Step 19: Game Theory
- Game theory studies situations where your best choice depends on what someone else chooses, such as two firms setting prices.
- Every game has three parts: players, strategies and payoffs, which you show in a payoff matrix.
- A dominant strategy is best whatever the rival does. A Nash equilibrium is an outcome where no player can do better by changing their own choice alone.
- The prisoner’s dilemma shows that sensible self-interest can leave everyone worse off. Repeating the game, with tit-for-tat and credible threats, can rescue co-operation.
What is a “game”?
Imagine you and a classmate each decide tonight whether to revise or watch a film, and tomorrow’s test is marked on a curve. Your grade depends on your choice and on hers, so each of you tries to predict the other. That is a game.
In Step 13 you saw that oligopolists are like this: Coca-Cola’s best price depends on Pepsi’s. Game theory, founded by von Neumann and Morgenstern (1944) and transformed by John Nash (1950), gives you the tools to analyse these situations.
Every game has three parts: players (firms, countries or people), strategies (the actions they can choose, such as “high price” or “low price”) and payoffs (what each player gets, such as profit or years in prison, for each combination of strategies).
A payoff matrix shows the payoffs for every combination of strategies. The row player’s payoff is listed first.
Dominant strategies
Two phone-repair shops, A and B, each decide whether to advertise. Monthly profits are in $000s, A’s first.
| B advertises | B does not advertise | |
|---|---|---|
| A advertises | 4, 4 | 8, 2 |
| A does not advertise | 2, 8 | 6, 6 |
Does shop A have a dominant strategy in the table above?
Both advertise and earn $4,000 each, yet if neither advertised both would earn $6,000. Keep that pattern in mind: it is the prisoner’s dilemma.
The prisoner’s dilemma
This is the most famous game in game theory. Two suspects are arrested and questioned in separate rooms. Each is offered the same deal:
- If you confess and your partner stays silent, you go free (0 years) and your partner gets 10 years.
- If you both confess, you each get 5 years.
- If you both stay silent, you each get 1 year on a lesser charge.
Here the payoffs are years in prison, so lower is better.
| B silent | B confesses | |
|---|---|---|
| A silent | 1, 1 | 10, 0 |
| A confesses | 0, 10 | 5, 5 |
Prisoner A thinks: “If my partner stays silent, confessing gives me 0 years, not 1. If my partner confesses, it gives me 5, not 10.” Confessing is dominant for both, so both get 5 years, even though silence would have given each just 1.
Why not agree to stay silent? Because the deal cannot be enforced: each would still be tempted to confess and walk free. That is exactly why cartels break down.
Nash equilibrium: the stable outcome
A Nash equilibrium is a combination of strategies, one for each player, where no player can do better by changing their own strategy alone, given what the others are doing. Once you reach it, nobody wants to move.
In the prisoner’s dilemma, (Confess, Confess) is the Nash equilibrium: if B confesses, A switching to silence means 10 years instead of 5, and the same holds for B. It is stable but not the best outcome. (Silent, Silent) is better for both, but each player wants to move away from it.
Game theory and oligopoly pricing
Swap “stay silent” for “keep prices high” and “confess” for “cut prices”, and you have oligopoly pricing.
Firms X and Y choose High or Low prices. Profits in $ millions (X first): both High (8, 8); X High, Y Low (2, 12); X Low, Y High (12, 2); both Low (4, 4). Find the dominant strategies, the Nash equilibrium and the best joint outcome.
Both firms would earn twice as much by keeping prices high, but each is tempted to undercut, so both cut. Oligopoly prices therefore tend to sit between the monopoly and perfectly competitive levels.
Repeated games: the shadow of the future
Most real rivals meet again and again: supermarkets set prices every week. In a repeated game, a player who cheats today can be punished tomorrow, and that threat can keep everyone co-operating.
In Robert Axelrod’s 1984 computer tournament, tit-for-tat beat every other strategy. It is simple, punishes cheating at once and forgives when the rival co-operates again. A punishment only works if it is a credible threat: one the rival believes you will carry out.
Firms A and B both charge high prices and earn $10M each per year. If A cuts its price and B does not respond, A earns $15M and B $3M. If B matches the cut, both earn $6M. Should A cut?
Because A knows B will match, cutting turns $10M into $6M a year, so both keep prices high without any formal agreement. This is why many oligopolies avoid price wars.
There is a catch. If the game has a known last round, nobody fears punishment then, so both cheat; knowing that, both cheat in the round before, and so on back to the start. This is backward induction, and it makes co-operation harder when the end is known.
Beyond firms. The same logic appears in many other places:
- Trade: both countries gain from free trade, but each is tempted to impose tariffs. The WTO works like a repeated game with rules and retaliation.
- Climate change: all gain if all cut emissions, but each is tempted to free-ride. The Paris Agreement relies on pledges and peer pressure.
- Arms races: deterrence (“if you launch, we launch back”) works only if the threat is credible.
- Advertising wars: the UK’s ban on cigarette TV adverts took tobacco firms out of this game together.
OPEC is a repeated prisoner’s dilemma. Members often exceed their quotas, but the cartel has survived over 60 years because Saudi Arabia can punish overproducers by flooding the market (as in 1985, 2014 and 2020). The result is partial co-operation: neither full collusion nor collapse.
Practise: 6 questions
Try each one on paper first, then check the worked answer.
EasyQ1. Name the three components of a game and give an example of each for two airlines setting fares. Show worked answer
Strategies: for example “keep fares high” or “cut fares”.
Payoffs: the profit each airline earns for each combination of choices, shown in a payoff matrix.
EasyQ2. What is the difference between a dominant strategy and a Nash equilibrium? Show worked answer
A Nash equilibrium is an outcome of the whole game: a pair of strategies where neither player gains by changing their own choice alone.
A Nash equilibrium can exist even when nobody has a dominant strategy.
MediumQ3. Two countries choose to Reduce Emissions or Keep Polluting. Welfare (A first): both Reduce (7, 7); A Reduces, B Pollutes (2, 10); A Pollutes, B Reduces (10, 2); both Pollute (4, 4). Is this a prisoner’s dilemma? Show worked answer
Nash equilibrium: (Pollute, Pollute) = (4, 4).
Co-operative outcome: (Reduce, Reduce) = (7, 7), which is better for both.
Each player’s dominant strategy leads to a worse result for both, so yes, it is a prisoner’s dilemma.
MediumQ4. Two friends want to meet up but cannot call each other. Payoffs (Ali first): both go to the Cinema (3, 2); both go to the Café (2, 3); they go to different places (0, 0). Find all Nash equilibria. Does either player have a dominant strategy? Show worked answer
(Café, Café): switching gives 0 < 2 for Ali and 0 < 3 for his friend. Also a Nash equilibrium.
The mismatched cells give 0 each, and either player could gain by switching, so they are not equilibria.
No dominant strategy: each player’s best choice depends on where the other goes. So there are two Nash equilibria and no dominant strategies.
MediumQ5. Explain why co-operation is easier to sustain when a game is expected to continue indefinitely than when it has a known final round. Show worked answer
With a known final round, nothing can punish cheating in that round, so both defect. By backward induction the same applies to the round before, and co-operation unravels from the end.
ExamQ6. Two supermarkets choose High or Low prices. Profits in $ millions (P first): both High (9, 9); P High, Q Low (3, 13); P Low, Q High (13, 3); both Low (5, 5). (a) Identify P’s dominant strategy. (b) State the Nash equilibrium. (c) Explain why both firms might still charge High prices in practice. [6 marks] Show worked answer
(b) By symmetry Low is also Q’s dominant strategy, so the Nash equilibrium is (Low, Low) with $5M each. [1]
(c) The supermarkets compete week after week, so this is a repeated game. [1] If each follows tit-for-tat, a price cut is matched next round, so cutting gives a one-off $13M and then $5M per round, instead of $9M per round. [1] Because matching is credible (once a rival cuts, matching gives 5 rather than 3), the threat sustains (High, High) without a formal agreement. [1]
Check yourself
1. A Nash equilibrium is an outcome where:
2. Tit-for-tat means:
3. Which situation is best modelled as a prisoner’s dilemma?
Your score: 0 out of 3
The short version
- A game has players, strategies and payoffs, shown in a payoff matrix.
- A dominant strategy is best whatever the rival does. Check it against every possible move by the rival.
- A Nash equilibrium is stable because no one gains by changing their own choice alone, but it is not necessarily the best outcome.
- In the prisoner’s dilemma, both players defect although co-operating would be better. It explains cartel cheating, trade wars and stalled climate deals.
- Repeated games, tit-for-tat and credible threats can sustain co-operation, as OPEC shows. The full chapter in Economics Made Simple has 15 exam-style questions with complete answers.
Further reading on EconTutorials
Older in-depth articles on this topic from our blog: