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On economic games which are not necessarily superadditive
Economics Letters 3 (1979) 301-306 0 North-Holland Publishing Company
ON ECONOMIC GAMES WHICH ARE NOT NECESSARILY SUPERADDITIVE Solution Concepts and Application to a Local Public Good Problem with Few Agents R. GUESNERIE CEPREMAP
Received
and C. ODDOU
and GRASCE,
31 July
France
1979
Core-like solution concepts based on stable coalition structures are studied in the context of games which are not necessarily superadditive. It is shown that a simple economic model involving local public goods financed by wealth taxes has this characteristic.
1. Introduction The modelling of economic situations in a game theoretical framework has resulted most of the time in considering superadditive games. Away the best known in the economic literature let us quote the market games associated with an exchange economy, the games associated with superadditive coalition production or with public good economies in a first best context. We present in this brief note, an example of an economic game, which is not necessarily superadditive, associated with a second best problem in the theory of local public goods. The examination of various second best problems has convinced us that the absence of superadditivity in the associated economic games was not at all pathological. Non-superadditive games have some unfamiliar characteristics, an appropriate reconsideration of standard concepts of efficiency and stability is required (see section 2). The application of these modified concepts will prove fruitful for the local public good game with few agents (see section 3).
2. Efficiency
and stability in games which are not necessarily superadditive
Through the concepts defined in this section, we want mainly to take into account two facts: - Efficiency may require that the grand coalition N breaks down. - Stability requirements which have the same strength as those involved in the core 301
302
R. Guesnerie, C. Oddou /Economic games which are not necessarily superadditive
concept, may be obtained for some particularly when the core stricto-sensu is empty.
appropriate
grouping of agents even
Let us consider then a game in characteristic form: N denotes the set { 1, 2, . . .. m],Z denotes the set of all non-empty subsets of N and u denotes a function from Z to, non-empty, compact subsets of @ that satisfies: for all SE 2, if CY E u(J’) and oS = pS, then /3E v(s). ’ In the standard interpretation, the elements of N are players, the elements of Z coaZitionsand the elements of IRN payoff or utility vectors. The elements of u(S) represent feasible outcomes for the coalition S and we denote D(S) the set of outcomes that S can improve upon, i.e., the set of elements u E IRN such that there is a vector u E u(S) for which us << us [see Shapley (1972)]. Definitions. A structure of coalitions is a partition II = (,S&= of the set of agents. The set of partitions of N is denoted TO. For any II = (Sk)ka, II E T(N) we denote u(II) = f?,, u(Sk). An efficient outcome of the game is a vector ii E w
such that
(a) there is Il E T(N) for which U E u(II), (b) there is no II’ and U’ E u(II’) such that U’ >> U. An efficient structure is a structure II such that there exists an efficient outcome U E u(n). A structure II is universaZZy efficient if any efficient outcome U E u(II). Game theoretical concepts of stability can now be introduced. The Core C(u) is the set of U* E u(N) which can be improved upon by no coalition - C(u) = u(N) \ “,,z D(J) -. A Core like stable solution or C-stable solution is a vector u* E Km which satisfies the following properties: - there is a II E TQ such that U* E u(II), -forallSEZ,u* $08. The reader will immediately check the following properties: (i) a C-stable solution is an efficient outcome, (ii) the core is a set of particular C-stable solutions, (iii) if N is universally efficient, any C-stable solution is necessarily in the core, (iv) II = {N} is a stable structure if and only if the core is non-empty. For games which are not superadditive, {N} is not necessarily universally effi-
’ Let lRN denote the n-dimensional euclidean space with coordinates indexed by the elements let IRS denote the corresponding ISI-dimensional subspace of IRN. If ofN,andforSE’. then as will denote the projection of LYon IRS, i.e., the restriction of OL oEIRNandSEZ, to the coordinates indexed by the elements of S.
