Microeconomics · Complete guide
How to solve a game theory question
Almost every game theory question in these papers is a payoff matrix with one of four questions attached: is there a dominant strategy, what survives elimination, where is the Nash equilibrium, and what is the mixed equilibrium. This guide turns those into one routine you can run on any matrix, and adds the backward-induction method for games played in turns.
Start here
What a game is, and how to read a payoff matrix
Players
The decision makers: two firms, two prisoners, a buyer and a seller.
Strategies
What each player can choose: confess or stay silent, high or low price, enter or stay out.
Payoffs
What each player gets for every combination of choices. The key point: your payoff depends on what the other player does too.
Reading the matrix
One player (the row player) picks a row, the other (the column player) picks a column, at the same time. The cell where they meet lists both payoffs, row player's first.
In the matrix alongside, if Firm A plays High and Firm B plays Low, A gets 1 and B gets 8. Get this order right before anything else; it is the most common slip under exam pressure.
| B: High | B: Low | |
|---|---|---|
| A: High | 5,5 | 1,8 |
| A: Low | 8,1 | 2,2 |
The method
The four-step recipe
- Look for dominant strategies. A strategy that is best whatever the opponent does. If both players have one, you are done.
- Eliminate strictly dominated strategies, repeatedly. Cross out anything that is always worse than another option, then look again at the smaller game.
- Underline best responses. For each column, underline the row player's best payoff; for each row, the column player's best. A cell with both underlined is a pure Nash equilibrium.
- If there is no pure equilibrium (or the question asks), find the mixed one. Each player randomises to make the other indifferent.
Key idea
Steps 1 and 2 are shortcuts: they make the game smaller and never remove a Nash equilibrium. Step 3 always works for pure equilibria. Step 4 is guaranteed to succeed: Nash proved every finite game has at least one equilibrium, possibly in mixed strategies.
Interactive
The game solver
Pick a classic game or type any payoffs (up to 3×3). The solver runs all four steps live. Underlined numbers are best responses, and gold cells are Nash equilibria. Step through the elimination one strategy at a time and watch rows and columns grey out.
Start from a classic
Prisoner 1's strategies
Prisoner 2's strategies
Prisoner 1 picks a row; Prisoner 2 picks a column. Each cell is (Prisoner 1, Prisoner 2). Click a number to change it.
| Stay silent | Confess | |
|---|---|---|
| Stay silent | , | , |
| Confess | , | ,Nash |
Payoffs are years in prison (negative). Confessing is better whatever the other does, so both confess, though both staying silent would be better for both.
1
Dominant strategies
Prisoner 1: Confess is strictly dominant: best whatever Prisoner 2 does.
Prisoner 2: Confess is strictly dominant.
Both have strictly dominant strategies, so (Confess, Confess) is a dominant-strategy equilibrium: the game is solved without thinking about the other player at all.
2
Iterated elimination of strictly dominated strategies
- Prisoner 1: remove Stay silent, strictly worse than Confess against everything still left.
- Prisoner 2: remove Stay silent, strictly worse than Confess against everything still left.
0 of 2 removed
3
Pure-strategy Nash equilibria
One cell where both payoffs are underlined: (Confess, Confess) with payoffs (−5, −5). At each, neither player can do better by changing strategy alone.
The equilibrium (Confess, Confess) is not Pareto optimal: another outcome would make both better off. Individually rational choices lead to a collectively worse result, the signature of a prisoner's dilemma.
4
Mixed-strategy equilibrium (2×2)
This game has no fully mixed equilibrium: its equilibrium is in pure strategies (a dominant strategy makes mixing pointless).
Step 1
Step 1: dominant strategies
A strategy is strictly dominant if it gives a strictly higher payoff than every other strategy, against every choice of the opponent. Weakly dominant means never worse and sometimes better. If each player has a strictly dominant strategy, playing them is a dominant-strategy equilibrium, and it is always a Nash equilibrium too.
Worked question · the prisoner's dilemma
Two suspects are questioned separately. Payoffs are years in prison, written as negatives: both silent (−1, −1); one confesses and the other stays silent (0, −10); both confess (−5, −5). What do they do?
- Prisoner 1, if 2 stays silent.Confess gives 0, silence gives −1. Confess is better.
- Prisoner 1, if 2 confesses.Confess gives −5, silence gives −10. Confess is better again.
- So Confess strictly dominates.By symmetry the same holds for prisoner 2.
Answer
Both confess: (−5, −5). Yet both staying silent, (−1, −1), would be better for both. The equilibrium is not Pareto optimal, which is what makes it a dilemma: cartels breaking down, arms races and overfishing all share this structure.
Step 2
Step 2: eliminate dominated strategies
A rational player never plays a strictly dominated strategy, and knows the other player won't either. So cross dominated strategies out, and in the smaller game new ones may become dominated. Repeat until nothing more can go. If a single cell survives, the game is dominance solvable.
Worked question · iterated elimination
Row chooses Up or Down; Column chooses Left, Middle or Right. Payoffs (Row, Column): Up: (1, 0), (1, 2), (0, 1); Down: (0, 3), (0, 1), (2, 0). Solve by elimination.
- Nothing is dominant at the start.Row prefers Up against Left or Middle but Down against Right; Column's best reply also depends on Row.
- Column: Right is dominated by Middle.Middle gives 2 > 1 against Up and 1 > 0 against Down. Remove Right.
- Row: now Down is dominated by Up.With only Left and Middle left, Up gives 1 > 0 in both. Remove Down.
- Column: Left is dominated by Middle.Against Up alone, Middle gives 2 > 0. Remove Left.
Answer
Only (Up, Middle) survives, with payoffs (1, 2). Load the “Elimination example” preset in the solver to replay it step by step.
Strict, not weak
Only eliminate strictly dominated strategies. Removing weakly dominated ones can depend on the order you remove them in, and can delete a Nash equilibrium.
Step 3
Step 3: Nash equilibrium by underlining
Nash equilibrium
A pair of strategies where each player's strategy is a best response to the other's: no one can gain by changing only their own choice.
Named after John Nash (1950). No player regrets their choice once they see what the other did.
- For each column, find the row player's highest payoff (the first number) and underline it.
- For each row, find the column player's highest payoff (the second number) and underline it.
- Every cell with both numbers underlined is a pure-strategy Nash equilibrium.
Worked question · battle of the sexes
Asha and Ravi want to spend the evening together. Asha prefers cricket, Ravi prefers a film. Payoffs (Asha, Ravi): both at cricket (2, 1); both at the film (1, 2); apart (0, 0). Find the pure Nash equilibria.
- Asha's best responses.If Ravi picks cricket, cricket (2 > 0). If Ravi picks film, film (1 > 0).
- Ravi's best responses.If Asha picks cricket, cricket (1 > 0). If Asha picks film, film (2 > 0).
- Match the underlines.Both are underlined in (Cricket, Cricket) and in (Film, Film).
Answer
Two pure Nash equilibria: (Cricket, Cricket) and (Film, Film). Nash equilibrium doesn't say which one happens; that is why coordination problems are hard.
Step 4
Step 4: mixed strategies
In matching pennies, whatever you do, the other player wants to switch, so there is no pure equilibrium. The answer is to be unpredictable: play each option with some probability. The trick for 2×2 games is counter-intuitive but always the same:
The indifference principle
You choose YOUR probabilities to make the OTHER player indifferent, so they are happy to randomise too.
Worked question · the mixed equilibrium of battle of the sexes
In the game above, find the mixed equilibrium: the probability p that Asha goes to cricket and q that Ravi does.
- Asha's p makes Ravi indifferent.
- Ravi's q makes Asha indifferent.
- Expected payoffs.Asha: 2 × ⅓ = ⅔. Ravi: 1 × ⅔ = ⅔. Both worse off than in either pure equilibrium, because they often miss each other.
Answer
Asha picks cricket with probability 2/3, Ravi with 1/3. With the two pure equilibria, battle of the sexes has three Nash equilibria in total.
Why it looks backwards
Asha's probability depends only on Ravi's payoffs. If Ravi's payoff from the film rose, Asha, not Ravi, would change her mix. A classic exam twist.
Taking turns
Sequential games and backward induction
When players move in turn, draw a game tree and solve it backwards: start at the last decision, work out what that player would actually choose there, replace that branch by its outcome, and move up a level.
The entry game
A challenger decides whether to enter a market. If it stays out, it gets 0 and the incumbent keeps a monopoly profit of 10. If it enters, the incumbent either fights a price war (−2 for the entrant, 1 for itself) or accommodates (3 and 4).
- Last move: the incumbent compares fight (1) with accommodate (4), and accommodates.
- So entering really means getting 3. The entrant compares 3 with 0, and enters.
The subgame perfect equilibrium is (Enter, Accommodate). The incumbent may threaten to fight, but the threat is not credible: once entry has happened, fighting would hurt it.
First-mover advantage
Moving first can be valuable because it commits you: the Stackelberg leader sets a high output that the follower then has to accept. Credible commitment (capacity built in advance, a contract, burned bridges) is how threats become believable.
Know these
The classic games at a glance
| Game | Story | Pure Nash equilibria | Mixed? | Lesson |
|---|---|---|---|---|
| Prisoner's dilemma | Confess or stay silent | 1 (both confess), from dominant strategies | No | Individually rational, collectively worse: not Pareto optimal |
| Battle of the sexes | Meet up, different favourites | 2 | Yes (3 equilibria in all) | Coordination with a conflict over which outcome |
| Matching pennies | One wants to match, one to mismatch | 0 | Yes, 50–50 each | Pure conflict: be unpredictable (zero-sum) |
| Chicken (hawk–dove) | Who swerves first | 2, one player backs down in each | Yes | Commitment and brinkmanship |
| Stag hunt | Cooperate for a big prize or play safe | 2, one better for both, one safer | Yes | Trust and coordination failure |
Practice
Exercises, with answers
A. Solve the game
Fractions like 2/3 are fine.
Problem 1 · dominant strategies
Two firms choose a High or Low price. Payoffs (A, B): both High (3, 3); A High, B Low (0, 5); A Low, B High (5, 0); both Low (1, 1). What are the payoffs in the Nash equilibrium?
Problem 2 · a mixed equilibrium
Payoffs (Row, Column): (Top, Left) = (4, 2); (Top, Right) = (0, 0); (Bottom, Left) = (0, 0); (Bottom, Right) = (2, 4). Find the probability p that Row plays Top and q that Column plays Left in the mixed equilibrium.
Problem 3 · no pure equilibrium
Row wins 3 if both choose Heads, wins 1 if both choose Tails, and loses 1 otherwise; Column's payoff is the negative of Row's. Find p, the probability Row plays Heads, in equilibrium.
Problem 4 · backward induction
An entrant chooses Enter or Stay out (Stay out gives (0, 8) to entrant and incumbent). After entry the incumbent chooses Fight, giving (−1, 2), or Accommodate, giving (4, 3). Find the payoffs in the subgame perfect equilibrium.
B. Concept checks
One attempt each. Read the explanation even when you get it right.
1A Nash equilibrium is a set of strategies in which:
2Which is true of the prisoner's dilemma?
3Matching pennies has:
4In a mixed-strategy equilibrium of a 2×2 game, a player's probabilities are chosen to:
5Backward induction selects the subgame perfect equilibrium, which rules out:
6Including mixed strategies, battle of the sexes has how many Nash equilibria?
Revise
One-page revision sheet
Reading the matrix
Row player chooses rows; each cell is (row payoff, column payoff), row first.
Dominant strategy
Best against everything the opponent might do. Both dominant ⇒ dominant-strategy equilibrium (also Nash).
Elimination
Cross out strictly dominated strategies, repeat. Never removes a Nash equilibrium. Only strict dominance.
Nash equilibrium
Mutual best responses. Underline each player's best payoff against each opponent move; double underline = equilibrium.
Mixed strategy
Choose your probabilities to make the opponent indifferent. Every finite game has an equilibrium, possibly mixed.
Sequential games
Backward induction from the last move. Subgame perfect equilibrium rules out non-credible threats.
Watch out for
The mistakes examiners are fishing for
- Reading payoffs in the wrong order. The first number is always the row player's.
- Comparing along the wrong line. For the row player compare down a column (their options, the opponent's choice fixed); for the column player compare along a row.
- Calling a Nash equilibrium “the best outcome”. It only means nobody wants to deviate alone; the prisoner's dilemma equilibrium is worse for both.
- Using your own payoffs to find your own mix. Your probabilities come from making the other player indifferent.
- Stopping at “no pure equilibrium”. There is still a mixed one.
Try it on real questions
Fourteen past-paper questions on this
From UGC NET, IIT JAM, GATE and CUET-PG: payoff matrices to solve for pure and mixed equilibria, games with no pure equilibrium, the prisoner's dilemma, battle of the sexes and a sequential entry game.
These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.
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