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4PAM1042 Applications of Mathematics

Question:

ˆIf you have already received a pass mark (40% or higher) for your non-technical report on Game Theory, or if you received an overall pass mark for the module, then you do not need to attempt this assignment again.

ˆIf you submitted a non-technical report on Game Theory (but obtained a grade lower than 40%), then you must choose a different topic than the one you chose originally. For his assignment, you will write a non-technical report about a topic in Game Theory. You will choose one of the following three topics, described in more detail in the next section.

1. Sequential games & Logic

2. Lizard & Chicken

3. Being a rock & prisoner’s dilemma

Each topic gives you two games and some questions. The given games and questions will form fixed points that you have to address and around which you can plan and structure your report. Much of the rest of the content, structure and style of the report is up to you and should be chosen with the overall aim and target audience of the report in mind.

Questions

Your project has to involve discussing the following questions.

1. Explain and discuss the term “winning strategy”.

2. What is the winning strategy for 333-Cesarball?

3. How do sequential games relate to logical statement involving quantifiers?

4. Who has a winning strategy for the sequence game?

Lizard & Chicken

Rock-Paper-Scissors-Lizard-Spock (RPSLSp) is a variant of the well-known Rock-Paper-Scissors (RPS) and originated in the surprisingly popular TV-show “The Big Bang Theory”. The game plays just like regular RPS, except that there are two additional moves available to both players, Lizard and Spock, with the following rules.

ˆLizard beats Paper and Spock (“eats” and “poisons”, respectively) but is beaten by Rock and Scissors (“crushed” and “decapitated”, respectively). Spock beats Rock and Scissors (“vapourizes” and “crushes”, respectively) but is beaten by Paper and Lizard (“disproved” and “poisoned”, respectively). Rock, Paper and Scissors relate to each other as before and if both players play the same move, the game is a draw.

“Chicken” is a single round, 2-player game. Each player either picks “Safe” or “Risk”.

ˆ If both players play Safe, then they both obtain 1 util.

ˆ If both play “Risk”, then both receive 0 utils.

ˆ If they do not play the same thing, then the player who played “Risk” obtains 2 utils and the player who played “Safe” receives 1 util.

Questions

Your project has to involve discussing the following questions.

1. Explain and discuss the terms “mixed strategy” and “Nash equilibrium”.

2. What is the Nash equilibrium for RPSLSp?

3. Suppose you are playing RPSLSp with a friend who dislikes “geek culture” in general and “The Big Bang Theory” in particular and so will only ever play the classical moves Rock, Paper, or Scissors. You are free to choose any moves. What is the Nash equilibrium now? Is this a fair game?

4. What is the Nash equilibrium for Chicken? Is there a unique equilibrium?

5. Does “chicken” occur in real life? How should real-life players behave?

At the beginning of this variant of Rock-Paper-Scissors, each player dresses up as either a rock, some paper or a pair of scissors. The players may not choose the same costume. The game is played just like regular RPS with the following difference.

ˆIf both players choose the same move and one of the players is dressed up as this move, then this player wins. For example, if both players choose Rock, then the player dressed as Rock (if there is one) wins.

The prisoner’s dilemma is a single round, 2-player game. Each player either picks “Solidarity” or “Betrayal”.

ˆ If both players play Solidarity, then they both obtain 2 utils.

ˆ If both play “Betrayal”, then both receive 1 util.

ˆ If they do not play the same thing, then the player who played “Betrayal” obtains 3 utils and the player who played “Solidarity” receives 0 utils.

Your project has to involve discussing the following questions.

1. Explain and discuss the terms “mixed strategy” and “Nash equilibrium”.

2. What is the Nash equilibrium for Costume party RPS?

3. Suppose you are playing Costume party RPS with a friend who dressed up as a rock. Should you choose a Scissors costume or a Paper costume?

4. What is the Nash equilibrium for the prisoner’s dilemma? Is there a unique equilibrium?

5. Does “the prisoner’s dilemma” occur in real life? How should real-life players behave?

Structure and overall aims

Your task is to write an informative, clear and enjoyable report for a non-expert audience that addresses the games and questions you were given. In particular, this means that:

ˆYour report should flow naturally and be structures as one piece of work. Do not think of the given questions as problems on a tutorial sheet to be addressed one after the other.

ˆYou need to address all the given questions, but do not mimic the structure of this “question sheet”.

You can analyze the games and address the questions in any order you think is best. You can choose your own title for the report.

ˆYour report is aimed at a non-expert audience (see target audience below). So you need to carefully explain all the background to the questions you are addressing and the concepts (things like strategy, best response, Nash equilibrium, zero-sum) that you are using. This means that significant parts of your report may be spent on background rather than directly answering the given questions.

Think about ways to make the report easy and enjoyable to read whilst being informative. Use example, diagrams, historical background, etc.

Your target audience is interested but non-expert. Do not assume that they know any advanced maths at all. Think of a friend of yours who does not do maths but might be interested in listening to something like “Inside Science” on Radio 4. (If you don’t know what I’m talking about, think of someone who might enjoy “popular science”. Also, Inside Science is pretty good.)

Your audience is reasonably patient but needs to be kept “on board” with examples, diagrams and engaging writing. They are not used to mathematical definitions and arguments and find them hard to absorb. Unlike mathematicians, they do not always prefer the most abstract and general version of a statement. However, they do want to understand and so overly vague and misleading statements are no good to them. They need clear and to-the-point explanations that are as simple as possible (but no simpler).

When structuring and writing your report, put yourself in your audience’s shoes and ask

ˆWhat do I need the audience to understand here? What do they need to have already understood at this point and have I adequately explained this earlier in the report? Is it clear to the audience what I am trying to explain and why? If not, can I state more clearly

what I’m about to do and motivate why I’m doing it?

ˆHow can I make the important and difficult points as clear as possible?

ˆWhere would some more examples be helpful?

ˆWhere would some diagrams be helpful?

ˆAre any parts of the report more hard work than they need to be? Can I make them simpler?

You report will be marked according to the following criteria.

Correctness Are the claims and reasoning in the report mathematically correct? Have the questions of the topic been addressed? Has any referencing been done appropriately?

Clarity Is the writing clear? Can the report be followed easily by the target audience? Has effort been made to provide helpful diagrams, examples, etc?

Originality and Flair Is the report enjoyable to read? Have interesting and original examples been

picked? Has the student read (and referenced!) external sources?