Conditions for the uniqueness of the Gately point for cooperative games
aa r X i v : . [ ec on . T H ] J a n Conditions for the uniqueness of the Gately pointfor cooperative games
Jochen Staudacher (cid:12) , Johannes AnwanderFakult¨at InformatikHochschule Kempten87435 Kempten, GermanyEmail (Corresponding Author): [email protected] 8, 2019
Abstract
We are studying the Gately point, an established solution concept for co-operative games. We point out that there are superadditive games for whichthe Gately point is not unique, i.e. in general the concept is rather set-valuedthan an actual point. We derive conditions under which the Gately point isguaranteed to be a unique imputation and provide a geometric interpreta-tion. The Gately point can be understood as the intersection of a line de-fined by two points with the set of imputations. Our uniqueness conditionsguarantee that these two points do not coincide. We provide demonstrativeinterpretations for negative propensities to disrupt. We briefly show thatour uniqueness conditions for the Gately point include quasibalanced gamesand discuss the relation of the Gately point to the τ -value in this context.Finally, we point out relations to cost games and the ACA method andend upon a few remarks on the implementation of the Gately point and anupcoming software package for cooperative game theory. JEL-classification:
C71
Keywords:
TU games; solution concept; quasibalanced games; utopia payoff;cost games; ACA method
Dermot Gately introduced a new solution concept for cooperative games withtransferable utility in Gately (1974) based on minimizing the temptation to leave1he grand coalition for individual players. In the original paper Gately (1974)the problem of sharing the gains from a joint investment in an electric powergrid in India between the participating regions is resolved with the help of theconcept “equal propensity to disrupt”. Since the publication of Gately (1974), theso-called Gately point has become a well-established solution concept taught inbooks by Straffin (1996) and Narahari (2014) and mentioned in highly regardedsurvey articles, like e.g. Sandler and Tschirhart (1980) and Young (1994).As of 6 January 2019, 211 quotes of Gately (1974) can be found on GoogleScholar.From its name Gately point one is tempted to assume that the solution concept inquestion was always unique. In this paper we point out that this is not actually thecase. We strive to answer the following question: Under which conditions is theGately point a unique imputation? Along the way, we also discuss what negativepropensities to disrupt tell us about a cooperative game.
We are studying a transferable utility game (TU game) in characteristic functionform consisting of the player set N = { , . . . , n } and the characteristic function v : 2 N → R with v ( ∅ ) = 0. We are using the shorthand notations v i = v ( { i } ) for i = 1 , . . . , n, for the worths of the singleton coalitions. Definition 1. (see Branzei et al (2008), p. 20) The so-called utopia payoff ofplayer i is given by M i = v ( N ) − v ( N \{ i } ) for i = 1 , . . . , n, i.e. M i is the marginal contribution of player i to the grand coalition. In this article we will only study games satisfying essentiality in the sense ofChakravarty et al (2015), p. 23.
Definition 2. (see Chakravarty et al (2015), p. 23) We call a transferable utilitygame with player set N = { , . . . , n } and characteristic function v : 2 N → R essential if n X j =1 v j < v ( N ) . (1)The imputation set of any essential TU game is guaranteed to consist of more thana single point. For a solution concept in cooperative game theory one would nor-mally prefer the solution vector x ∈ R n to be an imputation, i.e. both individually2ational x i ≥ v i for all i = 1 , . . . , n and efficient P nj =1 x j = v ( N ). For a formaldefintion of the imputation set we refer to Peleg and Sudh¨olter (2007), p. 20, orNarahari (2014), p. 407.Note that any cooperative game satisfying (1) is strategically equivalent to a 0-1-normalized game, see Maschler et al (2013), p. 670, or Chakravarty et al (2015),p. 24. Definition 3. (see e.g. Peleg and Sudh¨olter (2007), p. 10) We call a transferableutility game with player set N = { , . . . , n } and characteristic function v : 2 N → R weakly superadditive if v ( S ∪ { i } ) ≥ v ( S ) + v i for all S ⊆ N and i / ∈ S. (2)Note that weak superadditivity (2) guarantees v i ≤ M i for i = 1 , . . . , n. (3)For later convenience we repeat the following Definition 4. (see e.g. Straffin (1996), p. 131, or Narahari (2014), p. 408) Wecall a transferable utility game with player set N = { , . . . , n } and characteristicfunction v : 2 N → R superadditive if v ( S ∪ T ) ≥ v ( S ) + v ( T ) for all S, T ⊆ N with S ∩ T = ∅ . (4)Finally, we would like to introduce the following game property. Definition 5.
