Business Management · Decision-making process, attitude to risk and competition

Game Theory and Competitive Decisions: Payoff Matrices and Nash Equilibrium

Updated 11 October 2026 · Fact-checked

Game theory studies decisions where your best choice depends on what rivals choose. You write the choices and payoffs in a matrix, remove dominated strategies, then find the Nash equilibrium: the outcome where no player gains by switching alone. The prisoner's dilemma shows why rivals can end up worse off.

Understand Game Theory and Competitive Decisions

Game theory is a way to model decisions when the result depends on more than one decision-maker. In business, your pricing, advertising or product launch affects rivals, and their reply affects you. Game theory makes you think through that reply before you act.

A simple game has players, strategies (the choices open to each player) and payoffs (the result for each player for each pair of choices). With two players and two strategies each, you show this in a payoff matrix. Each cell gives both players' payoffs.

A dominant strategy is a strategy that gives a player a better payoff whatever the rival does. If you have one, a rational player uses it. A Nash equilibrium is a set of strategies, one per player, where no player can improve their own payoff by changing strategy while the others stay put. A dominant strategy is not the same thing: a player may have none, yet a Nash equilibrium can still exist. If both players have a dominant strategy, the pair of them is a Nash equilibrium.

The prisoner's dilemma is the classic case. Two firms can each keep prices high or cut prices. Whatever the rival does, cutting gives a firm more profit. So both cut and both earn less than if both had kept prices high. The equilibrium is not the best joint outcome. This explains why cartels and price agreements tend to break down, and why price wars occur in markets such as insurance.

In a repeated game, firms meet again and again. Threat of punishment next round can support cooperation, so tacit collusion is more likely. Also, a simultaneous game (choices made at the same time) differs from a sequential game, where one firm moves first and the other sees it. In your answer, link the model to a business setting and say what the firm should do, and note the limits: payoffs are estimates, rivals may not be rational, and there may be more than two players.

Key rules to remember

Dominant strategy test: Strategy A is dominant for a player if payoff(A) ≥ payoff(any other) against every rival choice, and > in at least one case; A strictly dominant strategy is better against every rival choice. Compare payoffs column by column (or row by row), using only your own payoffs.
Nash equilibrium test: For each player: payoff at the cell ≥ payoff from switching, with the rival's choice fixed; Check every cell. A cell can be an equilibrium even if neither player has a dominant strategy. There can be more than one, or none in pure strategies.
Best response method: Mark each player's best payoff against each rival choice; a cell with both marks is a Nash equilibrium; This is the safest routine for any matrix size.
Prisoner's dilemma pattern: Temptation > Reward (both cooperate) > Punishment (both defect) > Sucker's payoff; Ranking of one player's payoffs. Defecting dominates, but both defecting is worse than both cooperating.

How to solve Game Theory and Competitive Decisions questions

Use this routine for any game theory question. It works for a prisoner's dilemma and for larger matrices.

1Identify the players, their strategies and the payoffs. Redraw the matrix neatly if it is given in words.
2Say whose payoff is listed first in each cell. Mark this clearly to avoid mixing players up.
3For the row player, fix each column in turn and underline the row player's best payoff in that column.
4For the column player, fix each row in turn and underline the column player's best payoff in that row.
5A cell with both payoffs underlined is a Nash equilibrium. List all of them.
6Check for dominant strategies: does the same strategy have an underline in every column (or row)? If so, it is dominant.
7Compare the equilibrium with the best joint outcome. If they differ, name the prisoner's dilemma.
8Give the business conclusion: what each firm should do, whether cooperation can last (repeated play, penalties, trust), and the limits of the model.

Quickest way: Underline best responses

When to use it: Use for any two-player matrix in the MCQ section or when time is short in a written question.

Go column by column and mark the row player's higher payoff.
Go row by row and mark the column player's higher payoff.
The cell with both marks is the Nash equilibrium.
If one strategy is marked in every column, it is dominant.
Add one line: equilibrium versus best joint outcome.

Common mistakes in Game Theory and Competitive Decisions

Comparing a player's payoffs across the wrong direction, for example the row player comparing numbers along a row.
The cell holds two numbers and it is easy to compare the wrong pair.
Fix: The row player compares only their own payoffs within the same column. The column player compares their own payoffs within the same row.
Saying Nash equilibrium and dominant strategy are the same thing.
In the prisoner's dilemma, they coincide, so students assume they always do.
Fix: Dominant means best against every rival choice. Nash means best against the rival's actual choice at that cell. A game can have a Nash equilibrium with no dominant strategy.
Assuming the equilibrium is the best outcome for both players.
The word equilibrium sounds like an optimum.
Fix: Compare the equilibrium with the joint best cell. In a prisoner's dilemma the equilibrium is worse for both.
Stating only one equilibrium when the matrix has two or none.
Students stop when they find the first cell that works.
Fix: Check every cell using best-response marks and list all equilibria.
Giving a theory answer with no link to the business in the question.
Students recall definitions but do not apply them.
Fix: Name the firms, the strategies and the likely market result, such as a price war, and give a recommendation.
Ignoring repeated interaction when the question describes an ongoing market.
Textbook examples are one-off games.
Fix: If firms compete repeatedly, say that the threat of retaliation can sustain cooperation, so the one-off result may not hold.

Worked examples

Example 1

Two insurers, X and Y, choose High premium or Low premium. Payoffs (annual profit in ₹ crore) are written as (X, Y). Both High: (8, 8). X High, Y Low: (2, 10). X Low, Y High: (10, 2). Both Low: (4, 4). Find any dominant strategies and the Nash equilibrium, and comment.

Show the solution

X's payoffs: if Y is High, X gets 8 from High and 10 from Low, so Low is better. If Y is Low, X gets 2 from High and 4 from Low, so Low is better.
Low is better for X whatever Y does, so Low is X's dominant strategy.
Y's payoffs: if X is High, Y gets 8 from High and 10 from Low. If X is Low, Y gets 2 from High and 4 from Low. Low is better both times, so Low is Y's dominant strategy.
Both play Low. Check: at (Low, Low), X gets 4 and switching to High gives 2. Y also gets 4 and switching gives 2. So neither wants to switch.
Joint best is both High, giving 8 each, which beats 4 each.

Answer: Both insurers have Low premium as a dominant strategy. The Nash equilibrium is (Low, Low) with ₹4 crore each. Both would be better off at (High, High) with ₹8 crore each, so this is a prisoner's dilemma. Cooperation is hard to sustain in a one-off game because each insurer gains by undercutting. In repeated play, the threat of retaliation could support high premiums.

Example 2

Firms A and B each choose Launch or Hold a new product. Payoffs are (A, B). Launch/Launch: (2, 2). Launch/Hold: (6, 1). Hold/Launch: (1, 6). Hold/Hold: (3, 3). Find the Nash equilibria. Does either firm have a dominant strategy?

Show the solution

A's best response: if B launches, A gets 2 from Launch and 1 from Hold, so Launch. If B holds, A gets 6 from Launch and 3 from Hold, so Launch.
So Launch is dominant for A. Check B: if A launches, B gets 2 from Launch and 1 from Hold, so Launch. If A holds, B gets 6 from Launch and 3 from Hold, so Launch.
Launch is dominant for B too.
Hence (Launch, Launch) is a Nash equilibrium. Check other cells: at (Hold, Hold), A gains by switching to Launch (6 > 3), so it is not an equilibrium. At (Launch, Hold), B gains by switching to Launch (2 > 1). At (Hold, Launch), A gains by switching to Launch (2 > 1).
So there is only one equilibrium.

Answer: Both firms have Launch as a dominant strategy. The only Nash equilibrium is (Launch, Launch) with payoffs (2, 2). Both would prefer (Hold, Hold) at (3, 3), so the structure is a prisoner's dilemma. Each firm launches for fear that the rival will.

Exam tips

In MCQs, write best-response marks on the matrix. It takes seconds and avoids sign and direction slips.
State clearly whose payoff comes first in each cell before you start your working.
In written answers, always add the business interpretation: what the result means for pricing, advertising or entry, and the limits of the model.
Contrast one-off and repeated games when the question mentions an ongoing market or several periods.
Define dominant strategy and Nash equilibrium in one line each. Marks are often given for correct definitions.

Practice questions from Decision-making process, attitude to risk and competition

Game Theory and Competitive Decisions in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CMA FinalIntroduction to Game Theory and Basic Concepts

Game Theory and Competitive Decisions: frequently asked questions

What is the prisoner's dilemma in simple terms?

It is a game where each player does better by acting in their own interest whatever the other does, but when both do so, both end up worse off than if they had cooperated. A price war between two firms is a typical example.

How do I find a Nash equilibrium in a payoff matrix?

For each rival choice, mark your best response, and do the same for the rival. A cell where both players' payoffs are marked is a Nash equilibrium. Then check all cells, as there may be more than one.

What is the difference between a dominant strategy and a Nash equilibrium?

A dominant strategy is best against every choice the rival might make. A Nash equilibrium is a combination of strategies where each is best against the other's choice in that combination. Dominant strategies for both players give a Nash equilibrium, but a Nash equilibrium can exist without any dominant strategy.

Does game theory appear in CB3?

Game theory fits the CB3 Business Management material on competition and decision-making. Expect it in MCQs on matrices and in case-study answers where you must predict rival responses. Check the current IAI syllabus for exact coverage.

Can firms avoid the prisoner's dilemma?

Sometimes. In repeated games, the threat of punishment can sustain cooperation. Contracts, reputation and regulation can also change payoffs. Note that formal price-fixing agreements may be illegal under competition law.