R. Guesnerie, C. Oddou /Economic
games which are not necessarily superadditive
303
cient. In this case the concept of C-stable solution reflects the same ideas of stability as those underlying the core concept and appears as a natural generaliza. tion. To some extent, as made clear below, it is merely an adaptation of the core concept. ’ Let us define ?, the smallest superadditive game associated with u,
u
Q) =
rIET(S)
u
TErl
Gwill be intermediate u”(s) = 49
for all
G?. between u and gand it is defined as follows: G(N) = $(m, S #N.
Then Lemma 1. u* is a C-stable solution of v if and only if it belongs to the core of g and if and only if it belongs to the core of 6 Take U* a strongly stable solution, there is a II E T(N) such that U* E u(II) Boot so u* E 6(/V) = z(N), if u* does not belong to the core of v”(respectively g) there is SE Z and u E $S) (respectively u E E(S)) such that U’ >> Use. For i?, if S # N, U* E 08, we obtain a contradiction. If S = N, or for G, there is a II E TQ such that U E nTEn u(T) and for some T belonging to II, U* ED(T), which is impossible. On the other hand, if U* E C@) (respectively U* E C(G)), U* E i$V) = +V), so there is a II E T(N) such that U* E u(n). If there is an S E Z such that U* ED(J), then there is an U E u(S) with $ >> u*s. u E u(S) c G(s) c G(S), so S can improve upon U* and we obtain a contradiction. Q.E.D.
3. A local public good game which is not necessarily superadditive Consider the following simple economy: there are two goods, the first is a private good, the second is a pure public good. There are m consumers indexed by i = 1, . .. . m having preferences represented by a utility function Ui defined on @ (Ui(0,0) = 0). Moreover, consumers are initially endowed with oi (q > 0) units of the private good. Preferences and initial endowments being given, we define the rules of the game as follows: each coalition S which is a subset of N = (1, .... m) can use the same simple constant returns to scale technology which transforms one unit of private good into unit of public good. The public good can be financed only through
’ In particular, we discovered that a similar concept was used by Shaked (1978) who called it simply ‘core’. Even if Lemma 1 gives some justification for this usage, it seems to us that it would be a source of confusion not to distinguish core from stable solutions, at least in the context of the second best games we are looking at.
304
R. Guesnerie, C. Oddou /Economic
games which are not necessarily superadditive
a wealth tax: each coalition S can choose a rate of tax r E [0, 11, which applies to all members of the coalition. When the tax rate chosen is t, the tax revenue raised is t ZZiEs Wi and the amount of public good produced is q(t, 5) = t Tics Wi. Given these assumptions on the technology of production and on the institutional constraints for the financing of the public good, one can associate the problem of the allocation of the public good in this simple economy with a game, whose characteristic function is u(J), defined as follows: u(L!q= {uE@JtE
[o,l],U%u(t,ls)},
where u(t, s> is the vector of IR’.whose components Ui(t,s)=Ui{(l
-t)Wi*q(t,S)},
are
I’ES.
Fig. 1 shows that in this case, as stated in the introduction, superadditivity does not necessarily hold. Two individuals have the same initial endowments. When they are isolated, the agent 1 obtains UI and the agent 2 obtains & . When they are together the attainable states are associated with the segment AC and it is clear that (a1 , ii*) is not realisable. 3 4. Stable solutions and stable structures number of agents
in a public good economy with a small
In this section preferences are only assumed to be convex and initial endowments of private goods can be distributed among agents in any way. 3 The assumption concerning the financing of the public good is here crucial. If coalitions are able to raise personal taxes and to relate exactly marginal willingnesses to pay and individual contributions, it is well known (under standard assumptions) that the core is non-empty and even large. The use of a wealth tax leads us to a second best situation. Our model has some connection with the theory of local public goods initiated by Tiebout (1956). In particular, our general assumptions are similar to those of Westhof (1975) who considers the organization of communities producing a pure public good.
R. Guesnerie, C. Oddou /Economic
games which are not necessarily superadditive
305
The analysis is based on the application of Scarfs theorem according to which any balanced game has a non-empty core. We show that when the number of agents is small (m < 4) the systematic use of Scarfs sufficient condition allows us to make conclusive statements. Precisely we have When the number of agents is smaller or equal to three, m < 3, the Proposition 1. game v has a C-stable solution. Proof: The statement is trivial for m = 1, and straightforward for m = 2. However, it is not at all obvious for m = 3. We show in this case that the game u”is balanced in the sense of Scarf. The only minimal balanced family for which the property is not trivially satisfied is T = ({AB}, {BC}, {CA }) and it is enough to prove that if (U,I, us) E u(CABI), (U,, UC) E u({BC)), (UC, us) E u(ICAI) then (u_4, UB, UC> {ABC}). 4 For that, consider A(A) (respectively A(B), A(C)) the interval of public good q, which, if implemented in the grand coalition, would give more than U, to A (respectively UB to B, UC to C). Taken 2 by 2, these intervals have a non-empty intersection [for example A(A) fl A(B) > qAB, where qAB is the public good level in AB]. Then A(A) f~ A(B) fl A(C) # 4 (this is Helly’s theorem in dimension 1). Q.E.D. [Cf. Berge (1966, p. 173).] E v(
From a previous remark, one can straightforwardly Corollary.
assert
For m < 3, if v is superadditive, then it has a non-empty core.
We can actually prove that a similar result holds for m = 4 by using the same techniques. Proposition 2. empty core.
Form = 4, if v is superadditive, it is Scarf balanced and has a non-
Proof: According to Shapley (1967) the minimal balanced families of a 4 players game are, apart from the partitions, the following families: ~~ =({123], rz = ({12],
11241,134]), c131,c231,
{41),
~~=({123},{14},{24},{3}), T4
=
((1
TS =({I
2
31, Cl 41, 12 41, 13 411,
231,
11 241,
and all families obtained
Cl 341,
{2341),
by permutation.
The game being superadditive
4 If this holds for u, it is easy to check that it also holds for iT
we can
306
R. Guesnerie, C. Oddou /Economic
games which are not necessarily superadditive
prove a theorem similar to theorem 3 of Shapley (1967). It is thus sufficient to check that the Scarf condition is verified for proper minimal balanced families. For such families, every couple of agents are part of (at least) one coalition. It is only a matter of routine to check that the previous argument (on the intersection of intervals) still applies. Corollary. If m = 4, if all agents have the same resources, there is a C-stable solution and hence a stable structure. The argument is here more complicated and will only be briefly sketched: Using the proof of Proposition 2, we notice that it is enough to show that Scarfs condition is verified for the families T, and T3 (relative to G). (ii) Using Proposition 1, we prove the property for T2. (iii) Exhibiting the public good levels (associated with the utility vector under consideration) in the three first coalitions of T3, we look at 4 their median level (which may be either 4 1z 3, q, 4 or q2 4). Discussing case by taking into account the equality of endowments, we can conclude. (i)
We have considered in a simple model the existence problem for the core and its most obvious generalization when the game is not superadditive the C-stable solution. We attacked existence problems with a method which looks natural, trying to prove that the game, or its superadditive extension, is Scarf balanced. We obtained significantly positive results for m < 4. One would expect that the approach remains fruitful when a greater number of agents is involved. One might hope for example that Proposition 2 holds for any m. We were not able to prove that and it is unlikely to be true for m > 6 (although we have no counterexample). However, we prove in a forthcoming paper, using quite different techniques, that whatever m superadditive local public goods games have non-empty core, although as we just argued they might not be Scarf balanced.
References Berge, C., 1959, Espaces topologiques: Fonctions multivoques (Dunod, Paris). Shaked, A., 1978, Human environment as a local public good (London School of Economics, London). Shapley, L.S., 1967, On balanced sets and cores, Nav. Res. Long. R. 14,453-460. Shapley, L.S., 1972, On balanced games without side payments (The Rand Corporation, Santa Monica, CA) 4910. Tiebout, M., 1956, A pure theory of local expenditures, Journal of Political Economy 64,416424. Westhof, F., 1975, Existence of equilibria in economics with a local public good, Journal of Economic Theory 14, 84-l 12.
