Applications of TU Games Vito Fragnelli University of Eastern Piedmont Politecnico di Torino 24 April 2002 Outline Cost Allocation Problems Infrastructure Cost Games Short Survey

Cost Allocation Problems Alan and Bob want to rent a car they can rent the car for one day each (Saturday for Alan at 70 €, Sunday for Bob at 40 €) or they can rent the car for the weekend (at 90 €) Allocate among the agents the total cost of the joint project The equal sharing (45 € each) will be rejected by Bob Allocation should be fair Several agents have to carry out a common project they can perform it separately or they can do it jointly (if the total cost decreases)

Cost Allocation Problems It is a pair  N, c , where N = {1,..., n} set of agents c cost function c (S ) is the cost of the project if it is jointly performed by the agents in S, satisfying their necessities The agents not included in S cannot benefit from this project, so that they could have to carry out their own separate ones N = {Alan, Bob} c (Alan) = 70, c (Bob) = 40, c (Alan, Bob) = 90 The solution is a vector x in R n ; x should be efficient :  i  N x i = c (N )

Cost Allocation Problems The Tennessee Valley Authority Problem (’30) provide water for energy control flooding improve navigation Some interventions (dams and reservoirs) were planned in the Tennessee River valley in order to encourage the economy: Stand-alone cost test no subset of agents S should be charged more than the cost of an alternative project specialized for S  i  S x i  c (S ) Incremental cost test no subset of agents S should be charged less than the additional cost including them in the project  i  S x i  c (N ) - c (N \ S ) The fair allocations should satisfy: By efficiency the two properties are equivalent

Cost Allocation Problems The Tennessee Valley Authority Problem (’30) Does there exists a vector that satisfies the previous conditions? How do select a single vector? Point solutions: Separable costs methods ECA: Equal Charge Allocation ACA: Alternate Costs Avoided CGA: Cost Gap Allocation (  -value)

Cost Allocation Problems Separable costs methods Separable cost of player i (marginal cost) m i = c (N ) - c (N \i ) Non separable cost g(N ) = c (N ) -  i  N m i

Cost Allocation Problems Cost Allocations Problems and Game Theory A cost allocation problem  N, c  induces a cooperative cost game  N, c  The cost game  N, c  has non-empty core? How do select a single core allocation?

Infrastructure Cost Games Some users need an infrastructure They are grouped according to different requests, corresponding to increasing levels of the infrastructure or of the facilities that build up the infrastructure, as g 1,..., g k An infrastructure has costs independent from the number of users (Building costs) and costs depending on the number of users (Maintenance costs) Infrastructure cost game = Building cost game + Maintenance cost game The Shapley Value is additive

Infrastructure Cost Games It is precisely an Airport Game (Littlechild – Thompson, 1973): assign to a subset of planes the cost of the larger strip required c (S ) = C j(S) where j (S ) = max { j | S  g j   } Building Cost Game c1c1 c2c2 c3c3 Sh(1) = c 1 /4 Sh(2) = Sh(3) = c 1 /4 + c 2 /3 Sh(4) = c 1 /4 + c 2 /3 + c 3 N = g 1  g 2  g 3 ; g 1 = {1}; g 2 = {2, 3}; g 3 = {4};

Infrastructure Cost Games If a player of kind i uses (and damages) a facility of level j he has to refund an amount A ij =  k=i,j  ij, where  ii is the cost for a player of kind i to repair the facility to the level i and  ik (k > i ) is the cost for a player of kind i to repair the facility from the level k - 1 to the level k Maintenance Cost Game

Infrastructure Cost Games Maintenance Cost Game  11  12  13 2  22 2  23  33 Sh(1) =  /4  /2  13 Sh(2) = Sh(3) =  /2  /4 1/3  12 Sh(4) =  /4 1/3  /2  /2 2  23 N = {1, 2, 3, 4}

Other Applications Short survey Voting games (power indices) Bankruptcy games Operations Research games Production & Linear Transformation games Production Network games - Flow gamesFlow games - Connection games (Spanning tree games, Forest games, Extension games, Fixed tree games)Spanning tree games - Shortest Path games Sequencing, Assignment & Permutation games Sequencing Inventory games Location games Transportation games

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Bankruptcy Games A set of agents N = {1,..., n} has claims d = (d 1,..., d n ) over an estate E with:  i=1,n d i > E A feasible division x has to satisfy: 0  x i  d i  i=1,n d i = E It is possible to define a (pessimistic) game  N, v  whose core coincides with the set of feasible divisions: v (S ) = max {0, E -  i  S d i } back

Linear Production Games A set of agents N = {1,..., n} has a set of resources b = (b 1,..., b n ); they can produce a set of m goods according to a (fixed) technology represented by the matrix A; each unit of good j produce an income c j, j = 1,..., m A group of agents S  N can use their resources b S =  i  S b i to maximize their income It is possible to define a TU game as: v (S ) = max {c T x s.t. Ax  b S ; x  0 } back

Flow Games A set of agents N = {1,..., n} operates on a transportation network; each agent control a set of arcs A group of agents can use only the arcs they control in order to maximize the flow from the source s to the sink t back N = {1, 2, 3} st v (1) = 0; v (2) = 1; v (3) = 0 v (1, 2) = 3; v (1, 3) = 0; v (2, 3) = 2; v (1, 2, 3) = 4

Minimum Cost Spanning Tree Games A set of agents N = {1,..., n} wants to be connected to a common source {0} building suitable connections at minimum cost back N = {1, 2, 3} Monotonic game v (1) = 4; v (2) = 1; v (3) = 3 v (1, 2) = 4; v (1, 3) = 6; v (2, 3) = 3; v (1, 2, 3) = 6 Non monotonic game v (1) = 5; v (2) = 1; v (3) = 4 v (1, 2) = 4; v (1, 3) = 8; v (2, 3) = 3; v (1, 2, 3) = 6

Sequencing Games A set of agents N = {1,..., n} waits for a service; each agent i has a fixed service time t i at a fixed cost per unit of time c i Adjacent agents can reorder themselves for improving their global service time back N = {I, II, III} t = (5, 3, 4) c = (5, 9, 8) u = (1, 3, 2)  optimal order (Smith): II, III, I v (I) = v (II) = v (III) = 0 v (I, II) = 30 v (I, III) = 0 v (II, III) = 0 v (I, II, III) = 50 A single agent cannot switch = (5x9 – 3x5) + = (3x8 – 9x4) +  :Agent II prefers the switch = (5x9 – 3x5) + + (5x8 – 4x5) + The two agents are not adjacent