Alvaro Rodrigues-Neto Research School of Economics, Australian National University. ANU Working Papers in Economics and Econometrics # 587

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1 Cycles of length two in monotonic models José Alvaro Rodrigues-Neto Research School of Economics, Australian National University ANU Working Papers in Economics and Econometrics # 587 October JEL: C02, D80, D82, D83 ISBN:

2 Cycles of length two in monotonic models José Alvaro Rodrigues-Neto October 16, 2012 Abstract In the context of partitional knowledge models, we prove that, in a monotonic model, any cycle equation can be obtained as the product of cycle equations corresponding to cycles of length two. Hence, if a model is monotonic, has nite states, and players posteriors satisfy all cycle equations corresponding to cycles of length two, then these posteriors are consistent (i.e., there is a common prior). We also propose a new and elegant proof for one of the main results of Rodrigues-Neto [9]. Key words: acyclic, cycle, knowledge, monotonicity, order, partition, posterior, prior. JEL Classi cation: C02, D80, D82, D83. Research School of Economics, Australian National University, Canberra ACT 0200, Australia. I would like to thank Scott McCracken and James Taylor for comments, and Merrilyn Lárusson for the English editing. All remaining errors are my own responsibility. URL: jarnwi@yahoo.com.

3 1 Introduction This paper investigates a particular implication of imposing a new condition on players knowledge partitions, named monotonicity. A companion paper introduces the concept of monotonic model. 1 A "model" refers to a state space, a nite set of players, and their respective partitions of the state space. 2 A model is monotonic if its state space admits a linear order such that all players partitions respect this linear order; i.e., for every player, every element of her partition consists of consecutive states. Harsanyi [4] develops the basic ideas that allow us to study games of incomplete information. Harsanyi s consistency problem asks if there is a common prior when we can only observe the posteriors of all players. One approach to answering this problem is via cycles and cycle equations. 3 We prove that, in monotonic models, any arbitrary cycle equation can be deduced from cycle equations corresponding to cycles of length two. In other words, if all cycle equations corresponding to cycles of length two hold, all other cycle equations also hold. In this case, if there are only nitely many states, then posteriors must be consistent (i.e., there is a common prior). This result simpli es the procedure proposed by Rodrigues-Neto [7] to check for consistency in nite state spaces. In the original procedure, each cycle of the so-called meet-join diagram is associated with a cycle equation, and posteriors are consistent if and only if all cycle equations are satis ed. In monotonic models, it su ces to check the cycle equations corresponding to the shortest cycles: those of length two. Rodrigues-Neto [8] de nes cycle spaces for generic knowledge models. If a monotonic model is not acyclic, we provide here a procedure for nding a basis for its cycle space that contains only cycles of length two. Hence, every cycle can be decomposed as a linear combination of cycles belonging to this particular collection. The cycles of this collection are made of edges linking consecutive states. This construction sheds more light on the mathematical structure that makes it possible to decompose any cycle into a linear combination of cycles of length two, if the model is monotonic. Moreover, this new construction also allows us to provide a new and simpler proof for one of the main results 1 See Rodrigues-Neto [9]. 2 For details of how partitions can represent the knowledge of players, see Aumann [1]. 3 See Rodrigues-Neto [7] and [8] or Hellwig [5] for more details on cycles and cycle equations. 1

4 of Rodrigues-Neto [9]. The "price" for this simpler proof is that it is based on additional mathematical concepts. Indeed, the original proof is longer, but it is self-contained, and does not require the concept of cycle space. Hellwig [5] proposes a di erent hypothesis on players partitions under which, to check for consistency, one needs to inspect only the cycle equations corresponding to cycles of length four or less. 4 Rodrigues-Neto [7] introduces the idea and formalization of cycles to study information structures and Harsanyi s consistency problem. Rodrigues-Neto [8] introduces the cycle spaces, formalizing the ideas of basis and dimension for vector spaces generated by cycles. Rodrigues-Neto [9] introduces the class of monotonic models and relates acyclicity to joins of pairs of players partitions in monotonic models. Morris [6], Feinberg [3], and Samet [11] and [12] provide alternative approaches to solve Harsanyi s consistency problem. Section 2 contains the main result. Section 3 creates a special basis for cycle spaces of monotonic models. The Appendix presents the proofs. 2 Monotonic Models and Cycles A model M = (; J; f j j j 2 Jg) is a triple with a state space, which may be nite with n 2 elements, or countably in nite, for instance = f1; 2; 3; g; a non-empty, nite set of players J; and a collection of partitions f j j j 2 Jg, one partition j = j 1; j 2; ; j L j for each player j. A partial order % on an arbitrary set A is any binary relation on A that is re exive, antisymmetric and transitive. A linear order % on A is a partial order on A such that we can always compare any two points of A. A strict linear order on A refers to the strict part of a linear order %; i.e., for any a; b 2 A, a b if and only if a % b and it is not the case that b % a. 4 Harsanyi type spaces are models such that each player has one of many possible types, the state space is the Cartesian product of all sets of possible types, and partitions are such that each player knows only her own type. The condition imposed by Hellwig [5] makes his class of models more general than Harsanyi spaces because he requires only that for every pair of players i; j and every element i of i s partition and j of j s partition, the intersection i \ j is non-empty. Intuitively speaking, Harsanyi spaces could be imagined as hyperrectangles (generalization of a rectangle for higher dimensions), while a monotonic model could be imagined as a line (one dimensional). 2

5 Let (A; ) be a strictly linearly ordered set. Fix arbitrary a; b 2 A. Elements a and b are said to be consecutive according to if either: (i) a b and there is no other element g 2 A such that a g b; or (ii) b a and there is no other element g 2 A such that b g a. Being consecutive is a symmetric relation. 5 Let j (!) denote the element of j s partition containing state!. The model M = (; J; f j j j 2 Jg) is monotonic if there is a strict linear order on such that for any j 2 J and pair of states ^! and! such that ^! 2 j (!), there is a nite sequence of states! 1 ;! 2 ; ;! k 2 j (!) satisfying: (i)! 1 = ^!; (ii)! k =!; and (iii) for every i 2 f1; ; k 1g, states! i and! i+1 are consecutive. Any model of a single player is monotonic. Any model of two players, where one of the players has full information, is an example of a monotonic model because the partition of the fully informed player does not impose any restriction on the relative order of the states. The information structure of Rubinstein s game is an example of a monotonic model. 6 Example 2 presents a model that is not monotonic. A posterior for player j 2 J is a collection j = ( j 1; ; j n), if = f1; 2; ; ng, and j = ( j 1; j 2; ), if = f1; 2; g, where, for each state! 2, j! > 0 denotes player j s posterior belief that the true state is! given that the true state belongs to j (!). At X! 2, player j knows that the true state belongs to j (! ); that is, j! = 1.!2 j (! ) A pro le of posteriors is a collection = ( 1 ; ; jjj ) of posteriors, one for each player. Consider a probability measure j = j 1; ; j n or j = j 1; j 2; over the space, and let j (B) = X!2B j!, for any event B. Measure j is a (full support) prior for player j if j! > 0 and j! = j!= j ( j (!)), for every! 2. De ne an oriented multigraph, the meet-join diagram, as follows: the set of nodes is. An oriented j-edge hj;! 1 ;! 2 i is a triple composed of a player j 2 J and an ordered pair of distinct states! 1 6=! 2 belonging to the same element of j s partition. For any i; j 2 J, the edge hj;! 3 ;! 4 i is said to be consecutive to the edge hi;! 1 ;! 2 i if and only if! 3 =! 2 or! 1 =! 4. De ne the opposite of the j-edge hj;! 1 ;! 2 i, denoted hj;! 1 ;! 2 i, as the j-edge that switches the orders of the states; that is, hj;! 1 ;! 2 i = 5 For details on relations and orders, see Davey and Priestley [2]. 6 See Rubinstein [10] for more details of the game. 3

