Game theory is a field of mathematics that is used to analyse the strategies used by decision makers in competitive situations. It can apply to humans, animals, and computers in various situations but is commonly used in AI research to study “multi-agent” environments where there is more than one system, for example several household robots cooperating to clean the house. Traditionally, the evolutionary dynamics of multi-agent systems have been analysed using simple, symmetric games, such as the classic Prisoner’s Dilemma, where each player has access to the same set of actions. Although these games can provide useful insights into how multi-agent systems work and tell us how to achieve a desirable outcome for all players - known as the Nash equilibrium - they cannot model all situations.

Our new technique allows us to quickly and easily identify the strategies used to find the Nash equilibrium in more complex asymmetric games - characterised as games where each player has different strategies, goals and rewards. These games - and the new technique we use to understand them - can be illustrated using an example from ‘Battle of the Sexes’, a coordination game commonly used in game theory research.

Here, two players have to coordinate a night out to either the opera or the movies. One of the players has a slight preference for the opera and one of them has a slight preference for the movies. The game is asymmetric because, while both players have access to the same options, the corresponding rewards for each are different based on the players preferences. In order to maintain their friendship - or equilibrium - the players should choose the same activity (hence the zero payoff for separate activities).