SOCS0079--Game Theory for the Social Sciences, Second Summative Assessment, 2024/2025

INSTRUCTIONS:

- The assessment is due on 8 January 2025, 1pm, and shall be submitted via Turnitin.

- Late      submission       results       in       penalties,       see: https://www.ucl.ac.uk/academic-manual/chapters/chapter-4- assessmentframework-taught-programmes/section-3-module-assessment#3.12. There is no exception to late submission penalties,  unless  a  Delayed  Assessment  Permit  has  been  used  or  Extenuating  Circumstances  application  has  been successfully made.

- Submit on Turnitin a single document that includes the main body of your report (i.e., answers to the questions), and any tables and figures you may use in your report. If you prefer to do so, you can draw figures or write down algebra by hand. In that case, paste a photo/scan of the hand-drawn figure or algebra into your report, and make sure that your handwriting is legible.   However, do not paste the photo of handwritten text (i.e. anything beyond algebra, graphs, or tables consisting of numbers). The number of words in the handwritten text copied into the solution are to be included in the word limit.

- In the cover page include the number of words of your report, excluding the non-textual elements in tables and figures (if you used any).

- Make sure that you answer the question on AI use on the cover sheet.

- Word limit is 1,500. This excludes numbers in tables, figures, equations, algebra, and references, but includes footnotes, endnotes and all text in tables or figures. The number of words in tables/figures and handwritten text copied into the solution are included in the word limit. Exceeding this limit will result in penalties.

- This is an assessed piece of coursework for the SOCS0079 module; collaboration and/or discussion of the assessment with anyone is strictly prohibited. The rules for plagiarism apply and any cases of suspected plagiarism of published work or the work of classmates will betaken seriously.

- If you rely on any sources in your solution (books, articles, etc) provide reference for each. List full bibliographic details of references used at the end of your report. Any referencing style (ASA, APA, Harvard, Chicago etc.) is fine, provided that the style. is used consistently.

- The coursework will be assessed against the criteria set in the UCL UG-ESSAY GRADING SCHEME,a pdf of which could be seen in the assessment submission area of the course on Moodle. In addition to those general guidelines, further specific factors will affect the marks. Correctness of the solutions (in case the solutions are incorrect or no solution is found the effort made in trying to find the solution), clarity of arguments, rigour in presenting and analysing the games, creativity and novelty in your answers, and the ability to demonstrate that key concepts treated in the course are understood well.

- When answering the questions, always include an explanation (in terms of algebra or verbal explanation) about how you reached the solution. Solutions without a clear explanation about how you got them will not be accepted.

1. [50 Points] Describe briefly a situation, either from your daily life or from a book or movie, which can be modelled as a game, but you think that the actual players’ behaviour in this situation differed from what game theory predicts. You may not use Prisoner’s Dilemma in this question. Use exactly N=2 players. Keep the game as simple as possible.

a)          Model this situation as a simple game, indicate clearly who the players are, what strategies the players have and the associated payoffs. Use numerical payoffs only. Here, assume that all players are selfish and rational (in the game-theoretic sense). Make sure to give enough details within the word limit on key features of the situation and how you modelled it as a game. Analyse this game formally and find its Nash Equilibrium/Equilibria, and all Evolutionarily Stable Strategies (ifESS makes   sense   in   the   game   you   describe).   If  it   is   a   sequential   game,   also  report   the   subgame  perfect  Nash Equilibrium/Equilibria. Discuss how the actual players’ behaviour in the example is different from what game theory predicts.

Then:

b)         Extend your analysis in 1-a and find the Nash Equilibrium/equilibriaof the game by modifying the payoffs assuming two types of social preferences we discussed in class (and still assuming actors are fully rational):

i) Social Orientation Model (where Ui  = utility for player i, x = outcome for self, y = outcome for other, θi  is the individual “altruism” parameter):

ui (x, y; θi) = x + θiy   with  − 1 < θ < 1

for the two cases:

1) actors know each other’s θ values.

2) actors know their own θ, don’t know each other’s θ value, but know that players are random draws from a population where 50% of actors have θ = 0 and 50% have θ = 0.80.

ii)         Inequality Aversion Model (where Ui  = utility for player i, x = outcome for self, y outcome for other,  βi   (αi) is an individual parameter capturing i’s dislike for advantageous (disadvantageous) inequality):

for the two cases:

1) actors know each other’s β and α values.

2) actors know their own β and α, don’t know each other’s β and α values but players are random draws from a population where half of the actors are fully selfish (β = 0 and α = 0) and the other half of the actors are inequality averse with β = 0.5 and α = 2?

Discuss how your results in 1-b differ from the analysis in 1-a and summarise your main insights. Do social preferences help explain the gap between the Nash equilibrium/equilibria in 1-a, and the actual behaviour?

c) In around 200-250 words, discuss if you would need to modify the “rationality” assumption of game theory, too, to explain actual human behaviour in the game you described above; and what type of cognitive biases players may possibly display in your example above.

2. [50 points] Here we ask you to design an experiment to understand a specific aspect of human cooperation. In your experiment, you’ll need to use a game that captures a social dilemma. In particular, you’ll need to use at least one of the following games in your experiment: Prisoner’s Dilemma, Volunteer’s Dilemma, Public Goods Game, Weakest Link Game, or a similar game that addresses the problem of cooperation. We allow any variant of those games, that is, your game could be a game of social dilemma with complete information, incomplete information, non-repeated or repeated, sequential or simultaneous play, or in fact a combination of one or more of these types.

You may use some other game (ultimatum, dictator, etc.) in conjunction with your game of cooperation, so long as you make an explicit connection to the aspect of human cooperation you are studying:

- A form. of embeddedness (temporal, network, institutional), and/or

- Social preferences, and/or

- Bounded rationality.

a)         Theory and hypotheses: Describe your research question and the particular type/types of game you want to use in your experiment and why. Then describe what prediction(s) you aim to test in your experiment (i.e. your hypothesis). Here, your prediction(s) (hypothesis) can be based on a formal analysis of the game, or you can formulate your hypotheses informally, inspired by your intuition you developed during the game theory course. Remember that you will be studying one or more of the above issues (embeddedness, social preferences, bounded rationality), so make this aspect of your study clear

(i.e. make it clear as to what research question you are studying).

b)         Sample and design: Describe the participant sample you want to use and how you would gather this sample. Then describe your experimental procedure (what the subjects will do during the experiment, what treatments you will be using in your experiment) and justify this procedure (that is explain why you will be implementing your particular treatments).

c)          Expected results: Finally describe what you expect to find in your experiment and how this finding will relate to your research question and hypothesis that you started with.

Be creative and imaginative in your experiment, while keeping your experiment feasible and ethical to implement with a reasonable research budget. Also, your experimental design should be original, that is, it should not simply replicate an experiment that has been implemented in the literature (while you maybe inspired by a published study and build on it with proper citations). Any form. of experiment (lab, online, field, etc.) is allowed.