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Forward-Forward Mean Field Games in mathematical modeling with application to opinion formation and voting models

Adriano Festa, Simone Gottlich, Michele Ricciardi

Abstract

While the general theory for the terminal-initial value problem in mean-field games is widely used in many models of applied mathematics, the modeling potential of the corresponding forward-forward version is still under-considered. In this work, we study the well-posedness of the problem in a quite general setting and explain how it is appropriate to model a system of players that have a complete knowledge of the past states of the system and are adapting to new information without any knowledge about the future. Then we show how forward-forward mean field games can be effectively used in mathematical models for opinion formation and other social phenomena.

Forward-Forward Mean Field Games in mathematical modeling with application to opinion formation and voting models

Abstract

While the general theory for the terminal-initial value problem in mean-field games is widely used in many models of applied mathematics, the modeling potential of the corresponding forward-forward version is still under-considered. In this work, we study the well-posedness of the problem in a quite general setting and explain how it is appropriate to model a system of players that have a complete knowledge of the past states of the system and are adapting to new information without any knowledge about the future. Then we show how forward-forward mean field games can be effectively used in mathematical models for opinion formation and other social phenomena.
Paper Structure (22 sections, 2 theorems, 93 equations, 10 figures)

This paper contains 22 sections, 2 theorems, 93 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Derivation and interpretation of the evolution process
  3. Some results about Forward-Forward Mean Field Games
  4. Hypotheses:
  5. Two Forward-Forward models
  6. A 1D opinion formation model
  7. FF-Opinion formation: Tests
  8. Test 1. A Forward-Backward/Forward-Forward comparison.
  9. Test 2. The effect of the discount variable $\lambda$.
  10. Test 3. Cluster formation and control.
  11. A dynamic 2D vote model
  12. FF-dynamic vote model: Tests
  13. Test 4: Stable Cluster Formation between Two Candidates
  14. Test 5. Overcoming the impasse via different political strategies.
  15. Conclusions
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let Hypotheses $(H1)-(H3)$ hold true. Then there exists a unique solution $(u,m)\in\mathcal{C}^{1+\frac{\gamma}{2},2+\gamma}([0,T]\times \overline\Omega)\times\mathcal{C}^{1+\frac{\gamma}{2},2+\gamma}([0,T]\times \overline\Omega)$ for the FF-MFG system MFG.

Figures (10)

  • Figure 1: Test 1. Forward-Forward model: evolution of the potential function $u$ and density distribution $m$ (top), control map $-\nabla u/a_2$ and mean opinion $\int_{\Omega} xm(t,x)dx$.
  • Figure 2: Test 1. Forward-Backward model: evolution of the potential function $u$ and density distribution $m$ (top), control map $-\nabla u/a_2$ and mean opinion $\int_{\Omega} xm(t,x)dx$.
  • Figure 3: Test 1. Evolution of the density of opinion in the FF and FB model.
  • Figure 4: Test 2. Forward-Forward model: evolution of the potential function $u$ (left) and density distribution $m$ (top) with various values of the discount coefficient $\lambda$.
  • Figure 5: Test 3. Forward-Forward model: evolution of the potential function $u$ and density distribution $m$ (top), control map $-\nabla u/a_2$ and mean opinion $\int_{\Omega} xm(t,x)dx$.
  • ...and 5 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof