50MINUTES - Game Theory: The art of thinking strategically

The beginnings of game theory, strictly speaking, are found in the works of mathematicians from the first half of the 19 th century.

The first person to study the strategic aspects of interactions between economic agents was Antoine Augustin Cournot (French mathematician, philosopher and economist, 1801-1877). His 1838 book Researches into the Mathematical Principles of the Theory of Wealth contains the beginnings of game theory, which was later developed in the 1950s. He analyses the different forms of competition in duopolies (a market with two competing sellers) and in the specific context of the Nash equilibrium (between manufacturers), for which he gives the first formulations.

Good to know: Researches into the Mathematical Principles of the Theory of Wealth , 1838

Although it was completely ignored when it was first published, this book emerged from obscurity through the work of John Forbes Nash (American economist and mathematician, 1928-2015) on repeated game theory in 1950. Today, the Cournot competition is a model based on the analysis of imperfect competition in industrial economics.

While Cournot analysed the strategic interactions between two productive companies, the Anglo-Irish economist and lawyer Francis Ysidro Edgeworth (1845-1926) expanded this reasoning and applied the model to cases of economies without production. In Mathematical Physics: An Essay on the Application of Mathematics to the Moral Sciences (1881), he developed a tool to represent the interactions between two non-productive economic agents: the Edgeworth box. This book marked the introduction of mathematics into economics.

Good to know: the Edgeworth box

This box allows users to both analyse the possibilities for allocating resources between two entities and to see if this allocation is ideal according to Pareto optimality, i.e. if is possible to improve the situation of one agent without damaging that of the other.

Modern literature on game theory fully recognises that the first formal theorem of game theory was produced by Ernst Friedrich Ferdinand Zermelo (German mathematician, 1871-1953) in 1913. This theorem has been taken up by many authors and interpreted in several different ways. Mas Colell et al.s version from 1995 essentially states that in any perfect information (each player knows all the strategies and payoff functions of all the other players) fixed game (where the number of rounds is known in advance), there is an equilibrium that would later become known as the Nash equilibrium.

The Nash equilibrium is made up of pure strategies sequences of actions that a player is known to choose each time they are likely to play and is obtained by backward induction. This involves determining the optimal strategies of the players in the last round of the game. In other words, we reason by working back from the last round of the game to the first, determining the best strategies of the players at each stage of the game. This concept will be illustrated later.

Whereas all of the previous contributions allowed simple games (meaning those with pure strategies) to be solved, the contribution of the French mathematician mile Borel (1871-1956) marks a turning point for game theory from 1921 onwards. In Volume IV of his book Treaty of the Calculation of Probabilities and its Applications (1924-1934), the author introduces probabilities in games of chance and recommends the minimax theorem for zero-sum games, where gains for one player mean losses for another. In the same book, the author also distinguishes between two different categories of games of chance: