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Department of Mathematics and Applications
"Renato Caccioppoli"


Game Theory

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Game Theory

Credits

6

Prerequisites

None.

Scientific-disciplinary sector (SSD)

SECS-S/06 Mathematical Methods for Economics and Actuarial and Financial Sciences.

Examination method

Written exam (open-ended questions and numerical exercises) and oral exam.

Learning
objectives

The course provides the mathematical foundations of non-cooperative theory, thereby illustrating the level of rigour in applications typically developed in advanced economics courses. The cooperative approach is also analysed through natural applications of the main solution concepts proposed in the literature.

Syllabus

  • Review of analysis and topology of finite-dimensional spaces, with insights into infinite-dimensional analysis. Separation theorems and a brief theory of correspondences. Kakutani’s theorem. Mixed strategies from finite and infinite pure strategy spaces. Elements of measure theory: in finite-dimensional spaces and regular measures on topological spaces.
  • Conflict in the presence of strategic interaction. Nash equilibrium. Determination with differentiable payoffs. Analysis of the Bertrand and Cournot models. An example of incentive compatibility in the funding of a public project.
  • Existence of equilibria, with an outline of the case of discontinuous payoffs (Nash, Glicksberg, Dasgupta-Maskin).
  • Nash solution for bargaining problems. Outline of the Kalai-Smorodinski solution.
  • TU games. Nucleolus, Shapley value: existence and uniqueness theorems. Core, Bondareva’s theorem.
  • Bankruptcy problem, cost allocation problem, majority games.
  • Core of an NTU game: Scarf’s existence theorem. Core of an exchange economy, Debreu-Scarf theorem.

Expected learning
outcomes

At the end of the course students must demonstrate that they

  • know and understand the issues relating to the topics covered in the course and are able to extract, from the phenomena analysed, the characteristic elements of the strategic interaction involved;
  • are able to apply the knowledge acquired to arrive with mathematical rigour at the definition of possible solutions, clearly highlighting the assumptions on which they rest. Proving statements is an essential tool;
  • are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
  • are able to identify the most appropriate methods to analyse and solve a problem related to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

The exam assesses the skills acquired and the ability to apply them by discussing, also through solving exercises or proving theorems, the solution concepts of a game and their range of applicability.