From my doctoral course
This note comes from the doctoral course “Game Theory and Bargaining”. I rewrote the explanations in my own words and recomputed the numerical examples.
From a lone decision to a decision against others
In probabilistic decisions, “nature” was our opponent and did not deliberately play against us. Game theory becomes necessary when the opponent is a rational person or organisation who also wants a result for themselves; so the outcome of my action depends on another’s choice too, and they know it.
A short history
- 1921: Émile Borel writes papers on strategic games.
- 1928: John von Neumann proves the minimax theorem for two-person zero-sum games.
- 1944: von Neumann and Morgenstern publish Theory of Games and Economic Behavior and the field takes shape.
- 1950 and 1951: John Nash introduces the concept of equilibrium for general games.
- 1994: Nash, Harsanyi and Selten receive the Nobel Prize in economics.
What is a game?
In this course a “game” is any situation where several people (players), each by choosing a strategy, receive an outcome that also depends on the others’ choices. A few definitions:
- Player: a rational decision maker; an individual, a company or a country.
- Move: a choice made at one point of the game (personal or chance).
- Strategy: a complete plan for all possible situations. An important difference: a move is a choice, while a strategy is a decision rule.
- Rules of the game: who knows what, when, and what they can do.
- Payoff: the utility number each player receives at the end.
- Finite/infinite game: the number of strategies or stages is limited or not.
The basic assumption: rationality and common knowledge
Every player is rational (maximises their own utility) and this is also common knowledge: everyone knows everyone is rational and knows that everyone knows it. Even a little doubt about this assumption changes the result. (Critique: real people are not always like this; see Ultimatum, Stackelberg and sequential games.)
Classification
- Static or dynamic: players choose simultaneously or in turns.
- Complete or incomplete information: are everyone’s payoffs and preferences known to everyone?
- Zero-sum or non-zero-sum: is one’s gain the other’s loss? (Two firms fighting over a fixed market share: zero-sum; a contract negotiation where both profit: non-zero-sum.)
- Cooperative or non-cooperative: is there a binding contract?
Why does it matter to me?
In systems analysis, stakeholders are “players” too: finance and marketing each optimise for themselves. I think game theory is a precise language for seeing conflict and for mechanism design (what do we do so individual incentives align with the organisation’s goal?). See Conflict and conflicting systems.
Related: Static games: the prisoner’s dilemma and Nash equilibrium.