Monotone equilibria in nonatomic supermodular games. A comment
Monotone equilibria in nonatomic supermodular games. A comment∗ Lukasz Balbus†
Kevin Reffett‡
Lukasz Wo´zny§
April 2014
Abstract Recently Yang and Qi (2013) stated an interesting theorem on existence of complete lattice of equilibria in a particular class of large nonatomics supermodular games for general action and types spaces. Unfortunately, their result is incorrect. In this note, we detail the nature of the problem with the stated theorem, provide a counterexample, and then correct the result under additional assumptions. We also provide a number of new results per existence of equilibria in more general classes of games of strategic complementarities that are monotone in players’ types, as well as provide a class of constructive methods for computing monotone equilibrium comparative statics. keywords:
large games, supermodular games, games with strategic complemen-
tarities, distributional equilibria, JEL codes: C72
1
Introduction
We follow the notation in Yang and Qi (2013) in this note. A semi-anonymous game with an ordered set of player types is a tuple Γ = (T, p, S, f ), where T is a partially ordered ∗
We thank Amanda Fridenberg and Ali Khan for very helpful discussions on various issues related to this paper. Lukasz Balbus and Lukasz Wo´zny also thanks the NCN grant No.UMO2012/07/D/HS4/01393 allowing to finance our visit at the Department of Economics, Arizona State University in February 2014. All the usual caveats apply. † Faculty of Mathematics, Computer Sciences and Econometrics, University of Zielona G´ora, Poland. ‡ Department of Economics, Arizona State University, USA. § Department of Quantitative Economics, Warsaw School of Economics, Warsaw, Poland. Address: al. Niepodleglo´sci 162, 02-554 Warszawa, Poland. E-mail: [email protected].
1
set (a poset) of players’ types, p ∈ P(T ) a probability distribution on the measurable space (T, B(T )) where B(T ) is a Borel σ−field generated by the interval topology, S is the action space of the players, f : S × P(T × S) × T → R is a payoff function where f (s, r, θ) denotes the payoff for player θ ∈ T, using action s ∈ S, while facing a joint type-action distribution given by r ∈ P(T × S). Denote by M(T, S) the set of Borel measurable functions from T to S endowed with the pointwise order, iT the identity mapping. Then a Nash equilibrium of the game Γ is given by measure x ∈ M(T, S) that satisfies the following: ∀θ ∈ T, ∀s ∈ S we have f (x(θ), p ◦ (iT , x)−1 , θ) ≥ f (s, p ◦ (iT , x)−1 , θ),
(1)
where p ◦ (iT , x)−1 is a joint type-action distribution on T × S implied by x. In Yang and Qi (2013), they assume that T is a complete chain1 and S a complete lattice. Endow P(T × S) with the first stochastic dominance order. The assumptions Yang and Qi (2013) impose are the following: Assumption 1 Assume: • s → f (s, r, θ) is order upper semi-continuous for each θ ∈ T and r ∈ P(T × S), • s → f (s, r, θ) is supermodular for each θ ∈ T and r ∈ P(T × S), • f has increasing differences with s and (θ, r). The main existence result of Yang and Qi (2013) states that under assumption (1), the set of monotone Nash equilibria in M(T, S) of the game Γ is a nonempty complete lattice. This result in obtained essentially as an application of the fixed point results in Veinott (1992) or Zhou (1994) (which in turn are generalizations of Tarski (1955) fixed point theorem) for strong set order ascending best response correspondences that map the set I(T, S) ∩ M(T, S) to itself, where I(T, S) denote the set of monotone (measurable) mappings from T to S. To prove their main theorem, and apply these well-known tools, Yang and Qi (2013) claim the following key lemma is true: Lemma 1 Suppose T is a complete chain and S is a complete lattice, then I(T, S) ⊂ M(T, S). 1 We say a poset X is a complete chain if X is a chain and for all C ⊂ X we have ∨C ∈ X and ∧C ∈ X.
2
This lemma is critical for the existence result in Yang and Qi (2013), it claims that any monotone increasing function on the complete chain is measurable. In fact, if correct, them lemma implies that the set of monotone and measurable functions: I(T, S)∩M(T, S) is a complete lattice under pointwise partial orders. Therefore, if additionally the best response correspondence transforms the space I(T, S) ∩ M(T, S) into I(T, S), then it necessarily maps into I(T, S) ∩ M(T, S). Unfortunately, lemma 1 is not correct, and as a consequence, the main existence theorem in the paper is incorrect as well. Specifically, and related to this issues raised in a recent paper Balbus, Dziewulski, Reffett, and Wo´zny (2014), not only is the proof of the existence in this game via an simple application of Veinott (1992) or Zhou (1994) generally nappropriate; it is in fact wrong (as the best response correspondence could map outside the set of measurable functions). In the remainder of paper, we present a counterexample to lemma 1 in the next section of the paper in theorem 1. In the next section of the paper, we discuss sufficient conditions for existence of equilibrium in a version of the actual game considering in Yang and Qi (2013). In the final section of the paper, we consider the question of monotone equilibria in a more general class of large games with strategic complementarities.
2
A counterexample
Consider some interval I of ordinal numbers2 , and let ω1 by the least number of the set {x ∈ I : #([0, x]) ≥ ℵ1 }. Clearly every initial segment of the subset [0, ω1 ) is countable but Ω := [0, ω1 ] has cardinality ℵ1 . Let T be a topology generated by open intervals. We claim that Lemma 2 σ({[0, x) ∪ (y, ω1 ], x, y ∈ Ω}) 6= σ(T ). Proof: Let A := {A ∈ Ω : (#A ≤ ℵ0 ) ∨ (#Ac ≤ ℵ0 )}. That is A is a set of all countable or co-countable sets. Clearly it is a σ-field. Moreover each initial segment [0, x) is countable if x < ω1 , and [0, ω1 ) is co-countable since its complement is {ω1 }. Moreover, (x, ω1 ] is co-countable since its complement is [0, x). As a result A = σ {A ∈ 2Ω : (A = [0, x)) ∨ (A = (x, ω1 ]) for some x ∈ Ω} . 2
Our construction in this section is similar to that presented in Theorem 1.14 page 19 and Example 12.9 page 439 in Aliprantis and Border (2006).
3
We will construct an open set outside A. Observe that successors set S is equinumerous as Ω. Let φ : Ω → Ω be defined as follows: φ(ω1 ) = 0 and φ(x) = x + 1 for x < ω1 . Clearly φ is bijection between Ω and S ∪ {0}. As a result Ω is equinumerous as S, hence S S is uncountable. But S ∈ T since S = (x − 1, x + 1) ∈ T . Clearly by Theorem 1.14, x∈S
page 19 in Aliprantis and Border (2006) L = S c is uncountable because is closed and contains {ω1 }. Hence S ∈ / A.
Definition 1 A set A ⊂ Ω is big if A ∪ {ω1 } includes an uncountable and T -closed set. A set is small if its complement is big. Lemma 3 There exists a no-Borel set D. That is there exists D ∈ / σ(T ). Proof: Observe that σ(T ) is a collection of the sets including big and small sets only. To construct non-Borel set, we know it needs to be neither big nor small. Let F be a collection of closed uncountable sets. Clearly for all F ∈ F, ω1 ∈ F . Let T F0 = F . Obviously ω1 ∈ F0 and F0 is closed. Applying Theorem 1.14 in Aliprantis F ∈F
and Border (2006) F0 is uncountable. Let D be uncountable proper subset of F0 . We claim that D is neither big nor small. To show D is not big let us take any F ⊂ D and F ∈ F. Then F must be countable. Hence D is not big. On a contrary suppose D is small. Then Dc is big. Then F0 ⊂ Dc . Since also D ⊂ F0 , hence D ⊂ Dc . This contradiction proves D is not small. Consequently D is not Borel set.
Theorem 1 There exists an increasing function f : Ω → Ω that is unmeasurable in σ(T ). Proof: From Lemma 3 there is a set D ⊂ Ω such that D is not σ(T ) measurable. Without loss of generality assume D does not contain ω1 . Define f (ω) = ω + 1 for ω ∈ [0, ω1 ) and f (ω1 ) = ω1 . Clearly f maps into S ∪ {ω1 }. Clearly S is an open set and all subsets of S are open. Then D + 1 := {x + 1 : x ∈ D} ⊂ S, hence it is open. Then f −1 (D + 1) = D. Hence f is no Borel.
Theorem 1 shows that Lemma 1 in Yang and Qi (2013) is false. The point is the authors show measurability of all isotone function f : (X, ≤) → (Y, ≤), but for Y equipped with σ field generated by intervals only. As we have shown in Lemma 2 the σ-field generated by open (or closed) intervals may be different from the σ-field generated by topology generated by open intervals. 4
3
Discussion and corrected result
There are various ways the lemma 1 (and hence the existence result of Yang and Qi (2013)) can be corrected by adding the additional assumptions, and/or changing the definition of the game. We discuss a few of them at this stage. First, observe the counterexample is based on the uncountability of the complete chain Ω := [0, ω1 ] . In fact, the statement of lemma 1 remains valid under the additional assumption for the type space T, the interval topology is second-countable. That is, in the case of a countable basis, σ-field generated by the open (or closed) intervals coincides with those generated by open intervals (and hence, coincides with its interval topology). Then, in fact, every monotone increasing function on such a complete chain will be (Borel) measurable (and hence, if the best response correspondence transforms the space I(T, S)∩ M(T, S) into I(T, S), it also maps into I(T, S) ∩ M(T, S), where I(T, S) ∩ M(T, S) is a complete lattice. Then, the existence result follows from a standard application of Veinott (1992) or Zhou (1994) . This shows that the strength of Yang and Qi (2013) theorem is limited to type-spaces can be embedded in Hilbert cube3 (See Theorem 4 in Appendix). In fact, order topology need not be embedded in real line (See Theorem 5 in Appendix) This limitation is significant, as recently many theoretical and applied papers in the literature of large games have assumed a saturated measure space of players types/traits (which clearly is not second-countable). For example, assumption that players’ types space is saturated is a necessary one if one wants to use the large game in a context of a Bayesian information game and apply the Law of Large Numbers (see Sun, 2006). Second, using very different methods, Balbus, Dziewulski, Reffett, and Wo´zny (2014) prove the existence of a distributional equilibrium in a class of large games with strategic complementarities (which includes large superrmodular games) using Markowsky (1976) fixed point theorem4 . That is, under similar assumptions as as those in Yang and Qi (2013), but allowing for more general spaces of agents types/traits, they manage to establish conditions under which that set of distributional equilibria possess the greatest and the least Nash equilibria where players use only monotone strategies (and equilibria are not necessarily monotone in types).5 Interestingly, the set of distributional equilibria is 3 ∞ P
Hilbert cube is [0, 1]N endowed with product topology. Clearly it is metrizable with metric d(x, y) :=
1 |x 2k k k=1 4
− yk | where x = (xk )k∈N , and y = (yk )k∈N
Markowsky (1976) presents the following theorem (see theorem 9 in his paper). Let F : X → X be increasing, and X aWchain complete poset. Then, the set of fixed points of F is a chain complete poset. V Moreover, we have {x : x ≤ F (x)} the greatest fixed point, and {x : x ≥ F (x)} the least fixed point of F . 5 That is, importantly, neither monotonicity of strategies, nor increasing differences between the action
5
not a complete lattice. This is due to the fact that the set of probability distributions on T × S order first order stochastic dominance is in general not even a lattice (let alone a complete lattice). Third, another way to re-establish a result on a complete lattice of Nash equilibria in a large supermodular game is to use the definition of equilibria in a large game due to Schmeidler (1973), where in his definition of Cournot-Nash equilibrium, players only a.e. optimize. This construction is also presented in Balbus, Dziewulski, Reffett, and Wo´zny (2014), where authors analyze the set of equivalence classes of measurable functions and apply Veinott (1992) fixed point result to a best response correspondence mapping such equivalence classes of measurable functions. To apply it to the Yang and Qi (2013) setting, one needs to modify their Nash equilibrium definition requiring Nash equilibrium definition 1 be satisfied for a.e. θ ∈ Θ. For more results along these lines but in the context of Bayesian games, see Vives (1990). Clearly, in such cases, the special case of monotone equilibrium can be established when suitable added restrictions are placed on the game.
