coalition game

Consider a game with three players, noted 1,2 and 3, such that v ({1,2}) = 90, v ({1,3}) = v (N) = 100 and v ({i}) = v (2, 3) = 0 for all i = 1, 2, 3. Let (x1, x2, x3) be any element of the core.

1. Show that x2 can take a unique value that we will determine.
2. Show that x1 ∈ [90, 100].
3. Determine the range of values to which x3 belongs.

Exercise 2:

Consider a coalition game with 4 players. v (S) = 0 if the cardinal of S is 1, v (S) = a if the cardinal of S is 2, v (S) = b if the cardinal of S is 3 and v (N) = 1.
1. Show that the core is not empty if a≤1 / 2 and b≤3 / 4.
2. Show that the core is empty if a> 1/2 or b> 3/4.

Exercise 3:

1. Let be a collection {λS} ∈ S ∈ C (where C is the set of coalitions) such that λ {1} = 1, λ {3} = λ {2} = 1/3, λ {2,3} = x and λS = 0 for the other coalitions. Determine x such that this collection is balanced.

2. Let v (S) = – 2 if | S | = 1, v (S) = – 1 if | S | = 2 and v (N) = 0. Show in two different ways that the game is balanced.

3. Let v (S) = 1 if | S | = 1, v (S) = 2 if | S | = 2 and v (N) = 2.5. Show in two different ways that the game is unbalanced.

Exercise 4

We place ourselves in a framework of negotiation games. Find:

1. A solution which satisfies all the axioms of Nash’s solution except that of efficiency.

2. A solution which satisfies all the axioms of Nash’s solution except that of symmetry.

Exercise 5

We consider a firm and an employee. The employee is characterized by a productivity ρ> 0 and a disutility of labor D> 0. The transition from remuneration to disposable income is summarized by a function R which associates the cost of labor w borne by the employer with income net R (w) of the household of the person concerned. The function R depends on the spouse’s salary, social contributions … It also depends implicitly on the characteristics of the household: number and age of children, type of housing … We suppose that R is continuously differentiable, strictly increasing and concave in w. The utility functions of both parties are such that a paid job w earns ρ – w for the employer and R (w) – D for the employee. In the absence of a wage agreement, the worker receives a utility given by R (0) = 0 and the company receives zero profit. We assume that the minimum wage is w such that
w ∈ [0, ρ] , ρ > 0
R(ρ) − D > 0
Interpret the following assumptions:

w ∈ [0, ρ] , ρ > 0
R(ρ) − D > 0

2. Model the situation using a negotiation game (we will determine U such that d ∈ U). Determine the equation for the Pareto frontier.
3. Let us fix the following parameters, D = 1, ρ = 5 and R (w) = . Determine the salary to the Nash solution.
4. Determine the salary for the Kalai Smorodinski solution.

Exercise 6
We consider the following TU (transferable utility) game: N = {1, 2, 3, 4}

Show that the core is the set {(γ, −γ, γ, −γ): −1 ≤ γ ≤ 1}.

Exercise 7:

1. Find an example of a three-player add-on game with an empty core.
2. Find an example of a monotonous three-player game with an empty core.

Exercise 8:

A criminal, referred to as 1, is studying the possibility of stealing an amount M> 0 euros. If he commits the offense, the individual knows that he will be arrested by a police officer, referred to as 2, with a probability p ∈ [0.1]. This probability reflects the statistics established by the courts concerning thefts of the same type. The policeman is corruptible and negotiates a bribe L ∈ [0, M] (a share of the stolen loot) that the criminal gives him in exchange for his silence. If no agreement is reached then the police officer makes his report concerning the theft and the thief must pay a fine proportional to the amount of the theft at the legal rate γ ∈ (0, 1] set by the court. It is assumed that the utility of each player is the amount of money they hold.

1. Model this situation by a two-player negotiation game (U, d) (U will be the convex envelope of all the points associated with the possible agreements).
2. Determine Nash’s solution.
3. Is the fine a deterrent? How to modify the penal system so that the fine is dissuasive?

Exercise 9:

Consider a simple game with 7 players such that v (S) = 1 if
S ∈
and v (S) = 0 for all other coalitions. Show that this game cannot be represented by a weighted majority game.

Exercise 10:

Consider a set of sellers: N = {1, 2, 3, 4, 5}. We denote by Ei = (gi, ti) the initial endowment of player i: g represents the amount of gin and t the amount of tonic they initially have. We assume that Ei = (0.1 / 2) if i = 1.2 and Ei = (1.0) if i = 3,4,5. They can form coalitions and pool their resources. Thus a coalition S is a subset of N and the resources possessed by the coalition S is equal to where and i: is the total quantity of gin owned by coalition S and is the total amount of tonic owned by coalition S. Consumers want to buy only cocktails that contain equal parts of gin and tonic. The net profit from selling α units of gin and tonic is α dollars. Describe this situation as a coalition game and write in detail the coalition function v.