Foundationsmicroeconomics
Game Theory Introduction: Players, Strategies, and Payoffs
Overview
Game theory is the mathematical study of strategic interaction — situations where each person's outcome depends on what others do. Economists use it to analyze oligopoly pricing, auctions, bargaining, voting, and even evolutionary biology. This lesson introduces the building blocks: players, strategies, and payoffs; simultaneous versus sequential games; the payoff matrix; dominant strategies; and the difference between zero-sum and non-zero-sum games.
Learning objectives
- Define a game using players, strategies, and payoffs
- Read and interpret a two-player payoff matrix
- Distinguish simultaneous-move from sequential-move games
- Identify a dominant strategy and a dominated strategy
- Distinguish zero-sum from non-zero-sum and cooperative from non-cooperative games
What is a game?
In game theory, a game is any strategic situation with three ingredients. Players are the decision-makers — firms, consumers, countries, or even animals. Strategies are the complete plans of action available to each player — for example, a firm might choose a high price or a low price. Payoffs are the outcomes each player receives from every possible combination of strategies, usually measured as profit, utility, or years in prison.
Game theory assumes players are rational: each chooses the strategy that maximizes their own payoff, given what they expect others to do. The theory does not assume people are selfish in a moral sense — payoffs can include altruism — only that behavior is consistent with pursuing one's objectives. This framework turns vague ideas about 'competition' into precise predictions.
The payoff matrix: reading a simultaneous-move game
The simplest games have two players, each with two strategies, shown in a payoff matrix. Rows are one player's strategies, columns are the other's, and each cell lists the payoffs as an ordered pair: (row player's payoff, column player's payoff). By convention, the first number always belongs to the row player.
Consider two smartphone makers choosing between a high price and a low price. If both price high, each earns 10. If both price low, each earns 4. If one prices low while the other prices high, the discounter earns 14 and the other earns 2. Reading the matrix cell by cell reveals each player's incentives — the foundation for predicting what rational players will actually do.
In each cell, the first payoff belongs to the row player and the second to the column player. Always check which player is which before interpreting a matrix.
Dominant strategies: best no matter what
A dominant strategy is one that gives a player a higher payoff than any alternative, regardless of what the other player does. In the smartphone example, pricing low earns 14 versus 10 if the rival prices high, and 4 versus 2 if the rival prices low — low price is better in both cases, so it is a dominant strategy. A rational player never needs to guess the opponent's move when they have a dominant strategy.
The flip side is a dominated strategy: one that is never the best choice, no matter what others do. Dominated strategies can be eliminated from analysis, which sometimes simplifies a game enough to reveal a clear prediction. When every player has a dominant strategy, the predicted outcome is immediate — though, as the prisoner's dilemma shows, that outcome need not be good for anyone.
Simultaneous vs. sequential games
In simultaneous-move games, players choose without knowing the other's choice — like firms setting prices at the same time, or the prisoner's dilemma suspects interrogated separately. Strategy means picking the best action given beliefs about the rival's pick.
In sequential games, players move in order and later movers observe earlier choices — like a firm deciding whether to enter a market after observing the incumbent's capacity investment. These games are analyzed by backward induction: start at the end, figure out what the last mover will do in each situation, then work backward to the first move. Sequential structure creates possibilities like credible threats and first-mover advantages that simultaneous games lack, and it is the basis of the Stackelberg oligopoly model.
Zero-sum, non-zero-sum, cooperative, and non-cooperative games
Games are classified along two independent dimensions. In zero-sum games, one player's gain is exactly another's loss — the payoffs in each cell sum to zero. Poker and chess are zero-sum; trade negotiations are not. Most economic games are non-zero-sum: cooperation can make both players better off, and conflict can make both worse off.
Separately, cooperative game theory studies situations where players can make binding agreements and asks how the gains from cooperation get divided. Non-cooperative game theory studies situations without enforceable agreements, where each player acts independently — the setting for oligopoly pricing, where explicit collusion is illegal, and for the prisoner's dilemma. Do not confuse 'non-cooperative' with 'players never cooperate': cooperation can still emerge as an equilibrium outcome, as repeated-game analysis shows.