6 hj;! 2 ;! 1 i. A path is a list of consecutive edges of the form p = fhj 1 ;! 1 ;! 2 i, hj 2 ;! 2 ;! 3 i,, hj k ;! k ;! k+1 ig. Let p denote the opposite path of p; that is, the path obtained from p by considering the opposites of all of its edges. A path p = fhj 1 ;! 1 ;! 2 i, hj 2 ;! 2 ;! 3 i,, hj k ;! k ;! k+1 ig is said to be a cycle of length k if and only if: (i) p is a closed path; that is,! k+1 =! 1 ; and (ii) if hj;!;! i 2 p, then hj;!;! i =2 p. Each cycle has a cycle equation associated to it. Let p be an arbitrary cycle: p = fhj 1 ;! 1 ;! 2 i ; hj 2 ;! 2 ;! 3 i ; hj 3 ;! 3 ;! 4 i ; ; hj k 1 ;! k 1 ;! k i ; hj k ;! k ;! 1 ig. The cycle equation associated with (or corresponding to) cycle p is: j 1!2 j 2!3 j k 2! k 1 j k 1! k j k!1 = j 1!1 j 2!2 j 3!3 j k 1! k 1 j k!k. Given a set of posteriors, one for each player, the consistency problem asks if there is a common prior; i.e., if any single prior can generate all posteriors. Posteriors are consistent if and only if all cycle equations are satis ed. 7 Depending on players partitions, there may be redundant cycle equations. Hence, it may not be necessary to check all cycle equations. In monotonic models, to obtain consistency it su ces to check the cycle equations associated with cycles of length two. When these cycle equations are satis ed, then, necessarily, all other cycle equations must also be satis ed. Proposition 1 Suppose that is nite. Assume that M = (; J; f j j j 2 Jg) is a monotonic model with two or more players. Fix players posteriors j, j 2 J. If all cycle equations corresponding to cycles of length two hold, then all other cycle equations must hold as well. Hence, there is a common prior. Example 1 Let = f1; 2; 3; 4; 5; 6; 7g, J = fa; B; C; Eg, A = ff1g, f2, 3, 4, 5, 6g, f7gg, B = ff1; 2; 3g, f4; 5g, f6; 7gg, C = ff1; 2; 3g ; f4g; f5; 6; 7gg, and E = ff1g, f2; 3; 4; 5; 6; 7gg. This is a monotonic model under the usual order of the real numbers. Consider the following cycle of length ve: 7 See Rodrigues-Neto [7]. p = fha; 2; 6i ; hb; 6; 7i ; hc; 7; 5i ; hb; 5; 4i ; he; 4; 2ig. 4

7 Its cycle equation is: Consider the following sets of edges: A 6 B 7 C 5 B 4 E 2 = A 2 B 6 C 7 B 5 E 4. U 0 = fha; 2; 6i ; hb; 6; 7ig, D 0 = fhc; 7; 5i ; hb; 5; 4i ; he; 4; 2ig. Set U 0 contains all edges that "move up", i.e., edges hj;!;! i such that state! is larger than state! according to the order on. Set D 0 has all edges that "move down". Next, decompose each edge in U 0 so that we end up with edges connecting consecutive states only. For instance, instead of ha; 2; 6i, we use the edges ha; 2; 3i, ha; 3; 4i, ha; 4; 5i, ha; 5; 6i. Then, we obtain sets U and D, respectively, from sets U 0 and D 0 as follows: U = fha; 2; 3i ; ha; 3; 4i ; ha; 4; 5i ; ha; 5; 6i ; hb; 6; 7ig, D = fhc; 7; 6i ; hc; 6; 5i ; hb; 5; 4i ; he; 4; 3i ; he; 3; 2ig. There is a bijective function f : U!D, de ned by f(ha; 2; 3i)= he; 3; 2i, f(ha; 3; 4i)= he; 4; 3i, f(ha; 4; 5i) = hb; 5; 4i, f(ha; 5; 6i) = hc; 6; 5i, and f(hb; 6; 7i) = hc; 7; 6i. Consider ve di erent cycles of length two and their corresponding cycle equations: p 1 = fha; 2; 3i ; he; 3; 2ig, A 3 E 2 = A 2 E 3, p 2 = fha; 3; 4i ; he; 4; 3ig, A 4 E 3 = A 3 E 4, p 3 = fha; 4; 5i ; hb; 5; 4ig, A 5 B 4 = A 4 B 5, p 4 = fha; 5; 6i ; hc; 6; 5ig, A 6 C 5 = A 5 C 6, p 5 = fhb; 6; 7i ; hc; 7; 6ig, B 7 C 6 = B 6 C 7. Multiplying the ve cycle equations corresponding to the cycles p 1, p 2, p 3, p 4, and p 5, we obtain the cycle equation corresponding to cycle p, after we cancel out the common factors. Example 2 Does the converse of Proposition 1 hold? In other words, suppose that all cycles can be decomposed in cycles of length two; would this imply that the model is 5

8 monotonic? The answer is no. To see why, consider the following counter example: = fa; b; c; x; y; zg, J =f1; 2g, 1 = ffa; b; cg, fxg, fyg, fzgg and 2 = ffa; xg; fb; yg; fc; zgg. This model satis es the property that every cycle can be decomposed in cycles of length two because it is acyclic. However, the model is not monotonic. 3 A Basis for the Cycle Space 3.1 Cycle Space of a Monotonic Model Fix a nite set and a model M = (; J; f j j j 2 Jg). Suppose that M is monotonic under the strict linear order. Without loss of generality, assume that = f1; 2; ; ng, and n n De ne a graph G, ~ named a directed version of M, as follows: (i) the nodes of G ~ are the states in ; and (ii) for every j 2 J, a j-edge is a triple hj;! p ;! q i composed of player j and two consecutive states! q! p belonging to the same element j r of partition j. State! p is the beginning point of the j-edge and! q is the end point of this j-edge. The j-edge hj;! p ;! q i links two consecutive states! q ;! p, with! q! p, belonging to the same element of j. The set E ~ of edges of G ~ is the set of all j-edges of G, ~ for all j 2 J. Path p = fhj 1 ;! 1 ;! 2 i, hj 2 ;! 2 ;! 3 i,, hj k ;! k ;! k+1 ig is increasing if! k+1! k! 2! 1. Path p is decreasing if path p is increasing. The j-edges of G ~ inside each j r 2 j connect, directly or indirectly, all nodes in j r. For every pair of distinct nodes of j r, there is unique increasing path connecting them. De ne the edge space of G, ~ denoted V ( G), ~ to be the real vector space spanned by ~E. An edge x 2 E ~ can be interpreted as a vector of V ( G). ~ For this, think of x as the vector X y y, with coe cients y = 1, if y = x; and y = 0, if y 6= x. Multiplication of y2 ~E the j-edge x = hj;! p ;! q i by the scalar 1 is associated with x = hj;! q ;! p i. 8 Set E ~ is linearly independent because we can never express any edge of G ~ as a linear combination of other edges of E. ~ By de nition, E ~ spans V ( G). ~ Hence, E ~ is a basis for V ( G), ~ and dim V ( G) ~ = E ~. Every vector v 2 V ( G) ~ is a linear combination of edges with real 8 Formally, x is not a j-edge of graph G, ~ but it is a j-edge of the meet-join diagram, and it belongs to the edge space, V ( G). ~ 6

9 coe cients, i.e., there are unique scalars x 2 R, with x 2 ~ E, such that v = X x2 ~E x x. For every v 1 = X x2 ~E x x and v 2 = X x2 ~E x x, with x ; x 2 R, de ne the sum in V ( ~ G) as v 1 + v 2 = X x2 ~E ( x + x )x, and the product of v 1 2 V ( ~ G) by 2 R as v 1 = X x2 ~E( x )x. De ne the cycle space, denoted W ( ~ G), as the sub-vector space of V ( ~ G) spanned by all cycles of ~ G. De ne the cyclomatic number of ~ G as the dimension of its cycle space, to be denoted C(J). 3.2 A Special Collection of Cycles of Length Two For each index i 2 f1; 2; ; n 1g, and each pair of distinct players j 6= j 0 2 J, de ne the cycle of length two c! (j; j 0 ) = fhj;!;! + 1i ; hj 0 ;! + 1;!ig, if state! +1 2 j (!)\ j0 (!). If state! + 1 =2 j (!) \ j0 (!), then c! (j; j 0 ) is not de ned. Let B denote the collection of all such cycles of length two. If! + 1 =2 j (!) \ j0 (!), neither of the cycles c! (j; j 0 ) or c! (j 0 ; j) are de ned. If state! j (!) \ j0 (!), both cycles are de ned, and c! (j; j 0 ) = c! (j 0 ; j). In particular, they have the same cycle equation. The cycles in collection B span the cycle space W ( G). ~ However, the collection B may be linearly dependent, as the following example illustrates. Example 3 Let = f1; 2g, 2 1, J = fj 1 ; j 2 ; j 3 g, and j 1 = j 2 = j 3 = fg. Then, c 1 (j 1 ; j 2 )= fhj 1 ; 1; 2i; hj 2 ; 2; 1ig, c 1 (j 1 ; j 3 )= fhj 1 ; 1; 2i; hj 3 ; 2; 1ig and c 1 (j 2 ; j 3 ) = fhj 2 ; 1; 2i ; hj 3 ; 2; 1ig. Because hj 2 ; 1; 2i = hj 2 ; 2; 1i, then c 1 (j 1 ; j 2 )+c 1 (j 2 ; j 3 ) = c 1 (j 1 ; j 3 ). Thus, the collection fc 1 (j 1 ; j 2 ); c 1 (j 2 ; j 3 ); c 1 (j 1 ; j 3 )g B is linearly dependent. It is possible that the collection B is empty, depending on the model. This is the case if and only if i + 1 =2 j (i) \ j0 (i), for any i 2 f1; 2; ; n 1g and any pair of distinct players j 6= j 0 2 J. This means that every pair of partitions j and j0 has a join made only of singletons (this implies that the model is acyclic; the next section presents a new proof). Proposition 2 In a monotonic model, every cycle can be decomposed as a linear combination of cycles in the collection B. If the collection B is non-empty, then B contains 7

10 a non-empty subset B of linearly independent cycles that span the cycle space; i.e., a basis for the cycle space. 3.3 A Simple Proof The argument in the next paragraph provides us with a new short proof for the main result of Rodrigues-Neto [9]: every monotonic model, in which all the joins of two players at a time contain only singletons, is necessarily acyclic. In a monotonic model, suppose that for every pair of di erent partitions, say j 6= j0, their join j _ j0 contains only singletons. Then, there is never a pair of di erent states belonging to the same element of j and also to the same element of j0. In particular, there are no two consecutive states belonging to the same element of j and, at the same time, belonging to the same element of j0. This means that there is no cycle of length two; i.e., B is empty. Hence, this monotonic model is acyclic because, by Proposition 2, the existence of any cycle would imply the existence of a cycle of length two. A Appendix: Proofs Proof of Proposition 1: If there is no cycle, the result is trivial. Consider any arbitrary cycle p of arbitrary length k 2. Fix any arbitrary strict linear order on that makes the model monotonic. Let U 0 be the set of edges hj;! ;! i of cycle p such that!!. Let D 0 be the set of edges hj;! ;! i of cycle p such that!!. Because cycle p is a closed path, U 0 6=? and D 0 6=?. Let U be the set of edges obtained from the edges of U 0 in the following way: for each edge hj;! ;! i 2 U 0, include in U the edges hj;! 1 ;! 2 i, hj;! 2 ;! 3 i,, hj;! k 1 ;! k i, such that! 1 =!,! k =!, and for every i 2 f1; ; k 1g,! i+1 and! i are consecutive according to, with! i+1! i. The monotonicity of the model implies that! 1,! 2,,! k belong to the same element of player j s partition. Hence, every hj;! i ;! i+1 i belongs to the meet-join diagram. Construct a set D from D 0 in an analogous way. Because U 0 6=? and D 0 6=?, then U 6=? and D 6=?. Because p is a closed path, there is a bijection f from U to D such that each edge u = hj U ;! i ;! i+1 i of U is associated to edge f(u) = hj D ;! i+1 ;! i i of D, j U 6= j D, 8

11 (edge u "moves up" from! i to! i+1 ; edge f(u) "goes down" from! i+1 to! i ). 9 Thus, every fu; f(u)g, for u 2 U, is a cycle of length two. The collection of cycle equations corresponding to the collection of cycles ffu; f(u)g j u 2 Ug generates the cycle equation corresponding to cycle p. Hence, if all cycle equations corresponding to the collection of cycles ffu; f(u)g j u 2 Ug are satis ed, then the cycle equation corresponding to cycle p must also be satis ed. Thus, all cycle equations hold. Because is nite, then, by the main result of Rodrigues-Neto [7], there is a common prior. Proof of Proposition 2: This proof follows similar steps to those in the previous argument. Given any arbitrary cycle c, the decomposition of c proposed in the proof of Proposition 1 leads to cycles of length two belonging to the collection B. If the collection B is non-empty, it must contain a basis for the cycle space because the previous part of this proposition states that B spans the cycle space. Every maximal linearly independent subset of B is a basis for the cycle space. There is at least one maximal linearly independent subset of B because B is non-empty and nite. References [1] Aumann, R., "Agreeing to Disagree," The Annals of Statistics 4 (1976), [2] Davey, B.A. and Priestley, H.A., "Introduction to lattices and order," Second Edition, Cambridge University Press, [3] Feinberg, Y., "Characterizing Common Priors in the Form of Posteriors," Journal of Economic Theory 91 (2000), [4] Harsanyi, J.C., "Games with Incomplete Information Played by Bayesian Players, I, II, III," Management Science 14 ( ), , , [5] Hellwig, M., "From Posteriors to Priors via Cycles: An Addendum," Working Paper Series of the Max Planck Institute for Research on Collective Goods 7, [6] Morris, S., "Trade with heterogeneous prior beliefs and asymmetric information," Econometrica 62 (1994), Function f might not be unique, but the argument stands. 9

12 [7] Rodrigues-Neto, J.A., "From Posteriors to Priors via Cycles," Journal of Economic Theory 144 (2009), [8] Rodrigues-Neto, J.A., "The Cycles Approach," Journal of Mathematical Economics 48 (2012), [9] Rodrigues-Neto, J.A., "Monotonic models and cycles," Working Papers in Economics and Econometrics, Research School of Economics, Australian National University, [10] Rubinstein, A., "The electronic mail game: Strategic Behavior Under Almost Common Knowledge," American Economic Review 79 (1989), [11] Samet, D., "Common Priors and the Separation of Convex Sets," Games and Economic Behavior 24 (1998), [12] Samet, D., "Iterated Expectations and Common Priors," Games and Economic Behavior 24 (1998),