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###### 日期：2021-07-09 09:22

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)

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.

COMP524-JAN21 Safety and Dependability University of Liverpool

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.

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

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.