Strategic Cost Management · Game Theory

Introduction to Game Theory and Basic Concepts for CMA Final

Updated 11 October 2026 · Fact-checked

Game theory is a mathematical way to choose the best strategy when your result depends on what a competitor also chooses. You list the players, their strategies and the payoff matrix. Then you classify the game, for example two-person zero-sum, and apply a solution rule such as maximin and minimax.

Understand Introduction to Game Theory and Basic Concepts

Game theory studies decisions taken in competition. In many business situations your profit does not depend only on your own choice. It also depends on what rivals do. Two firms setting prices, bidding for a tender or planning an advertising campaign are typical cases. Game theory gives a structured way to pick a course of action in such cases.

A player is a decision maker, such as a firm. A strategy is a complete plan of action that a player can follow. A player may have a finite or an infinite number of strategies. The payoff is the outcome, such as profit, market share or cost, that results when each player has chosen a strategy. The payoff matrix is a table that shows the payoff for every combination of strategies. Rows normally show the strategies of player A and columns show those of player B. Each entry is usually the payoff to A.

The value of the game is the expected payoff to a player when both play their best strategies. A game is fair if its value is zero. It is strictly determinable if the maximin value equals the minimax value. An optimal strategy is the one that gives a player the most favourable result under the rules of the game.

The usual assumptions are these. Each player is rational and wants to maximise his own gain. Each player knows all strategies available to both sides and the payoffs of each combination. Players choose at the same time, without knowing the rival's choice. The payoff matrix is fixed and known to both players.

Types of games are classified in several ways:
- By number of players: two-person and n-person games.
- By sum of payoffs: in a zero-sum game one player's gain is exactly the other's loss, so the total is zero. In a non-zero-sum game the total is not zero.
- By strategy: pure strategy means a player always plays one fixed course. Mixed strategy means a player plays different courses with fixed probabilities.
- By number of strategies: finite or infinite games.

The exam focus is the two-person zero-sum game. It is the base for the saddle point, mixed strategy and dominance topics that follow.

Key rules to remember

Maximin value (row player): Maximin = maximum of the row minimums; Player A (maximiser, rows) finds the minimum of each row and picks the largest of these. It is A's lower value of the game.
Minimax value (column player): Minimax = minimum of the column maximums; Player B (minimiser, columns) finds the maximum of each column and picks the smallest of these. It is B's upper value of the game.
Saddle point condition: Maximin = Minimax = Value of the game; If they are equal, the game has a saddle point and is strictly determinable. Pure strategies are then optimal.
Zero-sum condition: Gain of one player + Gain of other player = 0; A payoff matrix shows only the payoff to A. B's payoff is the negative of each entry.
Fair game: Value of the game = 0; A game with value above 0 favours A. A game with value below 0 favours B.

How to solve Introduction to Game Theory and Basic Concepts questions

Use this order for any introductory question on game theory terms, classification or a simple payoff matrix.

1Identify the players and what each wants, such as profit or market share.
2List the strategies available to each player. Put player A in rows and player B in columns.
3Check the payoffs. Confirm they are shown as payoffs to A, and whether A maximises or minimises.
4Classify the game: number of players, zero-sum or not, pure or mixed, finite or infinite.
5Write the minimum of each row and the maximum of each column next to the matrix.
6Find maximin and minimax. If equal, state the saddle point, the optimal pure strategies and the value.
7State the conclusion in words: who is favoured and whether the game is fair.
8If the question asks for assumptions or limitations, link each point to the business setting given.

Quickest way: Row-min, column-max check

When to use it: Use it when a 2-mark MCQ or short question gives a payoff matrix and asks for the value, saddle point or type of game.

Check the sum of payoffs first. If the two players' payoffs cancel out, it is zero-sum.
Write the row minimums on the right and the column maximums at the bottom.
Take the largest row minimum and the smallest column maximum.
If they match, that number is the value. The row and column give the optimal strategies.
If they do not match, say there is no saddle point and the game needs a mixed strategy.

Common mistakes in Introduction to Game Theory and Basic Concepts

Calling every competitive situation a zero-sum game.
Students assume that one firm's gain always comes from the rival's loss.
Fix: Check that the payoffs of both players add to zero. If both can gain or lose together, it is non-zero-sum.
Mixing up maximin and minimax.
Both words sound alike and students forget which player uses which rule.
Fix: A (rows) takes row minimums then the maximum of them. B (columns) takes column maximums then the minimum of them.
Treating the strategy as a single move.
In daily use strategy means a general approach.
Fix: A strategy is a complete plan of action. Each row or column of the matrix is one strategy.
Reading the matrix payoffs as B's gains.
Students forget the convention that entries show A's payoff.
Fix: Unless stated otherwise, entries are A's gains and B's losses. B wants the smallest entry.
Saying a game with a positive value is fair.
Students confuse a determinable game with a fair game.
Fix: Fair means value is zero. A positive value favours A and a negative value favours B.
Forgetting the assumptions when asked for them.
Students focus on numerical work only.
Fix: Learn four: rational players, full knowledge of strategies and payoffs, simultaneous choice, and a fixed known payoff matrix.

Worked examples

Example 1

Two firms, A and B, compete. A has two strategies and B has three. The payoffs to A (in ₹ lakh) are: A1: 3, 5, 4; A2: 2, 1, 6. Identify the players, strategies and type of game, and find the value of the game.

Show the solution

Players: two firms, A and B. Strategies: A has A1 and A2. B has B1, B2 and B3. This is a 2 × 3 finite game.
Payoffs are given to A, and B's payoffs are their negatives, so the game is two-person zero-sum.
Row minimums: A1 = min(3, 5, 4) = 3. A2 = min(2, 1, 6) = 1.
Maximin = max(3, 1) = 3.
Column maximums: B1 = max(3, 2) = 3. B2 = max(5, 1) = 5. B3 = max(4, 6) = 6.
Minimax = min(3, 5, 6) = 3.
Maximin = Minimax = 3, so a saddle point exists at A1 and B1.

Answer: This is a finite two-person zero-sum game. The saddle point is at (A1, B1). Optimal strategies are A1 for A and B1 for B. The value is ₹3 lakh in favour of A, so the game is not fair.

Example 2

State whether each is a zero-sum game: (a) Two firms split a fixed market of 100 units. Firm X's share rises by 10 units, so Firm Y's share falls by 10 units. (b) Two firms run a joint advertisement and the market grows, so both earn ₹5 lakh more. Also name the assumption that fails if Firm X does not know Y's options.

Show the solution

In (a) the total market is fixed at 100 units. X's gain of 10 units equals Y's loss of 10 units.
The sum of the two changes is +10 + (-10) = 0, so (a) is zero-sum.
In (b) both firms gain ₹5 lakh. The sum is ₹10 lakh, not zero, so (b) is non-zero-sum.
If X does not know Y's available strategies, the assumption of full knowledge of strategies and payoffs fails. The matrix cannot then be built reliably.

Answer: (a) is a zero-sum game. (b) is a non-zero-sum game. The assumption of complete knowledge of both players' strategies and payoffs fails.

Exam tips

Expect 2-mark MCQs on definitions: zero-sum, fair game, saddle point, pure strategy. Learn each in one line.
Always write row minimums and column maximums beside the matrix. Examiners give marks for visible working.
State the conclusion in words with units, such as the value in ₹ lakh and who it favours.
For theory questions, list assumptions and types of games in short numbered points. Add a business example for each type.
Check first whether a saddle point exists. If not, the question needs the mixed strategy method from the next topics.

Practice questions from Game Theory

Introduction to Game Theory and Basic Concepts in other exams

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

Introduction to Game Theory and Basic Concepts: frequently asked questions

What is game theory in cost management?

It is a decision tool for competitive situations where your result depends on a rival's choice. Firms use it for pricing, advertising and bidding. You model the choices in a payoff matrix and pick the best strategy.

What is the difference between a pure and a mixed strategy?

In a pure strategy a player always plays one fixed course of action. In a mixed strategy the player plays two or more courses with set probabilities. Mixed strategies are used when there is no saddle point.

What is a payoff matrix?

It is a table showing the outcome for every combination of strategies of the two players. Rows are A's strategies, columns are B's. Each entry is normally the payoff to A.

What is a two-person zero-sum game?

It is a game with two players where one player's gain is exactly the other's loss. The payoffs always add up to zero. That is why one matrix, showing A's payoff, is enough.