Criminal Network Formation and Optimal Detection Policy: The Role of Cascade of Detection Liuchun Deng 1 and Yufeng Sun 2 Johns Hopkins University Chinese University of Hong Kong Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 1 / 30
Motivation Criminal networks are widely observed Mafia Terrorist networks Corruption networks Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 2 / 30
Research Question What is the optimal detection policy in the presence of endogenous network formation among criminals? How does the cascade of detection affect criminal network formation and social welfare? Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 3 / 30
Preview of the Results We consider two dimensions of detection policy Allocation of detection resource Degree of cascade Higher degree of cascade of detection may backfire Optimal budget allocation is highly asymmetric among ex ante identical agents Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 4 / 30
Time Line Our timing structure follows Baccara and Bar-Issac (2008) Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 5 / 30
Model Set of players: N = { 1 , 2 , ..., n } 1 Probability of player i being directly detected: β i ∈ [ 0 , 1 ] 2 The government allocates a fixed detection budget B ∈ R + 3 n � β i ≤ B i = 1 Players are ranked such that 4 β 1 ≤ β 2 ≤ ... ≤ β n β ≡ ( β 1 , β 2 , ..., β n ) 5 Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 6 / 30
Link formation G : set of n -by- n ( 0 , 1 ) -matrices with zeros on the diagonal G i : set of n -by-1 ( 0 , 1 ) -vectors with i -th element to be zero g i ∈ G i : linking decision by player i g ∈ G : A collection of linking choices by all players G : set of n -by- n symmetric ( 0 , 1 ) -matrices with zeros on the diagonal Link formation requires bilateral agreement. For any g ∈ G , g induces a criminal network g ( g ) ∈ G such that g ( g ) = min ( g , g ′ ) No explicit linking cost Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 7 / 30
Degree of Cascade Distance d ij between player i and j is the length of the shortest path connecting i and j Probability of player i not being detected p i ( g ; β, d ) = Π j ∈ N , d ij ≤ d ( 1 − β j ) with d = 0 → no cascade of detection d = 1 → limited cascade of detection d = n → full cascade of detection Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 8 / 30
Degree of Cascade: Example p 1 ( g ; β β β, 0 ) = 1 − β 1 β, 1 ) = Π 4 p 1 ( g ; β β i = 1 ( 1 − β i ) β, 6 ) = Π 6 p 1 ( g ; β β i = 1 ( 1 − β i ) Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 9 / 30
Strategic Complementarity Given g ∈ G , player i chooses effort level x i ∈ R + Player i ’s payoff n x i − 1 � 2 x 2 π i ( x , g ; β, λ, d ) = p i ( g ; β, d ) · i + λ g ij x i x j j = 1 1 where λ ∈ ( 0 , n − 1 ) and x ≡ ( x 1 , x 2 , ..., x n ) Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 10 / 30
Strategy and Strategy Profile X : set of all mappings from G to R + Player i ’s strategy is a pair of a linking choice g i ∈ G i and an effort mapping x i ( · ) ∈ X Given a strategy profile ( x ( · ) , g ) , player i ’s payoff Π i ( x ( · ) , g ) ≡ π i ( x ( g ( g )) , g ( g )) Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 11 / 30
Equilibrium Definition I Definition A Nash equilibrium is a strategy profile ( x ∗ ( · ) , g ∗ ) such that Π i ( x ∗ ( · ) , g ∗ ) ≥ Π i ( x i ( · ) , x ∗ − i ( · ) , g i , g ∗ − i ) , ∀ i ∈ N , x i ( · ) ∈ X , g i ∈ G i . Definition A subgame-perfect Nash equilibrium is a strategy profile ( x ∗ ( · ) , g ∗ ) such that a Nash equilibrium is played for every subgame. Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 12 / 30
Equilibrium Definition II Definition (Hiller, 2014) A pairwise stable Nash equilibrium is a strategy profile ( x ∗ ( · ) , g ∗ ) such that ( x ∗ ( · ) , g ∗ ) is a subgame-perfect Nash equilibrium 1 There is no profitable bilateral deviation at the stage of link 2 formation . For any ( i , j ) -pair such that g ( g ∗ ) ij = 0 (i � = j), Π i ( x ∗ ( · ) , g ∗ ⊕ ( i , j ) ⊕ ( j , i )) > Π i ( x ∗ ( · ) , g ∗ ) implies Π j ( x ∗ ( · ) , g ∗ ⊕ ( i , j ) ⊕ ( j , i )) < Π j ( x ∗ ( · ) , g ∗ ) . Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 13 / 30
Obtainability Definition (Jackson and van den Nouweland, 2005) A network g ′ ∈ G is obtainable from g ∈ G via deviations by a nonempty S ⊂ N if g ij = 0 and g ′ ij = 1 implies i , j ∈ S; 1 g ij = 1 and g ′ ij = 0 implies { i , j } ∩ S � = ∅ . 2 Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 14 / 30
