Game theory is what economics uses when your best move depends on what someone else does. Two firms setting prices, two countries setting tariffs, two students deciding whether to revise — each has to reason about the other’s choice before making their own. The tools that tame this are the dominant strategy and the Nash equilibrium, and they are among the highest-yield ideas in first-year micro, which is why they anchor so much London economics tutoring at exam time.
1 · What a game is
A game has three ingredients: players, the strategies each can choose, and the payoffs each earns from every combination of choices. When players move at the same time without seeing each other’s decision, we lay the game out in a payoff matrix — one player’s strategies down the rows, the other’s across the columns, and each cell holding a pair of payoffs.
The convention is that the first number in a cell belongs to the row player, the second to the column player. Read a cell as: “if the row player picks this row and the column player picks this column, here is what each gets.” Everything below is just careful reading of that grid.
2 · Dominant strategies
A strategy is dominant if it gives a player their best payoff no matter what the other player does. To test whether a row is dominant, compare it against the alternative rows column by column: if it wins in every column, it dominates.
When a player has a dominant strategy, prediction is easy — a rational player plays it, full stop, because deliberating about the opponent is pointless. And when both players have a dominant strategy, the outcome is nailed down: each simply plays theirs.
The catch, which the worked example makes vivid, is that two players each doing the individually smart thing can land them both somewhere worse than if they had cooperated. That is the famous prisoners’ dilemma, and it explains why cartels collapse, why price wars break out, and why unregulated pollution is overproduced.
3 · Nash equilibrium
Most games have no dominant strategy, so we need a broader idea. A Nash equilibrium is a combination of strategies where each player is playing a best response to the other’s choice — so no one can do better by switching alone. It is a resting point: given what everyone else is doing, nobody wants to move.
Three facts to carry into an exam:
- A game can have a Nash equilibrium without anyone having a dominant strategy.
- A game can have more than one Nash equilibrium — coordination problems typically do.
- A game can have no Nash equilibrium in pure strategies at all, in which case players must mix — randomise over their options.
To find the pure-strategy equilibria, mark each player’s best response in every situation; a cell where both marks land is a Nash equilibrium.
Worked example — two cafés set prices
Two cafés on the same street each choose a High or a Low price. Weekly profits, in thousands of pounds, are laid out with Café A’s payoff first:
| B: High | B: Low | |
|---|---|---|
| A: High | 10, 10 | 2, 12 |
| A: Low | 12, 2 | 5, 5 |
Step 1 — Read the incentives. If both keep prices High, they split the street calmly at 10 each. If one undercuts, it steals the crowd: the low-price café gets 12, the high-price café just 2. If both go Low, a price war leaves each with 5.
Step 2 — Café A’s best responses. Suppose B prices High. A earns 10 by matching, 12 by undercutting — so Low is better. Suppose B prices Low. A earns 2 by staying High, 5 by matching Low — so Low is better again. Low is a dominant strategy for A.
Step 3 — Café B’s best responses. The game is symmetric, so the same logic gives B a payoff of 12 against a High rival and 5 against a Low one — Low is dominant for B too.
Step 4 — The Nash equilibrium. The only cell where both are best-responding is (Low, Low), paying 5 each. Neither café can improve by raising its price alone — it would just lose customers — so this is the equilibrium.
Step 5 — The dilemma. Both cafés would earn 10 at (High, High) — double the equilibrium payoff. Cooperation is better for everyone, yet it is not stable: from (High, High) either café can grab 12 by cutting its price, so the tempting high-price deal unravels.
Step 6 — Interpret. Each café, acting in its own interest, drives them both to the worse outcome. Individually rational, collectively poor. That single tension is why a price-fixing agreement is so hard to sustain without enforcement, and it recurs whenever short-run self-interest cuts against a shared payoff.
Sure you would spot every equilibrium, and never call a cell one that isn’t? The best-response method is quick once it is second nature, but reading the payoffs the wrong way round or mistaking a Pareto-better outcome for an equilibrium is the classic slip that costs marks — and exactly what a one-on-one economics tutor works through with you. Book a trial session.
