[PPT] - Strategic interaction in interacting particle systems Elena Sartori PowerPoint Presentation

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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

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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

SLIDE 3

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

SLIDE 4

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

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The framework

N interacting agents with interaction of mean-field type; binary setting: state variables σi ∈ {−1, 1}, i = 1, . . . , N; microscopic interaction − → macroscopic phenomena: evolution in time of aggregate variables, such as mN(t) = 1 N

N

i=1

σi(t) ; decision problem: at each time, agents have to choose between {-1,1} maximizing their utility Ui = 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

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The framework

N interacting agents with interaction of mean-field type; binary setting: state variables σi ∈ {−1, 1}, i = 1, . . . , N; microscopic interaction − → macroscopic phenomena: evolution in time of aggregate variables, such as mN(t) = 1 N

N

i=1

σi(t) ; decision problem: at each time, agents have to choose between {-1,1} maximizing their utility Ui = 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

SLIDE 7

The framework

N interacting agents with interaction of mean-field type; binary setting: state variables σi ∈ {−1, 1}, i = 1, . . . , N; microscopic interaction − → macroscopic phenomena: evolution in time of aggregate variables, such as mN(t) = 1 N

N

i=1

σi(t) ; decision problem: at each time, agents have to choose between {-1,1} maximizing their utility Ui = 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

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The framework

N interacting agents with interaction of mean-field type; binary setting: state variables σi ∈ {−1, 1}, i = 1, . . . , N; microscopic interaction − → macroscopic phenomena: evolution in time of aggregate variables, such as mN(t) = 1 N

N

i=1

σi(t) ; decision problem: at each time, agents have to choose between {-1,1} maximizing their utility Ui = 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

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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

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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

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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

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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

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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. P(σ(t) = s | σ(t − 1) = ξ) =

N

i=1

P(σi(t) = si | σ(t − 1) = ξ), for instance, for β, µ ≥ 0, P(σi(t) = si | σ(t − 1) = ξ) = eβsi [µξi+f(mN(ξ))] eβsi [µξi+f(mN(ξ))] + e−βsi [µξi+f(mN(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

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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. P(σ(t) = s | σ(t − 1) = ξ) =

N

i=1

P(σi(t) = si | σ(t − 1) = ξ), for instance, for β, µ ≥ 0, P(σi(t) = si | σ(t − 1) = ξ) = eβsi [µξi+f(mN(ξ))] eβsi [µξi+f(mN(ξ))] + e−βsi [µξi+f(mN(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

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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. P(σ(t) = s | σ(t − 1) = ξ) =

N

i=1

P(σi(t) = si | σ(t − 1) = ξ), for instance, for β, µ ≥ 0, P(σi(t) = si | σ(t − 1) = ξ) = eβsi [µξi+f(mN(ξ))] eβsi [µξi+f(mN(ξ))] + e−βsi [µξi+f(mN(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

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An optimization problem

N agents face a binary decision problem at each t: σi(t) = 1 or −1? The aim of the i-th agent is to maximize a random utility function Ui(si; σ(t − 1)) = si [µσi(t − 1) + f(mN(t − 1)) + ǫi(t))] where (ǫi(t))i=1,...,N; t≥1 are i.i.d. real r.v. with distribution η(x) := P(ǫi(t) ≤ x) = 1 1 + e−2βx (logit distr.). Agents carry out simultaneously their optimization: σi(t) = 1 ⇔ Ui(1) > Ui(−1) ⇔ µσi(t − 1) + f(mN(t − 1)) + ǫi(t) > 0, which happens with probability η [µσi(t − 1) + f(mN(t − 1))] = 1 1 + e−2β[µσi(t−1)+f(mN(t−1))] .

E.Sartori (University of Padova) Strategic interaction in particle systems 6 / 20

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An optimization problem

N agents face a binary decision problem at each t: σi(t) = 1 or −1? The aim of the i-th agent is to maximize a random utility function Ui(si; σ(t − 1)) = si [µσi(t − 1) + f(mN(t − 1)) + ǫi(t))] where (ǫi(t))i=1,...,N; t≥1 are i.i.d. real r.v. with distribution η(x) := P(ǫi(t) ≤ x) = 1 1 + e−2βx (logit distr.). Agents carry out simultaneously their optimization: σi(t) = 1 ⇔ Ui(1) > Ui(−1) ⇔ µσi(t − 1) + f(mN(t − 1)) + ǫi(t) > 0, which happens with probability η [µσi(t − 1) + f(mN(t − 1))] = 1 1 + e−2β[µσi(t−1)+f(mN(t−1))] .

E.Sartori (University of Padova) Strategic interaction in particle systems 6 / 20

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An optimization problem

N agents face a binary decision problem at each t: σi(t) = 1 or −1? The aim of the i-th agent is to maximize a random utility function Ui(si; σ(t − 1)) = si [µσi(t − 1) + f(mN(t − 1)) + ǫi(t))] where (ǫi(t))i=1,...,N; t≥1 are i.i.d. real r.v. with distribution η(x) := P(ǫi(t) ≤ x) = 1 1 + e−2βx (logit distr.). Agents carry out simultaneously their optimization: σi(t) = 1 ⇔ Ui(1) > Ui(−1) ⇔ µσi(t − 1) + f(mN(t − 1)) + ǫi(t) > 0, which happens with probability η [µσi(t − 1) + f(mN(t − 1))] = 1 1 + e−2β[µσi(t−1)+f(mN(t−1))] .

E.Sartori (University of Padova) Strategic interaction in particle systems 6 / 20

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Rationality

The kind of rationality implicitly defined in PCA is not typical of human individuals: in interacting with others, any individual tries to forecast what the others will be doing in the near future. ↓ Different type of interaction: strategic interaction. Compare models, starting from the one inspired by PCA, with others, where we introduce and combine: the presence of strategic interaction, memory of past actions, i.e., trend.

E.Sartori (University of Padova) Strategic interaction in particle systems 7 / 20

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Rationality

The kind of rationality implicitly defined in PCA is not typical of human individuals: in interacting with others, any individual tries to forecast what the others will be doing in the near future. ↓ Different type of interaction: strategic interaction. Compare models, starting from the one inspired by PCA, with others, where we introduce and combine: the presence of strategic interaction, memory of past actions, i.e., trend.

E.Sartori (University of Padova) Strategic interaction in particle systems 7 / 20

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Rationality

The kind of rationality implicitly defined in PCA is not typical of human individuals: in interacting with others, any individual tries to forecast what the others will be doing in the near future. ↓ Different type of interaction: strategic interaction. Compare models, starting from the one inspired by PCA, with others, where we introduce and combine: the presence of strategic interaction, memory of past actions, i.e., trend.

E.Sartori (University of Padova) Strategic interaction in particle systems 7 / 20

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Approach

In all 4 models: introduce utility functions; for N < ∞, characterize choices of the static optimization; for N → ∞, describe limiting dynamics via an appropriate LLN, if possible; study the long time behavior: steady states and phase transition. ... get four different stationary scenarios!

E.Sartori (University of Padova) Strategic interaction in particle systems 8 / 20

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Approach

In all 4 models: introduce utility functions; for N < ∞, characterize choices of the static optimization; for N → ∞, describe limiting dynamics via an appropriate LLN, if possible; study the long time behavior: steady states and phase transition. ... get four different stationary scenarios!

E.Sartori (University of Padova) Strategic interaction in particle systems 8 / 20

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Approach

In all 4 models: introduce utility functions; for N < ∞, characterize choices of the static optimization; for N → ∞, describe limiting dynamics via an appropriate LLN, if possible; study the long time behavior: steady states and phase transition. ... get four different stationary scenarios!

E.Sartori (University of Padova) Strategic interaction in particle systems 8 / 20

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Approach

In all 4 models: introduce utility functions; for N < ∞, characterize choices of the static optimization; for N → ∞, describe limiting dynamics via an appropriate LLN, if possible; study the long time behavior: steady states and phase transition. ... get four different stationary scenarios!

E.Sartori (University of Padova) Strategic interaction in particle systems 8 / 20

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Non-strategic case

No-trend: Ui(si; σ(t − 1)) = si [k mN(t − 1) + µσi(t − 1) + ǫi(t)] k ≥ 0 measures the strength of interaction between agents, µ ≥ 0 models friction, (ǫi)i=1,...,N are i.i.d. local random noises with logit distr. η(x) =

1 1+e−2βx .

Dynamics: σi(t) = sign[k mN(t − 1) + µσi(t − 1) + ǫi(t)]. Limiting dynamics: mN(t) − → m(t) in probability, m(t) = [1 + m(t − 1)] η(k m(t − 1) + µ) + [1 − m(t − 1)] η(k m(t − 1) − µ) − 1. Long time behavior: ∃ kc(βµ) > 0 such that k ≤ kc(βµ) ⇒ unique fixed point and global attractor; k > kc(βµ) ⇒ two locally stable fixed points, no other attractors.

E.Sartori (University of Padova) Strategic interaction in particle systems 9 / 20

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Non-strategic case

No-trend: Ui(si; σ(t − 1)) = si [k mN(t − 1) + µσi(t − 1) + ǫi(t)] k ≥ 0 measures the strength of interaction between agents, µ ≥ 0 models friction, (ǫi)i=1,...,N are i.i.d. local random noises with logit distr. η(x) =

1 1+e−2βx .

Dynamics: σi(t) = sign[k mN(t − 1) + µσi(t − 1) + ǫi(t)]. Limiting dynamics: mN(t) − → m(t) in probability, m(t) = [1 + m(t − 1)] η(k m(t − 1) + µ) + [1 − m(t − 1)] η(k m(t − 1) − µ) − 1. Long time behavior: ∃ kc(βµ) > 0 such that k ≤ kc(βµ) ⇒ unique fixed point and global attractor; k > kc(βµ) ⇒ two locally stable fixed points, no other attractors.

E.Sartori (University of Padova) Strategic interaction in particle systems 9 / 20

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Non-strategic case

No-trend: Ui(si; σ(t − 1)) = si [k mN(t − 1) + µσi(t − 1) + ǫi(t)] k ≥ 0 measures the strength of interaction between agents, µ ≥ 0 models friction, (ǫi)i=1,...,N are i.i.d. local random noises with logit distr. η(x) =

1 1+e−2βx .

Dynamics: σi(t) = sign[k mN(t − 1) + µσi(t − 1) + ǫi(t)]. Limiting dynamics: mN(t) − → m(t) in probability, m(t) = [1 + m(t − 1)] η(k m(t − 1) + µ) + [1 − m(t − 1)] η(k m(t − 1) − µ) − 1. Long time behavior: ∃ kc(βµ) > 0 such that k ≤ kc(βµ) ⇒ unique fixed point and global attractor; k > kc(βµ) ⇒ two locally stable fixed points, no other attractors.

E.Sartori (University of Padova) Strategic interaction in particle systems 9 / 20

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Non-strategic case

No-trend: Ui(si; σ(t − 1)) = si [k mN(t − 1) + µσi(t − 1) + ǫi(t)] k ≥ 0 measures the strength of interaction between agents, µ ≥ 0 models friction, (ǫi)i=1,...,N are i.i.d. local random noises with logit distr. η(x) =

1 1+e−2βx .

Dynamics: σi(t) = sign[k mN(t − 1) + µσi(t − 1) + ǫi(t)]. Limiting dynamics: mN(t) − → m(t) in probability, m(t) = [1 + m(t − 1)] η(k m(t − 1) + µ) + [1 − m(t − 1)] η(k m(t − 1) − µ) − 1. Long time behavior: ∃ kc(βµ) > 0 such that k ≤ kc(βµ) ⇒ unique fixed point and global attractor; k > kc(βµ) ⇒ two locally stable fixed points, no other attractors.

