Introduction to Game Theory and Strategic Games
Game theory is the mathematical study of decisions in which the outcome for each participant depends on the choices of all; a strategic game is described by its players, the strategies available to each, and the payoffs that each combination of strategies produces. In brief: in a two-player game with payoffs written as pairs (row player, column player), the cell (3, 3) means that both players receive 3. In the classic prisoner’s dilemma each player does better by defecting whatever the other does, yet mutual defection (1, 1) is worse for both than mutual cooperation (3, 3).
Firms setting prices, drivers choosing routes, countries negotiating treaties and bidders in an auction all face strategic interaction. This guide introduces the language of game theory: players, strategies, payoffs, the normal form (payoff matrix) and the extensive form (game tree), simultaneous and sequential moves, zero-sum and non-zero-sum games, and the assumption of rationality, with worked examples and a table.
What are the elements of a game?
A game consists of:
- the players, the decision makers (individuals, firms, countries);
- for each player, a set of strategies, complete plans of action;
- the payoffs, numbers describing how much each player values the outcome produced by each profile (combination) of strategies.
A strategy profile lists one strategy for each player, so a game with 2 players with 3 strategies each has 3 × 3 = 9 profiles. Payoffs usually represent preferences: money, utility or probability of winning. Only the ranking matters for many results, but in games with randomization the numbers must be consistent with expected values.
How is a game written in normal form?
For two players, the normal (strategic) form is a table: the rows are the strategies of the first player, the columns those of the second, and each cell contains the pair of payoffs (first player, second player). The players choose simultaneously, or at least without knowing the other’s choice.
Example 1: the prisoner’s dilemma
Two suspects are interrogated separately. Each can stay silent (cooperate, C) or confess (defect, D). Payoffs are higher for shorter prison terms:
| C (silent) | D (confess) | |
|---|---|---|
| C (silent) | (3, 3) | (0, 5) |
| D (confess) | (5, 0) | (1, 1) |
Whatever the column player does, the row player gets more from D (5 > 3 against C, 1 > 0 against D), and the same holds for the column player. So both defect and receive (1, 1), although (3, 3) was available.
Example 2: matching pennies
Each player shows heads or tails. The first player wins 1 if the coins match, and the second wins 1 if they differ. The payoffs (+1, −1) and (−1, +1) always sum to zero. Whatever pure choice one makes, the other can exploit it, which leads to randomization.
Example 3: rock-paper-scissors
Three strategies each, 9 profiles. A win gives +1, a loss −1 and a tie 0. It is a symmetric zero-sum game with no best pure strategy: each strategy beats one and loses to another.
What types of games are there?
- Simultaneous games (choices made without knowledge of others’ choices) and sequential games (players move in turns and observe earlier moves).
- Zero-sum games (one player’s gain is the other’s loss, as in matching pennies) and non-zero-sum games, where cooperation can benefit everyone, as in the prisoner’s dilemma.
- Cooperative games (players can make binding agreements and form coalitions) and non-cooperative games (no binding agreements), the main subject here.
- One-shot and repeated games, and games of complete or incomplete information (whether the payoffs of others are known).
What is the extensive form?
A sequential game is shown as a game tree: the nodes are decision points, the branches are actions, and the payoffs are at the ends. Such a game is solved by backward induction: start from the last decision and assume that each player chooses the best action there, then move back up the tree.
Example 4: market entry
An entrant decides whether to enter a market held by an incumbent. If it stays out, the payoffs are (0, 2) (entrant, incumbent). If it enters, the incumbent decides whether to fight (a price war, payoffs (−1, −1)) or to accommodate (payoffs (1, 1)). Backward induction: after entry, the incumbent prefers 1 to −1 and accommodates, so the entrant, anticipating this, enters and gets 1 > 0. The outcome is (1, 1). The incumbent’s threat to fight is not credible, since fighting hurts it once entry has happened.
Example 5: the same game in normal form
The incumbent’s strategies are “fight if entry” and “accommodate if entry”. The strategy profiles (Enter, Accommodate) and (Out, Fight) are both stable in the sense that neither player gains by changing alone, but only the first survives backward induction. This is why the normal form can hide timing and credibility.
What does rationality mean in game theory?
Players are assumed to be rational: each maximizes the expected payoff given their beliefs about the others’ actions. Often it is also assumed to be common knowledge that everyone is rational, that everyone knows that everyone is rational, and so on, which allows reasoning such as the iterated elimination of dominated strategies. Experiments show that people often deviate from these predictions (for example, they cooperate more in the prisoner’s dilemma than the theory predicts), which is studied in behavioral game theory.
Table of the main concepts
| Concept | Meaning | Example |
|---|---|---|
| player | a decision maker | a firm, a prisoner |
| strategy | a complete plan of action | C or D |
| strategy profile | one strategy per player | (D, D) |
| payoff | value of an outcome to a player | (1, 1) |
| zero-sum game | payoffs always sum to 0 | matching pennies |
| backward induction | solving a game tree from the end | market entry |
Where does game theory come from?
Early analyses of strategic situations include Cournot’s study of duopoly (1838) and Zermelo’s theorem on chess (1913). Émile Borel and John von Neumann studied zero-sum games in the 1920s, and von Neumann proved the minimax theorem in 1928. The field was established by the book Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern (1944). John Nash introduced his equilibrium concept in 1950, and the 1994 Nobel Prize in economics was shared by Nash, Harsanyi and Selten.
Which mistakes are common?
Confusing payoffs with moral value
Payoffs describe each player’s own preferences. A “selfish” payoff model can still describe altruistic behavior if the payoffs include concern for others.
Reading cells in the wrong order
In a payoff matrix, the first number belongs to the row player and the second to the column player. Mixing them up changes the whole analysis.
Assuming the best joint outcome will be played
Rational individual choices need not produce the best collective result, as the prisoner’s dilemma shows.
Ignoring the order of moves
A sequential game and a simultaneous game with the same payoffs can have different solutions. Always decide who moves when, and who knows what.