4
General existence results
We now present in detail some additional results on monotone equilibrium existence using the construction found in Balbus, Dziewulski, Reffett, and Wo´zny (2014). Let T be a compact metric space of players characteristics/traits. Endow T with σ-field T , as well as a positive nonatomic σ-finite measure p. Mathematically, (T, T , p) is a measure space. Let S ⊂ RM (m ∈ N) be the player’s action space endowed with its natural product order and Euclidean topology. Endow the product space T × S with an order ≥p . Assume this order satisfies an implication (θ0 , s0 ) ≥p (θ, s) ⇒ s0 ≥ s. In what follows, we make an additional assumption. Assumption 2 Assume: • s → f (s, r, θ) is order continuous for each θ ∈ T and r ∈ P(T × S), • θ → f (s, r, θ) is measurable for each s ∈ S and r ∈ P(T × S). and types are required in the construction of Balbus, Dziewulski, Reffett, and Wo´zny (2014). This is important as ”monotonicity in names” is often a very restrictive form of equilibria to seek. Still, using their methods, results on equilibria that are monotone in types/names can be obtained if one assumes increasing differences between the action and types (as Yang and Qi (2013) do in their paper). We discuss this in the next section of the paper.
6
Theorem 2 Assume 1 and 2, then the set of monotone Nash equilibria of Γ is nonempty and possesses a greatest and least element. Proof: By theorem 3.1. of Balbus, Dziewulski, Reffett, and Wo´zny (2014), the set P(T × S) is a chain complete poset. Moreover, the greatest best response operator m(θ, r) = ∨ arg maxs∈S f (s, r, θ) is monotone by Topkis’ theorem. We next show that the θ → m(θ, r) is measurable. As f is a Carath´eodory for all r , by Theorem 18.19 in Aliprantis and Border (2006), we have Rr (θ) := max f (s, r, θ) s∈S
measurable, and the arg max correspondence m(θ, r) also measurable. As m(·, r) maps a measurable space into metrizable space, it is weakly measurable (see Aliprantis and Border, 2006, Theorem 18.2). Further, observe that m(θ, r) = (ˆ s1 , ..., sˆm ), where sˆi = max P roji (s) and P roji denotes a projection of a vector on its ith coordinate.6 Since a
s∈m(θ,r)
projection is a continuous function in this context, by the Measurable Maximum Theorem (see Aliprantis and Border (2006), Theorem 18.19), θ → m(θ, r) is p-measurable function. Next, consider an operator B(r)(G) = p({θ : (θ, m(θ, r)) ∈ G}) that maps P(T × S) into P(T × S). Clearly, B is monotone in the sense of first order stochastic dominance.As P(T × S) is chain complete, by Markowsky (1976) fixed point theorem, B possesses the greatest fixed point theorem, say r∗ , for which we construct the greatest Nash equilibrium x∗ with x∗ (θ) = m(θ, r∗ ). The existence of a least Nash equilibrium follows from a similar construction.
Few comments in this result are in order. First, this result establishes the existence of the greatest and the least Nash equilibrium of F in monotone strategies. In fact, if we drop increasing differencs between (s, θ) in our assumptions, our result still states existence of the greatest and the least Nash equilibrium of F . So, as opposed to Yang and Qi (2013), we obtain our result for general type spaces T , but at the cost of restricting S ⊂ RM . This latter restriction is necessary in our current construction to obtain measurability of the best response selector. Moreover, similar to corollary 3.1 in Balbus, Dziewulski, Reffett, and Wo´zny (2014) or theorem 2 in Yang and Qi (2013), we can establish the monotone comparative statics results. Finally, our result does not claim the set of Nash equilibria is a complete lattice; rather, at best, it is merely a chain complete poset. This happens, in particular, if B = B (i.e, when the best reply map is a function). 6
For si ∈ RM by max we denote a max of all coordinates.
7
Next, to re-establish complete lattice of Nash equilibria, we need to slightly modify the definition of the Nash equilibrium to be in line with that found in Schmeidler (1973): i.e., for p − a.e. θ ∈ T, ∀s ∈ S we have f (x(θ), p ◦ (iT , x)−1 , θ) ≥ f (s, p ◦ (iT , x)−1 , θ),
(2)
Now, endow the set M (T, S) with the following order for x0 , x ∈ M (T, S) we say x0 ≥ x iff x0 (θ) ≥ x(θ) p − a.e. θ. We can then we state the following result: Theorem 3 Assume 1 and 2. Then, the set of monotone Schmeidler-Nash equilibria of the game Γ is a nonempty complete lattice. Proof: By a result in Vives (1990), the set (M (T, S), ≥) is a complete lattice; hence, as I(T, S) subcomplete in (M (T, S), ≥), M (T, S) ∩ I(T, S) a complete lattice. We, next consider the operator B(x) = {x0 ∈ M (T, S) ∩ I(T, S) : x0 (θ) = arg max f (s, p ◦ (iT , x)−1 , θ) p − a.e. θ}. s∈S
Clearly, B(x) is nonempty (by the argument analogous to that in the proof of theorem 2). Moreover, B is monotone and maps M (T, S) ∩ I(T, S) into itself. Therefore, by the theorems of Veinott (1992) or Zhou (1994), the game Γ has a complete lattice of fixed points, and hence Schmeidler-Nash equilibria.
5
Appendix
In this appendix, we provide an answer to the question of what properties of type space T are sufficient for the key lemma in Yang and Qi (2013) to be correct. In particular, we discuss the case that of type spaces where the interval topology is second countable. Having this additional condition in place, lemma 1 will be correct. We begin with the following theorem: Theorem 4 Suppose T is a complete chain and order topology T≤ is second countable. Then, T≤ is separable metric space and can be embedded in a Hilbert cube. Proof: We first claim the base can be formed from countable union of open intervals (θk , θk )k∈N for some dense collection {θk , θk : k ∈ R}, where θk < θk (and, indeed, the 8
collection of the unions of intervals are closed for finite intersections). Take for any uncountable set T, Nt [[ t (θti , θi ). t∈T i=1 t
where for each t ∈ T {θti : i ∈ N} ⊂ {θi : i ∈ N} and {θi : i ∈ N} ⊂ {θi : i ∈ N}. Hence, t we have possibly uncountable union of the sets (θti , θi ); but for each i, there is index ki such that θti = θki for all but countably many t. By a similar argument, we have for t θi . Hence this union is in fact countable. Hence, the base is built from countably many intervals. Now, we show T≤ is a regular topology. Let x ∈ ∂F1 . Then, take neighbourhood of x, S Ex . For all y ∈ ∂F2 , we construct say Ex ∈ E , disjoint with F2 . Put G1 := F1c ∪ x∈∂F1 W neighbourhood of Dy ∈ E to be disjoint with G1 . Set ξy := {x ∈ F2 : x < y} and V ηy := {x ∈ F2 : x > y}. Clearly ξy , ηy ∈ F2 . By definition Eξy and Eηy do not include W V S y. Set Dy = Eξy , Eηy . Then , we set G2 := F2o ∪ Dy . Clearly, G1 ∩ G2 = ∅. y∈∂F2
Hence, T≤ is regular and Hausdorff. As a result by Urysohn Metrization Theorem (see Theorem 3.40 in Aliprantis and Border (2006)) (Θ, T≤ ) is metrizable space and can be embedded a in Hilbert cube.
Define Θ := Q × [0, 1] with Q as rational numbers. Define ≺ as lexicographical order i.e. (a, b) ≺ (x, y) ⇔ (a < x) ∨ (a = x, b ≤ y). Let T≺ be its order topology. Clearly base of T≺ is second countable with base sets Bq,I := {q} × I, where I ⊂ [0, 1] is interval with rational center and rational length. By Theorem 4, Θ can be embedded in Hilbert cube. For all nets {(xα , yα ) : α ∈ D} , we write (xα , yα ) →≺ (x, y) to denote that (x, y) is limit of (xα , yα ) in topology T≺ . As we show in next Theorem, it is not embedded in R. Theorem 5 There exists a complete chain T such that its order topology is second countable, and T is not embedded in R. Proof: Put T := Θ with ≺ its order topology. On a contrary, suppose u : T → R is continuous function. We show that it is not injection. For all y ∈ [0, 1], define f (x) = u(x, y) − u(x, 0). Then, f is continuous as well. Let xα ↑ x be some net, α ∈ D and D is some directed set. Suppose xα is not eventually constant. We show that (xα , y) →≺ (x, 0). 9
Set (x0 , y 0 ) ≺ (x, 0). If xα is not eventually constant. Then x0 < xα eventually and consequently (x0 , y 0 ) ≺ (xα , y) ≺ (x, 0). As a result net (xα , y) →≺ (x, 0). Hence f (xα ) = (xα , y) − u(xα , 0) → 0 for all y ∈ [0, 1]. Since f is continuous, hence f (xα ) → f (x), as a result f (x) ≡ 0. Consequently u(x, y) = u(x, 0) for all y ∈ [0, 1]. As a result u can not be injection, and T is not embedded in R.
References Aliprantis, C. D., and K. C. Border (2006): Infinite Dimentional Analysis. A Hitchhiker’s Guide. Springer Verlag: Heilbelberg, 3rd edn. Balbus, L., P. Dziewulski, K. Reffett, and L. Wo´ zny (2014): “A qualititive theory of large games with strategic complementarities,” Mimeo. Markowsky, G. (1976): “Chain-complete posets and directed sets with applications,” Algebra Universitae, 6, 53–68. Schmeidler, D. (1973): “Equilibrium points of nonatomic games,” Journal of Statisitical Physics, 17(4), 295–300. Sun, Y. (2006): “The exact law of large numbers via Fubini extension and characterization of insurable risks,” Journal of Economic Theory, 126(1), 31–69. Tarski, A. (1955): “A lattice-theoretical fixpoint theorem and its applications,” Pacific Journal of Mathematics, 5, 285–309. Veinott (1992): Lattice programming: qualitative optimization and equilibria. MS Standford. Vives, X. (1990): “Nash equilibrium with strategic complementarites,” Journal of Mathematical Economics, 19, 305–321. Yang, J., and X. Qi (2013): “The nonatomic supermodular game,” Games and Economic Behavior, 82(C), 609–620. Zhou, L. (1994): “The set of Nash equilibria of a supermodular game is a complete lattice,” Games and Economic Behavior, 7, 295–300.
