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 class we played "Will it end?": pairs played the contractor stage game
(first action High, second action Low), and after each round rolled a die, continuing unless a 1 was rolled, so the game continued with probability \(\delta = 5/6\).
(a) Interpret \(\delta = 5/6\) and state the expected number of rounds. [2]
(b) Both players use the Grudger strategy (play High until the opponent plays Low, then Low for ever). Holding the opponent's Grudger fixed, write down the discounted payoff to a player who conforms and plays High in every round, and to one who deviates to Low in the first round. [6]
(c) Determine whether mutual cooperation is sustained at \(\delta = 5/6\). [4]
(d) Find the smallest \(\delta\) for which cooperation is sustained, and hence say whether our die made cooperation easy or hard to sustain. [5]
(e) Suppose the players replace Grudger with a forgiving trigger: a single Low is punished by mutual Low for two rounds, after which both return to High. Holding the opponent to this strategy, determine whether cooperation is sustained with our die (\(\delta = 5/6\)). Find the smallest \(\delta\) for which it is sustained, and compare with the Grudger threshold of part (d). [7]
(a) Provide definitions for the following terms:
(b) Two players repeatedly play the Prisoner's Dilemma with row-player stage payoffs
cooperate first, defect second, infinitely repeated and discounted by \(\delta \in (0, 1)\). Both use the Grudger strategy.
(i) Holding the opponent's Grudger fixed, write down the discounted payoff to a player who conforms and cooperates in every round, and to one who defects in the first round. [3]
(ii) Hence find the smallest \(\delta\) for which cooperation is sustained. [3]
(iii) Show that Grudger is a subgame perfect equilibrium for \(\delta\) above this threshold, by checking that no player can gain by deviating either on the cooperative path or in the punishment phase. [5]
(c) State the Folk Theorem. [3]
(d) Define the individually rational payoff and state its value for this Prisoner's Dilemma. [3]
(e) Explain what the Folk Theorem implies for cooperation in this game. [4]
(a) Define a subgame perfect equilibrium of a repeated game. [2]
(b) The Prisoner's Dilemma above is now repeated a finite and commonly known number of times \(N\).
(i) Use backward induction to determine the subgame perfect equilibrium. [4]
(ii) Explain why cooperation cannot be sustained in any subgame perfect equilibrium. [4]
(c) Explain why the backward-induction argument from part (b) does not apply to an infinitely repeated Prisoner's Dilemma. [3]
(d) Return to the infinitely repeated game with Grudger, but with temptation payoff \(T = 4\) in place of \(5\). Recompute the smallest \(\delta\) for which cooperation is sustained. [4]
(e) Comment on how the threshold for \(\delta\) changes with the temptation payoff. [3]
(f) With stage payoffs \(R = 3\), \(P = 1\), \(S = 0\) and general temptation \(T\), show that the threshold is \(\delta^{*} = \dfrac{T - 3}{T - 1}\), and state what happens to it as \(T\) grows. [5]
Two players repeatedly play the Prisoner's Dilemma with stage payoffs \(R = 3\) for mutual cooperation, \(T = 5\) for a unilateral defection, \(P = 1\) for mutual defection and \(S = 0\) for being defected against, discounted by \(\delta \in (0, 1)\). They use a forgiving trigger of length \(T\): both cooperate until someone defects in a cooperative round, then both defect for exactly \(T\) rounds, after which they return to cooperation and ignore the deviation.
(a) Holding the opponent to this strategy, write down the discounted payoff to a player who conforms and cooperates in every round, and to one who defects once in the first round and is then punished. [4]
(b) Show that cooperation is sustained if and only if \(2\delta - \delta^{\,T+1} \ge 1\). [8]
(c) Show that the resulting threshold \(\delta^{*}(T)\) is decreasing in \(T\), and that as \(T \to \infty\) it tends to \(\tfrac{1}{2}\), the grim-trigger threshold. [6]
(d) Prove that, for \(\delta\) above the threshold, the forgiving trigger is a subgame perfect equilibrium, by checking with the one-shot deviation principle that no player can gain from a single deviation on the cooperative path or during the punishment phase. [7]
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:
Conforming: both players cooperate for ever, so the payoff is
3 + 3δ + 3δ² + ... = 3 / (1 - δ)
Deviating: the deviator earns 5 in the first round; after that both
players defect for ever, which yields nothing, so the payoff is 5.
Threshold: cooperation is sustained when 3 / (1 - δ) >= 5, that is
3 >= 5 - 5δ, giving
δ >= 2/5
A student pastes Question 2(c) and (e) into a free AI chatbot, which replies:
The Folk Theorem states that in an infinitely repeated game, ANY payoff
pair can be achieved in a subgame perfect equilibrium, for ANY discount
factor δ > 0. In particular, cooperation (3, 3) is always an
equilibrium outcome of the repeated Prisoner's Dilemma, however
impatient the players are. This is why we see cooperation everywhere in
real life! 🤝
You do not need any of this to follow the topic, but the following chapters of the textbook may help if you would like more background:
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