Order on Types based on Monotone Comparative Statics

Transcription

1 Order on Types based on Monotone Comparative Statics Takashi Kunimoto Takuro Yamashita July 10, 2015

2 Monotone comparative statics Comparative statics is important in Economics E.g., Wealth Consumption Cost Production in Cournot Econ fundamental Investment Implicit function theorem Monotone comparative statics No concavity or differentiability assumptions Single-crossing / Supermodularity Supermodular games ( strategic complementarity ) Our goal: To enlarge applicability of MCS.

3 E.g., Coordinating investment You invest in a project, with your partner. Invest Not Invest θ +b θ Not 0 0 θ R is the investment s profitability. Monotone comparative statics (Milgrom and Roberts (1990)): Suppose θ is common knowledge. As θ increases, the set of equilibrium investments (weakly) increases. What if θ is unobservable?

4 Incomplete information about θ Popular approach (e.g., global games, Athey (2002)): Each player i observes si R, a noisy signal about θ. Common prior on (θ,s1,s 2 ). i s posterior on (θ,s i ) s i is based on Bayesian updating. Monotone comparative statics (Athey (2002)): Suppose the common prior on (θ,s) exhibits affiliation. As s i increases, the set of i s equilibrium investment (weakly) increases. Are they really necessary?: One-dim signal structure (cf. dynamics/info acquisition)? Common prior (cf. agree to disagree /overconfidence)? Bayesian updating (cf. frequentist/inattention/naivete)? None of them is necessary...! What do we really need?

5 Intuition News on TV: Merkel is optimistic about Greece. Do you invest more? 1. Suppose the news makes you optimistic about θ. Is it enough for you to invest more? Not necessarily. You partner may become rather pessimistic! 2. Suppose the news makes you (i) optimistic about θ, and (ii) optimistic about your partner s optimism about θ. Is it enough for you to invest more? Not necessarily. For you to make invest more, it should be (i) θ is higher, and (ii ) your partner invests more. However, your partner being optimistic about θ is not enough for him/her to invest. The logic can continue up to arbitrary high levels.

6 Answer 1: Sufficiency A1. i takes a higher equilibrium action (in any supermodular game parametrized by Θ) if his belief becomes higher in the order of common certainty (knowledge) of optimism. In words, with the news, you become confident that it s good news for everyone, and this becomes common knowledge. So everyone must agree that it is a good news. But how good? You don t need to agree.

7 Answer 2: Necessity A2. If i s belief changes in a way that does not satisfy common certainty of optimism, then there exists a supermodular game (parametreized by Θ) such that i does not play a higher equilibrium action. Belief-elicitation game. (High-order version of) beauty contest game.

8 Roadmap Model (High-order) beliefs Supermodular games and Equilibria. Sufficient condition for MCS Common Certainty of Optimism Necessity Belief-elicitation game Applications (mostly under construction)

9 FOSD Let X be a complete and separable metric space (Polish), endowed with a (closed) partial order (e.g., Θ). E.g., finite, [0,1], R d, (well-behaved) function space... Definition A closed set Y X is called an upper event if [y Y and z y] implies z Y. Let U(X) denote the set of all upper events of X. Definition Let β,β (X). β (first-order) stochastically dominates β (denoted β SD β ) if, Y U(X), β(y) β (Y). β is more optimistic than β.

10 Model N(= 2) players. θ Θ: payoff-relevant state Θ is a complete and separable metric space (Polish). Endowed with Borel σ-algebra. Endowed with a (closed) partial order Θ. Harsanyi type space T = (T 1,T 2,b 1,b 2 ), where Ti is a (measurable) set of i s types bi : T i (Θ T j ) is i s (measurable) belief map.

11 Belief hierarchy From b i (t i ) (Θ T j ), we can compute i s entire belief hierarchy, h i (t i ) = (h 1 i (t i),h 2 i (t i),...): 1st-order belief: h 1 i(t i) (Θ) = H 1 is the marginal of b i(t i) on Θ. 2nd-order belief: h 2 i(t i) (Θ H 1 ) = (Θ (Θ)) = H 2 is the joint distribution for θ and j s 1st-order belief. k-th-order belief: h k i (t i) (Θ H 1 H k 1 ). Definition t i is higher than t i in the order of common certainty of optimism (denoted by t i CCO t i ) if, for all k, h k i (t i ) SD h k i (t i). 1st: You become optimistic about Θ. 2nd: You also become optimistic about the optimism of your partner about Θ. 3rd: You also become optimistic about the optimism of your partner about your optimism (about Θ), and so on.

12 Supermodular game Definition (N,(A i,u i ) i=1,2,θ) is a supermodular game (parametrized by Θ) if A i is a (measurable) set of i s actions. A i is a complete lattice with a partial order Ai. i.e., any subset Ãi of A i has the smallest upper bound (supã i ) and the greatest lower bound (infãi) in A i (e.g., A i = [0,1]). Payoff (first imagine A i R): For each (ai,a j,θ) (a i,a j,θ ), u i (a i,a j,θ) u i (a i,a j,θ) u i (a i,a j,θ ) u i (a i,a j,θ ). Strategic complementarity For the case with general A i : for each (a,θ),(a,θ ) with (a j,θ) (a j,θ ), u i (sup{(a,θ),(a,θ )})+u i (inf{(a,θ),(a,θ )}) u i (a,θ)+u i (a,θ )

13 Equilibria i s (pure) strategy is a measurable map σ i : T i A i. Definition σ is a (pure-strategy) Bayesian equilibrium if for each i,t i,a i, E ti u i (σ (t),θ) u i (σi (t i),σj (t j),θ)db i (θ,t j t i ) (θ,t j ) (θ,t j ) u i (a i,σ j (t j ),θ)db i (θ,t j t i ). Possibly not exist: Assume it exists (e.g., compactness/continuity/etc in Van Zandt and Vives, 2007). Possibly multiple equilibria: Talk about the set of equilibria, Σ.

14 Our question Q. Suppose that news makes your type (belief) changes from t i to t i. What are the conditions on t i and t i such that Σ i (t i) Ai Σ i (t i )? What news is a good (or higher-action-enhancing ) news? (Behaviorally meaningful) partial order on types? A. t i CCO t i Σ i (t i) Ai Σ i (t i ), supermodular games. Σ has a lattice structure: the least eqm, minσ, and the greatest eqm, maxσ, s.t. any σ Σ satisfies minσ σ maxσ. In the following, we focus on σ = minσ and show [t i CCO t i ] [σ i (t i) Ai σi (t i ), supermodular games]. WLOG, because the same result is true for maxσ.

15 Interim correlated rationalizability σ is obtained by iterative elimination of dominated actions (Interim correlated rationalizability of Dekel, Fudenberg, and Morris (2007)), from below. (Step 1) Let a 0 j (t j) = mina j for all t j, and let A 1 i (t i ) = argmaxe ti u i (a i,a 0 a i j(t j ),θ), and a 1 i (t i) = infa 1 i (t i). By supermodularity, ai Ai a 1 i (t i) is strictly dominated for t i. Remark: If h 1 i (t i ) SD h 1 i (t i ), then a1 i (t i) Ai a 1 i (t i ).

16 Iterative elimination of dominated actions (Step 2) Let A 2 i (t i ) = argmaxe ti u i (a i,a 1 a i j(t j ),θ), and a 2 i (t i) = infa 2 i (t i). By supermodularity, ai Ai a 2 i (t i) is iteratively strictly dominated for t i. Recall [h 1 j (t j ) SD h 1 j (t j ) a1 j (t j) Aj a 1 j (t j )]. Thus, if h 2 i (t i ) SD h 2 i (t i ), then t i puts more weight on higher (a j,θ). Therefore, a 2 i (t i) Ai a 2 i (t i ). And so on, for each step. Define a i (t i ) = sup{a 1 i (t i ),a 2 i(t i ),...}. Then, ti CCO t i implies a i (t i ) Ai a i (t i ).

17 Sufficiency Assumption For each i and t i, (i) each A k i (t i) is well-defined, and (ii) a i (t i ) is a best response to a j. (E.g., Van Zandt and Vives (2007)) Proposition σ = a. Theorem t i CCO t i σ i (t i) Ai σ i (t i ).

18 Toward necessity What if t i CCO t i? Theorem If t i CCO t i, then there exists a supermodular game such that (σ exists and) σi (t i) Ai σi (t i ). In fact, there exists a single game, the belief-elicitation game, that works for all i, t i and t i with t i CCO t i. Before describing the general belief-elicitation game, we first provide some intuition through examples.

19 Example 1: Violation in 1st-order Suppose h 1 i (t i) SD h 1 i (t i ). There is an upper event Y U(Θ) such that h 1 i (Y t i) > h 1 i (Y t i ). E.g., h 1 i (Y t i ) = 0.7 and h 1 i (Y t i ) = 0.3. Consider the game such that i buys (or not) a lottery that yields 1 if θ Y. Price 0.5. j does nothing. ai {0,1}, and u i (a i,θ) = a i (1 {θ Y} 0.5). So Eti (u i (a i,θ)) = a i (h 1 i (Y t i) 0.5). It is a supermodular game ( buy is the higher action), and we have σ i (t i) = 1 Ai σ i (t i ) = 0. So far: Given Y where violation occurs, a supermodular game is contructed. In fact, we can construct a single supermodular game that works for any Y U(Θ). Useful when we consider higher orders.

