Difficulty: Intermediate | Prerequisites: Basic understanding of strategic interaction, payoff matrices
Game theory models strategic interaction: situations where your best choice depends on what the other player does. This section covers the core solution concepts you need for intermediate micro, from reading a payoff matrix through to solving sequential games. You should already be comfortable with the idea that firms make decisions to maximise profit, and that those decisions can depend on rivals' choices.
In a simultaneous game, look for pure-strategy Nash Equilibria by checking best responses, then solve for mixed strategies by setting each player indifferent between their options. In a sequential game, draw the extensive form and use backward induction to find the Subgame Perfect Nash Equilibrium, which eliminates non-credible threats.
Nash Equilibrium (NE)
A set of strategies, one for each player, where no player can improve their payoff by unilaterally changing their own strategy. Think of it as a stable resting point: once everyone is playing their NE strategy, nobody has a reason to deviate on their own.
Pure strategy
A strategy where a player chooses one specific action with certainty (e.g. "always produce Large"). In simple terms, a definite choice rather than a coin flip.
Mixed strategy
A strategy where a player randomises over actions according to specific probabilities (e.g. "play Large with probability 1/2, Small with probability 1/2"). Players mix when no pure strategy is a clear best response to all possibilities.
Best response
The strategy that gives a player the highest payoff, given what the other player is doing. Finding NE amounts to finding strategy profiles where every player is playing a best response to the others.
Dominant strategy
A strategy that gives a higher payoff than any alternative regardless of what the other player does. If it exists, it simplifies the analysis considerably, but not every game has one.
Expected payoff
The probability-weighted average of the payoffs across all possible outcomes. Used when players randomise or when you are calculating what a mixed strategy is worth.
Extensive form
A game tree that shows the sequence of moves, who moves when, what they know at the time, and the payoffs at each terminal node. Used for sequential (not simultaneous) games.
Subgame
A portion of the extensive-form game that starts at a decision node where the player knows exactly where they are, and includes everything that follows. Each subgame is itself a complete game.
Subgame Perfect Nash Equilibrium (SPNE)
A Nash Equilibrium that is also a Nash Equilibrium in every subgame of the extensive form. It eliminates strategies that rely on non-credible threats. Found by backward induction.
Backward induction
Solving a sequential game by starting at the final decision nodes, determining what the last mover would do, then working backwards. Each player anticipates the future moves and chooses accordingly.
Two firms (Prest and Tollgate) each choose to produce Large or Small. The payoff matrix:
Prest: Large | Prest: Small | |
|---|---|---|
Tollgate: Large | Prest: $24, Tollgate: $6 | Prest: $26, Tollgate: $12 |
Tollgate: Small | Prest: $30, Tollgate: $8 | Prest: $25, Tollgate: $10 |
The method: for each cell, check whether either player could improve by unilaterally switching.
(Small, Large): Prest gets $26. If Prest switches to Large, Prest gets $24 (worse). Tollgate gets $12. If Tollgate switches to Small, Tollgate gets $10 (worse). Neither wants to deviate. This is a NE.
(Large, Small): Prest gets $30. If Prest switches to Small, Prest gets $25 (worse). Tollgate gets $8. If Tollgate switches to Large, Tollgate gets $6 (worse). Neither wants to deviate. This is a NE.
(Large, Large): Prest gets $24. Prest could switch to Small and get $26 (better). Not a NE.
(Small, Small): Prest gets $25. Prest could switch to Large and get $30 (better). Not a NE.
Pure-strategy Nash Equilibria: {Prest: Small, Tollgate: Large} and {Prest: Large, Tollgate: Small}.
Let alpha = probability Prest plays Large, and beta = probability Tollgate plays Large.