We call a transferable utility game with player set N = { , . . . , n } and characteristic function v : 2 N → R weakly constant-sum if v i + v ( N \{ i } ) = v ( N ) for all i = 1 , . . . , n. (5)Note that weakly constant-sum games v can equivalently be characterized by v i = M i for i = 1 , . . . , n. (6) In this section we will introduce the Gately point as a solution concept for coop-erative games along the lines of the article by Littlechild and Vaidya (1976).The following definition is central to understanding the Gately point as a solutionconcept for cooperative games. 3 efinition 6. (see Littlechild and Vaidya (1976), p. 152) For a given transferableutility game with player set N = { , . . . , n } and characteristic function v : 2 N → R the expression d ( i, x ) = v ( N ) − v ( N \{ i } ) − x i x i − v i = M i − x i x i − v i (7) quantifies the propensity to disrupt of player i for a payoff vector x ∈ R n in theinterior of the imputation set, i.e. P nj =1 x j = v ( N ) with x i > v i for all i = 1 , . . . , n . Expression (7) quantifies the disruption caused if player i breaks away from thegrand coalition. Within (7) the denominator stands for the loss incurred by player i for breaking away from the grand coalition, whereas the numerator stands forthe joint loss of the rest of the players due to the breakup caused by player i .The original approach in Gately (1974) for three-person games was generalized to n -person games by Littlechild and Vaidya (1976), p. 152. The idea is simply tofind an imputation x ∈ R n with minimal propensity to disrupt. It can be shownthat this minimal propensity to disrupt can be found by equating the propensityto disrupt over all players, i.e. d ( i, x ) = d ∗ for i = 1 , . . . , n. As pointed out by Littlechild and Vaidya (1976), p. 153, using (7) one can easilyfind the following closed-form expression d ∗ = ( n − v ( N ) − P nj =1 v ( N \{ j } ) v ( N ) − P nj =1 v j = P nj =1 M j − v ( N ) v ( N ) − P nj =1 v j (8)which also highlights the fact that the Gately point is a solution concept dependingsolely on the values of the coalitions of sizes 1, n − n .Looking at (7) one recognizes that for d ∗ = − Theorem 1.
For an essential transferable utility game with player set N = { , . . . , n } and characteristic function v : 2 N → R the Gately point is well-defined unless theequal propensity to disrupt d ∗ = − . We can find the Gately point as the uniqueimputation x ∈ R n with the components x i = v i + ( v ( N ) − n X j =1 v j ) M i − v i P nj =1 M j − P nj =1 v j (9) for i = 1 , . . . , n , if one of the following two conditions holds:a) For games satisfying (3) there needs to hold v i < M i for at least one i ∈ { , . . . , n } , (10)4 .e. (3) is satisfied with strict inequality for at least one i ∈ { , . . . , n } .b) We also obtain the Gately point x ∈ R n as a unique imputation if v i ≥ M i for i = 1 , . . . , n, (11) as long as (11) is satisfied with strict inequality for at least one i ∈ { , . . . , n } ,i.e. as long as the game is not weakly constant-sum (6). Proof: As long as d ∗ = − d ∗ = − x i = v i for those i ∈ { , . . . , n } with v i = M i . Looking at the expression (9), essentiality (1) implies that x ∈ R n is animputation if and only if M i − v i P nj =1 M j − P nj =1 v j ≥ i ∈ { , . . . , n } . The latter condition is fulfilled for both games satisfying (3)and games satisfying (11) as long as these games are not weakly constant-sum (6). Remark 1.
The case d ∗ < can be interpretated as enthusiasm of each playernot to be the one left out of the grand coalition. In other words: d ∗ < indicatesthat coalitions of size n − are preferred over the grand coalition. In the case of(11) being satisfied with strict inequality for at least one i ∈ { , . . . , n } this fact isparticularly striking as there even holds d ∗ < − . Remark 2.
Geometrically, (9) allows us to interpret the Gately point as the in-tersection of the imputation set with the half-line drawn from the point ( v , . . . , v n ) with directional vector ( M − v , . . . , M n − v n ) . Remark 3.
For -normalized games (9) simplifies to x i = v ( N ) M i P nj =1 M j (12) for i = 1 , . . . , n . We finally consider
Example 1.
Let the three-person game v be given by v = 3 , v = 4 , v = 5 , v ( { , } ) = 9 , v ( { , } ) = 10 , v ( { , } ) = 11 , v ( N ) = 14 . The above game is clearly superadditive (4) and essential (1), but the propen-sity to disrupt equals − x . In a sense, the Gately point for v would be the complete imputation set. Naturally, one would make the identicalobservation considering the 0-normalization of v , i.e. the coalitional game w with w = w = w = 0 , w ( { , } ) = w ( { , } ) = w ( { , } ) = w ( N ) = 2, or the 0-1-normalization of v , i.e. the coalitional game u with u = u = u = 0 , u ( { , } ) = u ( { , } ) = u ( { , } ) = u ( N ) = 1. Note that the latter could also be interpretedas a weighted voting game. 5 Relations to the τ -value In the previous section we have seen that the Gately point is the intersection ofthe imputation set with a line connecting the points ( v , . . . , v n ) and ( M , . . . , M n )and pointed out a problem for the case that these two points coincide (6). Thereis another well-established solution concept in cooperative game theory computingthe intersection of a line connecting two points with the imputation set, i.e. the τ -value proposed by Tijs (1981). Definition 7. (see Branzei et al (2008), p. 20) The remainder R ( S, i ) of player i in coalition S is the amount which remains for player i if coalition S forms andthe rest of the players in coalition S all obtain their individual utopia payoffs, i.e. R ( S, i ) = v ( S ) − X j ∈ S,j = i M j . We can define a vector of minimal rights with components m i = max S : i ∈ S R ( S, i ) , for i = 1 , . . . , n, since player i has a justification to ask at least m i in the grand coalition. The τ -value is defined only for quasibalanced games. Definition 8. (see e.g. Branzei et al (2008), pp. 31) We call a transferable utilitygame with player set N = { , . . . , n } and characteristic function v : 2 N → R quasibalanced if m i ≤ M i for all i ∈ { , . . . , n } (13) and n X j =1 m j ≤ v ( N ) ≤ n X j =1 M j . (14) For a quasibalanced game v the τ -value is defined as the intersection of the impu-tation set with the line from the minimal rights vector ( m , . . . , m n ) to the utopiapayoff vector ( M , . . . , M n ) . Remark 4. (see e.g. Branzei et al (2008), p. 32) We can find the τ -value withthe components τ i = αm i + (1 − α ) M i , where α ∈ [0 , is uniquely determined by the condition P ni =1 τ i = v ( N ) . Combining (8) and condition (14) we find that for quasibalanced games d ∗ ≥ orollary 1. The Gately point is always unique for quasibalanced games.
Note that the conditions we formulated for the Gately point to be a unique im-putation are more general than quasibalancedness, i.e. there are games for whichthe τ -value is not defined whereas the Gately point is. Consider Example 2.
