Coordination on Networks

Transcription

1 Coordination on Networks Matt Leister 1 Yves Zenou 1,2 Junjie Zhou 3 1 Monash, 2 IFN, 3 National University of Singapore Workshop on Game Theory at NUS June 4, 2018

2 Motivating examples Games with complements, binary actions and incomplete information: 1 Cryptocurrency adoption the more others adopt, the more viable as medium of exchange, value (exchange rates) of currency in the future unknown. 2 Crime the more neighbors partake in crime, (i) the more help you have, (ii) the less likely you will be caught, presence / strength of police force unknown. 3 Immigration / refugee policy bordering countries open-door-policies imply less refugee flow to your country, state of economy / outcome of war unknown.

3 Who coordinates with whom? Non-coordination

4 Who coordinates with whom? Non-coordination

5 Who coordinates with whom? Non-coordination

6 Who coordinates with whom? Non-coordination

7 Who coordinates with whom? Non-coordination

8 Who coordinates with whom? Coordination sets

9 Who coordinates with whom? Coordination sets

10 Who coordinates with whom? Coordination sets

11 Who coordinates with whom? Coordination sets

12 Who coordinates with whom? Common coordination

13 Who coordinates with whom? Common coordination

14 Global games: review Carlsson and van Damme (1993), Morris and Shin (2006): Invest NotInvest Invest θ, θ θ 1,0 NotInvest 0, θ 1 0, 0 If θ > 1: Invest dominant strategy If θ [0,1]: two symmetric Nash. If θ < 0: NotInvest dominant strategy

15 Global games: review Carlsson and van Damme (1993), Morris and Shin (2006): Invest NotInvest Invest θ, θ θ 1,0 NotInvest 0, θ 1 0, 0 Assume each i = 1,2 observe s i = θ+ ǫ i, ǫ i N(0, σ), θ uniform on R. Cutoff strategy of j = i: { Invest if sj s NotInvest if s j < s probability j NotInvest given s i is Φ( s 2σ s i ) (stand. norm. cdf Φ).

16 Global games: review Carlsson and van Damme (1993), Morris and Shin (2006): Given s i,s, value to Invest is: (1 Φ( s s i 2σ ))s i + Φ( s s i 2σ )(s i 1) = s i Φ( s s i 2σ ) 1 Decreasing σ s i s j

17 Summary of results Coordination sets characterize limit-equilibrium properties: solve for unique set of limiting coordination sets, easy to obtain common coordination: e.g., trees, bipartite graphs. Contagion contained within coordination sets: all agents within sets respond uniformly to targeted adoption subsidy, strategic affect falls discontinuously across coordination sets. Welfare and policy implications: in the limit, optimal policies reduce to targeting coordination sets, externalities wedge between adoption vs. welfare policies.

18 Literature Global Games (Morris and Shin 2006 for review) Theory: Carlsson and van Damme (1993), Ui (2001), Frankel, Morris and Pauzner (2003), Morris and Ui (2005), Sákovics and Jakub Steiner (2012), more. Applications: Morris and Shin (2003), Goldstein and Pauzner (2004, 2005), Dasgupta (2004), Rochet and Vives (2004), Edmond (2013), many more. Network Games Ballester et al. (2006), Galeotti et al. (2010), Bramoullé et al. (2014), many more. With incomplete information: Calvó-Armengol, and de Martí (2007), Hagenbach and Koessler (2010), Galeotti et al. (2013), Calvó-Armengol et al. (2015), de Martí and Zenou (2016), Leister (2017), Myatt and Wallace (2017), others.

19 Model setup (I) Connected network G = (N,E), nodes i N, edges (i,j) E. Assume undirected graph: (i,j) E iff (j,i) E. N i := {j : (i,j) E}; d i := N i ; d i (S) := S N i, S N. i s action a i, binary action space {0,1}: adopt (a i = 1) and not adopt (a i = 0) technology. Payoff-relevant state θ Θ, interval support Θ R. Payoff from adopting (a i = 1): u i (a i θ) = v i + σ(θ)+φ j N i a j, v i R, σ : Θ R is C 1 and strictly increasing, φ > 0. Payoff from not adopting (a i = 0): zero.

20 Model setup (II) Existence of dominance regions For each i, assume v i, σ and φ are s.t. θ i, θ i Θ: v i + σ(θ)+φd i < 0 when θ < θ i, and v i + σ(θ) > 0 when θ > θ i. Information structure (perturbed game G(ν)) θ is observed with noise by all agents. Signal s i = θ+ νǫ i, ν > 0, ǫ i i.i.d., pdf f, cdf F i, with support [ 1,1].

21 Model setup (II) Existence of dominance regions For each i, assume v i, σ and φ are s.t. θ i, θ i Θ: v i + σ(θ)+φd i < 0 when θ < θ i, and v i + σ(θ) > 0 when θ > θ i. Information structure (perturbed game G(ν)) θ is observed with noise by all agents. Signal s i = θ+ νǫ i, ν > 0, ǫ i i.i.d., pdf f, cdf F i, with support [ 1,1].

22 Conditional expectations { 1 if Cutoff strategies: π i (s si s i) := i 0 if s i < s i Lower cutoff more adoption.. Expected payoff to i adopting conditional on s i : [ ] ] U i (π i s i ) := E θ E s i [u i (a i θ) π i, θ si [ ] ] = v i + E θ [σ(θ) s i ]+φ E θ [E sj π j si (s j ) θ. j N i

23 Equilibrium in the noiseless limit ν 0 U i (π i s i ) = 0, i N define a Bayesian Nash Equilibrium of perturbed game G(ν). Lemma 1 Bayesian Nash Equilibrium π of G(ν) in cutoff strategies exists. Frankel et al. (2006) Theorem 1, in this setting: Proposition 1 There exists a unique strategy profile π, which is in cutoff strategies, such that any π surviving iterative elimination of strictly dominated strategies in G(ν) satisfies lim ν 0 π = π. π defined by limit cutoffs θ ; limiting probability weightings w ij := lim ν 0 E sj [π j (s j) s i = s i ].

24 Limit equilibrium Define feasible weighting functions: W = {w = (w ij,(i,j) E) w ij 0,w ji 0,w ij +w ji = 1; (i,j) E}, Recall w ij := lim ν 0 E sj [π j (s j) s i = s i ]. Lemma (Limit-equilibrium weights) For each ij E, Moreover, if θ i < θ j, then Note that this gives w W. w ij +w ji = 1. w ij = 0, and w ji = 1.

25 Limit equilibrium Define affine mapping (with image Φ(W)): Φ i (w) := v i + φ j N i w ij, i N. Theorem (Limit equilibrium) For any v, φ, and network G, the equilibrium limit cutoffs θ are given by: σ(θ i )+q i = 0, i, where q = (q 1,,q n) is the unique solution to: q = argmin z. z Φ(W)

26 Limit equilibrium Define affine mapping (with image Φ(W)): Φ i (w) := v i + φ j N i w ij, i N. Theorem (Limit equilibrium) For any v, φ, and network G, the equilibrium limit cutoffs θ are given by: σ(θ i )+q i = 0, i, where q = (q 1,,q n) is the unique solution to: q = argmin z. z Φ(W) Define T := i v i + φ E. For dyad, T = v 1 +v 2 + φ.

27 Limit equilibrium solution: dyad q 2 T 2 (v 1,v 2 ) q Φ(W) T 2 q 1 v 1 v 2 < φ; T = v 1 +v 2 + φ.

28 Limit equilibrium solution: dyad q 2 T 2 (v 1,v 2 ) q Φ(W) T 2 q 1 φ v 1 v 2 φ; T = v 1 +v 2 + φ.

29 Limit equilibrium solution: dyad q 2 T 2 q (v 1,v 2 ) Φ(W) T 2 v 1 v 2 > φ; T = v 1 +v 2 + φ. q 1

30 Limit partition C For S N, denote E S E corresponding to subgraph G S := (S,E S ) of G restricted to vertices S. Definition (limit partition) The limit equilibrium π maps to an ordered partition C := (C1,...C m ) of N satisfying: 1 For each C m, all members have the same cutoff θ m (equiv., same q m). 2 For each C m, subgraph G C m is connected. 3 For each C m = C m such that θ m = θ m, C m and C m are not directly connects in G. For each C m C, denote C m := m <mc m and C m := m >mc m. Denote i s coordination set by m(i).

31 From coordination sets to cutoffs For any S,S N, S S =, denote: v(s) := v i, i S e(s,s ) := d i (S ), i S e(s) := 1 2 d i (S). i S Proposition (Coordination-set cutoffs) Conditional on C m: θm = σ 1 ( v(c m)+φ(e(cm, C m)+e(c m)) C }{{ m ). } qm

32 Characterizations & Comparative Statics: Homogenous Values Assume: v i = v for each i N.

33 Coordination sets and equilibrium structure Corollary (Limit partition homogeneity) Under homogeneous valuations, C is independent of v and of φ. Moreover, q = v1 n + φˆq, where ˆq := q under v = 0 and φ = 1.

34 Coordination sets and equilibrium structure Corollary (Limit partition homogeneity) Under homogeneous valuations, C is independent of v and of φ. Moreover, q = v1 n + φˆq, where ˆq := q under v = 0 and φ = 1. If we scale (i) common certain value v, (ii) slope of σ, (iii) fundamental uncertainty of θ, and/or (iv) value to neighbors adoption, who coordinates with whom does not change.

35 Coordination sets and equilibrium structure Corollary (Limit partition homogeneity) Under homogeneous valuations, C is independent of v and of φ. Moreover, q = v1 n + φˆq, where ˆq := q under v = 0 and φ = 1. If we scale (i) common certain value v, (ii) slope of σ, (iii) fundamental uncertainty of θ, and/or (iv) value to neighbors adoption, who coordinates with whom does not change. Without loss, set v = 0 and φ = 1 in the following examples...

36 Example 1 1p c 3p 2p Star network Why periphery s cutoff not below center s cutoff: d ip ({ip}) = 0 < d c ({1p,2p,3p})+d c ({c}) = 3+0, Why center s cutoff not below periphery s cutoff: d c ({c}) = 0 < d ip ({c})+d ip ({ip}) = 1+0.

37 Example 2 1p 1c 3c 2c 3p 2p Triad-core-periphery network

38 Example 3 4p 1p 4c 1c 3c 2c 3p 2p Quad-core-periphery network Why core nodes have strictly lower cutoff than periphery: e(c m, )+e(c m) 4 = for C m = {1c,2c,3c,4c} C 1 > e({jp},c m)+e({jp}) = 1+0, = {1c,...,4c}, C j = {jp}, j = 1,...,4.

39 Example 4 6c 1c 5c 2c 4c 3c Large core-periphery network

40 Example 4 r 6c 1c 5c 2c 4c 3c Large core-periphery network

41 Example 4 r 2q 1q 6c 1c 5c 2c 4c 3c Large core-periphery network

42 Example 4 r 2q 1p 1q 6c 1c 2p 5c 2c 3p 4c 3c 4p Large core-periphery network

43 Example 4 r 2q 1p 1q 6c 1c 2p 5c 2c 3p 4c 3c 4p Large core-periphery network Limit partition: C1 = {ic : i = 1,...,6}, C 2 = {r}, C3 = {1q,2q}, {C 4,...,C 7 } = {{1p}...{4p}}.

44 Single coordination set We can characterize when a single coordination set on G obtains: Proposition (Single coordination set) Under homogeneous valuations, a single coordination set exists (i.e. C = {C 1 }) if and only if G is balanced, in the sense that for every nonempty S N, e(s) e(n) S N. When this condition is satisfied, the common cutoff in the network is θ 1 = σ 1 ( v φ e(n) N ).

45 Single coordination set: examples Regular-bipartite network: B 1 and B 2, with B 1 B 2 = N and of sizes n s := B s and degrees d s := d i, i B s, for sides s = 1,2. Proposition 2 (Single coordination set: examples) Under homogeneous valuations, there exists a single coordination set if G: 1 is a regular network, or 2 is a tree network, or 3 is a regular-bipartite network, or 4 has a unique cycle, or 5 has at most four nodes.

46 Examples: real-world networks q 1 = 4 q 2 = 1.66 Figure: Banerjee et al. (2013) network 1 q 3 = 1.5

47 Examples: real-world networks q 5 = 1.33 q 3 = 2.5 q 6 = 1 q 2 = 2.5 q 1 = 3 q 4 = 2 Figure: Banerjee et al. (2013) network 2

48 Examples: real-world networks q 1 = 1 Figure: Add Health friendship network

49 Comparative statics: linkage G +ij : supergraph of G including link (i,j). C +ij : limit partition under G +ij. q m,+ij corresponds to C m under network G +ij. Proposition (Linkage) Take i,j with m(i) m(j), (i,j) / E, such that C +ij = C. If: 1 θ m(i) > θ m(j), then: q m(i),+ij q m(i) = φ 1 C m(i), and q m(j),+ij q m(j) = 0; 2 m(i) = m(j) =: m, then: qm,+ij qm 1 = φ Cm(i).

50 Comparative statics: heterogeneous values Arbitrary (v i ).

51 Heterogeneous values: example 1p vary v 1p c 3p 2p Star network u i (a i θ) = v i 3 1 θ θ + j N i a j. Exercise: vary v 1p, hold v i = 1 for other agents.

52 Heterogeneous values: example 0.68 u i (a i θ) = v i 3 1 θ θ + j N i a j ; ν = s 2p,s 3p 0.64 s i, θ i 2ν sc s1p θ2p, θ 3p, θ c θ1p 0.6 2ν v 1p

53 Comparative statics: sticky coordination For each i C m denote: ˆv i := argmax{v i : θ i = θ j,j C m\{i};v i }, ˇv i := argmin{v i : θi = θj,j Cm\{i};v i }. Proposition (Sticky coordination) Take coordination set C m C with C m > 1. Then for each i C m: ˆv i ˇv i φd i (C m). When C is constant for v i (ˆv i,ˇv i ), then: ˆv i ˇv i = C m C m 1 φd i(c m).

54 Comparative statics: local contagion Proposition (Local contagion) In the limit, the mapping q (v) is piecewise linear, Lipschitz continuous, and monotone. Generically, q v exists. Generically, when i,j C m and k / C m, then: q m v i = 1 C m, and q k v i = 0. Note: θ m v i = q m v i 1 σ (θ j ), by σ(θ j )+q j = 0.

55 Comparative statics: local contagion Proposition (Local contagion) In the limit, the mapping q (v) is piecewise linear, Lipschitz continuous, and monotone. Generically, q v exists. Generically, when i,j C m and k / C m, then: Note: θ m v i = q m v i q m v i = 1 C m, and q k v i = 0. 1 σ (θ j ), by σ(θ j )+q j = 0. Contained contagion: perturbations spread within coordination sets, evenly so, but discontinuously drops to zero across coordination sets.

56 Welfare and Policy Implications

57 Benchmarks Benchmark 1 (adoption maximization): ma i := j v i E sj [ π j ]. Benchmark 2 (welfare maximization): mwi [ := E sj Uj (π j v j s j ) ]. i

58 Policy implications Proposition (Policy impact) Denote H the marginal cdf of θ. For each C m C and i C m: 1 lim ν 0 ma i = H (θm) σ (θm), 2 lim ν 0 mw i = (1 H(θ m))+φ ( e(c m, C m)+e(c m) C m ) H (θ m) σ (θ m).