20 More general version that works for any Y U(Θ) Agent i reports non-decreasing a i : U(Θ) [0,1]. For each Y U(θ), ai (Y) means the maximum price i would pay for a lottery that yields 1 if θ Y. i bets on all Y s. Designer randomly chooses Y full-supp µ (U(Θ)), and price p U(0,1). Payoff: u i (a i,θ) = = E ti (u i (a i,θ)) = Y Y Y ai (Y) p=0 (1 {θ Y} p)dpdµ (1 {θ Y} a i (Y) (a i(y)) 2 )dµ. 2 (hi 1 (Y t i )a i (Y) (a i(y)) 2 )dµ. 2 Truth-telling is optimal. Even if Θ is uncountable, truth-telling can be made uniquely optimal with some modification of the game. Let G 1 denote the set of all a i.

21 Example 2: Violation in 2nd-order h 2 i (t i) SD h 2 i (t i ) (though h1 i (t i) SD h 1 i (t i )). There is an upper event Y U(Θ G 1 ) such that h 2 i (Y t i) > h 2 i (Y t i ). E.g., h 2 i (Y t i ) = 0.7 and h 2 i (Y t i ) = 0.3. Now we consider the following beauty-contest game: j reports aj G 1 (payoff is as in the last slide). i buys (or not) a lottery that yields 1 if (θ,aj ) Y. Price 0.5. ai {0,1}. u i (a i,a j,θ) = a i (1 {(θ,aj) Y} 0.5). As long as j is truthful, only ti would buy it. i bets jointly on θ and j s 1st-order belief.

22 Belief elicitation game: Strategy space G 1 = Set of all non-decreasing a 1 i : U(Θ) [0,1]. G 2 = Set of all non-decreasing a 2 i : U(Θ G 1 ) [0,1], and so on. Finally define A i = k=1 Gk. Lemma (Endowed with a component-wise order) A i is a complete lattice. Furthermore, under certain technical conditions on Θ, each G k (and hence A i ) can be made a Polish space. G 1 is metrized by L 1 (µ)-norm, where µ (U(Θ)) is a full-support distribution (last slide). Not a weak-* topology. and so on, for all G k.

23 Belief elicitation game: Payoff Payoffs satisfy supermodularity. where u i (a,θ) = u 0 i (a1 i,θ)+ k=1 δ k ui k (ak+1 i,aj k,θ), ui 0 (a1 i,θ) = (1 {θ Y 0 }a i (Y 0 ) (a i(y 0 )) 2 )dµ 0, Y 0 2 Yk ui k (a k+1 i,aj k,θ) = (1 {(θ,a kj ) Y k} a i(y k ) (a i(y k )) 2 )dµ k, 2 and µ 0 (Θ), µ k (Θ G k ) are full-support distributions.

24 Truth-telling in BE game Proposition Truth-telling is the unique equilibrium (and obtained by the iterative dominance). (Step 1) Truthful bidding on each upper event Y U(Θ) is the unique optimum. Remark: Not up to some measure-zero events. Θ is Polish, and hence have a countable base. The support of µ is defined based on this countable base (and hence µ has a countable support). This implies a strict incentive for truth-telling. (Step 2) Given this, truthful bidding on each upper event Y U(Θ G 1 ) is the unique optimum, etc.

25 Necessity of CCO Theorem If t i CCO t i, then σ i (t i) Ai σ i (t i ). Corollary t i CCO t i Σ i (t i) Ai Σ i (t i ). CCO order on types characterizes MCS.

26 Application 1 Investment game with additional information. Invest Not Invest θ +b θ Not 0 0 Each i observes (i) s i = θ +ε i, and (ii) r i = s j +η i. E.g., Information acquisition. Dynamics. A good news (i.e., investment-enhancing) is a condition jointly on (s i,r i ). Even if si decreases, if r i increases a lot, then CCO may hold.

27 Application 2 Non Bayesian updating A change in belief ti t i is not necessarily driven by Bayesian updating. E.g., Frequentists. Each i publicly observes a sequence of signals s 1,...,s T iid(mean= θ). All agents are frequentists: ˆθ = t st T. Inattention, Bounded memory, Ambiguity aversion.

28 Application 3 Trade in market with systematically biased beliefs. Q. A good news (i.e., trade-enhancing)? Seller with a unit of asset. Buyer without asset. Common value θ Θ R. No-trade theorem with common prior

29 Other applications... Technology adoption on a network N agents over a network, each adopting a new technology. Each i observes only direct neighbors. Belief about the structure of non-neighbors. More closely connected more incentive to coordinate. Herding, Mechanism design, Information disclosure...?

30 Conclusion CCO MCS in any supermodular game. Next? Applications? More stringent conditions on restricted class of supermodular games (e.g., binary investment games)?