Why economists care: applications everywhere
Game theory is the language of modern microeconomics wherever small numbers of agents interact strategically. Oligopoly pricing and output games determine whether industries look competitive or monopolistic. Auction theory — a branch of game theory — designs the spectrum auctions and ad auctions worth billions. Bargaining models explain wage negotiations and merger terms. Mechanism design, which won multiple Nobel prizes, engineers the rules of games (like matching medical residents to hospitals) to produce desirable outcomes.
Beyond economics, the same mathematics models arms races, animal conflict, traffic routing, and cybersecurity. Learning to think in terms of players, strategies, and payoffs is learning to see the hidden structure of strategic situations — a skill that transfers far beyond the classroom.
Game theory always asks the same question: given that everyone is rational and everyone knows everyone else is rational, what strategies will the players choose? Every solution concept — dominant strategies, Nash equilibrium, backward induction — is an answer to that question under different assumptions.
Key points
- A game consists of players, strategies, and payoffs, with rationality assumed
- Payoff matrices list (row player, column player) payoffs for each strategy pair
- A dominant strategy is best regardless of the opponent's choice
- Simultaneous games are solved by best responses; sequential games by backward induction
- Most economic games are non-zero-sum: mutual gain and mutual loss are both possible
- Non-cooperative game theory assumes no binding agreements, not no cooperation
Common misconceptions
✓ Truth: It assumes people act consistently to maximize their objectives, whatever those are. Payoffs can represent altruistic, social, or moral goals just as easily as money.
✓ Truth: Only zero-sum games work that way. Most economic games are non-zero-sum, where all players can gain together (successful trade) or lose together (a price war).
✓ Truth: Its main applications are economic: oligopoly competition, auctions, bargaining, contract design, and market regulation.
Practice questions
Q1. In a two-player payoff matrix, the numbers in each cell represent:
- the probability of each outcome
- the payoffs to the row player and column player, in that order
- the number of strategies available
- the players' budgets
By convention each cell contains an ordered pair: the first number is the row player's payoff and the second is the column player's payoff for that strategy combination.
Q2. A dominant strategy is a strategy that:
- wins the game for sure
- gives the highest payoff no matter what the other player does
- is chosen only after seeing the opponent's move
- requires cooperation with the opponent
A strategy dominates when it yields a strictly higher payoff than every alternative against each possible strategy of the opponent — so a rational player chooses it without needing to predict the opponent.
Q3. Which pair of classifications can both describe the same game?
- Zero-sum and cooperative
- Sequential and simultaneous
- Non-zero-sum and non-cooperative
- Dominant and dominated
Zero-sum/non-zero-sum and cooperative/non-cooperative are independent dimensions. A non-zero-sum game (mutual gains possible) can be non-cooperative (no binding agreements) — the prisoner's dilemma is exactly this combination.
Q4. Sequential-move games are typically analyzed using:
- dominant strategies only
- backward induction from the final move
- random choice
- ignoring the order of moves
Because later movers observe earlier choices, the game is solved backward: determine what the last mover will do in each contingency, then fold that knowledge back to the first mover's decision.
Click a question to reveal the answer and explanation.
Frequently asked questions
What is the difference between a strategy and a single action?
A strategy is a complete plan specifying what a player does in every situation that could arise, including responses to others' moves. In a simple simultaneous game the strategy is just one action, but in sequential games a strategy covers every contingency.
How do I find a dominant strategy in a payoff matrix?
Compare a player's payoffs across their own strategies, holding the opponent's strategy fixed, then repeat for each opponent strategy. If one of the player's strategies gives a higher payoff in every column (for the row player), it dominates.
What does 'common knowledge of rationality' mean?
It means everyone is rational, everyone knows everyone is rational, everyone knows that everyone knows, and so on. Many equilibrium predictions rely on this infinite chain of mutual expectations.
Is game theory math-heavy? Do I need calculus?
Basic game theory needs only arithmetic and careful logic — reading matrices and comparing payoffs. Advanced topics like mixed strategies use probability, and continuous-strategy models use calculus, but the core ideas are accessible without them.