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 revision-aid activity, three students make a one-page aid for a game theory topic: the Theorist does the maths, the definitions and a worked example (player 1); the Artist draws the picture, a diagram (player 2); and the Storyteller writes the story, the plain-English intuition (player 3). The class scored the winning aid out of ten for each combination of the parts on show, giving a characteristic function game on \(N = \{1, 2, 3\}\) with \(v(\emptyset) = 0\), \(v(\{1\}) = 2\), \(v(\{2\}) = 1\), \(v(\{3\}) = 2\), \(v(\{1, 2\}) = 5\), \(v(\{1, 3\}) = 6\), \(v(\{2, 3\}) = 4\) and \(v(\{1, 2, 3\}) = 10\).
(a) Define a characteristic function game and the Shapley value. [4]
(b) Compute the marginal contribution of each player in each of the six orderings of the players. [5]
(c) Hence compute the Shapley value. [2]
(d) Verify that the Shapley value is efficient, and interpret the result: the three parts are worth only \(2 + 1 + 2 = 5\) on their own, so half of the aid's value comes from the parts reinforcing one another. Explain how the Shapley value shares this synergy. [4]
(a) Provide definitions for the following terms:
(b) Consider the game on \(N = \{1, 2, 3\}\) with \(v(\emptyset) = 0\), \(v(\{1\}) = v(\{2\}) = v(\{3\}) = 0\), \(v(\{1, 2\}) = 90\), \(v(\{1, 3\}) = 80\), \(v(\{2, 3\}) = 70\), \(v(\{1, 2, 3\}) = 120\).
(i) Compute the marginal contribution vector for each of the six orderings of the players. [5]
(ii) Hence compute the Shapley value. [2]
(iii) Verify that the Shapley value is efficient. [2]
(c) Explain what the Shapley value represents, and state which of its defining properties (efficiency, null player, symmetry, additivity) justify calling it a fair division. [4]
(a) State the efficiency, null player, symmetry and additivity properties of the Shapley value. [4]
(b) Consider the game on \(N = \{1, 2, 3\}\) with \(v(S) = 1\) if \(|S| \ge 2\) and \(v(S) = 0\) otherwise.
(i) Using symmetry and efficiency, write down the Shapley value. [3]
(ii) State, with reason, whether any player is a null player. [2]
(c) Consider the game on \(N = \{1, 2, 3\}\) with \(v(S) = 4\) if \(\{1, 2\} \subseteq S\) and \(v(S) = 0\) otherwise.
(i) Identify the null player. [2]
(ii) Compute the Shapley value using the six orderings. [5]
(iii) Verify efficiency, confirm that players 1 and 2 are symmetric and receive equal payoffs, and interpret the result. [4]
This question uses the additivity of the Shapley value to compute it without summing over all orderings.
(a) State the additivity property of the Shapley value, and explain what the sum \(v + w\) of two characteristic function games on the same player set means. [4]
(b) For a non-empty coalition \(T \subseteq N\), the unanimity game \(u_T\) is defined by \(u_T(S) = 1\) if \(T \subseteq S\) and \(u_T(S) = 0\) otherwise. Using the symmetry, null player and efficiency properties, show that the Shapley value of \(u_T\) gives \(1/|T|\) to each member of \(T\) and \(0\) to every other player. [6]
(c) Consider the game on \(N = \{1, 2, 3\}\) given by \(v = 2\, u_{\{1,2\}} + 3\, u_{\{1,2,3\}}\). Write down \(v(S)\) for every coalition \(S\), and use additivity with part (b) to compute the Shapley value. Verify that it is efficient. [9]
(d) Explain how the decomposition into unanimity games shows where each player's share comes from, and why additivity is a useful shortcut compared with summing marginal contributions over all \(n!\) orderings. [6]
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:
Step 1: Marginal contributions for each ordering.
(1, 2, 3): 1 adds 0, 2 adds 90, 3 adds 30 -> (0, 90, 30)
(1, 3, 2): 1 adds 0, 3 adds 80, 2 adds 40 -> (0, 40, 80)
(2, 1, 3): 2 adds 0, 1 adds 90, 3 adds 30 -> (90, 0, 30)
(2, 3, 1): 2 adds 0, 3 adds 70, 1 adds 120-90 = 30 -> (30, 0, 70)
(3, 1, 2): 3 adds 0, 1 adds 80, 2 adds 40 -> (80, 40, 0)
(3, 2, 1): 3 adds 0, 2 adds 70, 1 adds 50 -> (50, 70, 0)
Step 2: Average over the six orderings.
phi_1 = (0 + 0 + 90 + 30 + 80 + 50) / 6 = 250/6 ≈ 41.67
phi_2 = 240/6 = 40
phi_3 = 210/6 = 35
Step 3: Check efficiency.
41.67 + 40 + 35 = 116.67 ≈ 120 ✅
(small rounding differences are expected when averaging)
A student pastes Question 3(b) into a free AI chatbot, which replies:
Player 3 is a null player, because v({3}) = 0: on their own they create
no value. By the null player property, phi_3 = 0. Players 1 and 2 are
symmetric, so by efficiency they share v(N) = 1 equally:
phi = (1/2, 1/2, 0)
No player is left out: the value of the grand coalition is fully
distributed, so this allocation is fair and efficient.
You do not need this to follow the topic, but the chapter on The Core may help if you would like more background on stable allocations alongside the Shapley value.
Worked solutions to the example questions →
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