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, and are intended to be a little harder than the examination itself, 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 ran a Rock-Paper-Scissors tournament. The row player's payoff matrix, with actions ordered (Rock, Paper, Scissors), using \(+1\) for a win, \(-1\) for a loss and \(0\) for a draw, is
(a) Explain why this is a symmetric zero-sum game, write down the column player's payoff matrix. [3]
(b) Obtain the utilities to both players for the following pairs of strategies:
[3]
(c) Using the best response condition, prove that both players playing each action with probability \(\tfrac{1}{3}\) is a Nash equilibrium. [5]
(d) Prove that this is the unique Nash equilibrium: show that in any Nash equilibrium each player is indifferent between all three actions, and deduce the mixing probabilities. [5]
(e) For Rock-Paper-Scissors-Lizard-Spock, in which each of five actions beats two others and loses to two, state the Nash equilibrium and justify it by the same indifference argument. [3]
(a) Provide definitions for the following terms:
(b) Consider the normal form game defined by
where \(M_r\) gives the payoffs of the row player and \(M_c\) those of the column player.
(i) Obtain all pure Nash equilibria. [4]
(ii) By sketching the row player's expected utilities against \(\sigma_2 = (y, 1 - y)\) and the column player's against \(\sigma_1 = (x, 1 - x)\), find the value of \(y\) at which the row player is indifferent and the value of \(x\) at which the column player is indifferent. [6]
(iii) State the best response condition, and use it (or the support enumeration algorithm) to obtain the mixed Nash equilibrium. [5]
(iv) Compute each player's expected payoff at the mixed Nash equilibrium, and determine whether each player would prefer it to the pure Nash equilibria. [4]
(a) Define a strictly dominated strategy and a weakly dominated strategy, and give an example of a game with an action that is weakly but not strictly dominated. [5]
(b) Consider the Prisoner's Dilemma defined by
(i) Using iterated elimination of strictly dominated strategies, obtain the Nash equilibrium, justifying each elimination. [4]
(ii) Explain why the equilibrium is worse for both players than the outcome in which each plays their first action, and why that better outcome cannot be sustained in a one-shot game. [3]
(c) Consider the coordination game defined by
(i) Obtain all pure Nash equilibria. [3]
(ii) Using the best response condition, obtain the mixed Nash equilibrium. [5]
(iii) Compute each player's expected payoff at the mixed Nash equilibrium, and show that it is worse for both players than either pure equilibrium. [3]
A two-player symmetric game has row-player payoff matrix \(A\); the column player has payoff matrix \(A^T\). The game is symmetric and zero-sum when \(A\) is antisymmetric, \(A^T = -A\).
(a) Show that for an antisymmetric \(A\), every strategy \(\sigma\) satisfies \(\sigma A \sigma^T = 0\). Interpret this: a player using any strategy against an identical opponent has expected payoff zero. [5]
(b) Using the best response condition, show that \((\sigma^*, \sigma^*)\) is a symmetric Nash equilibrium if and only if \((A \sigma^{*T})_i \le 0\) for every action \(i\), with equality on the support of \(\sigma^*\). Deduce that the value of the game is zero. [6]
(c) Consider the weighted Rock-Paper-Scissors game with
By solving \(A \sigma^{*T} = 0\), find the full-support symmetric Nash equilibrium, and confirm it reduces to \(\bigl(\tfrac{1}{3}, \tfrac{1}{3}, \tfrac{1}{3}\bigr)\) when \(\beta = 1\). [8]
(d) Describe how the equilibrium changes as \(\beta\) varies between \(0\) and \(\infty\), and interpret what the weight \(\beta\) does to the players' use of each action. [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)(iii) into a free AI chatbot, which replies:
Step 1: Set up the indifference conditions.
In a mixed Nash equilibrium, each player must be indifferent between
their two strategies; otherwise they would simply play the better one!
Step 2: Find the row player's mix.
Let the row player play their first action with probability x. Using
the row player's matrix M_r, indifference between their two actions
gives
4x + 1(1 - x) = 0x + 3(1 - x)
so 3x + 1 = 3 - 3x, which solves to give x = 1/3. ✅
Step 3: Find the column player's mix.
Let the column player play their first action with probability y.
Using the column player's matrix, indifference between their two
actions gives
3y + 1(1 - y) = 0y + 4(1 - y)
so 2y + 1 = 4 - 4y, which solves to give y = 1/2. ✅
Conclusion.
The unique Nash equilibrium of this game is:
σ1 = (1/3, 2/3), σ2 = (1/2, 1/2)
Let me know if you would like me to walk through any of the steps in
more detail!
A student pastes Question 3(b) into a free AI chatbot, which replies:
Finding the dominated strategies.
Compare the two rows of the row player's matrix M_r. The second row
gives 5 > 3 against the first column and 1 > 0 against the second, so
the first action is weakly dominated by the second and can be
eliminated. By the symmetry of the game the column player's first
action is also weakly dominated, and it can be eliminated too.
The equilibrium.
The only cell that survives is both players choosing their second
action, with payoffs (1, 1). This is the Nash equilibrium obtained by
iterated elimination.
Why can the players not do better?
Although (3, 3) looks better for both players, the equilibrium (1, 1)
is Pareto efficient, because neither player can improve their payoff by
changing their own strategy. Cooperation cannot be sustained in a
one-shot game because each player has an incentive to deviate. This is
the famous Prisoner's Dilemma!
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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