Obtainability: Example Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 15 / 30
Equilibrium Definition III Definition (Jackson and van den Nouweland, 2005) A subgame-perfect Nash equilibrium ( x ∗ ( · ) , g ∗ ) is strongly stable if for any nonempty S ⊂ N, h ∈ G that is obtainable from g ( g ∗ ) via deviations by S, and i ∈ S such that π i ( x ∗ ( h ) , h ) > π i ( x ∗ ( g ( g ∗ )) , g ( g ∗ )) , there exists j ∈ S such that π j ( x ∗ ( h ) , h ) < Π j ( x ∗ ( g ( g ∗ )) , g ( g ∗ )) . Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 16 / 30
Equilibrium Characterization Lemma (Ballester, et al., 2006) Given a criminal network g ∈ G, if λ ∈ ( 0 , 1 / ( n − 1 )) , there exists a unique interior Nash equilibrium for the stage game at the second period. In particular, x ( g ) = ( I − λ g ) − 1 · 1 , where I is an n-dimensional identity matrix and 1 is a 1 -by-n vector with all elements equal to one. Moreover, player i’s equilibrium payoff is given by p i ( g ) x 2 i ( g ) / 2 . Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 17 / 30
Equilibrium Characterization: No Cascade Proposition If there is no cascade of detection (d = 0 ), there exists a generically unique pairwise stable Nash equilibrium in which agents form a complete network. Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 18 / 30
Equilibrium Characterization: Full Cascade Lemma Under full cascade of detection (d = n), each component of the criminal network is complete in a pairwise stable Nash equilibrium. Lemma In any strongly stable Nash equilibrium, the equilibrium partition of agents “preserves” the order of detection probability � k − 1 k � � � { 1 , ..., n 1 } , { n 1 + 1 , ..., n 1 + n 2 } , ... { n i + 1 , ..., n i } i = 1 i = 1 where n ≡ � k i = 1 n i and agents are labeled such that β 1 ≤ ... ≤ β n . Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 19 / 30
Equilibrium Characterization: Full Cascade Proposition There exists a generically unique strongly stable Nash equilibrium with the equilibrium partition {{ 1 , 2 , ..., n 0 } , { n 0 + 1 } , { n 0 + 2 } , ..., { n }} and � � k ∈ N π k n 0 = max arg max , where π k is the individual payoff of a complete component formed by the first k agents, � 2 � π k = 1 1 Π k i = 1 ( 1 − β i ) . 2 1 − ( k − 1 ) λ Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 20 / 30
Equilibrium Characterization: Full Cascade A numerical example: n = 10 ; β k = k / 20 ; λ = 0 . 08 Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 21 / 30
Equilibrium Characterization: Partial Cascade Proposition Those players who are isolated in the strongly stable Nash equilibrium under full cascade of detection remain isolated in any pairwise stable Nash equilibrium under partial cascade of detection. Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 22 / 30
Detection Policy: Full Cascade The government’s decision problem n n x i ( β, λ ) − 1 � � 2 x 2 min i ( β, λ ) + λ g ij ( β, λ ) x i ( β, λ ) x j ( β, λ ) β ∈ R n + : � n i = 1 β i ≤ B i = 1 j = 1 Equivalently, min i = 1 β i ≤ B n 0 ( β ) + : � n β ∈ R n Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 23 / 30
Detection Policy: Full Cascade Proposition Under full cascade of detection, the government can keep each agent isolated in the strongly stable Nash equilibrium if and only if n � 2 � 1 − ( k − 1 ) λ � B > B 1 ≡ n − 1 − , 1 − ( k − 2 ) λ k = 2 and the optimal allocation of the detection budget is given by β 1 = 0 and � 2 � 1 − ( k − 1 ) λ + B − B 1 β k = 1 − n − 1 , k = 2 , 3 , ..., n . 1 − ( k − 2 ) λ Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 24 / 30
Detection Policy: Full Cascade Corollary Under full cascade of detection, the government can keep the size of the largest component of the criminal network in the strongly stable Nash equilibrium to be S ∈ { 2 , 3 , ..., n − 1 } if and only if n � 2 � 1 − ( k − 1 ) λ � B > B S ≡ n − S − , 1 − ( k − 2 ) λ k = S + 1 and the optimal allocation of the detection budget is given by β k = 0 for k ≤ S and � 2 � 1 − ( k − 1 ) λ + B − B S β k = 1 − n − S , k = S + 1 , S + 2 , ..., n . 1 − ( k − 2 ) λ Deng & Sun (JHU & CUHK) Criminal Network 2/27/2016 25 / 30
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