10
Kevin Reffett‡
Lukasz Wo´zny§
April 2014
Abstract Recently Yang and Qi (2013) stated an interesting theorem on existence of complete lattice of equilibria in a particular class of large nonatomics supermodular games for general action and types spaces. Unfortunately, their result is incorrect. In this note, we detail the nature of the problem with the stated theorem, provide a counterexample, and then correct the result under additional assumptions. We also provide a number of new results per existence of equilibria in more general classes of games of strategic complementarities that are monotone in players’ types, as well as provide a class of constructive methods for computing monotone equilibrium comparative statics. keywords:
large games, supermodular games, games with strategic complemen-
tarities, distributional equilibria, JEL codes: C72
1
Introduction
We follow the notation in Yang and Qi (2013) in this note. A semi-anonymous game with an ordered set of player types is a tuple Γ = (T, p, S, f ), where T is a partially ordered ∗
We thank Amanda Fridenberg and Ali Khan for very helpful discussions on various issues related to this paper. Lukasz Balbus and Lukasz Wo´zny also thanks the NCN grant No.UMO2012/07/D/HS4/01393 allowing to finance our visit at the Department of Economics, Arizona State University in February 2014. All the usual caveats apply. † Faculty of Mathematics, Computer Sciences and Econometrics, University of Zielona G´ora, Poland. ‡ Department of Economics, Arizona State University, USA. § Department of Quantitative Economics, Warsaw School of Economics, Warsaw, Poland. Address: al. Niepodleglo´sci 162, 02-554 Warszawa, Poland. E-mail: [email protected].
1
set (a poset) of players’ types, p ∈ P(T ) a probability distribution on the measurable space (T, B(T )) where B(T ) is a Borel σ−field generated by the interval topology, S is the action space of the players, f : S × P(T × S) × T → R is a payoff function where f (s, r, θ) denotes the payoff for player θ ∈ T, using action s ∈ S, while facing a joint type-action distribution given by r ∈ P(T × S). Denote by M(T, S) the set of Borel measurable functions from T to S endowed with the pointwise order, iT the identity mapping. Then a Nash equilibrium of the game Γ is given by measure x ∈ M(T, S) that satisfies the following: ∀θ ∈ T, ∀s ∈ S we have f (x(θ), p ◦ (iT , x)−1 , θ) ≥ f (s, p ◦ (iT , x)−1 , θ),
(1)
where p ◦ (iT , x)−1 is a joint type-action distribution on T × S implied by x. In Yang and Qi (2013), they assume that T is a complete chain1 and S a complete lattice. Endow P(T × S) with the first stochastic dominance order. The assumptions Yang and Qi (2013) impose are the following: Assumption 1 Assume: • s → f (s, r, θ) is order upper semi-continuous for each θ ∈ T and r ∈ P(T × S), • s → f (s, r, θ) is supermodular for each θ ∈ T and r ∈ P(T × S), • f has increasing differences with s and (θ, r). The main existence result of Yang and Qi (2013) states that under assumption (1), the set of monotone Nash equilibria in M(T, S) of the game Γ is a nonempty complete lattice. This result in obtained essentially as an application of the fixed point results in Veinott (1992) or Zhou (1994) (which in turn are generalizations of Tarski (1955) fixed point theorem) for strong set order ascending best response correspondences that map the set I(T, S) ∩ M(T, S) to itself, where I(T, S) denote the set of monotone (measurable) mappings from T to S. To prove their main theorem, and apply these well-known tools, Yang and Qi (2013) claim the following key lemma is true: Lemma 1 Suppose T is a complete chain and S is a complete lattice, then I(T, S) ⊂ M(T, S). 1 We say a poset X is a complete chain if X is a chain and for all C ⊂ X we have ∨C ∈ X and ∧C ∈ X.
2
This lemma is critical for the existence result in Yang and Qi (2013), it claims that any monotone increasing function on the complete chain is measurable. In fact, if correct, them lemma implies that the set of monotone and measurable functions: I(T, S)∩M(T, S) is a complete lattice under pointwise partial orders. Therefore, if additionally the best response correspondence transforms the space I(T, S) ∩ M(T, S) into I(T, S), then it necessarily maps into I(T, S) ∩ M(T, S). Unfortunately, lemma 1 is not correct, and as a consequence, the main existence theorem in the paper is incorrect as well. Specifically, and related to this issues raised in a recent paper Balbus, Dziewulski, Reffett, and Wo´zny (2014), not only is the proof of the existence in this game via an simple application of Veinott (1992) or Zhou (1994) generally nappropriate; it is in fact wrong (as the best response correspondence could map outside the set of measurable functions). In the remainder of paper, we present a counterexample to lemma 1 in the next section of the paper in theorem 1. In the next section of the paper, we discuss sufficient conditions for existence of equilibrium in a version of the actual game considering in Yang and Qi (2013). In the final section of the paper, we consider the question of monotone equilibria in a more general class of large games with strategic complementarities.
2
A counterexample
Consider some interval I of ordinal numbers2 , and let ω1 by the least number of the set {x ∈ I : #([0, x]) ≥ ℵ1 }. Clearly every initial segment of the subset [0, ω1 ) is countable but Ω := [0, ω1 ] has cardinality ℵ1 . Let T be a topology generated by open intervals. We claim that Lemma 2 σ({[0, x) ∪ (y, ω1 ], x, y ∈ Ω}) 6= σ(T ). Proof: Let A := {A ∈ Ω : (#A ≤ ℵ0 ) ∨ (#Ac ≤ ℵ0 )}. That is A is a set of all countable or co-countable sets. Clearly it is a σ-field. Moreover each initial segment [0, x) is countable if x < ω1 , and [0, ω1 ) is co-countable since its complement is {ω1 }. Moreover, (x, ω1 ] is co-countable since its complement is [0, x). As a result A = σ {A ∈ 2Ω : (A = [0, x)) ∨ (A = (x, ω1 ]) for some x ∈ Ω} . 2
Our construction in this section is similar to that presented in Theorem 1.14 page 19 and Example 12.9 page 439 in Aliprantis and Border (2006).
3
We will construct an open set outside A. Observe that successors set S is equinumerous as Ω. Let φ : Ω → Ω be defined as follows: φ(ω1 ) = 0 and φ(x) = x + 1 for x < ω1 . Clearly φ is bijection between Ω and S ∪ {0}. As a result Ω is equinumerous as S, hence S S is uncountable. But S ∈ T since S = (x − 1, x + 1) ∈ T . Clearly by Theorem 1.14, x∈S
page 19 in Aliprantis and Border (2006) L = S c is uncountable because is closed and contains {ω1 }. Hence S ∈ / A.
Definition 1 A set A ⊂ Ω is big if A ∪ {ω1 } includes an uncountable and T -closed set. A set is small if its complement is big. Lemma 3 There exists a no-Borel set D. That is there exists D ∈ / σ(T ). Proof: Observe that σ(T ) is a collection of the sets including big and small sets only. To construct non-Borel set, we know it needs to be neither big nor small. Let F be a collection of closed uncountable sets. Clearly for all F ∈ F, ω1 ∈ F . Let T F0 = F . Obviously ω1 ∈ F0 and F0 is closed. Applying Theorem 1.14 in Aliprantis F ∈F
and Border (2006) F0 is uncountable. Let D be uncountable proper subset of F0 . We claim that D is neither big nor small. To show D is not big let us take any F ⊂ D and F ∈ F. Then F must be countable. Hence D is not big. On a contrary suppose D is small. Then Dc is big. Then F0 ⊂ Dc . Since also D ⊂ F0 , hence D ⊂ Dc . This contradiction proves D is not small. Consequently D is not Borel set.
Theorem 1 There exists an increasing function f : Ω → Ω that is unmeasurable in σ(T ). Proof: From Lemma 3 there is a set D ⊂ Ω such that D is not σ(T ) measurable. Without loss of generality assume D does not contain ω1 . Define f (ω) = ω + 1 for ω ∈ [0, ω1 ) and f (ω1 ) = ω1 . Clearly f maps into S ∪ {ω1 }. Clearly S is an open set and all subsets of S are open. Then D + 1 := {x + 1 : x ∈ D} ⊂ S, hence it is open. Then f −1 (D + 1) = D. Hence f is no Borel.
Theorem 1 shows that Lemma 1 in Yang and Qi (2013) is false. The point is the authors show measurability of all isotone function f : (X, ≤) → (Y, ≤), but for Y equipped with σ field generated by intervals only. As we have shown in Lemma 2 the σ-field generated by open (or closed) intervals may be different from the σ-field generated by topology generated by open intervals. 4
3
Discussion and corrected result
There are various ways the lemma 1 (and hence the existence result of Yang and Qi (2013)) can be corrected by adding the additional assumptions, and/or changing the definition of the game. We discuss a few of them at this stage. First, observe the counterexample is based on the uncountability of the complete chain Ω := [0, ω1 ] . In fact, the statement of lemma 1 remains valid under the additional assumption for the type space T, the interval topology is second-countable. That is, in the case of a countable basis, σ-field generated by the open (or closed) intervals coincides with those generated by open intervals (and hence, coincides with its interval topology). Then, in fact, every monotone increasing function on such a complete chain will be (Borel) measurable (and hence, if the best response correspondence transforms the space I(T, S)∩ M(T, S) into I(T, S), it also maps into I(T, S) ∩ M(T, S), where I(T, S) ∩ M(T, S) is a complete lattice. Then, the existence result follows from a standard application of Veinott (1992) or Zhou (1994) . This shows that the strength of Yang and Qi (2013) theorem is limited to type-spaces can be embedded in Hilbert cube3 (See Theorem 4 in Appendix). In fact, order topology need not be embedded in real line (See Theorem 5 in Appendix) This limitation is significant, as recently many theoretical and applied papers in the literature of large games have assumed a saturated measure space of players types/traits (which clearly is not second-countable). For example, assumption that players’ types space is saturated is a necessary one if one wants to use the large game in a context of a Bayesian information game and apply the Law of Large Numbers (see Sun, 2006). Second, using very different methods, Balbus, Dziewulski, Reffett, and Wo´zny (2014) prove the existence of a distributional equilibrium in a class of large games with strategic complementarities (which includes large superrmodular games) using Markowsky (1976) fixed point theorem4 . That is, under similar assumptions as as those in Yang and Qi (2013), but allowing for more general spaces of agents types/traits, they manage to establish conditions under which that set of distributional equilibria possess the greatest and the least Nash equilibria where players use only monotone strategies (and equilibria are not necessarily monotone in types).5 Interestingly, the set of distributional equilibria is 3 ∞ P
Hilbert cube is [0, 1]N endowed with product topology. Clearly it is metrizable with metric d(x, y) :=
1 |x 2k k k=1 4
− yk | where x = (xk )k∈N , and y = (yk )k∈N
Markowsky (1976) presents the following theorem (see theorem 9 in his paper). Let F : X → X be increasing, and X aWchain complete poset. Then, the set of fixed points of F is a chain complete poset. V Moreover, we have {x : x ≤ F (x)} the greatest fixed point, and {x : x ≥ F (x)} the least fixed point of F . 5 That is, importantly, neither monotonicity of strategies, nor increasing differences between the action
5
not a complete lattice. This is due to the fact that the set of probability distributions on T × S order first order stochastic dominance is in general not even a lattice (let alone a complete lattice). Third, another way to re-establish a result on a complete lattice of Nash equilibria in a large supermodular game is to use the definition of equilibria in a large game due to Schmeidler (1973), where in his definition of Cournot-Nash equilibrium, players only a.e. optimize. This construction is also presented in Balbus, Dziewulski, Reffett, and Wo´zny (2014), where authors analyze the set of equivalence classes of measurable functions and apply Veinott (1992) fixed point result to a best response correspondence mapping such equivalence classes of measurable functions. To apply it to the Yang and Qi (2013) setting, one needs to modify their Nash equilibrium definition requiring Nash equilibrium definition 1 be satisfied for a.e. θ ∈ Θ. For more results along these lines but in the context of Bayesian games, see Vives (1990). Clearly, in such cases, the special case of monotone equilibrium can be established when suitable added restrictions are placed on the game.
4
General existence results
We now present in detail some additional results on monotone equilibrium existence using the construction found in Balbus, Dziewulski, Reffett, and Wo´zny (2014). Let T be a compact metric space of players characteristics/traits. Endow T with σ-field T , as well as a positive nonatomic σ-finite measure p. Mathematically, (T, T , p) is a measure space. Let S ⊂ RM (m ∈ N) be the player’s action space endowed with its natural product order and Euclidean topology. Endow the product space T × S with an order ≥p . Assume this order satisfies an implication (θ0 , s0 ) ≥p (θ, s) ⇒ s0 ≥ s. In what follows, we make an additional assumption. Assumption 2 Assume: • s → f (s, r, θ) is order continuous for each θ ∈ T and r ∈ P(T × S), • θ → f (s, r, θ) is measurable for each s ∈ S and r ∈ P(T × S). and types are required in the construction of Balbus, Dziewulski, Reffett, and Wo´zny (2014). This is important as ”monotonicity in names” is often a very restrictive form of equilibria to seek. Still, using their methods, results on equilibria that are monotone in types/names can be obtained if one assumes increasing differences between the action and types (as Yang and Qi (2013) do in their paper). We discuss this in the next section of the paper.
6
Theorem 2 Assume 1 and 2, then the set of monotone Nash equilibria of Γ is nonempty and possesses a greatest and least element. Proof: By theorem 3.1. of Balbus, Dziewulski, Reffett, and Wo´zny (2014), the set P(T × S) is a chain complete poset. Moreover, the greatest best response operator m(θ, r) = ∨ arg maxs∈S f (s, r, θ) is monotone by Topkis’ theorem. We next show that the θ → m(θ, r) is measurable. As f is a Carath´eodory for all r , by Theorem 18.19 in Aliprantis and Border (2006), we have Rr (θ) := max f (s, r, θ) s∈S
measurable, and the arg max correspondence m(θ, r) also measurable. As m(·, r) maps a measurable space into metrizable space, it is weakly measurable (see Aliprantis and Border, 2006, Theorem 18.2). Further, observe that m(θ, r) = (ˆ s1 , ..., sˆm ), where sˆi = max P roji (s) and P roji denotes a projection of a vector on its ith coordinate.6 Since a
s∈m(θ,r)
projection is a continuous function in this context, by the Measurable Maximum Theorem (see Aliprantis and Border (2006), Theorem 18.19), θ → m(θ, r) is p-measurable function. Next, consider an operator B(r)(G) = p({θ : (θ, m(θ, r)) ∈ G}) that maps P(T × S) into P(T × S). Clearly, B is monotone in the sense of first order stochastic dominance.As P(T × S) is chain complete, by Markowsky (1976) fixed point theorem, B possesses the greatest fixed point theorem, say r∗ , for which we construct the greatest Nash equilibrium x∗ with x∗ (θ) = m(θ, r∗ ). The existence of a least Nash equilibrium follows from a similar construction.
Few comments in this result are in order. First, this result establishes the existence of the greatest and the least Nash equilibrium of F in monotone strategies. In fact, if we drop increasing differencs between (s, θ) in our assumptions, our result still states existence of the greatest and the least Nash equilibrium of F . So, as opposed to Yang and Qi (2013), we obtain our result for general type spaces T , but at the cost of restricting S ⊂ RM . This latter restriction is necessary in our current construction to obtain measurability of the best response selector. Moreover, similar to corollary 3.1 in Balbus, Dziewulski, Reffett, and Wo´zny (2014) or theorem 2 in Yang and Qi (2013), we can establish the monotone comparative statics results. Finally, our result does not claim the set of Nash equilibria is a complete lattice; rather, at best, it is merely a chain complete poset. This happens, in particular, if B = B (i.e, when the best reply map is a function). 6
For si ∈ RM by max we denote a max of all coordinates.
7
Next, to re-establish complete lattice of Nash equilibria, we need to slightly modify the definition of the Nash equilibrium to be in line with that found in Schmeidler (1973): i.e., for p − a.e. θ ∈ T, ∀s ∈ S we have f (x(θ), p ◦ (iT , x)−1 , θ) ≥ f (s, p ◦ (iT , x)−1 , θ),
(2)
Now, endow the set M (T, S) with the following order for x0 , x ∈ M (T, S) we say x0 ≥ x iff x0 (θ) ≥ x(θ) p − a.e. θ. We can then we state the following result: Theorem 3 Assume 1 and 2. Then, the set of monotone Schmeidler-Nash equilibria of the game Γ is a nonempty complete lattice. Proof: By a result in Vives (1990), the set (M (T, S), ≥) is a complete lattice; hence, as I(T, S) subcomplete in (M (T, S), ≥), M (T, S) ∩ I(T, S) a complete lattice. We, next consider the operator B(x) = {x0 ∈ M (T, S) ∩ I(T, S) : x0 (θ) = arg max f (s, p ◦ (iT , x)−1 , θ) p − a.e. θ}. s∈S
Clearly, B(x) is nonempty (by the argument analogous to that in the proof of theorem 2). Moreover, B is monotone and maps M (T, S) ∩ I(T, S) into itself. Therefore, by the theorems of Veinott (1992) or Zhou (1994), the game Γ has a complete lattice of fixed points, and hence Schmeidler-Nash equilibria.
