Strategic interaction in interacting particle systems Elena Sartori Department of Mathematics · University of Padova Joint work with Paolo Dai Pra (Padova) and Marco Tolotti (Ca’ Foscari) Stochastic Analysis and applications in Biology, Finance and Physics Berlin - Potsdam · October 23 - 25, 2014
Aim Describe collective behavior of a large system of agents... I. including heterogeneity and interaction in decision problems to model herding and, in case, contagion, [go back to the seventies Föllmer, Random economies with many interacting agents , JME 1974; Schelling, Micromotives and Macrobehavior , Norton NY 1978; Granovetter, Threshold models of collective behavior , AJS 1978]; II. introducing trend (dependence on past) and, at last, strategic interaction (forecasting of other individuals’ behavior). E.Sartori (University of Padova) Strategic interaction in particle systems 2 / 20
Aim Describe collective behavior of a large system of agents... I. including heterogeneity and interaction in decision problems to model herding and, in case, contagion, [go back to the seventies Föllmer, Random economies with many interacting agents , JME 1974; Schelling, Micromotives and Macrobehavior , Norton NY 1978; Granovetter, Threshold models of collective behavior , AJS 1978]; II. introducing trend (dependence on past) and, at last, strategic interaction (forecasting of other individuals’ behavior). E.Sartori (University of Padova) Strategic interaction in particle systems 2 / 20
Aim Describe collective behavior of a large system of agents... I. including heterogeneity and interaction in decision problems to model herding and, in case, contagion, [go back to the seventies Föllmer, Random economies with many interacting agents , JME 1974; Schelling, Micromotives and Macrobehavior , Norton NY 1978; Granovetter, Threshold models of collective behavior , AJS 1978]; II. introducing trend (dependence on past) and, at last, strategic interaction (forecasting of other individuals’ behavior). E.Sartori (University of Padova) Strategic interaction in particle systems 2 / 20
The framework N interacting agents with interaction of mean-field type; binary setting: state variables σ i ∈ {− 1 , 1 } , i = 1 , . . . , N ; microscopic interaction − → macroscopic phenomena: evolution in time of aggregate variables, such as N m N ( t ) = 1 � σ i ( t ) ; N i = 1 decision problem : at each time, agents have to choose between {-1,1} maximizing their utility U i = private term + social component + random noise . [Brock, Durlauf (RES01); Barucci, Tolotti (JEDC12); Bouchaud (JSP13)]. E.Sartori (University of Padova) Strategic interaction in particle systems 3 / 20
The framework N interacting agents with interaction of mean-field type; binary setting: state variables σ i ∈ {− 1 , 1 } , i = 1 , . . . , N ; microscopic interaction − → macroscopic phenomena: evolution in time of aggregate variables, such as N m N ( t ) = 1 � σ i ( t ) ; N i = 1 decision problem : at each time, agents have to choose between {-1,1} maximizing their utility U i = private term + social component + random noise . [Brock, Durlauf (RES01); Barucci, Tolotti (JEDC12); Bouchaud (JSP13)]. E.Sartori (University of Padova) Strategic interaction in particle systems 3 / 20
The framework N interacting agents with interaction of mean-field type; binary setting: state variables σ i ∈ {− 1 , 1 } , i = 1 , . . . , N ; microscopic interaction − → macroscopic phenomena: evolution in time of aggregate variables, such as N m N ( t ) = 1 � σ i ( t ) ; N i = 1 decision problem : at each time, agents have to choose between {-1,1} maximizing their utility U i = private term + social component + random noise . [Brock, Durlauf (RES01); Barucci, Tolotti (JEDC12); Bouchaud (JSP13)]. E.Sartori (University of Padova) Strategic interaction in particle systems 3 / 20
The framework N interacting agents with interaction of mean-field type; binary setting: state variables σ i ∈ {− 1 , 1 } , i = 1 , . . . , N ; microscopic interaction − → macroscopic phenomena: evolution in time of aggregate variables, such as N m N ( t ) = 1 � σ i ( t ) ; N i = 1 decision problem : at each time, agents have to choose between {-1,1} maximizing their utility U i = private term + social component + random noise . [Brock, Durlauf (RES01); Barucci, Tolotti (JEDC12); Bouchaud (JSP13)]. E.Sartori (University of Padova) Strategic interaction in particle systems 3 / 20
Updating Consider discrete time. What about the mechanism for updating ? sequential: spins may flip at most one per time; parallel: spins may flip, possibly, all together. Parallel dynamics − → a sequence of optimizations : at each time particles choose maximizing their own utility. An example: probabilistic cellular automata ( PCA ). E.Sartori (University of Padova) Strategic interaction in particle systems 4 / 20
Updating Consider discrete time. What about the mechanism for updating ? sequential: spins may flip at most one per time; parallel: spins may flip, possibly, all together. Parallel dynamics − → a sequence of optimizations : at each time particles choose maximizing their own utility. An example: probabilistic cellular automata ( PCA ). E.Sartori (University of Padova) Strategic interaction in particle systems 4 / 20
Updating Consider discrete time. What about the mechanism for updating ? sequential: spins may flip at most one per time; parallel: spins may flip, possibly, all together. Parallel dynamics − → a sequence of optimizations : at each time particles choose maximizing their own utility. An example: probabilistic cellular automata ( PCA ). E.Sartori (University of Padova) Strategic interaction in particle systems 4 / 20
Updating Consider discrete time. What about the mechanism for updating ? sequential: spins may flip at most one per time; parallel: spins may flip, possibly, all together. Parallel dynamics − → a sequence of optimizations : at each time particles choose maximizing their own utility. An example: probabilistic cellular automata ( PCA ). E.Sartori (University of Padova) Strategic interaction in particle systems 4 / 20
Back to statistical mechanics A PCA is a discrete time Markov chain, whose transition probabilities are product measure, i.e., different components update simultaneously and independently . N � P ( σ ( t ) = s | σ ( t − 1 ) = ξ ) = P ( σ i ( t ) = s i | σ ( t − 1 ) = ξ ) , i = 1 for instance, for β, µ ≥ 0, e β s i [ µξ i + f ( m N ( ξ ))] P ( σ i ( t ) = s i | σ ( t − 1 ) = ξ ) = e β s i [ µξ i + f ( m N ( ξ ))] + e − β s i [ µξ i + f ( m N ( ξ ))] . How to link the stochastic evolution of this PCA with a decision problem ? E.Sartori (University of Padova) Strategic interaction in particle systems 5 / 20
Back to statistical mechanics A PCA is a discrete time Markov chain, whose transition probabilities are product measure, i.e., different components update simultaneously and independently . N � P ( σ ( t ) = s | σ ( t − 1 ) = ξ ) = P ( σ i ( t ) = s i | σ ( t − 1 ) = ξ ) , i = 1 for instance, for β, µ ≥ 0, e β s i [ µξ i + f ( m N ( ξ ))] P ( σ i ( t ) = s i | σ ( t − 1 ) = ξ ) = e β s i [ µξ i + f ( m N ( ξ ))] + e − β s i [ µξ i + f ( m N ( ξ ))] . How to link the stochastic evolution of this PCA with a decision problem ? E.Sartori (University of Padova) Strategic interaction in particle systems 5 / 20
Back to statistical mechanics A PCA is a discrete time Markov chain, whose transition probabilities are product measure, i.e., different components update simultaneously and independently . N � P ( σ ( t ) = s | σ ( t − 1 ) = ξ ) = P ( σ i ( t ) = s i | σ ( t − 1 ) = ξ ) , i = 1 for instance, for β, µ ≥ 0, e β s i [ µξ i + f ( m N ( ξ ))] P ( σ i ( t ) = s i | σ ( t − 1 ) = ξ ) = e β s i [ µξ i + f ( m N ( ξ ))] + e − β s i [ µξ i + f ( m N ( ξ ))] . How to link the stochastic evolution of this PCA with a decision problem ? E.Sartori (University of Padova) Strategic interaction in particle systems 5 / 20
An optimization problem N agents face a binary decision problem at each t : σ i ( t ) = 1 or − 1? The aim of the i -th agent is to maximize a random utility function U i ( s i ; σ ( t − 1 )) = s i [ µσ i ( t − 1 ) + f ( m N ( t − 1 )) + ǫ i ( t ))] where ( ǫ i ( t )) i = 1 ,..., N ; t ≥ 1 are i.i.d. real r.v. with distribution 1 η ( x ) := P ( ǫ i ( t ) ≤ x ) = ( logit distr . ) . 1 + e − 2 β x Agents carry out simultaneously their optimization: σ i ( t ) = 1 ⇔ U i ( 1 ) > U i ( − 1 ) ⇔ µσ i ( t − 1 ) + f ( m N ( t − 1 )) + ǫ i ( t ) > 0 , which happens with probability 1 η [ µσ i ( t − 1 ) + f ( m N ( t − 1 ))] = 1 + e − 2 β [ µσ i ( t − 1 )+ f ( m N ( t − 1 ))] . E.Sartori (University of Padova) Strategic interaction in particle systems 6 / 20
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