Game Theory Autumn 2024
Problem Sheet 1
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Release date (by): Friday 25 October 2024
Submission date: 17:00, Wednesday 6 November 2024
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Pre-submission
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Post-submission
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• Your Guided Study Support Class in Week 5
• Office Hours: Monday 10:00-11:30, Watson 209
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• Model Solutions (Friday 8 November 2024)
• Generic feedback and individual feedback on your submission (by Wednesday 20 November 2024)
• Office Hours: Monday 10:00-11:30, Watson 209
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Penalties for late submissions:
• Penalty 5% for submissions by 17:00 on Thursday,
• Penalty 10% for submissions by 17:00 on Friday (assignment will be closed at this time),
• 100% thereafter.
Note that extenuating circumstances (ECs) cannot grant extensions but can remove late penalties. ECs can allow at most two sheets of the whole module (in our case, GTMDM) to be waived.
The questions follow on the next page in the PDF file.
Please scroll down in the HTML file for the questions.
Game Theory questions:
1. [LH,LM] Consider the following parametric matrix game:
where p is an arbitrary integer parameter.
(i) For each value of p check whether the game has a solution in pure strategies and, if this is the case, write out all such solutions. Write your arguments. [15]
(ii) For p = 1, find the gain-floor v 1(P S)(C) and loss-ceiling v2(P S)(C) of C in pure strategies. Is it true that v(C) > 0, where v(C) denotes the value of the game in mixed strategies? Give your argument using the value of v2(P S)(C). [5]
(iii) Setting p = 0 solve the matrix game C by dual simplex method in mixed strategies (show your working). Solving a matrix game means: finding an optimal mixed strategy of Player I, an optimal mixed strategy of Player II and the value of the game. [30]
Hint: Ifv(C) > 0 does not hold, then before applying the dual simplex method it is necessary to add a big enough positive constant to C to obtain a matrix game whose value is guaranteed to be positive.
(iv) (Optional, unmarked) Setting p = 0 solve the matrix game C using the graphical method and explicit formulas for 2 × 2 matrix games. Compare the result with that of part (iii).
2. [LH only] Consider the following matrix game:
(i) Suppose that x* is an arbitrary optimal strategy of Player I and y* is an arbitrary optimal strategy of Player II. Using appropriate dominance relations on A, find two components of x* and two components of y* that are equal to 0. Show your working. [16]
(ii) Using the graphical method and explicit formulas for 2 × 2 matrix games (as written in Lecture Notes or below) solve the matrix game
in mixed strategies. Show your working. [26]
(iii) (Optional, unmarked) Solve the same matrix game by dual simplex method and compare the
results.
(iv) Using your answer to (ii) write out an optimal mixed strategy of Player I and an optimal mixed strategy of Player II for the matrix game given by A. Are these optimal strategies unique? Give a short explanation. [8]
3. [LM only] Consider the following matrix game:
(i) Suppose that x* is an arbitrary optimal mixed strategy of Player I and y* is an arbitrary optimal mixed strategy of Player II. Using appropriate dominance relations on A, find two components of x* and two components of y* that are equal to 0. Show your working. [16]
(ii) Using the graphical method find the value and all optimal mixed strategies of Player I and Player II for the following matrix game
Show your working and your arguments.
Hint: You may be in doubt how to apply the graphical method in this case, as the examples presented at the lectures always gave a unique pair of optimal strategies (which is not the case for this problem). To improve your understanding consult Lecture Notes, especially Section 3.2.3 about the graphical method for two columns. [30]
(iii) (Optional, unmarked) Solve the same matrix game by dual simplex method. How does this compare with your answer to part (ii)?
(iv) Using your answer to (ii) write out all optimal strategies of Player I and all optimal strategies of Player II for the matrix game given by A (no explanation is necessary). [4]
Explicit formulas for optimal strategies and value of a 2 × 2 matrix game C are to be used only if C does not have solution in pure strategies. In this case, explicit formulas for the optimal strategies of such game denoted by x* and y* and the value of the game v(C) are:
Here 1 is the column vector consisting of two components both of which equal 1, and we have: