Games on Networks Yves Zenou (Stockholm University and IFN) Whos - PowerPoint PPT Presentation

Games on Networks Yves Zenou (Stockholm University and IFN)

Who’s Who in Networks. Wanted: the Key Player Coralio Ballester, Antoni Calvó- Armengol and Yves Zenou Econometrica 2006

When σ ij > 0 , an increase in the e ff ort x j of agent j creates an incentive for i to increase his level of activity x i . We then talk of strategic complementarity in e ff orts. When σ ij < 0 , instead, an extra e ff ort from j triggers a downards shift in i ’s e ff ort in response. We say that e ff orts are strategic substitutes .

The General Model • We can decompose bilateral influences like ∑  −  I −  U   G    Local Net Self- Global Complementarity Substitutability Substitutability where G represents a network of local complementarities, 0 ≤ g ij ≤ 1

I is the n − square identity matrix and U the n − square matrix of ones. Σ = − β I − γ U + λ G with β > 0 , γ ≥ 0 and λ > 0 . The pattern of bilateral in fl uences results from the com- bination of an idiosyncratic e ff ect, a global interaction e ff ect, and a local interdependence component. The idiosyncratic e ff ect re fl ects (part of) the concavity of the payo ff function in own e ff orts. The global interaction e ff ect is uniform across all players (matrix U ) and corresponds to a substitutability e ff ect across all pairs of players with value − γ ≤ 0 . The local interaction component captures the (relative) strategic complementarity in e ff orts that varies across pairs of players, with maximal strength λ and population pattern re fl ected by G .

The decomposition is depicted in Figure 1. This is just a centralization ( β and λ are de fi ned with respect to γ ) followed by a normalization (the g ij s are in [0 , 1] ) of the cross e ff ects. The fi gure in the upper panel corresponds to σ ij of either sign (the case σ ij ≤ 0 , for all i 6 = j is similar) while the fi gure in the lower panel corresponds to σ ij ≥ 0 , for all i 6 = j (recall that we assume σ < 0 ).

λ β γ σ σ σ ij σ 0 λ g ij γ = 0 β λ σ σ σ ij σ 0 λ g ij

The General Model • Three players 2  1 u 1  x 1 , x 2 , x 3   x 1 − 3 x 1 2 x 1 x 2 − x 1 x 3 − 6 1/2 − 1 2  1 ∑  u 2  x 1 , x 2 , x 3   x 2 − 3 x 2 2 x 1 x 2  x 2 x 3 1/2 − 6 1 2 − x 1 x 2  x 1 x 3 − 1 − 6 1 u 3  x 1 , x 2 , x 3   x 3 − 3 x 3 1 0 0 1 1 1 0 3/4 0 ∑  − 5 − 1  2 0 1 0 1 1 1 3/4 0 1    −  −    0 0 1 1 1 1 0 1 0 2 1 3

Explanation of this example: ⎛ ⎞ − 6 1 / 2 − 1 ⎜ ⎟ Σ = 1 / 2 − 6 1 ⎝ ⎠ − 1 1 − 6 σ = min { σ ij } = − 1 and σ = max { σ ij } = 1 OBS: σ and σ do not include σ ii = σ . σ ii = σ = − 6 γ = − min { σ, 0 } = − min { − 1 , 0 } = 1 λ = σ + γ = 1 + γ = 2

g ij = σ ij + γ but 0 ≤ g ij ≤ 1 λ Thus ⎛ n − 6+1 o ⎞ 1 / 2+1 = 3 − 1+1 max , 0 = 0 = 0 2 2 4 2 ⎜ ⎟ n − 6+1 o ⎜ ⎟ 1 / 2+1 = 3 1+1 G = ⎜ ⎟ max , 0 = 0 = 1 ⎝ 2 4 2 2 ⎠ n − 6+1 o − 1+1 1+1 = 0 = 1 max , 0 = 0 2 2 2 As a result Σ = − β I − γ U + λ G = − 5 × I − 1 × U + 2 × G

In our example, we have: α = 1 , γ = 1 , λ = 2 , β = 5 Since for all i = 1 , 2 , 3 3 3 X X u i = αx i − 1 2 ( β − γ ) x 2 i − γ x i x j + λ g ij x i x j j =1 j =1 we have: 3 3 X X = αx 1 − 1 2 ( β − γ ) x 2 u 1 1 − γ x 1 x j + λ g 1 j x 1 x j j =1 j =1 = x 1 − 1 1 − x 1 x 1 − x 1 x 2 − x 1 x 3 + 23 24 x 2 4 x 1 x 2 1 + 1 = x 1 − 3 x 2 2 x 1 x 2 − x 1 x 3 Similarly 2 + 1 u 2 = x 2 − 3 x 2 2 x 1 x 2 + x 2 x 3 u 3 = x 3 − 3 x 2 3 − x 1 x 2 + x 1 x 3

The network Bonacich centrality To each network g , we associate its adjacency matrix G = [ g ij ]. Symmetric zero-diagonal square matrix that keeps track of the direct connections in g . The k th power G k = G ( k times ) G of the adjacency matrix G keeps ... track of indirect connections in g . The coe ffi cient g [ k ] ij in the ( i, j ) cell of G k gives the number of paths of length k in g between i and j .

Example Network g with three individuals (star) t t t 2 1 3 Figure 1 Adjacency matrix : ⎡ ⎤ 0 1 1 ⎢ ⎥ G = 1 0 0 ⎣ ⎦ 1 0 0 ⎡ ⎤ ⎡ ⎤ 2 k 2 k 2 k 0 0 0 ⎢ ⎥ ⎢ ⎥ G 2 k = G 2 k +1 = 2 k − 1 2 k − 1 2 k and ⎦ , k ≥ 1 0 0 0 ⎣ ⎦ ⎣ 2 k − 1 2 k − 1 2 k 0 0 0

⎡ ⎤ 0 2 2 ⎢ ⎥ G 3 = 2 0 0 ⎣ ⎦ 2 0 0 G 3 : two paths of length three between 1 and 2: 12 → 21 → 12 and 12 → 23 → 32. no path of length three from i to i

For all integer k , de fi ne: n X g [ k ] b k i ( g ) = ij j =1 This is the sum of all paths of length k in g starting from i . Next, let φ ≥ 0, and de fi ne: + ∞ X φ k b k b i ( g ,φ ) = i ( g ) k =0 This is the sum of all paths in g starting from i , where paths of length k are weighted by the geometrically decaying factor φ k .

For φ small enough, this in fi nite sum takes on a fi nite value. + ∞ X φ k G k · 1 = [ I − φ G ] − 1 · 1 , b ( g , φ ) = (2) k =0 where 1 is the vector of ones. b ( g , φ ) Bonacich network centrality of parameter φ in g . b i ( g ,φ ) as the Bonacich centrality of agent i in g . To each agent, it associates a value that counts the total number of direct and indirect (weighted) paths in the network stemming from this agent.

Example Consider the network g in Figure 1. t t t 2 1 3 Figure 1 When φ is small enough, the vector of Bonacich network centralities is: ⎡ ⎤ ⎡ ⎤ b 1 ( g , φ ) 1 + 2 φ 1 ⎢ ⎥ ⎢ ⎥ b ( g , φ ) = b 2 ( g , φ ) ⎦ = 1 + φ ⎦ . ⎣ ⎣ 1 − 2 φ 2 b 3 ( g, φ ) 1 + φ

The Bonacich centrality of node i is b i ( g , a ) = P n j =1 m ij ( g , a ) , and counts the total number of paths in g starting from i . It is the sum of all loops starting from i and ending at i , and all outer paths that connect i to every other player j 6 = i : X b i ( g , a ) = m ii ( g , a ) + m ij ( g , a ) . | {z } | {z } j 6 = i self − loops out − paths Note that, by de fi nition, m ii ( g , a ) ≥ 1 , and thus b i ( g , a ) ≥ 1 .

Example 2. Consider the network g in Figure 1. t t t 2 1 3 Figure 1 ³ 2 1 / 2 ´ Largest eigenvalue: 2 1 / 2 . When d < c , the unique Nash equilib- rium is: 1 = a c + 2 d 3 = a c + d x ∗ x ∗ 2 = x ∗ and c 2 − 2 d 2 . c 2 − 2 d 2

Dyads No social interactions. Then, the utility of each agent i would be given by: u i ( x i ) = αx i − 1 2 x 2 i The unique symmetric equilibrium is: x ∗ ni = α

Now, in order to understand the general model and to see the role of λ and γ , let us take the simplest possible network, that is n = 2 and each player has a link with the other, that is g 12 = g 21 = 1 . The adjacency matrix à ! 0 1 G = 1 0 Two eigenvalues: 1 , − 1 . Thus µ 1 ( G ) = 1 . The network locations in g are interchangeable. In this case, the utility is now given by: ³ ´ u i ( x 1 , x 2 ) = αx i − 1 2 x 2 x 2 i − γ i + x i x j + λx i x j where 0 ≤ γ < 1 . Compared to our utility function β = 1 + γ .

The fi rst order condition are: ∂u i = α − (1 + 2 γ ) x i − ( γ − λ ) x j = 0 ∂x i Since we have a dyad, the unique symmetric equilibrium is given by: α x ∗ = 1 − λ + 3 γ Observe that since β = 1 + γ and µ 1 ( G ) = 1 , β > λµ 1 ( G ) ⇔ λ < 1 + γ Guarantees this solution to be strictly positive.

Check with Theorem α x ∗ = β + γb ( g , λ/β ) b ( g , λ/β ) Here " # − 1 I − λ b ( g , a ) = β G · 1 "Ã ! Ã !# − 1 Ã ! − λ 1 0 0 1 1 = 0 1 1 0 1 β Ã ! − 1 Ã ! 1 − λ/β 1 = − λ/β 1 1 ⎛ ⎞ ⎛ ⎞ β 2 Ã ! λβ β 1 ⎜ β 2 − λ 2 β 2 − λ 2 ⎟ β − λ ⎝ ⎠ = = ⎝ ⎠ β 2 β 1 λβ β − λ β 2 − λ 2 β 2 − λ 2

Thus ⎛ ⎞ à ! β x ∗ α β − λ ⎝ ⎠ 1 = x ∗ β β + γb ( g , λ/β ) 2 β − λ where 2 β b ( g , λ/β ) = b 1 ( g , λ/β ) + b 2 ( g , λ/β ) = β − λ We have à ! à ! α x ∗ β − λ +2 γ 1 = α x ∗ 2 β − λ +2 γ Now since β = 1 + γ , we have: à ! à ! α x ∗ 1 − λ +3 γ 1 = α x ∗ 2 1 − λ +3 γ

Suppose fi rst that γ = 0 , i.e. there is no global substitu- ability. We obtain α x ∗ = 1 − λ In the dyad, agents rip complementarities from their part- ner, and choose an e ff ort level above the optimal value for an isolated agent ( x ∗ = α ). The factor 1 / (1 − λ ) > 1 is often referred to as the social multiplier . Suppose now that λ = 0 . We obtain α x ∗ = 1 + 3 γ Equilibrium e ff orts are decreasing in γ . Indeed, global substituabilities add to the idiosyncratic concavity in one’s e ff orts, an exhaust decreasing marginal returns below the optimal value for an isolated agent. The general expres- sion results from a combination of both e ff ects.