A few questions to warm up and check the basics, before the exam-type questions below. They are reshuffled each time, so you can keep retrying.
The following are exam-type questions in the style of the examination paper, with marks at the rates used in the papers. A question totalling fewer than 25 marks would, in the examination, be combined with further parts, often one of the examinable proofs, to make a full 25-mark question. Attempt them in full before reading the worked solutions.
In the Traitors debrief we saw that the Faithful's threat to banish a deviator works only when they can be sure who deviated. Once players sometimes break Vote-Left for fun, a flagged suspect might be the colluding Traitor or merely a Faithful having fun, and the threat can lose its bite. We model this with a two-stage game. Let \(q\) be the Faithful's belief, the probability they assign, that the suspect they are poised to banish is the Traitor rather than someone breaking Vote-Left for fun. The Traitor chooses to comply with Vote-Left or deviate (keep colluding); the Faithful then choose to banish the suspect or carry on. Complying gives payoffs (Traitor, Faithful) of \((1, 2)\), and carrying on after a deviation gives \((3, 0)\). Banishing the suspect is a gamble: it yields \((-5, 3)\) if the suspect really is the Traitor, which happens with probability \(q\), and \((3, -5)\) if the suspect is an innocent Faithful, with probability \(1 - q\). Taking \(q = \tfrac{1}{2}\), the expected payoff to banishing is therefore \(\tfrac{1}{2}(-5, 3) + \tfrac{1}{2}(3, -5) = (-1, -1)\).
(a) Represent the game in extensive form (draw the tree). [3]
(b) Use backward induction to find the subgame perfect equilibrium and its payoffs. [5]
(c) Show that (comply, banish) is a Nash equilibrium of the game. [5]
(d) Explain why (comply, banish) is not subgame perfect, and how this mirrors the Traitors threat to banish a suspect that does not survive contact with the actual decision once the Faithful are unsure who deviated. [6]
(e) More generally, let the suspect be the Traitor with probability \(q\), so that banishing gives expected payoffs \((3 - 8q,\ 8q - 5)\). Determine the range of \(q\) for which the threat deters the Traitor, the range for which it is credible, and hence the range for which (comply, banish) is a Nash equilibrium that is not subgame perfect. [4]
(a) Provide definitions for the following terms:
(b) An entrant must decide whether to enter a market or stay out. If the entrant enters, the incumbent must decide whether to fight or accommodate. The payoffs (entrant, incumbent) are: stay out \((0, 2)\); enter then fight \((-1, -1)\); enter then accommodate \((1, 1)\).
(i) Draw the game in extensive form. [2]
(ii) Use backward induction to find the subgame perfect equilibrium and the resulting payoffs. [3]
(iii) Find a Nash equilibrium of the game that is not subgame perfect, and explain why it is not. [5]
(iv) Write the game in normal form and confirm the pure Nash equilibria. [3]
(c) Explain the difference between a Nash equilibrium and a subgame perfect equilibrium. [3]
(d) State the theorem on the existence of a subgame perfect equilibrium in finite games of perfect information. [4]
(a) Define sequential rationality. [2]
(b) Player 1 chooses \(L\) or \(R\). After \(L\), player 2 chooses \(\ell\) (payoffs \((3, 1)\)) or \(r\) (payoffs \((0, 0)\)). After \(R\), player 2 chooses \(\ell\) (payoffs \((2, 2)\)) or \(r\) (payoffs \((1, 3)\)).
(i) Use backward induction to find the subgame perfect equilibrium and its payoffs. [3]
(ii) Write the game in normal form. [4]
(iii) Obtain all pure Nash equilibria. [4]
(iv) State which Nash equilibrium is subgame perfect, and for each of the others identify the non-credible threat that makes it fail. [4]
(c) Consider the centipede game: player 1 may take, giving \((2, 0)\), or pass; then player 2 may take, giving \((1, 3)\), or pass; then player 1 may take, giving \((4, 2)\), or pass, giving the leaf \((3, 5)\).
(i) Solve the game by backward induction. [5]
(ii) State the subgame perfect outcome and payoffs, and comment on why the prediction is striking given that both players could do better. [3]
Two firms compete by choosing quantities, one after the other. Inverse demand is \(P(Q) = a - Q\), where \(Q = q_1 + q_2\) is total output and \(a > 0\); marginal costs are zero, so firm \(i\) earns profit \(q_i P(Q)\). Firm 1 (the leader) chooses \(q_1\) first; firm 2 (the follower) observes \(q_1\) and then chooses \(q_2\).
(a) Explain what a subgame perfect equilibrium is for a game with continuous action sets. [3]
(b) By backward induction, find the follower's best response \(q_2(q_1)\). [6]
(c) Hence find the leader's subgame perfect quantity, the follower's quantity, and the resulting profits. [8]
(d) Compare with the simultaneous-move (Cournot) equilibrium, in which \(q_1 = q_2 = a/3\), and explain the first-mover advantage in terms of commitment: why the leader gains by moving first even though the follower observes the choice. [8]
These exercises are entirely optional: they are extra practice, not extra content. The best way to use an AI assistant is to be its examiner, not its student: checking work is exactly the skill the examination rewards, and the skill you will need to use these tools well. Each exercise gives a transcript in the style of a widely used free AI chatbot answering one of the example questions above. Every transcript contains at least one error. Mark it as an examiner would: find each error, explain what goes wrong, and award a mark out of the marks of the original question. A marker's report for each transcript is on the solutions page.
A student pastes Question 2(b) into a free AI chatbot, which replies:
Backward induction: if the entrant enters, the incumbent compares
fighting (-1) with accommodating (1) and accommodates. Anticipating
this, the entrant compares staying out (0) with entering (1) and
enters. The subgame perfect equilibrium is (enter, accommodate), with
payoffs (1, 1). ✅
A Nash equilibrium that is not subgame perfect: consider (stay out,
accommodate): the entrant stays out, and the incumbent would
accommodate if entered. This is a Nash equilibrium, but it is not
subgame perfect, because the entrant's choice to stay out is not
optimal in every subgame.
A student pastes Question 4 into a free AI chatbot, which replies:
Each firm maximises q_i (a - q1 - q2). The first-order conditions are
a - 2 q1 - q2 = 0, a - q1 - 2 q2 = 0
Solving simultaneously gives q1 = q2 = a/3, with profits a²/9 each.
Firm 1 moves first, so it simply commits to a/3, and firm 2 follows
with a/3. Moving first makes no difference: both firms play the same
equilibrium either way. ♟️
You do not need this to follow the topic, but the Repeated Games chapter of the textbook may help if you would like more background on subgame perfect equilibria over a long horizon.
Worked solutions to the example questions →
Download the full quiz as a PDF (all questions, no solutions) →