Example 1: Key player but not alone

This example aims to describe the simple use of the package for a 3-user case. The major reward allocation techniques supported by the package are considered, using the enumerative technique.

The sample game considered in this example has the following characteristics:

• there are 3 players: 1, 2 and 3
• each user can join or not the coalition
• the utility value is:
  • always 0 if player 1 does not join the coalition
  • +1 if player 1 joins the coalition but it is not alone

In the following, the game is constructed.

First, the packages are imported

using TheoryOfGames
using JuMP, Ipopt

Initialization of the game

Define the set of the users that can join or not the coalition

player_set = [1, 2, 3]

3-element Vector{Int64}:
 1
 2
 3

Define the utility function that is a function that:

• takes as input an iterable that represents a coalition
• it returns a value in agreement to what discussed above and here summarized:
  • it returns +1 if player 1 joins the coalition but it is not alone $(1 in x) && length(x)$
  • it returns 0 if player 1 does not join the coalition (otherwise) `else condition

utility = x->((1 in x) && (length(x) > 1)) ? 1.0 : 0.0

#1 (generic function with 1 method)

Show the value of utility for the coalition where only user 1 joins

utility([1])

0.0

Show the value if also user 2 joins

utility([1, 2])

1.0

Calculation of selected reward allocation functions

Define the Enumerative object

enum_obj = EnumMode(player_set, utility)

EnumMode([1, 2, 3], Dict{Set{Int64}, Float64}(Set([3]) => 0.0, Set([1]) => 0.0, Set([2, 3, 1]) => 1.0, Set([2]) => 0.0, Set([3, 1]) => 1.0, Set([2, 3]) => 0.0, Set() => 0.0, Set([2, 1]) => 1.0))

Calculate shapley value

shapley_value(enum_obj)

Dict{Int64, Float64} with 3 entries:
  2 => 0.166667
  3 => 0.166667
  1 => 0.666667

Define the optimizer needed for nucleolus and var least core techniques. We use the default Ipopt and disable the output for simplicity ($print_level=>0$).

OPTIMIZER = optimizer_with_attributes(Ipopt.Optimizer, "print_level" => 0)

MathOptInterface.OptimizerWithAttributes(Ipopt.Optimizer, Pair{MathOptInterface.AbstractOptimizerAttribute, Any}[MathOptInterface.RawOptimizerAttribute("print_level") => 0])

Calculate nucleolus

nucleolus(enum_obj, OPTIMIZER)

Dict{Int64, Float64} with 3 entries:
  2 => 6.71073e-10
  3 => 4.21514e-10
  1 => 1.0

Calculate var least core

var_least_core(enum_obj, OPTIMIZER)

Dict{Int64, Float64} with 3 entries:
  2 => 1.26778e-8
  3 => 1.26521e-8
  1 => 1.0

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