ON ECONOMIC GAMES WHICH ARE NOT NECESSARILY SUPERADDITIVE Solution Concepts and Application to a Local Public Good Problem with Few Agents R. GUESNERIE CEPREMAP
Received
and C. ODDOU
and GRASCE,
31 July
France
1979
Core-like solution concepts based on stable coalition structures are studied in the context of games which are not necessarily superadditive. It is shown that a simple economic model involving local public goods financed by wealth taxes has this characteristic.
1. Introduction The modelling of economic situations in a game theoretical framework has resulted most of the time in considering superadditive games. Away the best known in the economic literature let us quote the market games associated with an exchange economy, the games associated with superadditive coalition production or with public good economies in a first best context. We present in this brief note, an example of an economic game, which is not necessarily superadditive, associated with a second best problem in the theory of local public goods. The examination of various second best problems has convinced us that the absence of superadditivity in the associated economic games was not at all pathological. Non-superadditive games have some unfamiliar characteristics, an appropriate reconsideration of standard concepts of efficiency and stability is required (see section 2). The application of these modified concepts will prove fruitful for the local public good game with few agents (see section 3).
2. Efficiency
and stability in games which are not necessarily superadditive
Through the concepts defined in this section, we want mainly to take into account two facts: - Efficiency may require that the grand coalition N breaks down. - Stability requirements which have the same strength as those involved in the core 301
302
R. Guesnerie, C. Oddou /Economic games which are not necessarily superadditive
concept, may be obtained for some particularly when the core stricto-sensu is empty.
appropriate
grouping of agents even
Let us consider then a game in characteristic form: N denotes the set { 1, 2, . . .. m],Z denotes the set of all non-empty subsets of N and u denotes a function from Z to, non-empty, compact subsets of @ that satisfies: for all SE 2, if CY E u(J’) and oS = pS, then /3E v(s). ’ In the standard interpretation, the elements of N are players, the elements of Z coaZitionsand the elements of IRN payoff or utility vectors. The elements of u(S) represent feasible outcomes for the coalition S and we denote D(S) the set of outcomes that S can improve upon, i.e., the set of elements u E IRN such that there is a vector u E u(S) for which us << us [see Shapley (1972)]. Definitions. A structure of coalitions is a partition II = (,S&= of the set of agents. The set of partitions of N is denoted TO. For any II = (Sk)ka, II E T(N) we denote u(II) = f?,, u(Sk). An efficient outcome of the game is a vector ii E w
such that
(a) there is Il E T(N) for which U E u(II), (b) there is no II’ and U’ E u(II’) such that U’ >> U. An efficient structure is a structure II such that there exists an efficient outcome U E u(n). A structure II is universaZZy efficient if any efficient outcome U E u(II). Game theoretical concepts of stability can now be introduced. The Core C(u) is the set of U* E u(N) which can be improved upon by no coalition - C(u) = u(N) \ “,,z D(J) -. A Core like stable solution or C-stable solution is a vector u* E Km which satisfies the following properties: - there is a II E TQ such that U* E u(II), -forallSEZ,u* $08. The reader will immediately check the following properties: (i) a C-stable solution is an efficient outcome, (ii) the core is a set of particular C-stable solutions, (iii) if N is universally efficient, any C-stable solution is necessarily in the core, (iv) II = {N} is a stable structure if and only if the core is non-empty. For games which are not superadditive, {N} is not necessarily universally effi-
’ Let lRN denote the n-dimensional euclidean space with coordinates indexed by the elements let IRS denote the corresponding ISI-dimensional subspace of IRN. If ofN,andforSE’. then as will denote the projection of LYon IRS, i.e., the restriction of OL oEIRNandSEZ, to the coordinates indexed by the elements of S.
R. Guesnerie, C. Oddou /Economic
games which are not necessarily superadditive
303
cient. In this case the concept of C-stable solution reflects the same ideas of stability as those underlying the core concept and appears as a natural generaliza. tion. To some extent, as made clear below, it is merely an adaptation of the core concept. ’ Let us define ?, the smallest superadditive game associated with u,
u
Q) =
rIET(S)
u
TErl
Gwill be intermediate u”(s) = 49
for all
G?. between u and gand it is defined as follows: G(N) = $(m, S #N.
Then Lemma 1. u* is a C-stable solution of v if and only if it belongs to the core of g and if and only if it belongs to the core of 6 Take U* a strongly stable solution, there is a II E T(N) such that U* E u(II) Boot so u* E 6(/V) = z(N), if u* does not belong to the core of v”(respectively g) there is SE Z and u E $S) (respectively u E E(S)) such that U’ >> Use. For i?, if S # N, U* E 08, we obtain a contradiction. If S = N, or for G, there is a II E TQ such that U E nTEn u(T) and for some T belonging to II, U* ED(T), which is impossible. On the other hand, if U* E C@) (respectively U* E C(G)), U* E i$V) = +V), so there is a II E T(N) such that U* E u(n). If there is an S E Z such that U* ED(J), then there is an U E u(S) with $ >> u*s. u E u(S) c G(s) c G(S), so S can improve upon U* and we obtain a contradiction. Q.E.D.
3. A local public good game which is not necessarily superadditive Consider the following simple economy: there are two goods, the first is a private good, the second is a pure public good. There are m consumers indexed by i = 1, . .. . m having preferences represented by a utility function Ui defined on @ (Ui(0,0) = 0). Moreover, consumers are initially endowed with oi (q > 0) units of the private good. Preferences and initial endowments being given, we define the rules of the game as follows: each coalition S which is a subset of N = (1, .... m) can use the same simple constant returns to scale technology which transforms one unit of private good into unit of public good. The public good can be financed only through
’ In particular, we discovered that a similar concept was used by Shaked (1978) who called it simply ‘core’. Even if Lemma 1 gives some justification for this usage, it seems to us that it would be a source of confusion not to distinguish core from stable solutions, at least in the context of the second best games we are looking at.
304
R. Guesnerie, C. Oddou /Economic
games which are not necessarily superadditive
a wealth tax: each coalition S can choose a rate of tax r E [0, 11, which applies to all members of the coalition. When the tax rate chosen is t, the tax revenue raised is t ZZiEs Wi and the amount of public good produced is q(t, 5) = t Tics Wi. Given these assumptions on the technology of production and on the institutional constraints for the financing of the public good, one can associate the problem of the allocation of the public good in this simple economy with a game, whose characteristic function is u(J), defined as follows: u(L!q= {uE@JtE
[o,l],U%u(t,ls)},
where u(t, s> is the vector of IR’.whose components Ui(t,s)=Ui{(l
-t)Wi*q(t,S)},
are
I’ES.
Fig. 1 shows that in this case, as stated in the introduction, superadditivity does not necessarily hold. Two individuals have the same initial endowments. When they are isolated, the agent 1 obtains UI and the agent 2 obtains & . When they are together the attainable states are associated with the segment AC and it is clear that (a1 , ii*) is not realisable. 3 4. Stable solutions and stable structures number of agents
in a public good economy with a small
In this section preferences are only assumed to be convex and initial endowments of private goods can be distributed among agents in any way. 3 The assumption concerning the financing of the public good is here crucial. If coalitions are able to raise personal taxes and to relate exactly marginal willingnesses to pay and individual contributions, it is well known (under standard assumptions) that the core is non-empty and even large. The use of a wealth tax leads us to a second best situation. Our model has some connection with the theory of local public goods initiated by Tiebout (1956). In particular, our general assumptions are similar to those of Westhof (1975) who considers the organization of communities producing a pure public good.
R. Guesnerie, C. Oddou /Economic
games which are not necessarily superadditive
305
The analysis is based on the application of Scarfs theorem according to which any balanced game has a non-empty core. We show that when the number of agents is small (m < 4) the systematic use of Scarfs sufficient condition allows us to make conclusive statements. Precisely we have When the number of agents is smaller or equal to three, m < 3, the Proposition 1. game v has a C-stable solution. Proof: The statement is trivial for m = 1, and straightforward for m = 2. However, it is not at all obvious for m = 3. We show in this case that the game u”is balanced in the sense of Scarf. The only minimal balanced family for which the property is not trivially satisfied is T = ({AB}, {BC}, {CA }) and it is enough to prove that if (U,I, us) E u(CABI), (U,, UC) E u({BC)), (UC, us) E u(ICAI) then (u_4, UB, UC> {ABC}). 4 For that, consider A(A) (respectively A(B), A(C)) the interval of public good q, which, if implemented in the grand coalition, would give more than U, to A (respectively UB to B, UC to C). Taken 2 by 2, these intervals have a non-empty intersection [for example A(A) fl A(B) > qAB, where qAB is the public good level in AB]. Then A(A) f~ A(B) fl A(C) # 4 (this is Helly’s theorem in dimension 1). Q.E.D. [Cf. Berge (1966, p. 173).] E v(
From a previous remark, one can straightforwardly Corollary.
assert
For m < 3, if v is superadditive, then it has a non-empty core.
We can actually prove that a similar result holds for m = 4 by using the same techniques. Proposition 2. empty core.
Form = 4, if v is superadditive, it is Scarf balanced and has a non-
Proof: According to Shapley (1967) the minimal balanced families of a 4 players game are, apart from the partitions, the following families: ~~ =({123], rz = ({12],
11241,134]), c131,c231,
{41),
~~=({123},{14},{24},{3}), T4
=
((1
TS =({I
2
31, Cl 41, 12 41, 13 411,
231,
11 241,
and all families obtained
Cl 341,
{2341),
by permutation.
The game being superadditive
4 If this holds for u, it is easy to check that it also holds for iT
we can
306
R. Guesnerie, C. Oddou /Economic
games which are not necessarily superadditive
prove a theorem similar to theorem 3 of Shapley (1967). It is thus sufficient to check that the Scarf condition is verified for proper minimal balanced families. For such families, every couple of agents are part of (at least) one coalition. It is only a matter of routine to check that the previous argument (on the intersection of intervals) still applies. Corollary. If m = 4, if all agents have the same resources, there is a C-stable solution and hence a stable structure. The argument is here more complicated and will only be briefly sketched: Using the proof of Proposition 2, we notice that it is enough to show that Scarfs condition is verified for the families T, and T3 (relative to G). (ii) Using Proposition 1, we prove the property for T2. (iii) Exhibiting the public good levels (associated with the utility vector under consideration) in the three first coalitions of T3, we look at 4 their median level (which may be either 4 1z 3, q, 4 or q2 4). Discussing case by taking into account the equality of endowments, we can conclude. (i)
We have considered in a simple model the existence problem for the core and its most obvious generalization when the game is not superadditive the C-stable solution. We attacked existence problems with a method which looks natural, trying to prove that the game, or its superadditive extension, is Scarf balanced. We obtained significantly positive results for m < 4. One would expect that the approach remains fruitful when a greater number of agents is involved. One might hope for example that Proposition 2 holds for any m. We were not able to prove that and it is unlikely to be true for m > 6 (although we have no counterexample). However, we prove in a forthcoming paper, using quite different techniques, that whatever m superadditive local public goods games have non-empty core, although as we just argued they might not be Scarf balanced.
References Berge, C., 1959, Espaces topologiques: Fonctions multivoques (Dunod, Paris). Shaked, A., 1978, Human environment as a local public good (London School of Economics, London). Shapley, L.S., 1967, On balanced sets and cores, Nav. Res. Long. R. 14,453-460. Shapley, L.S., 1972, On balanced games without side payments (The Rand Corporation, Santa Monica, CA) 4910. Tiebout, M., 1956, A pure theory of local expenditures, Journal of Political Economy 64,416424. Westhof, F., 1975, Existence of equilibria in economics with a local public good, Journal of Economic Theory 14, 84-l 12.