April 19, 2021

Webinar link

Webinar ID: 962 7774 9870

Passcode: 285404

In integer programming games, players' feasible strategies are described by lattice points inside polyhedra. This game representation is natural when players' decisions have integrality restrictions. In this talk, we will start by presenting practical examples of integer programming games. Then, we will focus on a particular "dynamic" integer programming game played over a graph, the Multilevel Critical Node problem. Besides a discussion on the problem difficulty, we will describe an exact cutting plane algorithm to determine the game equilibrium and a reinforcement learning based heuristic to approximate it.

April 12, 2021

**Webinar link**

Webinar ID: 910 7928 6959

Passcode: VISS

In this talk, we explore how networked compartmental models of epidemic processes combined with transportation data can be used to model the spread of COVID-19. We first employ a networked SEIR (susceptible-exposed-infected-recovered) model and present necessary and sufficient conditions for identifying the model parameters from data. We illustrate several shortcomings of traditional approaches by applying the identification results to COVID-19 testing and travel data from the Northeastern United States and use these inaccuracies as motivation for the latter two parts of the talk. One typical error is assuming that testing data perfectly capture the underlying epidemic states, which is not accurate due to delays in testing results, testing inaccuracies, and biased/partial population sampling. We present an algorithm for inferring the underlying epidemic states of an SIR model from testing data that accounts for heterogeneous delays and a closed-form expression for the error of the algorithm. The last part of the talk focuses on the recent development of a networked SEIR model that incorporates population flow as the viral spread mechanism to capture infection transmission between sub-populations. We show, under reasonable assumptions, that the dynamics have a consensus-type behavior where in steady-state each sub-population has the same amount of recovered individuals. Employing this model, we present several approaches for using travel restrictions as a control mechanism.

April 12, 2021

**Dynamic Games and Applications Seminar **

**Apr 15, 2021 11:00 AM – 12:00 PM **(Eastern Daylight Time, UTC -4)

Webinar link

Webinar ID: 962 7774 9870

Passcode: 285404

Within a non-cooperative transboundary pollution dynamic game, we study the strategic impact of a region's investment in the adoption of a cleaner technology, as embodied by a reduction in the emission per output ratio, on the equilibrium outcomes and regions' welfare. The ratio of emissions to output is endogenous and is a decreasing function of the level of the stock of clean technology. Each region can invest in a clean technology in addition to its control of emissions. Clean technology is assumed to be public knowledge so that both regions benefit from the investment in clean technology of an individual region. Pollution damage is modelled as a strictly convex function in the pollution stock. We analyze the feedback equilibrium of the non-cooperative game between two regions played over an infinite horizon. The formulation of the transboundary pollution dynamic game does not fit any special structure of analytically tractable games such as linear-state or linear-quadratic differential games. We develop numerical methods to characterize the feedback equilibrium of the non-cooperative game between two regions. The equilibrium trajectories of the stock of pollution and stock of clean technology as well the regions' welfare are compared under different scenarios.

(jointly with Javier de Frutos and Paula M. López-Pérez).

April 7, 2021

**ISS Informal Systems Seminar **

**Apr 9, 2021 10:00 AM – 11:00 AM **(Eastern Daylight Time, UTC -4)

**Webinar link**

Webinar ID: 910 7928 6959

Passcode: VISS

In this work, we study a model for opinion dynamics where the influence weights of agents evolve in time via an equation which is coupled with the opinions’ evolution. We explore the natural question of the large population limit with two approaches: the now classical mean-field limit and the more recent graph limit. After establishing the existence and uniqueness of solutions to the models that we will consider, we provide a rigorous mathematical justification for taking the graph limit in a general context. Then, establishing the key notion of indistinguishability, which is a necessary framework to consider the mean-field limit, we prove the subordination of the mean-field limit to the graph one in that context. This actually provides an alternative (but weaker) proof for the mean-field limit. We conclude by showing some numerical simulations to illustrate our results.

Bio: Nathalie Ayi is an Assistant Professor at the Jacques Louis Lions laboratory at Sorbonne University (Paris). Before joining this laboratory, she was a post-doctoral fellow in the IPSO INRIA team at Rennes. She received her PhD degree in Mathematics from the University of Nice Sophia-Antipolis in 2016. Her field of research is partial differential equations with a specific interest in kinetic theory and the study of scaling limits.

April 6, 2021

**Dynamic Games and Applications Seminar **

**Apr 8, 2021 11:00 AM – 12:00 PM **(Eastern Daylight Time, UTC -4)

Webinar link

Webinar ID: 962 7774 9870

Passcode: 285404

One year after the beginning of COVID-19 pandemic, vaccination seems to be the main instrument to block its spread, by raising the percentage of unsusceptible population. The best tool to overcome the pandemic is obviously the development of a sufficiently effective vaccine. Thus since the beginning of the pandemic considerable R&D effort is made. In the meantime, the lockdown still remains the most effective tool to contrast this virus. Assuming that the discovery time of such successful vaccine is not known a priori, we formalize a two-stage optimal control problem with stochastic switching time, where in the first period (before the vaccination entry) the control functions are the research effort and the lockdown, while in the second period both vaccination and lockdown tools act in synergy.

(with Stefan Wrzaczek, Maddalena Muttoni, Michael Freiberger)

March 29, 2021

**Dynamic Games and Applications Seminar **

**Apr 1, 2021 11:00 AM – 12:00 PM **(Eastern Daylight Time, UTC -4)

Webinar link

Webinar ID: 962 7774 9870

Passcode: 285404

We introduce a framework for new product diffusion that integrates consumer heterogeneity and strategic social influences at individual level. Forward-looking consumers belong to two segments: individualists, whose adoption decision is influenced by the price and firm’s goodwill, and conformists, whose adoption decision depends on social influences and the price. We use a mean-field game approach to translate consumer strategic interactions into aggregate social influences. We provide the conditions for existence and uniqueness of equilibrium. Our results suggest that the firm adopts a penetration pricing strategy when consumers are forward-looking, whereas it implements a penetration-skimming policy in face of myopic consumers.

(Jointly with Rabih Salhab and Georges Zaccour)