23506 Chap.1 Game Theory Foundations: Normal-Form Games, Payoff Matrices and the Classic 2×2 Games
Game Theory Foundations: Normal-Form Games, Payoff Matrices and the Classic 2×2 Games
Game theory studies situations in which each decision maker’s payoff depends on a profile of choices, not on one choice in isolation. A normal-form game states the players, each player’s strategy set and a payoff for every strategy profile. In a two-player matrix, Row selects a row, Column selects a column and the ordered pair in their cell is read in the declared player order. A unilateral deviation changes only the deviator’s coordinate. This representation discipline is the foundation for every later equilibrium calculation.
The classic 2×2 games organise recurring strategic tensions. In Prisoner’s Dilemma, each player has a strict incentive to defect, producing a unique equilibrium worse for both than mutual cooperation. Battle of the Sexes has two coordinated pure equilibria with a distributional conflict. Stag Hunt contrasts a high-payoff coordinated equilibrium with a safer alternative. Chicken produces asymmetric anti-coordination equilibria, while Rock–Paper–Scissors illustrates a response cycle without a pure equilibrium. These labels are recognition aids; the displayed payoffs decide the actual game.
Larger applications preserve the same primitives. An auction strategy can be a bid as a function of value. A demand game lets players claim from a fixed resource and uses a rule for compatible or excessive total demand. Before using a theorem, state who chooses, what each can choose, what is known at the decision and how outcomes are valued. Changing timing, information, an outside option or a payoff changes the strategic model and can change its prediction.
What this chapter covers
- 01Players, strategies, profiles and ordered payoffs
- 02Matrix coordinate and unilateral-deviation discipline
- 03Prisoner’s Dilemma and collective loss
- 04Battle of the Sexes and coordination selection
- 05Stag Hunt, payoff dominance and security
- 06Chicken, response cycles, auctions and demand
Build and diagnose a 2×2 game
- 2Declare players, actions and ordered-pair convention.
- 2Compare Row’s two actions against each Column action.
- 2Repeat for Column using the second payoff coordinate.
- 2Identify mutual best responses and compare with mutual protection.
- 2Name the pattern and explain the private incentive versus joint outcome.
Key terms
- Player
- A decision maker whose payoff can depend on the whole strategy profile.
- Strategy
- A complete choice or rule available to a player.
- Strategy profile
- One strategy for every player, written in a declared order.
- Payoff matrix
- A table assigning an ordered payoff vector to every normal-form profile.
- Unilateral deviation
- A change by one player while all other strategies are held fixed.
- Coordination game
- A game with gains from matching or otherwise aligning choices.
Game Theory Foundations: Normal-Form Games, Payoff Matrices and the Classic 2×2 Games FAQ
What is the first number in a payoff pair?
It is the payoff of the first player in the declared ordering, normally Row in this guide. Always check the matrix label.
Is an action the same as a strategy?
In a one-shot normal-form game a pure strategy can be a single action. In a dynamic game a strategy is a complete contingent plan and is larger than one observed action.
Why is Prisoner’s Dilemma a dilemma?
Individual strict incentives produce mutual defection even though mutual cooperation would give both players more.
How many pure equilibria does Battle of the Sexes have?
The standard game has two coordinated pure Nash equilibria and also a mixed equilibrium studied later.
Is payoff dominance the same as Nash equilibrium?
No. First establish the equilibrium set; payoff dominance compares equilibrium payoffs and may select one that benefits every player.
Can a familiar story guarantee a familiar answer?
No. One changed payoff can reverse a best response. Read the displayed game rather than importing a memorised label.
What changes if moves are observed sequentially?
The representation usually becomes an extensive-form tree and strategies include contingent responses after observed actions.
Why define the action domain?
Boundaries, integer restrictions and continuous choices determine available deviations and whether a threshold or table is appropriate.
Can equilibrium be unfair?
Yes. Nash equilibrium tests unilateral incentives, not fairness, equality or total welfare.
How should I check a cell?
Hold the opponent action fixed, compare only the deviator’s payoff coordinate across that player’s alternative actions, and repeat for every player.
How do I translate an unfamiliar story into a game without overfitting a label?
Write four primitives before naming the game. First list the decision makers and retain a consistent order. Second state every feasible action or contingent rule, including outside options and boundaries. Third state what each player observes when choosing; simultaneous unobserved moves call for normal form, while observed sequence usually calls for a tree. Fourth map every action profile or terminal path to ordered payoffs and say whether those numbers represent profit, utility or another objective. Now test unilateral comparisons while holding opponents fixed. Only after the response pattern is visible should you call the structure coordination, Prisoner’s Dilemma, Chicken, Stag Hunt or cyclic conflict. Check that the named pattern satisfies its defining incentive conditions rather than merely resembling its story. Change one primitive—timing, information, payoff or outside action—and explain which conclusion moves. This process makes the answer auditable and transfers to auctions, demand, entry and dynamic games where the narrative is less familiar.
Exam move
Create one matrix card for each classic pattern. On the front put only labelled payoffs; on the back write each best-response map, equilibrium set, payoff comparison and the sentence explaining the tension. Rebuild the answer rather than recognising it visually. Practise ordered-pair reading by narrating one cell and one unilateral move aloud. Then vary one payoff, add an outside option or change simultaneity to observation and predict which conclusion changes. Include auction and demand examples so strategy sets are not mistaken for always having two discrete actions. Finish every drill by stating the model boundary: players, feasible choices, information and payoff rule. A correct label without those primitives is not a transferable solution.
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