de nition. De nition 3.2 (Mutual Consistency/Compatibility) Let S f and S c be comparable topological settable systems with S f - S c. If (5) holds for all c b ; b = 1; :::; Bc ; then S f and S c are mutually consistent or compatible settable systems, and we write S f - c S c : 10 In a related context, Strotz and Wold (1960, p.423) state, "what are simultaneous equilibrium conditions ought not to be confused with causal relations." 11 In sharp contrast, causal discourse in the PCM is de ned only when the analog of (5) has a unique solution. 12 For the equilibrium bid representation, b g i = b i ( g i ; xi) = e( i (g i ); xi) = e(g i ; xi) = e(y i ; xi) = y i (xi): 13

14 An important consequence of the cross-partition coherence a orded by compatibility is that it makes possible valid statements about cause and e ect in unobservable partial equilibrium ( ne partition) structures using knowledge gained solely from observable full equilibrium (coarse partition) structures. Speci cally, by linking response functions across partitions, compatibility ensures functional dependence between ne and coarse partition e ects, generalizing the e ect linkages studied in the classical identi cation problem for linear structural systems (e.g., Fisher, 1966). The appendix contains an example designed to provide further insight. So far, we have simply assumed the existence of full equilibrium best responses. Before trying to analyze their causal content, we should be sure these exist. Ensuring equilibrium existence in Bayesian games is a fascinating and deep subject. Athey (2001) provided the pioneering result that a monotone pure-strategy equilibrium exists whenever a Bayesian game obeys a Spence-Mirlees single-crossing restriction. Using the powerful notion of contractability, Reny (2011, theorem 4.1) has extended Athey s results and related results of McAdams (2003) to give weaker conditions ensuring equilibrium existence. As Reny (2011, p.501) states: Suppose that action spaces are compact convex locally convex semilattices or compact locally complete metric semilattices, that type spaces are partially ordered probability spaces, that payo s are continuous in actions for each type vector, and that the joint distribution over types induces atomless marginals for each player assigning positive probability only to sets that can be order-separated by a xed countable set of his types. 13 If, whenever the others employ monotone pure strategies, each player s set of monotone pure-strategy best replies is non-empty and join-closed, 14 then a monotone pure-strategy equilibrium exists. This result applies directly to our auction game, as can be straightforwardly veri ed, ensuring the desired compatibility. The Appendix provides details. As this veri cation shows, existence of the monotone pure-strategy equilibrium requires restricting the best responses to belong to S g I N = S g [1] IN ; say, where S g is a speci c proper Borel subset of H N N i=1 H = i2 g 1 S i: Given equilibrium existence, we can examine its causal content. Inspecting eq. (6) and applying De nition 2.2, we see that only the bidder s own type (directly) causes the bidder s action (bid) in the global partition. But what about the bidder s full equilibrium strategy g i? What causes it? The answer illustrates a subtle insight elucidated by the settable systems framework: nothing causes g i : Nevertheless, the optimal strategy g i is formally explained by the principles of Bayes-Nash equilibrium, in which a bidder s rational (optimizing) behavior and rational beliefs 13 One set is order-separated by another if the one set contains two points between which lies a point in the other. 14 A subset of strategies is join-closed if the pointwise supremum of any pair of strategies in the set is also in the set. 14

15 about others behavior, operating on the speci cs of the game (the primitives of the system, as embodied in the attributes a), determine the bidder s full equilibrium strategy. In the deductivenomological theory of explanation (e.g., Hempel and Oppenheim, 1948; Popper, 1959), to explain a phenomenon is to show that it is the logical consequence of the application of a speci c set of governing principles to the primitives of a given system. In this sense, the explanatory answer to the question, Why is g i = y i? is that this is the logical consequence of the principles of rational action and belief underlying Bayes-Nash equilibrium applied to the system primitives, in particular N and F. Interestingly, this is an example of a non-causal explanation. 15 Correspondingly, the equation for g i is an example of a structural equation in which causality is absent. This shows that, within the settable systems paradigm, structural relations are more basic than causal relations, as causal relations are necessarily structural, but structural relations need not be causal. With this understanding, we can resolve our questions at the outset about the structural and causal content of the "reduced form": The reduced form is structural. particular, both g i and bg i have structural meaning as responses in a structural system; however, only b g i is causally determined. 3.2 Recursive and Canonical Topological Settable Systems Causal discourse is especially straightforward in recursive systems. In To describe these systems formally, for b > 0; we let [1:b] [ b a=1 a and [0:b] 0 [ [1:b] ; where 0 f(0; 1); :::; (0; m)g: By convention, [0:0] = 0 and Z [0:0] = Z 0 : De nition 3.3 (Recursivity) Let S be a partitioned topological settable system. For b = 0; 1; :::; B, let Z [0:b] denote the vector containing the settings Z i for i 2 [0:b] and taking values in S [0:b] S 0 i2[1:b] S i ; S [0:b] 6=?. For b = 1; :::; B; suppose that r fr[b] g is such that the responses Y [b] are jointly determined as Y [b] = r [b] (Z [0:b 1] ; a): Then we say that is a recursive partition, r is recursive, and that S is a recursive topological settable system or simply that S is recursive. Recursive systems are also called triangular or acyclical. The global partition is always recursive, as B = 1 and global partition responses are jointly determined as Y g = r g [1] (Z [0:0]; a): In the global partition, we see that only the fundamental settable variables X 0 are potential direct 15 There is an on-going debate in the philosophy of science about whether all explanations must be causal. Together, the deductive-nomological theory of explanation and the settable systems causal framework provide a particular context in which this issue can be resolved: non-causal explanations are possible. Further, the nomologicaldeductive theory provides a foundation for causal explanation. 15

16 causes of the settable variables X g : This matches what we saw in Bayes-Nash equilibrium for our auction game, where only types could potentially cause equilibrium strategies and bids. A somewhat richer example involves the strategy partition, s = f s 1 ; s 2 g; where s 1 = f1; :::; Ng and s 2 = fn + 1; :::; 2Ng: The joint responses for s 1 are the jointly determined strategies s i ; i = 1; :::; N; and those for s 2 are the jointly determined bids bs i ; i = 1; :::; N: Observe that the agent partition a is not nested in s ; as there is no way to form s 1 as a union of elements of a : But the elementary partition e = f e 1 ; :::; e 2N g; e a = fag; is nested in s ; as is easily checked; thus S e - S s : We determine the responses for the strategy partition by requiring that these be mutually consistent with the elementary partition responses. From eq.(1) and related discussion, the elementary partition strategy and bid responses are e i = r e i ( i ; b; x; a) = i ( i ) b e i = r e i+n(; b i ; x; a) = e( i ; x i ); i = 1; :::; N; where we have adapted the notation to make the elementary partition settings clear. As we have seen throughout, rationality in behavior and the underlying information constraints imply speci c restrictions on the response functions in the form of exclusion restrictions. Next, we impose compatibility between the strategy and elementary partitions. This gives s i = r e i ( s i; ~ b s ; x; a) = i ( s i) b s i = r e i+n( ~ s ; b s i; x; a) = e( ~ s i ; x i ); i = 1; :::; N; where we write ~ b s and ~ s to denote arbitrary admissible strategy partition settings for bids and strategies, distinguishing these from the strategy partition responses b s and s for bids and strategies. Whenever a speci es that responses for the strategy partition are governed by Bayes- Nash equilibrium, the system above gives s = y : The xed point for the bid responses is trivial. Thus, the strategy partition settable system S s compatible with the elementary system S e is s = y b s i = e( ~ s i ; x i ); i = 1; :::; N: Observe that although the strategy responses for the strategy partition are the same as those for the global partition, the bid response functions di er between the two. Only the own types can directly cause global partition bids. In contrast, both the own types and the own strategies can cause the strategy partition bids. Also observe that the strategy partition is recursive, as responses in each block depend at most only on settings in "predecessor" blocks. That is, responses in block 1 depend at most on settings in block 0; and responses in block 2 depend at most on settings in blocks 0 and 1: 16

17 In recursive settable systems, mutual causality is absent. It is then natural to consider the unimpeded evolution of the recursive system in the absence of intervention. This is implemented by equating the settings for a given block with the responses for that block. We formalize this with the notion of canonical topological settable systems. De nition 3.4 (Canonical Topological Settable System) Let S be a recursive topological settable system such that Z[b] = Y[b] ; b = 1; :::; B: Then S is a canonical topological settable system. The canonical version of the strategy partition is then s = y b s i = e( s i ; x i ); i = 1; :::; N: Observe that the canonical strategy partition yields the same strategies and bids as the global partition, but there is still a subtle distinction: strategies can directly cause bids in the strategy partition, but not in the global partition. Because mutual causality is absent and because settings and responses coincide in canonical settable systems, it is natural to simplify causal discourse by dropping explicit references to settable variables X and, for a < b; instead simply speaking about the direct e ects of, say, Y[a] on Y[b] or of y [a] on y [b]). For example, in the canonical strategy partition, it is sensible to say that s i and x i are direct causes of b s i ; whereas in the global partition only x i is a direct cause of b g i : Standard canonical settable systems support natural de nitions of indirect and total e ects of Y[a] on Y[b], as Chalak and White (2011) show. The patterns of total and indirect e ects can then imply speci c probabilistic conditional independence relations, supporting recovery of structural/causal information from observed data. The notions of indirect and total e ects extend to canonical topological settable systems, but we leave this analysis aside here for brevity. 4 Discussion As noted in the introduction, the original settable systems framework of CW is not able to support causal discourse in the N bidder private-information auction game. In contrast, topological settable systems are well suited to supporting causal discourse in this game. Further, since monotone pure strategies are elements of non-metrizable Helly space, versions of settable systems less general than topological settable systems will not be adequate to support causal discourse either in this game or in other important games of incomplete information. topological settable systems are not more general than necessary. In this respect, 17

18 There is, however, one feature of topological settable systems that we did not exploit in mapping the private-value auction game to our causal framework. This is the possibility of having a countable number of structurally exogenous (fundamental) variables or endogenous variables. This feature, however, is not super uous. It is required in a variety of economic contexts, not only in repeated games, time-series modeling, and adaptive learning (see CW, section 7; Chen and White, ), but also in Bayesian games of incomplete information with countably in nitedimensional type and action spaces, as in Reny s (2011) equilibrium existence theory for monotone pure-strategy Bayesian games. In fact, it appears straightforward to map topological settable systems to Reny s general setup, supporting causal discourse there. We forego this here, as the impressive generality of Reny s framework would interfere with our desire to make our discussion relatively transparent. Pure-strategy equilibrium exists when, among other things, types do not contain atoms that is, types do not take some value with positive probability. As Reny (2011) notes, when type spaces have atoms, pure-strategy monotone equilibrium need not exist; however, mixed-strategy equilibrium may exist under suitable conditions. Plausibly, topological settable systems also apply to mixed-strategy games. The key to showing this is to specify appropriate topological spaces to which the mixed strategy settings and responses belong. To see the basic idea, suppose that mixed-strategy bidders randomly choose among a countable number of pure strategies s ; s = 1; 2; :::. We can represent this random choice as 1X 1X S = s 1fS = sg = s 1f(X; ) = sg; s=1 where 1fg is the indicator function and S (X(); ) is a random integer whose realization s = (X(!);!) determines which strategy is played. A new feature here is the "strategy choice" function : I! N + ; here, we permit agent type 17 to in uence the probability of choosing a particular pure strategy. The mixed strategy settings B = S are thus random strategies, as de ned in Section 2. The individually rational partial equilibrium response B can be written 1X B = s 1f (X; ) = sg; s=1 where f 1; 2; :::g is the (selected) set of optimally chosen pure strategies and is the (selected) optimal strategy choice function. From this, we see that mixed strategy settings can be viewed as elements of ( 1 s=1h) S; where S is a collection of measurable mappings ; and that mixed strategy best responses take 16 Among their examples of recursive learning in Banach spaces, Chen and White (1998) consider a learning agent solving a stochastic dynamic programming problem and the game of ctitious play with continuum strategies, an in nitely repeated dynamic game of incomplete information. 17 The type could be countably dimensioned, but we suppress this possibility here for simplicity. s=1 18