5
Appendix
In this appendix, we provide an answer to the question of what properties of type space T are sufficient for the key lemma in Yang and Qi (2013) to be correct. In particular, we discuss the case that of type spaces where the interval topology is second countable. Having this additional condition in place, lemma 1 will be correct. We begin with the following theorem: Theorem 4 Suppose T is a complete chain and order topology T≤ is second countable. Then, T≤ is separable metric space and can be embedded in a Hilbert cube. Proof: We first claim the base can be formed from countable union of open intervals (θk , θk )k∈N for some dense collection {θk , θk : k ∈ R}, where θk < θk (and, indeed, the 8
collection of the unions of intervals are closed for finite intersections). Take for any uncountable set T, Nt [[ t (θti , θi ). t∈T i=1 t
where for each t ∈ T {θti : i ∈ N} ⊂ {θi : i ∈ N} and {θi : i ∈ N} ⊂ {θi : i ∈ N}. Hence, t we have possibly uncountable union of the sets (θti , θi ); but for each i, there is index ki such that θti = θki for all but countably many t. By a similar argument, we have for t θi . Hence this union is in fact countable. Hence, the base is built from countably many intervals. Now, we show T≤ is a regular topology. Let x ∈ ∂F1 . Then, take neighbourhood of x, S Ex . For all y ∈ ∂F2 , we construct say Ex ∈ E , disjoint with F2 . Put G1 := F1c ∪ x∈∂F1 W neighbourhood of Dy ∈ E to be disjoint with G1 . Set ξy := {x ∈ F2 : x < y} and V ηy := {x ∈ F2 : x > y}. Clearly ξy , ηy ∈ F2 . By definition Eξy and Eηy do not include W V S y. Set Dy = Eξy , Eηy . Then , we set G2 := F2o ∪ Dy . Clearly, G1 ∩ G2 = ∅. y∈∂F2
Hence, T≤ is regular and Hausdorff. As a result by Urysohn Metrization Theorem (see Theorem 3.40 in Aliprantis and Border (2006)) (Θ, T≤ ) is metrizable space and can be embedded a in Hilbert cube.
Define Θ := Q × [0, 1] with Q as rational numbers. Define ≺ as lexicographical order i.e. (a, b) ≺ (x, y) ⇔ (a < x) ∨ (a = x, b ≤ y). Let T≺ be its order topology. Clearly base of T≺ is second countable with base sets Bq,I := {q} × I, where I ⊂ [0, 1] is interval with rational center and rational length. By Theorem 4, Θ can be embedded in Hilbert cube. For all nets {(xα , yα ) : α ∈ D} , we write (xα , yα ) →≺ (x, y) to denote that (x, y) is limit of (xα , yα ) in topology T≺ . As we show in next Theorem, it is not embedded in R. Theorem 5 There exists a complete chain T such that its order topology is second countable, and T is not embedded in R. Proof: Put T := Θ with ≺ its order topology. On a contrary, suppose u : T → R is continuous function. We show that it is not injection. For all y ∈ [0, 1], define f (x) = u(x, y) − u(x, 0). Then, f is continuous as well. Let xα ↑ x be some net, α ∈ D and D is some directed set. Suppose xα is not eventually constant. We show that (xα , y) →≺ (x, 0). 9
Set (x0 , y 0 ) ≺ (x, 0). If xα is not eventually constant. Then x0 < xα eventually and consequently (x0 , y 0 ) ≺ (xα , y) ≺ (x, 0). As a result net (xα , y) →≺ (x, 0). Hence f (xα ) = (xα , y) − u(xα , 0) → 0 for all y ∈ [0, 1]. Since f is continuous, hence f (xα ) → f (x), as a result f (x) ≡ 0. Consequently u(x, y) = u(x, 0) for all y ∈ [0, 1]. As a result u can not be injection, and T is not embedded in R.
References Aliprantis, C. D., and K. C. Border (2006): Infinite Dimentional Analysis. A Hitchhiker’s Guide. Springer Verlag: Heilbelberg, 3rd edn. Balbus, L., P. Dziewulski, K. Reffett, and L. Wo´ zny (2014): “A qualititive theory of large games with strategic complementarities,” Mimeo. Markowsky, G. (1976): “Chain-complete posets and directed sets with applications,” Algebra Universitae, 6, 53–68. Schmeidler, D. (1973): “Equilibrium points of nonatomic games,” Journal of Statisitical Physics, 17(4), 295–300. Sun, Y. (2006): “The exact law of large numbers via Fubini extension and characterization of insurable risks,” Journal of Economic Theory, 126(1), 31–69. Tarski, A. (1955): “A lattice-theoretical fixpoint theorem and its applications,” Pacific Journal of Mathematics, 5, 285–309. Veinott (1992): Lattice programming: qualitative optimization and equilibria. MS Standford. Vives, X. (1990): “Nash equilibrium with strategic complementarites,” Journal of Mathematical Economics, 19, 305–321. Yang, J., and X. Qi (2013): “The nonatomic supermodular game,” Games and Economic Behavior, 82(C), 609–620. Zhou, L. (1994): “The set of Nash equilibria of a supermodular game is a complete lattice,” Games and Economic Behavior, 7, 295–300.
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