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Coalitional Games: Graphical representation and Core Condition for LSG, B1, and S2 start-up firms in

Graphical Representation of Coalitional Games

Section 1 (Coalitional games)Laura, Sheena, and Grace, three students at DMU, are the owners of the start-up firm LSG in Leicester. They will join forces with two other start-ups, B1 and S2, to expand their portfolio of products and increase their market share. LSG generates a total of £420 in profits on its own, while B1 produces £260 in profits alone and S2 reports £140 in firm’s profits. The three firms have collaborated before in different forms, generating joint profits of £680 for work done by LSG and B1 together, in a collaboration between LSG and S2 the firms produced £600 in joint profits, and B1 and S2 were able to make £420 together. After the merger happens, it is estimated that the three firms together will generate a total profit of £1400.

Use the notation indicated/used in each question in all your answers. Answers using a different notation will lose all or some marks.

Firms LSG and B1 are forming a coalition on their own. Produce a graph where you show the worth of each singleton coalition, the worth of the coalition formed by the two firms, and indicate the core. Use of labels, clearness, and tidiness will be considered as well.  (10 marks)

Let  be the payoff for  when the firm forms the singleton coalition  Let  be a payoff profile for the grand coalition , where  is the payoff for is the payoff for , and  is the payoff for . For coalition , let the payoff profile be  be a payoff profile for the coalition, with  the payoff for , and  the payoff for . Let  be the payoff profile for coalition , with  being the payoff for , and  the payoff for  in this coalition. Finally, let   be a payoff profile for coalition , with  the payoff for , and  the payoff for . What conditions should hold for the payoffs of firms , , and , such that the payoff profile  is in the Core. In your answer use the notation presented in this question only, answers using a different notation will get zero marks (10 marks)

To answer this question, use the equivalent of the following notation:

Let  be three players. Let  be the worth of singleton coalition , that is the coalition formed by player  only,  be the worth of coalition be the worth of coalition be the worth of coalition , and  be the worth of coalition Indicate the marginal contribution of LSG, B1, and S2 if the grand coalition is formed in the following way {B1, LSG, S2}

Core Condition for Payoff Profiles in Coalitional Games

Complete the table below.  indicates the coalition, and  the worth of the coalition.

2.Three clients seek to connect to a server that charges fees according to different plans. The clients are represented by letters , , and , and the server is represented by letter . The clients can connect to the server independently or jointly with other clients. The costs of connection are: connecting client  to the server costs , connecting client  to the server costs , connecting client  to the server costs , connecting client  to client  costs , connecting client  to client  costs , and connecting client  to client  costs .

2.1.Construct a graph showing the different ways in which the clients can connect with each other and the server, representing each client and the server with nodes, and the connections with edges, indicating the costs of each connection (as shown above in the form of  variables) on the side of the corresponding edges. (5 marks)

2.2.Let If connecting the three clients to the server in the following way  costs the same as connecting them , how much is. (5 marks)

2.3.Is the grand coalition  superadditive? (5 marks)

2.4.What are the Shapley values for each client when they connect to the server forming the grand coalition. You only need to present the Shapley values for this question. (10 marks)

3.Consider the two-sided platform model covered in the lectures. There, we found the following equations for the willingness to pay for each of the sides of the platform:

3.1.Let , what are the prices (willingness to pay) for each side of the platform? (5 marks)

3.2.If increases to , ceteris paribus, what will be the new value for  Interpret this result briefly explaining why is the new price equal, higher, or lower to the previous price level (10 marks)

4.For each question underline all those statements that are correct, more than one statement may be correct
4.1.Consider a two-sided platform , with sides  and . When the number of users in side  increase, ceteris paribus, this changes the valuation of the platform by side . What type of effect is this? (5 marks)
a.Network effect
b.Cross-network effect
c.Direct network effect
4.2.A positive network effect is (2.5 marks)
d.A positive externality caused by one side of the platform on the other side of the platform
e.A positive externality caused by any side of the platform on any side of the platform, that should not be considered for the platform’s pricing policy
f.A positive externality caused by any side of the platform on any side of the platform, that should be considered for the platform’s pricing policy
4.3. Which one is an example of a negative direct network effect (2.5 marks)
g.In a game console system, more gamers increase the number of people other gamers can play with
h.In a dating platform, more members on both sides increase due to a marketing campaign
i.A club becomes the most popular all-night club in the city thanks to a significant increase in the number of females attending. Before the current success, the club used charge an entry fee only to males to attract females, but with the current success this is not needed anymore and now a flat entry fee applies for all attendants
4.4. Consider a two-sided platform (5 marks)
j.Ceteris paribus, the side with the highest price elasticity will pay a higher price
k.Ceteris paribus, that side that produces a higher network externality will pay a lower price
l.Ceteris paribus, the side with a significant lower number of members will pay a higher price

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