Practice
Q1. Two firms choose a Standard or a New technology. Payoffs (row firm first): (Standard, Standard) = 4, 4; (Standard, New) = 2, 3; (New, Standard) = 3, 2; (New, New) = 6, 6. Does either firm have a dominant strategy? Find all pure-strategy Nash equilibria.
Q2. With payoffs (Top, Left) = 6, 3; (Top, Right) = 3, 2; (Bottom, Left) = 5, 4; (Bottom, Right) = 2, 1, identify each player’s dominant strategy and the Nash equilibrium.
Q3. In matching pennies, (Heads, Heads) = 1, −1; (Heads, Tails) = −1, 1; (Tails, Heads) = −1, 1; (Tails, Tails) = 1, −1. How many pure-strategy Nash equilibria are there?
Answers. Q1: neither firm has a dominant strategy — each prefers to match the other. There are two pure Nash equilibria, (Standard, Standard) = 4, 4 and (New, New) = 6, 6; this is a coordination game, and (New, New) Pareto-dominates. Q2: Top beats Bottom in both columns (6 > 5 and 3 > 2), so Top is dominant for the row player; Left beats Right in both rows (3 > 2 and 4 > 1), so Left is dominant for the column player. The unique Nash equilibrium is (Top, Left) = 6, 3. Q3: none. Whatever cell you pick, one player would deviate, so there is no pure-strategy equilibrium — the players mix 50/50, the mixed-strategy equilibrium.
Key takeaways
- A game is players, strategies and payoffs; a simultaneous game is read off a payoff matrix, row player’s payoff first.
- A dominant strategy is best whatever the opponent does. A rational player always plays it, and if both players have one the outcome is settled.
- A Nash equilibrium is a profile where each plays a best response to the other, so no one gains by deviating alone. Find it by marking best responses and looking for a cell both mark.
- Equilibria can be unique, multiple, or absent in pure strategies — and, as in the prisoners’ dilemma, the equilibrium can be worse for everyone than an unstable cooperative outcome.
Why London students choose our economics tutoring
- Exam-ready technique: whether you study at LSE, UCL, King’s or Queen Mary, sessions drill the best-response method on the matrices your paper actually uses, so you never miss an equilibrium.
- From matrix to meaning: our tutors connect the prisoners’ dilemma, coordination games and mixing to the real applications — cartels, standards wars, entry deterrence — that essay questions reward.
- One-on-one pace: a private tutor catches the classic slips (reading the payoffs the wrong way round, calling a Pareto-better cell an equilibrium) before they cost you marks.
FAQ
Q: What is the difference between a dominant strategy and a Nash equilibrium?
A: A dominant strategy is best for one player whatever the opponent does. A Nash equilibrium is a pair of strategies, one per player, where each is a best response to the other. Every dominant-strategy outcome is a Nash equilibrium, but many Nash equilibria involve no dominant strategy at all.
Q: How do I find a Nash equilibrium in a payoff matrix?
A: For each column, mark the row player’s highest payoff; for each row, mark the column player’s highest payoff. Any cell that carries both marks is a pure-strategy Nash equilibrium, because each player is then best-responding to the other.
Q: Can a game have more than one Nash equilibrium?
A: Yes. Coordination games often have two or more — for example, two firms both adopting the same technology standard. When several equilibria exist, game theory alone may not say which one occurs; that can depend on expectations, history or communication.
Q: What is the prisoners’ dilemma?
A: A game where each player has a dominant strategy, but both playing it leaves them worse off than if they had cooperated. The equilibrium is individually rational yet collectively poor — the reason cartels are unstable and pollution is overproduced without regulation.
Q: What happens when there is no pure-strategy equilibrium?
A: The players use mixed strategies — they randomise over their options with specific probabilities. In matching pennies, for instance, each player chooses heads or tails with probability one-half, and that randomisation is the equilibrium.
Book London economics tutoring, in person or online
Game theory rewards a clean method: find the best responses, spot the dominant strategies, read off the equilibrium. A one-on-one session builds that method on the exact matrices your course sets — prisoners’ dilemma, coordination, mixing — until they are quick and certain. Tell us your university and module, and we will match you with the right tutor this week, near London or online.