E.Sartori (University of Padova) Strategic interaction in particle systems 9 / 20

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Non-strategic case, cont’d

Trend: Ui(si; σ(t − 1), σ(t − 2)) = si [k (mN(t − 1)−mN(t − 2)) + µσi(t − 1) + ǫi(t)] k ≥ 0 measures the strength of interaction between agents and dependence by trend, µ ≥ 0 models friction, (ǫi)i=1,...,N are i.i.d. local random noises with logit distr. η(x) =

1 1+e−2βx .

Dynamics: σi(t) = sign [k (mN(t − 1)−mN(t − 2) + µσi(t − 1) + ǫi(t)]. Limiting dynamics: mN(t) − → m(t) in probability, m(t) = [1 + m(t − 1)] η(k (m(t − 1)−m(t − 2)) + µ) + [1 − m(t − 1)] η(k (m(t − 1)−m(t − 2)) − µ) − 1; Long time behavior: analysis of steady states harder!

E.Sartori (University of Padova) Strategic interaction in particle systems 10 / 20

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Non-strategic case, cont’d

Trend: Ui(si; σ(t − 1), σ(t − 2)) = si [k (mN(t − 1)−mN(t − 2)) + µσi(t − 1) + ǫi(t)] k ≥ 0 measures the strength of interaction between agents and dependence by trend, µ ≥ 0 models friction, (ǫi)i=1,...,N are i.i.d. local random noises with logit distr. η(x) =

1 1+e−2βx .

Dynamics: σi(t) = sign [k (mN(t − 1)−mN(t − 2) + µσi(t − 1) + ǫi(t)]. Limiting dynamics: mN(t) − → m(t) in probability, m(t) = [1 + m(t − 1)] η(k (m(t − 1)−m(t − 2)) + µ) + [1 − m(t − 1)] η(k (m(t − 1)−m(t − 2)) − µ) − 1; Long time behavior: analysis of steady states harder!

E.Sartori (University of Padova) Strategic interaction in particle systems 10 / 20

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Non-strategic case, cont’d

Trend: Ui(si; σ(t − 1), σ(t − 2)) = si [k (mN(t − 1)−mN(t − 2)) + µσi(t − 1) + ǫi(t)] k ≥ 0 measures the strength of interaction between agents and dependence by trend, µ ≥ 0 models friction, (ǫi)i=1,...,N are i.i.d. local random noises with logit distr. η(x) =

1 1+e−2βx .

Dynamics: σi(t) = sign [k (mN(t − 1)−mN(t − 2) + µσi(t − 1) + ǫi(t)]. Limiting dynamics: mN(t) − → m(t) in probability, m(t) = [1 + m(t − 1)] η(k (m(t − 1)−m(t − 2)) + µ) + [1 − m(t − 1)] η(k (m(t − 1)−m(t − 2)) − µ) − 1; Long time behavior: analysis of steady states harder!

E.Sartori (University of Padova) Strategic interaction in particle systems 10 / 20

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Non-strategic case, cont’d

Trend: Ui(si; σ(t − 1), σ(t − 2)) = si [k (mN(t − 1)−mN(t − 2)) + µσi(t − 1) + ǫi(t)] k ≥ 0 measures the strength of interaction between agents and dependence by trend, µ ≥ 0 models friction, (ǫi)i=1,...,N are i.i.d. local random noises with logit distr. η(x) =

1 1+e−2βx .

Dynamics: σi(t) = sign [k (mN(t − 1)−mN(t − 2) + µσi(t − 1) + ǫi(t)]. Limiting dynamics: mN(t) − → m(t) in probability, m(t) = [1 + m(t − 1)] η(k (m(t − 1)−m(t − 2)) + µ) + [1 − m(t − 1)] η(k (m(t − 1)−m(t − 2)) − µ) − 1; Long time behavior: analysis of steady states harder!

E.Sartori (University of Padova) Strategic interaction in particle systems 10 / 20

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Phase diagram: non-strategic case with trend

0.25 0.5 0.75 1 1.25 1.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 µ k Phase diagram (β=1) − non−strategic case fixed point coexist. chaos E.Sartori (University of Padova) Strategic interaction in particle systems 11 / 20

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Strategic interaction

What if agents’ utilities explicitly depend on other players’ actions at the same time? ... we move to a game-theoretic framework due to strategic type interactions. So simultaneous, but not independent updatings: the choice will be based on the forecast of the future value mN(t).

E.Sartori (University of Padova) Strategic interaction in particle systems 12 / 20

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Strategic interaction

What if agents’ utilities explicitly depend on other players’ actions at the same time? ... we move to a game-theoretic framework due to strategic type interactions. So simultaneous, but not independent updatings: the choice will be based on the forecast of the future value mN(t).

E.Sartori (University of Padova) Strategic interaction in particle systems 12 / 20

SLIDE 37

Strategic case

No-trend: Ui( si, si ; σ(t − 1)) = si

k Ei(mN(t)) + µσi(t − 1) + ǫi(t)
where Ei denotes the expectation w.r.t. the joint distribution of (ǫj(t))j=i (common knowledge).

Ui, now, depends on the actions of all the agents − → Nash equilibrium: “s is a Nash equilibrium (NE) iff it is a fixed point for the best-response map s → Φ(s) with Φi(s) := argmax Ui( ·, si ; σ(t − 1)).” Dynamics: σi(t) = sign[ k Ei(mN(t)) + µσi(t − 1) + ǫi(t)]; possible multiplicity for NE, dynamics multi-valued! Limiting dynamics: mN(t) − → m(t) in probability, where m(t) is a solution of the implicit equation m(t) = [1 + m(t − 1)] η(k m(t) + µ) + [1 − m(t − 1)] η(k m(t) − µ) − 1 =: G(m(t), m(t − 1)). Limiting dynamics exist, even if they can be multi-valued, so ill-defined.

E.Sartori (University of Padova) Strategic interaction in particle systems 13 / 20

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Strategic case

No-trend: Ui( si, si ; σ(t − 1)) = si

k Ei(mN(t)) + µσi(t − 1) + ǫi(t)
where Ei denotes the expectation w.r.t. the joint distribution of (ǫj(t))j=i (common knowledge).

Ui, now, depends on the actions of all the agents − → Nash equilibrium: “s is a Nash equilibrium (NE) iff it is a fixed point for the best-response map s → Φ(s) with Φi(s) := argmax Ui( ·, si ; σ(t − 1)).” Dynamics: σi(t) = sign[ k Ei(mN(t)) + µσi(t − 1) + ǫi(t)]; possible multiplicity for NE, dynamics multi-valued! Limiting dynamics: mN(t) − → m(t) in probability, where m(t) is a solution of the implicit equation m(t) = [1 + m(t − 1)] η(k m(t) + µ) + [1 − m(t − 1)] η(k m(t) − µ) − 1 =: G(m(t), m(t − 1)). Limiting dynamics exist, even if they can be multi-valued, so ill-defined.

E.Sartori (University of Padova) Strategic interaction in particle systems 13 / 20

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Strategic case, Cournot adjustment

A way to select one NE: the Cournot adjustment: a NE can be seen as not achieved instantaneously, but emerging as a result of a learning

mechanism. Then σ(t) is the limit of the iterates of the map G applied n times, where we

assume as a starting point σ(t − 1). This procedure selects exactly one NE. Long time behavior: βk ≤ 1 ⇒ dynamics well-defined. The stationary scenario is similar to the non-strategic case. βk > 1 ⇒ dynamics can be ill-defined → Cournot adjustment. Even if there are periodic orbits between possible dynamics, they are eliminated by the Cournot adjustment. The stationary scenario is similar to the non-strategic case.

E.Sartori (University of Padova) Strategic interaction in particle systems 14 / 20

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Strategic case, Cournot adjustment

A way to select one NE: the Cournot adjustment: a NE can be seen as not achieved instantaneously, but emerging as a result of a learning

mechanism. Then σ(t) is the limit of the iterates of the map G applied n times, where we

assume as a starting point σ(t − 1). This procedure selects exactly one NE. Long time behavior: βk ≤ 1 ⇒ dynamics well-defined. The stationary scenario is similar to the non-strategic case. βk > 1 ⇒ dynamics can be ill-defined → Cournot adjustment. Even if there are periodic orbits between possible dynamics, they are eliminated by the Cournot adjustment. The stationary scenario is similar to the non-strategic case.

E.Sartori (University of Padova) Strategic interaction in particle systems 14 / 20

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Strategic case, Cournot adjustment

A way to select one NE: the Cournot adjustment: a NE can be seen as not achieved instantaneously, but emerging as a result of a learning

mechanism. Then σ(t) is the limit of the iterates of the map G applied n times, where we

assume as a starting point σ(t − 1). This procedure selects exactly one NE. Long time behavior: βk ≤ 1 ⇒ dynamics well-defined. The stationary scenario is similar to the non-strategic case. βk > 1 ⇒ dynamics can be ill-defined → Cournot adjustment. Even if there are periodic orbits between possible dynamics, they are eliminated by the Cournot adjustment. The stationary scenario is similar to the non-strategic case.

E.Sartori (University of Padova) Strategic interaction in particle systems 14 / 20

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Strategic case, cont’d

Trend: Ui( si, si ; σ(t), σ(t − 1)) = si

k Ei(mN(t))−k mN(t − 1) + µσi(t − 1) + ǫi(t)
Dynamics:

σi(t) = sign

k Ei(mN(t)) − k mN(t − 1) + µσi(t − 1) + ǫi(t)
;

possible multiplicity for NE, dynamics multi-valued! Limiting dynamics: mN(t) − → m(t) in probability, where m(t) is a solution of the implicit equation m(t) = [1 + m(t − 1)] η(k (m(t)−m(t − 1)) + µ) + [1 − m(t − 1)] η(k (m(t)−m(t − 1)) − µ) − 1. Limiting dynamics exist, even if they can be multi-valued, so ill-defined → Cournot adjustment. Long time behavior: see Phase diagram, where a periodic orbit survives the Cournot adjustment!

E.Sartori (University of Padova) Strategic interaction in particle systems 15 / 20

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Strategic case, cont’d

Trend: Ui( si, si ; σ(t), σ(t − 1)) = si

k Ei(mN(t))−k mN(t − 1) + µσi(t − 1) + ǫi(t)
Dynamics:

σi(t) = sign

k Ei(mN(t)) − k mN(t − 1) + µσi(t − 1) + ǫi(t)
;

possible multiplicity for NE, dynamics multi-valued! Limiting dynamics: mN(t) − → m(t) in probability, where m(t) is a solution of the implicit equation m(t) = [1 + m(t − 1)] η(k (m(t)−m(t − 1)) + µ) + [1 − m(t − 1)] η(k (m(t)−m(t − 1)) − µ) − 1. Limiting dynamics exist, even if they can be multi-valued, so ill-defined → Cournot adjustment. Long time behavior: see Phase diagram, where a periodic orbit survives the Cournot adjustment!

E.Sartori (University of Padova) Strategic interaction in particle systems 15 / 20

SLIDE 44

Strategic case, cont’d

Trend: Ui( si, si ; σ(t), σ(t − 1)) = si

k Ei(mN(t))−k mN(t − 1) + µσi(t − 1) + ǫi(t)
Dynamics:

σi(t) = sign

k Ei(mN(t)) − k mN(t − 1) + µσi(t − 1) + ǫi(t)
;

possible multiplicity for NE, dynamics multi-valued! Limiting dynamics: mN(t) − → m(t) in probability, where m(t) is a solution of the implicit equation m(t) = [1 + m(t − 1)] η(k (m(t)−m(t − 1)) + µ) + [1 − m(t − 1)] η(k (m(t)−m(t − 1)) − µ) − 1. Limiting dynamics exist, even if they can be multi-valued, so ill-defined → Cournot adjustment. Long time behavior: see Phase diagram, where a periodic orbit survives the Cournot adjustment!

E.Sartori (University of Padova) Strategic interaction in particle systems 15 / 20

SLIDE 45

Strategic case, cont’d

Trend: Ui( si, si ; σ(t), σ(t − 1)) = si

k Ei(mN(t))−k mN(t − 1) + µσi(t − 1) + ǫi(t)
Dynamics:

σi(t) = sign

k Ei(mN(t)) − k mN(t − 1) + µσi(t − 1) + ǫi(t)
;

possible multiplicity for NE, dynamics multi-valued! Limiting dynamics: mN(t) − → m(t) in probability, where m(t) is a solution of the implicit equation m(t) = [1 + m(t − 1)] η(k (m(t)−m(t − 1)) + µ) + [1 − m(t − 1)] η(k (m(t)−m(t − 1)) − µ) − 1. Limiting dynamics exist, even if they can be multi-valued, so ill-defined → Cournot adjustment. Long time behavior: see Phase diagram, where a periodic orbit survives the Cournot adjustment!

E.Sartori (University of Padova) Strategic interaction in particle systems 15 / 20

SLIDE 46

Phase diagram: strategic case with trend

0.25 0.5 0.75 1 1.25 1.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 µ k Phase diagram (β=1) − strategic case fixed point coexist. 2−cycle E.Sartori (University of Padova) Strategic interaction in particle systems 16 / 20

SLIDE 47

Phase diagrams with trend: non-strategic vs strategic case

0.25 0.5 0.75 1 1.25 1.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 µ k Phase diagram (β=1) − non−strategic case fixed point coexist. chaos 0.25 0.5 0.75 1 1.25 1.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 µ k Phase diagram (β=1) − strategic case fixed point coexist. 2−cycle

E.Sartori (University of Padova) Strategic interaction in particle systems 17 / 20

SLIDE 48

“No trend no party!”

Non strategic int. - no trend: 1 global attractor or 2 locally stable ones. Non strategic int. with trend: 1 fixed point, a chaotic phase or coexistence. Strategic int. - no trend (+ Cournot adjustment): as non strategic case without trend. Strategic int. with trend: 1 fixed point, a 2-cycle or coexistence It seems that forecasting from one hand creates polarization, from the other

ne a more synchronous behavior.

E.Sartori (University of Padova) Strategic interaction in particle systems 18 / 20

SLIDE 49

“No trend no party!”

Non strategic int. - no trend: 1 global attractor or 2 locally stable ones. Non strategic int. with trend: 1 fixed point, a chaotic phase or coexistence. Strategic int. - no trend (+ Cournot adjustment): as non strategic case without trend. Strategic int. with trend: 1 fixed point, a 2-cycle or coexistence It seems that forecasting from one hand creates polarization, from the other

ne a more synchronous behavior.

E.Sartori (University of Padova) Strategic interaction in particle systems 18 / 20

SLIDE 50

Thanks... ...grazie!

E.Sartori (University of Padova) Strategic interaction in particle systems 19 / 20

SLIDE 51

Barucci, E. and Tolotti, M. Social interaction and conformism in a random utility model Journal of Economic Dynamics and Control 36: 1855–1866, 2012. Bouchaud, J.P. Crises and collective socio-economic phenomena: simple models and challenges, Journal of Statistical Physics, 151: 567–606, 2013. Brock, W. and Durlauf, S. Discrete choice with social interactions, Review of Economic Studies, 68: 235–260, 2001. Dai Pra, P., Sartori, E. and Tolotti, M. Strategic interaction in trend-driven dynamics, Journal of Statistical Physics, 152: 724–741, 2013. Dai Pra, P., Sartori, E. and Tolotti, M. Strategic interaction in interacting particle systems, forthcoming. Föllmer, H. Random economies with many interacting agents, Journal of Mathematical Economics, 1: 51–62, 1974; Granovetter, M. Threshold models of collective behavior, The American Journal of Sociology, 83: 1420–1443, 1978. Schelling, T. Micromotives and Macrobehavior, Norton New York, 1978.

E.Sartori (University of Padova) Strategic interaction in particle systems 20 / 20