Tollgate must be indifferent between Large and Small, so set Tollgate's expected payoff from Large equal to its expected payoff from Small:
Tollgate plays Large: 6(alpha) + 12(1 - alpha)
Tollgate plays Small: 8(alpha) + 10(1 - alpha)
Setting equal: 6alpha + 12 - 12alpha = 8alpha + 10 - 10alpha
Simplifying: 12 - 6alpha = 10 - 2alpha
2 = 4alpha
alpha = 1/2
Prest must be indifferent between Large and Small, so set Prest's expected payoff from Large equal to its expected payoff from Small:
Prest plays Large: 24(beta) + 30(1 - beta)
Prest plays Small: 26(beta) + 25(1 - beta)
Setting equal: 24beta + 30 - 30beta = 26beta + 25 - 25beta
Simplifying: 30 - 6beta = 25 + beta
5 = 7beta
beta = 5/7
Mixed-strategy NE: Prest plays Large with probability 1/2, Small with probability 1/2. Tollgate plays Large with probability 5/7, Small with probability 2/7.
Now Prest moves first, and Tollgate observes Prest's choice before deciding.
The extensive form is a game tree:
Prest chooses Large or Small at the root
Tollgate then chooses Large or Small at each of its two decision nodes
Terminal payoffs: (Prest, Tollgate)
Prest Large, Tollgate Large: (24, 6)
Prest Large, Tollgate Small: (30, 8)
Prest Small, Tollgate Large: (26, 12)
Prest Small, Tollgate Small: (25, 10)
Solve by backward induction:
If Prest plays Large: Tollgate chooses between $6 (Large) and $8 (Small). Tollgate picks Small.
If Prest plays Small: Tollgate chooses between $12 (Large) and $10 (Small). Tollgate picks Large.
Anticipating Tollgate's responses:
Prest plays Large, expecting Tollgate to respond with Small: Prest gets $30
Prest plays Small, expecting Tollgate to respond with Large: Prest gets $26
Prest picks Large ($30 > $26)
SPNE: Prest plays Large. Tollgate's strategy: play Small if Prest plays Large, play Large if Prest plays Small.
Outcome: (Prest: Large, Tollgate: Small) with payoffs ($30, $8).
Expected payoff (mixed strategies)
E[payoff from action A] = (prob opponent plays X) x (payoff if A,X) + (prob opponent plays Y) x (payoff if A,Y)
Indifference condition
To find the mixed-strategy NE probability for Player 1, set Player 2 indifferent. To find the probability for Player 2, set Player 1 indifferent. Each player's mixing probability is determined by making the other player indifferent, not by making themselves indifferent.
Backward induction (sequential games)
Start at the terminal nodes. At each final decision node, the player picks the action with the higher payoff. Replace that decision node with the chosen payoff. Move one level up and repeat until you reach the root.
The Prest-Tollgate setup is a simplified version of capacity competition: firms deciding how much to produce when rivals' choices affect everyone's profits. This logic appears in OPEC oil production decisions, airline seat capacity, and semiconductor fab investment.
Sequential games with observable moves model first-mover advantage. In practice, the firm that commits to a large investment first (building a factory, locking up a distribution channel) can shape the rival's best response, exactly as Prest does by moving first and securing the higher payoff.
Students often think a mixed-strategy NE means each player is randomising to maximise their own expected payoff. The mixing probabilities are actually chosen to make the other player indifferent. Your mixing makes your opponent not care, not you.
A frequent error in finding NE is checking only whether a cell is "good" for both players. The correct test is whether either player can unilaterally improve by deviating. A cell where both players get high payoffs is not a NE if one of them can do even better by switching.
Students sometimes confuse Nash Equilibrium with Subgame Perfect Nash Equilibrium. Every SPNE is a NE, but not every NE is subgame perfect. In the sequential game, {Prest: Small, Tollgate: Large} is a NE of the simultaneous game but is not part of the SPNE because Tollgate's threat to play Large when Prest plays Large is not credible.
When computing mixed strategies, students sometimes set their own payoffs equal rather than their opponent's. Remember: your mixing probability is found from the other player's indifference condition.
⚠️ You will be asked to find all NE, pure and mixed. Do not stop at the pure strategies. If the game has two pure-strategy NE, there is almost certainly a mixed-strategy NE as well.
⚠️ Expect to draw a game tree for a sequential version of a simultaneous game and solve for the SPNE using backward induction.
⚠️ The indifference condition is a very common exam calculation. Know the setup cold: to find alpha, write Tollgate's payoffs as a function of alpha and set them equal.
⚠️ Be ready to explain why the SPNE outcome may differ from the simultaneous NE. The key idea: sequential play with observable moves lets the first mover commit, changing the second mover's best response.
⚠️ Clearly write the SPNE as a complete strategy profile. For Tollgate, you must specify what Tollgate does in each subgame ("Small if Prest Large, Large if Prest Small"), not just the action on the equilibrium path.
True or false: In a Nash Equilibrium, every player is playing their best response to the other players' strategies. (True)
Fill in the blank: To find Player 1's mixed-strategy probability, you set Player ______ indifferent. (2)
True or false: Every SPNE is a Nash Equilibrium, but not every Nash Equilibrium is subgame perfect. (True)
Fill in the blank: Backward induction starts at the ______ decision nodes and works towards the root. (Final / terminal)
True or false: In the Prest-Tollgate game, both firms have a dominant strategy. (False)
Q: In the Prest-Tollgate simultaneous game, what are the pure-strategy Nash Equilibria?
A: {Prest: Small, Tollgate: Large} and {Prest: Large, Tollgate: Small}. In each, neither player can improve by unilaterally switching.
Q: What is the mixed-strategy Nash Equilibrium? Show the indifference conditions.
A: Prest plays Large with probability 1/2 and Small with probability 1/2. Tollgate plays Large with probability 5/7 and Small with probability 2/7. Prest's probability (alpha = 1/2) comes from setting Tollgate indifferent: 6alpha + 12(1 - alpha) = 8alpha + 10(1 - alpha). Tollgate's probability (beta = 5/7) comes from setting Prest indifferent: 24beta + 30(1 - beta) = 26beta + 25(1 - beta).
Q: Draw the extensive form of the sequential game where Prest moves first. What is the SPNE?
A: The game tree has Prest at the root choosing Large or Small, then Tollgate choosing Large or Small at each branch. By backward induction: if Prest plays Large, Tollgate picks Small ($8 > $6); if Prest plays Small, Tollgate picks Large ($12 > $10). Anticipating this, Prest plays Large ($30 > $26). SPNE: {Prest: Large; Tollgate: Small if Prest Large, Large if Prest Small}.
Q: Why does the SPNE outcome differ from one of the simultaneous-game NE?
A: In the simultaneous game, {Prest: Small, Tollgate: Large} is a NE because neither deviates given the other's choice. In the sequential game, Tollgate cannot credibly commit to playing Large when Prest plays Large, because once Prest has actually chosen Large, Tollgate's best response is Small. Backward induction eliminates the non-credible path.
Q: Does either firm have a dominant strategy in the simultaneous game?
A: No. Prest prefers Large when Tollgate plays Small ($30 > $25), but prefers Small when Tollgate plays Large ($26 > $24). Similarly, Tollgate's best response depends on Prest's action. Neither has a strategy that is best regardless of the rival's choice.
This material connects to oligopoly models you will see later in the course: Cournot competition (simultaneous quantity choice) and Stackelberg competition (sequential quantity choice) are direct applications of the simultaneous and sequential game frameworks covered here.
The mixed-strategy concept reappears in auction theory and in models of market entry, where firms randomise over whether to enter a market based on expected competitor behaviour.
The SPNE concept and backward induction are foundational for any multi-stage economic model, including bargaining theory (the Rubinstein model) and contract design.
Nash Equilibrium, NE, pure strategy, mixed strategy, best response, dominant strategy, expected payoff, payoff matrix, normal form, extensive form, game tree, sequential game, simultaneous game, backward induction, subgame perfect Nash equilibrium, SPNE, credible threat, non-credible threat, first-mover advantage, capacity competition, oligopoly, Cournot, Stackelberg, ECON 300, intermediate microeconomics, game theory