Let the three-person game v be given by v = 3 , v = 4 , v = 5 , v ( { , } ) = 9 , v ( { , } ) = 10 , v ( { , } ) = 11 , v ( N ) = 14 . . The game v is not quasibalanced and its Gately point can be computed to x = 3 , x = 4 , x = 5 .We finally observe that the problem we report for the Gately point never occurs forthe τ -value which we already mentioned to be the intersection of the imputationset with a line drawn from the point ( m , . . . , m n ) to the point ( M , . . . , M n ), seeTijs (1981). However, if these two points coincide, then (14) guarantees this pointto be an imputation and thus the τ -value of the game. Note that in this specialcase there is d ∗ = 0. We are looking at cost games in characteristic function form consisting of the set N = { , . . . , n } of agents (or purposes, projects or services) and the characteristicfunction c : 2 N → R with c ( ∅ ) = 0. We are using the shorthand notation c i = c ( { i } ) for i = 1 , . . . , n, for the costs of single agents. The connection to TU games is given by the associ-ated savings game v for N = { , . . . , n } defined by v ( S ) = X i ∈ S c i − c ( S )for every coalition S . Note that the associated savings game v is automatically0-normalized.We are now discussing the so-called ACA (Alternate Cost Avoided) method, i.e. anestablished method for cost allocation going back to Ransmeier (1942), along thelines of Straffin and Heaney (1981). The ACA method has been widely discussed,see also Otten (1993), Tijs and Driessen (1996) and Young (1994).The ACA method is based on the concept of allocating separable costs SC i = c ( N ) − c ( N \{ i } ) = c i − M i i = 1 , . . . , n . The remaining nonseparable costs N SC = c ( N ) − n X j =1 SC j = n X j =1 M j − v ( N ) (15)are assigned in proportion to c i − SC i , i.e. the final cost allocation for an individualagent i is y i = SC i + c i − SC i P nj =1 c j − SC j N SC.
As pointed out in Straffin and Heaney (1981), p. 40, the corresponding savingsallocation x ∈ R n is exactly the Gately point, i.e. x i = c i − y i = v ( N ) M i P nj =1 M j as seen in (12).It is very natural to understand why the problem of nonuniqueness of ACA nevercame up in the context of cost games. In practice, only subadditive cost gamesare studied, i.e. the corresponding savings game is superadditive (4), see Young(1994), p. 1197. The ACA method can only fail to deliver a unique cost allocationif M i = 0, or equivalently c i = SC i , for i = 1 , . . . , n . Then (1) implies N SC <
N SC < d ∗ <
0, see(8) and (15). We finally consider
Example 3.
Let the subadditive three-agent cost game c be given by c = 7 , c = 8 , c = 9 , c ( { , } ) = 14 , c ( { , } ) = 15 , c ( { , } ) = 16 , c ( N ) = 23 . The corresponding savings game u is the weighted voting game u = u = u =0 , u ( { , } ) = u ( { , } ) = u ( { , } ) = u ( N ) = 1 we already know from Example 1.The Gately point does not exist and so ACA fails to deliver a unique cost allocation.In general, ACA can only run into problems if all coalitions of size n − n − i to be left out. The main purpose of this article is to answer the question when it is at all sensibleto compute the Gately point of a TU game v . We derived very general condi-tions for the Gately point to be a unique imputation and pointed out why weakly8onstant-sum games lead to problems. We feel that our analysis underlines thecriticism of the Gately point made in Littlechild and Vaidya (1976), p. 153, thatthe solution concept only makes use of the values of the coalitions of sizes 1, n − n and completely ignores the rest of the information contained in the coalitionfunction v .The nonuniqueness of the Gately point was first discussed in Anwander (2017) andit was discovered during efforts to implement the Gately point in R. The authorsare currently finalizing an R-package named CoopGame (see Staudacher and Anwander(2019)) which the authors hope to make publicly available via CRAN, the Compre-hensive R Archive Network. Among various other solution concepts, the packageCoopGame will not only provide an implementation of the Gately point, but alsoprovide the user with possibilities to compute the equal propensity to disrupt d ∗ of a given cooperative game v . The scope of our Gately point implementation isslightly broader as for an inessential game v in the sense of Narahari (2014), p. 408,i.e. if (4) holds with P nj =1 v j = v ( N ), our code will simply return ( v , . . . , v n ).Otherwise, we make sure to check the conditions derived in this paper before thecomputation of the Gately point and to return a meaningful message in the casethe user specifies a TU game v with an equal propensity to disrupt d ∗ = − Bibliography
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