59 Policy implications Proposition (Policy impact) Denote H the marginal cdf of θ. For each C m C and i C m: 1 lim ν 0 ma i = H (θm) σ (θm), 2 lim ν 0 mw i = (1 H(θ m))+φ ( e(c m, C m)+e(c m) C m ) H (θ m) σ (θ m). Optimal policy problems reduce to targeting a coordination set. Wedge between ma and mw : subsidies to high-cutoff coordination sets impose positive externalities on low-cutoff coordination sets; low-cutoffs unresponsive (contained contagion).

60 Variation: miscoordination costs Under heterogeneous valuations, set v i to v i φd i to give: u i (a i θ) = v i + σ(θ) φ j N i (1 a j ). Interpretation: homogeneous values under miscoordination costs. If v i = v for each i, common coordination persists within trees, regular-bipartite networks, networks with a unique cycle, and networks with at most four nodes.

61 Miscoordination in immigration policy 2015: more than a million migrants and refugees crossed into Europe. Some European countries such as Germany and Sweden were positively inclined towards these migrants, whereas other countries such as Poland and Hungary took strong stances against regularizing them.

62 Miscoordination in immigration policy a i = 0: anti-immigration (i.e. isolationist ) stance. a i = 1: inclusive policy. v i + σ(θ): value of taking an inclusive policy (political support). Inflow of immigrants into country i: f + τ j Ni (1 a j ); τ > 0: overflow of migrants into i when j N i takes anti-immigration stance. u i (a i θ) = v i + σ(θ) c ( ) f + τ (1 a j ) j N i = v i cf + σ(θ) }{{} cτ (1 a j ). φ j N i

63 Miscoordination in immigration policy a i = 0: anti-immigration (i.e. isolationist ) stance. a i = 1: inclusive policy. v i + σ(θ): value of taking an inclusive policy (political support). Inflow of immigrants into country i: f + τ j Ni (1 a j ); τ > 0: overflow of migrants into i when j N i takes anti-immigration stance. u i (a i θ) = v i + σ(θ) c ( ) f + τ (1 a j ) j N i = v i cf + σ(θ) }{{} cτ (1 a j ). φ j N i v i v: countries with many borders take high θ i (isolationism). EU should have a common immigration policy: avoid miscoordination costs due to excessive immigrant flows to pro-immigration countries.

64 Conclusion Technical Contributions: Provide solution to limit cutoffs for general networks, incorporating multiple coordination sets in a global-game setting. Characterize ( ) network conditions for common coordination.

65 Conclusion Technical Contributions: Provide solution to limit cutoffs for general networks, incorporating multiple coordination sets in a global-game setting. Characterize ( ) network conditions for common coordination. Equilibrium characterizations unique to network-games literature: coordinated adoption cutoffs in noiseless limit. Homogeneous values: stratified coordination across network cliques/peripheries. Heterogeneous values: sticky coordination amongst interconnected agents. Local contagion: strategic spillovers contained within coordination sets.

66 Conclusion Common coordination not hard to obtain: Homogenous values: regular networks, trees, regular-bipartite networks, networks with a unique cycle, and networks with at most four nodes.

67 Conclusion Common coordination not hard to obtain: Homogenous values: regular networks, trees, regular-bipartite networks, networks with a unique cycle, and networks with at most four nodes. Comparative statics: Quantify effect of linkage on equilibrium cutoffs, Quantify marginal affect of adoption subsidization on equilibrium cutoffs.

68 Conclusion Common coordination not hard to obtain: Homogenous values: regular networks, trees, regular-bipartite networks, networks with a unique cycle, and networks with at most four nodes. Comparative statics: Quantify effect of linkage on equilibrium cutoffs, Quantify marginal affect of adoption subsidization on equilibrium cutoffs. Welfare implications: Optimal policy problems reduce to targeting a coordination set. Planner aiming to maximize adoption designs intervention to yield large strategic effects. Planner aiming to maximize welfare also accounts for direct (ex-ante) externalities on neighbors with strictly lower cutoffs.