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Game Theory

Game theory provides a formal language for the representation and analysis of interactive situations, that is, situations where several “entities”, called players, take actions that affect each other. The nature of the players varies depending on the context in which the game theoretic language is invoked: in evolutionary biology (see, for example, John Maynard Smith, 1982) players are non-thinking living organisms; in computer science (see, for example, Shoham-Leyton-Brown, 2008) players are artificial agents; in behavioral game theory (see, for example, Camerer, 2003) players are “ordinary” human beings, etc. Traditionally, however, game theory has focused on interaction among intelligent, sophisticated and rational individuals.

Game theory is divided into two main branches. The first is cooperative game theory, which assumes that the players can communicate, form coalitions and sign binding agreements. Cooperative game theory has been used, for example, to analyze voting behavior and other issues in political science and related fields.

The other main branch is non-cooperative game theory. Non-cooperative game theory models situations where the players are either unable to communicate or are able to communicate but cannot sign binding contracts. An example of the latter situation is the interaction among firms in an industry in an environment where antitrust laws make it illegal for firms to reach agreements concerning prices or production quotas or other forms of collusive behavior.

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Differential Game Thory & Optimal Control

In game theory, differential games are a group of problems related to the modeling and analysis of conflict in the context of a dynamical system. More specifically, a state variable or variables evolve over time according to a differential equation.

Differential games are related closely with optimal control problems. In an optimal control problem there is single control signal and a single criterion to be optimized; differential game theory generalizes this to two ore more control signals and two or more criteria, one for each player. Each player attempts to control the state of the system so as to achieve its goal; the system responds to the inputs of all players.

Application to HRI

Wherever there are two or more agents interacting with eachother, we can use game theory to model these interactions and to describe the underlying dynamics of these interactions. From Human-Human interactions to Robot-Robot interactions, from aoutonomus vehicles to rehabilitation robotics, from leader-follower tasks to cooperitive tasks. Below you can find several papers applying game theory methods to variety of problems:

• A framework to describe, analyze and generate interactive motor behaviors

• Differential game theory for versatile physical human–robot interaction

• A game theory of mind

• Nash Equilibria in Multi-Agent Motor Interactions

• A Review of Intent Detection, Arbitration, and Communication Aspects of Shared Control for Physical Human–Robot Interaction

• Cooperating with machines

• Communication and inference of intended movement direction during human–human physical interaction

• Physical Collaboration of Human-Human and Human-Robot Teams

• Physically interacting individuals estimate the partner’s goal to enhance their movements

## General Resources

• Binmore, Ken. Game theory: a very short introduction. OUP Oxford, 2007.

• Faruqi, Farhan A. Differential game theory with applications to missiles and autonomous systems guidance. John Wiley & Sons, 2017.

• Coursera: Game Theory

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