Climb on the Bandwagon: Consensus and Periodicity in a Lifetime Utility Model with Strategic Interactions

Pra, Paolo Dai; Sartori, Elena; Tolotti, Marco

doi:10.1007/s13235-019-00299-y

Cited by 6 publications

(5 citation statements)

References 41 publications

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“…This paper extends our unpublished preprint [36] in the direction of asymmetric interactions with rigorous analytics on the bifurcations of the system in the absence of delays. We note that [36] has been the basis of subsequent developments in various directions, including investigations of the role of complex networks topologies [23], asymmetric interactions between classes of individuals [7] (that we extend here), or socio-economic applications [26,19,22,11].…”

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confidence: 99%

The hipster effect: When anti-conformists all look the same

Touboul¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

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In such different domains as statistical physics, neurosciences, spin glasses, social science, economics and finance, large ensemble of interacting individuals evolving following (mainstream) or against (hipsters) the majority are ubiquitous. Moreover, in a variety of applications, interactions between agents occur after specific delays that depends on the time needed to transport, transmit or take into account information. This paper focuses on the role of opposition to majority and delays in the emerging dynamics in a population composed of mainstream and anti-conformist individuals. To this purpose, we introduce a class of simple statistical system of interacting agents taking into account (i) the presence of mainstream and anti-conformist individuals and (ii) delays, possibly heterogeneous, in the transmission of information. In this simple model, each agent can be in one of two states, and can change state in continuous time with a rate depending on the state of others in the past. We express the thermodynamic limit of these systems as the number of agents diverge, and investigate the solutions of the limit equation, with a particular focus on synchronized oscillations induced by delayed interactions. We show that when hipsters are too slow in detecting the trends, they will consistently make the same choice, and realizing this too late, they will switch, all together to another state where they remain alike. Another modality synchronizing hipsters are asymmetric interactions, particularly when the cross-interaction between hipsters and mainstreams aree prominent, i.e. when hipsters radically oppose to mainstream and mainstreams wish to follow the majority, even when led by hipsters. We demonstrate this phenomenon analytically using bifurcation theory and reduction to normal form. We find that, in the case of asymmetric interactions, the level of randomness in the decisions themselves also leads to synchronization of the hipsters. Beyond the choice of the best suit to wear this winter, this study may have important implications in understanding synchronization of nerve cells, investment strategies in finance, or emergent dynamics in social science, domains in which delays of communication and the geometry of information accessibility are prominent.2010 Mathematics Subject Classification. Primary: 82C22, 34D06; Secondary: 34K11.

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confidence: 99%

The hipster effect: When anti-conformists all look the same

Touboul¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

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“…At this point, we prefer to introduce the macroscopic model as the formal limit for N that tends to infinity the microscopic one, assuming that all companies are exchangeable. This approach is standard, but it is not trivial: when the model involves mean-field games, the passage to the limit is not straightforward, and additional equilibria may arise (see, e.g., [12,14]). We are not interested in this situation, and instead, we formalize the problem directly, corresponding to the formal limit for a representative player.…”

Section: The Macroscopic Modelmentioning

confidence: 99%

A Binary-State Continuous-Time Markov Chain Model for Offshoring and Reshoring

Brambilla,

Grosset,

Sartori

2024

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We present a two-country model (North and South) that describes the phenomenon of offshoring and reshoring. The model is a continuous time-controlled Markov chain with binary states. The main trade-off involves production costs and transaction costs between one country and another. In the first part of this paper, we identify the key parameters of the model: the difference in unit production costs between the two countries considered, the marginal cost of transitioning between countries, and the incentive paid by the North country to all companies that have not relocated at the end of the planning interval. The final goal of our paper is to understand how national tax incentives can influence this process.

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Section: Introductionmentioning

confidence: 99%

“…Such an example was first considered by Gomes, Velho, and Wolfram in [20,21], where numerical evidence on the convergence behavior was presented; it should also be compared to Lacker's "illuminating example" (Subsection 3.3 in [23]) and to the example in Subsection 3.3 of [1] by Bardi and Fischer, both in the diffusion setting. In the infinite time horizon and finite state case, an example of non-uniqueness is studied in [13], via numerical simulations, where periodic orbits emerge as solutions to the mean field game.For the two-state example studied here, the mean field game possesses exactly three solutions, given any initial distribution, as soon as the time horizon is large enough. Consequently, there is no regular solution to the master equation, while multiple weak solutions exist.…”

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confidence: 99%

On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness

Cecchin¹,

Pra²,

Fischer³

et al. 2019

SIAM J. Control Optim.

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We consider N -player and mean field games in continuous time over a finite horizon, where the position of each agent belongs to {−1, 1}. If there is uniqueness of mean field game solutions, e.g. under monotonicity assumptions, then the master equation possesses a smooth solution which can be used to prove convergence of the value functions and of the feedback Nash equilibria of the N -player game, as well as a propagation of chaos property for the associated optimal trajectories. We study here an example with anti-monotonous costs, and show that the mean field game has exactly three solutions. We prove that the value functions converge to the entropy solution of the master equation, which in this case can be written as a scalar conservation law in one space dimension, and that the optimal trajectories admit a limit: they select one mean field game soution, so there is propagation of chaos. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N -player game selects the optimizer of this problem. 1 2 ALEKOS CECCHIN, PAOLO DAI PRA, MARKUS FISCHER, AND GUGLIELMO PELINO are shown to be concentrated on weak solutions of the corresponding mean field game. This concept of solution is also used in another, more recent work by Lacker; see below.Here, we are interested in the convergence problem for Nash equilibria in Markov feedback strategies with full state information. A first result in this direction was given by Gomes, Mohr, and Souza [19] in the case of finite state dynamics. There, convergence of Markovian Nash equilibria to the mean field game limit is proved, but only if the time horizon is small enough. A breakthrough was achieved by Cardaliaguet, Delarue, Lasry, and Lions in [7]. In the setting of games with non-degenerate Brownian dynamics, possibly including common noise, those authors establish convergence to the mean field game limit, in the sense of convergence of value functions as well as propagation of chaos for the optimal state trajectories, for arbitrary time horizon provided the so-called master equation associated with the mean field game possesses a unique sufficiently regular solution. The master equation arises as the formal limit of the Hamilton-Jacobi-Bellman systems determining the Markov feedback Nash equilibria. It yields, if well-posed, the optimal value in the mean field game as a function of initial time, state and distribution. It thus also provides the optimal control action, again as a function of time, state, and measure variable. This allows, in particular, to compare the prelimit Nash equilibria to the solution of the limit model through coupling arguments.If the master equation possesses a unique regular solution, which is guaranteed under the Lasry-Lions monotonicity conditions, then the convergence analysis can be considerably refined. In this case, for games with finite state dynamics, Cecchin and Pelino [11] and, independently, Bayraktar and Cohen [3] obtain a central limit theorem and lar...

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Climb on the Bandwagon: Consensus and Periodicity in a Lifetime Utility Model with Strategic Interactions

Cited by 6 publications

References 41 publications

The hipster effect: When anti-conformists all look the same

The hipster effect: When anti-conformists all look the same

A Binary-State Continuous-Time Markov Chain Model for Offshoring and Reshoring

On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness

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