COMP524-JAN21 Safety and Dependability University of Liverpool COMP524-JAN21 Continuous Assessment 2 Coordinator: Fabio Papacchini Assessment Information Assignment Number 2 (of 2) Weighting 15% Assignment Circulated 21/06/2021 Deadline 14/07/2021 at 5pm BST (GMT +1) Submission Mode Please submit your solutions electronically on Canvas. You submission should have two files: (1) a ZIP file containing executables (the .pm and .pctl files) for the models and properties, and (2) a PDF/DOC/DOCX file that contains the results and explanations. Submission necessary in order to satisfy Module requirements? No Late Submission Penalty Standard UoL Late Penalties policy applies Plagiarism and Collusion Please be aware of the University guidelines on plagiarism and collusion. If you are unsure on what the University policy is, please take part to the tu- torial and quiz at https://liverpool.instructure.com/courses/28504/ modules/items/1028781 Fabio Papacchini:
[email protected] 1 COMP524-JAN21 Safety and Dependability University of Liverpool Task Description Eight and a Half – Simplified Variation of Seven and a Half. Eight and a half 1 is a spin-the-wheel game between a player and the bank. It is played with the wheel depicted below (i.e., the wheel is divided into 12 equal slices, each slice has a value between 1 and 8, or the value of 12 ). The game consists of two rounds: in a first round, the player can spin the wheel several times, in the second round the bank can spin the wheel several times. 1 21 2 1 2 3 4 1 2 5 6 1 2 7 8 Figure 1: The 8 + 12 wheel The player’s round. Starting with a score of 0, the player can repeatedly spin the wheel add the number pointed by the marker to their score. After adding the number to their score, the player can either finish their round, or repeat the above process. However, the player loses immediately if their score exceeds 8 + 12 . The bank’s round. Starting with a score of 0, the bank can (sim- ilar to the player in the previous round) repeatedly spin the wheel add the number pointed by the marker to the bank’s score. The bank has to keep on spinning the wheel until the bank’s score reaches or exceeds the player’s score. The bank has won if the bank’s score at this time does not exceed 8 + 12 and lost if the bank’s score at this time exceeds 8 + 12 . 1. Model the game as a Markov decision process using PRISM or ePMC. (50 marks) 2. Assume that we want to maximise the chance of winning. Write a PRISM property and determine the maximal chance to win. (10 marks) 3. Describe an optimal winning strategy of the player. (10 marks) 4. Assume that we want to minimise the chance of winning. Write a PRISM property and determine the minimal chance to win. (5 marks) Discuss why the chance of winning is like this when the player minimises their chances to win. (5 marks) 5. Change the model such that the bank has to exceed the score of the player to stop. (The player, however, still loses immediately when their score exceeds 8 + 12 .) Determine the maximial chance for the player to win in this case and discuss the question of whether or not their chance to win is fair (giving a brief justification for your answer). (10 marks) 6. Briefly (≤ 123 words) describe a contemporary research problem associated with Markov chains, Markov games, or Markov decision processes. Cite two recent (from 2016 or younger) articles or conference papers related to the problem you describe. (10 marks) 1Seven and a half is an Italian card game. Fabio Papacchini:
[email protected] 2 COMP524-JAN21 Safety and Dependability University of Liverpool You submission should have two files: (1) a ZIP file containing executables (the .pm and .pctl files) for the models and properties, and (2) a PDF/DOC/DOCX file that contains the results and explanations. You are welcome to form mini groups of two students, but it is not a strict requirement. If you decide to work in a group, then it is enough if one group member submits, but all relevant information on the group – group members (names and student IDs) – must be clearly stated on the first page of your submission. Fabio Papacchini:
[email protected] 3
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