The social choice theory offers models of decisions, which are relevant for the public administration. For, the decision is a part of the policy cycle. Notably the social choice theory studies the formation of coalitions. The coalition game can have stable outcomes, which are in the core. It is conceivable, that the actors in the coaliton can transfer utility. Often the actors will exchange votes (*logrolling*). The Shapley value is a way to integrate morals in the social choice theory. Finally the applicability of the social choice theory will be discussed.

A previous column has described four policy models in the public administration, which have as their starting point respectively the rationality, the political struggle for power, the social culture, and the institutions^{1}. This distinction is clarifying, but the terms deviate somewhat from the conventions, which are used by the Gazette. In the economic perspective the policy models refer to respectively the plan, the power, the communication, and the institution. In this row the communication is somewhat different, because it is actually more a skill than a social phenomenon. Yet it is essential for the success of policies. Managers must be able to convince, motivate, and be experienced in making propaganda. A decision is only executed, when it has sufficient support of all concerned. Until now the Gazette has usually ignored the effect of communication^{2}.

In the traditional policy approach (say, the plan-model and the power-model) a certain policy programme is introduced as a reaction to a social problem. It is commonly assumed, that the policy can be interpreted as a linear succession of phases. The decision is just a part of the whole. Moreover, policies are never complete, because the environment is dynamic. There is feedback from the society, and this results in continuous learning. Then the linear chain changes into a learning cycle, like in the model of Kolb. Therefore the policy cycle has the form of the figure 1. However, a striking hallmark of the policy cycle is, that the learning process is affected by conflicts between interest groups. Then the interpretation of the cycle of Kolb has two components, namely the political struggle about the policy agenda, and the design of the policy programme^{3}.

Almost all models in the Gazette assume, that the actor is a homo economicus. He decides in such a manner, that his utility or well-being is maximized. However, the well-known policy analyst Herbert Simon states, that decisions are based on a *bounded* rationality. In practice the maximization of utility is simply not possible. The homo economicus becomes modest, and is satisfied with decisions, which minimize the risks of loss^{4}. Such decisions can use a logic of goal realization, or a logic of *appropriateness*^{5}. Appropriateness implies, that the dominant norms and role expectations within the organizations are satisfied. The corresponding image of man is the homo sociologicus. These two types of logic can be related to respectively the Weberian instrumental rationality and value rationality.

In the logic of appropriateness the actors aim for a *satisficing* solution of the problem. The context of the decision is included in the evaluation. Often such a solution is incremental, because then the group interests will not be hurt significantly, and the risks of a disastrous failure are small. There is a continuous evalution, and the policy evolves. See the institution model of policies^{6}. In this regard the approach of *new public management* (in short NPM) is two-edged. On the one hand new public managent allows to rapidly react to external threats. On the other hand, innovative entrepreneurship is naturally risky. Decentralization of the policy decisions is a manner to maintain control in social pluralism^{7}. This view is for instance embodied in the *garbage can* model.

Decentralization implies, that the ambition of the central plan (plan-model) is abandoned, or at least has a lower priority. More room is given to the competition between interest groups. The agenda is determined by policy networks. Here the power model and the communication models of policy are applied. Economists and sociologists have developed the rational choice paradigm, with applications aiming at social decisions (*social choice*) and public policy (*public choice*)^{8}. The actors (individuals and organizations) primarily promote their own interest. All concerned groups are rent seeking. Models usually translate this interest in terms of income. In principle moral or psychological interests can also be taken into account, but this makes the interpretation of the model less tangible. Thus the power of the citizens can be studied, or even of a coalition of politicians and agencies (Leviathan model)^{9}.

When the state prefers decisions by means of network management, then this is called *public governance*^{10}. Use is made of partnerships, and of contracting^{11}. The state negotiates with the concerned groups, and compromises. The means of the concerned actors are bundled. Thus the decision is made by a coalition of groups. So within this coalition there is cooperation. Nevertheless in such a network entrepreneurship has an added value^{12}. The management of policies becomes polycentric^{13}.

Policy formation by means of networks is an attempt to increase the effectiveness and efficiency of policies. However, it can also be advocated on moral grounds, namely that it furthers the participation of citizens, and therefore reinforces the democracy. This does introduce the danger, that in this way the active citizens obtain an excessive amount of power.This would affect the legal security of the citizens. And the social benefits are vague^{14}. The argument for participation is mainly important in the communication model, and in the institution model as well. Networks in the institution model can have the form of corporatism. In (neo-)corporatism the state transfers power to some interest groups^{15}.

The science of public administration addresses the environment, where policy is formulated. This includes the procedures, the pertinent factors, the conditions, the point of decision, and the chosen instruments. But evidently a coalition of actors must also emerge, which is prepared to be the owner of the policy problem, and to decide about the best solution. In public administration little research is done with regard to the factors, which determine the composition of this coalition. This question has been addressed by the social choice theory.

The *decision* theory in the public administration pays little attention to the ideas of the social choice theory, which is methodologically related to economics. Here your columnist wants to try to fill the lacuna. A column of three years ago used an *exchange theory* in order to model decisions. But usually the social choice theory applies game theory to policy problems. However, the public administration sometimes also describes the policy decisions as a game^{16}. Each actor k in this game (with k = 1, ..., K) makes a choice from his set s_{j}(k) of behavioural strategies (j = 1, ..., J(k)), tailored to the given situation. In principle the number of strategies J(k) of each actor k can be infinitely large. Moreover, j could be a continuous variable within the interval [1, J(k)]. The combination of strategies leads to a policy decision.

Thanks to the decision the actor receives benefits in quantities x_{n}(k) (n = 1, ..., N). So there are N types of benefits. Each type n can be material, but also in kind, for instance information or a good reputation. The actor k must estimate, what the value of these benefits is for himself. Thus for instance money and reputation are mutually compared. It seems odd, but it is possible according to the model of Fao and Fao^{17}. The decision determines the totally available quantities x_{n} of each type n, as well as their distribution Σ_{k=1}^{K} x_{n}(k) among the actors.

utility possibilities (

The evaluation of the actor is naturally based on his preferences. In the rational choice paradigm the preferences of the actor k are represented by the utility function u_{k}(__x__(k)), where the vector __x__(k) consists of the elements x_{n}(k). Apparently the utility vector __u__ is in the end the criterion for selecting a decision. Therfore the social choice theory primarily studies the distribution of utilities, and not of benefits^{18}. It has just been remarked, that the decision is the result of the strategies s_{j}(k), which the K actors use. Therefore the set of all strategy combinations {s_{j1}(1), ...., s_{jK}(K)} (with jk=1, ..., J(k)), which is the set U of utility possibilities __u__. As an illustration the figure 2 shows the outer boundary of such a set U for K=3 ^{19}.

The figure 2 assumes, that all three actors participate in the decision about the policy. Each one contributes with his own means to the policy formation. But except for this there is not yet a cooperation and coordination between the actors. Each actor maximizes in isolation his utility, by means of his strategy. Note that in practice a large number of points __u__ within U will most likely not occur. For instance, in the figure 2 the actors 2 and 3 will simply not select a strategy, which gives the actor 1 much more utility than they acquire. Therefore the final decision __u__^{*} will probably lie somewhere in the middle on the outer layer. Only when the actor 1 has a large ascendancy of power, he can appropriate all benefits. Then he barely needs the actors 2 and 3 for the policy formation.

However, the present column discusses the cooperation of actors. In it simplest form this is a coalition C of two actors. They can benefit from C, because they can coordinate their strategies in this manner, so that another decision is possible than without cooperation. For instance, the coalition C, consisting of C = {1, 2} can enforce a decision, which increases u^{*}_{1} and u^{*}_{2}. This hurts actor 3, so that his u_{3} diminishes. Actor 3, the loser, will yet try to maximize his utility u_{3} in the given situation^{20}. As an illustration of this phenomenon the table 1 (originating from the policy analyst P.C. Ordeshook) shows a game with three actors, where each actor k disposes of two strategies s_{1}(k) and s_{2}(k) ^{21}. So the available strategies are not continuous, such as in the figure 2, but discrete. The cells show the utilities (u_{1}, u_{2}, u_{3}) of each strategy combination. The table 1 forms the basis of a network analysis^{22}.

s_{1}(3) | s_{2}(3) | |||
---|---|---|---|---|

s_{1}(2) | s_{2}(2) | s_{1}(2) | s_{2}(2) | |

s_{1}(1) | 57, 50, 15 | 0, 60, 35 | 55, 20, 0 | 30, 30, 40 |

s_{2}(1) | 70, 55, 30 | 0, 90, 60 | 50, 0, 80 | 20, 60, 70 |

Game theory calls the table 1 the *normal* form of the game. The outcome (30, 30, 40) is the result of the strategy combination (s_{1}(1), s_{2}(2), s_{2}(3)), and is the so-called Nash equilibrium. No actor {k} can unilaterally take a decision, which yields him a higher outcome. His first preference (respectively 70, 90, and 80 for k = 1, 2 and 3) is blocked by the other actors. It is also not advantageous for an actor k to form a coalition {h, k} (with h, k in {1, 2, 3}). Consider for instance the coalition C = {1, 2}. Thanks to C the actors 1 and 2 can block outcomes, which are favourable for actor 3. The actor 3 naturally does keep his freedom of acting s_{j}(3). Now he must undermine the coalition in negotiations by threatening the two actors with punishments.

C can choose from four strategies s_{j}(C) = (s_{1}(1), s_{1}(2)), (s_{2}(1), s_{1}(2)), (s_{1}(1), s_{2}(2)) and (s_{2}(1), s_{2}(2)), which yield respectively μ_{j} = (u_{1}, u_{2}) = (55, 20), (50, 0), (0, 30) and (0, 60) with certainty. Note that these are the *minimum* outcomes for the coalition, irrespective of the strategy s_{j}(3) ^{23}. However, there is a chance, that the actor 3 selects a strategy, which makes the outcome for C rise above the minimum. The behaviour of the actor 3 is not amenable to an exact analysis. Nevertheless, in the following some comments will be made about this aspect.

Apparently the actors 1 and 2 never have a *mutual* benefit from μ_{j}. The Nash equilibrium E = (30, 30) remains attractive. Nonetheless, the coalition C can yet be worthwhile. Namely, it can switch between the strategies s_{j}(C) with a fixed frequency. In this way a *mixed* strategy σ(C) results. An example of a mixed strategy is

(1) σ(C) = p × s_{1}(C) + (1 − p) × s_{4}(C) = p × (s_{1}(1), s_{1}(2)) + (1 − p) × (s_{2}(1), s_{2}(2))

and

In the formula 1, p is a real number in the interval [0, 1]. Note that in this approach a choice is made between an infinite number of strategies, just like in the figure 2. Now the certain outcomes in the formula 1 are μ_{1,4} = (u_{1}, u_{2}) = p × μ_{1} + (1 − p) × μ_{4} = p × (55, 20) + (1 − p) × (0, 60). Simple calculations show, that for ^{6}/_{11} < p < ^{3}/_{4} the coalition receives more utility than in the Nash equilibrium (30, 30). Other mixed strategies than σ(C) are naturally also possible, but these have a lower yield. This is shown in the figure 3, where the four "pure" outcomes μ_{j} are drawn, together with the mixed outcome μ_{1,4} (**red** line).

The preceding argument yields new insights in the decisions about policies. It is worthwhile to formulate this approach in general terms. For this, the application of mathematics is convenient. It is said that a game has its *characteristic* form, when there is a set of K actors, and a (mathematical) rule V, which connects each coalition (subset) C in K to a set V(C) of utility possibilities. The rule V(C) is called the characteristic function of C ^{24}. As an illustration, the figure 3 shows such a set V({1, 2}). It has been coloured **yellow** in the figure 3. The set V({1, 2}) not only includes the **red** boundary μ_{1,4} of utility possibilities, but also all points with lower values u_{1} and u_{2}. The rule V can also be applied to all other conceivable coalitions ({1, 2, 3}, {2, 3}, {1} etcetera), and then always yields a corresponding yellow area of utility possibilities.

It is instructive to again compare the figures 2 and 3. The **red** contour in the figure 2 shows all possibilities U = {__u__} in absence of cooperation. The figure 2 suggests, that the actors 1 and 2 can benefit from a decision, which reduces the utility of actor 3. However, without cooperation actor 3 will block such decisions by means of his strategy. The actors 1 and 2 must mutually form a coalition in order to limit the choices of the actor 3. Then it is still uncertain, whether the actors 1 and 2 will both benefit from the coalition. In the figure 3 they do succeed, thanks to the mixed strategy. Then the analysis does not study U, but a set V(C) of certain outcomes. However, coalitions in a game are not by definition beneficial. Sometimes the actor 3 can select s_{j}(3) in such a manner, that the set V({1, 2}) becomes unfavourable. Then the 3 actors must check, whether another coalition does yield benefits^{25}.

In the same way that the Nash equilibrium of a game is interesting, it is also useful to find stable utility vectors __u__ within the set U of utility possibilities. For, in a stable outcome it is no longer necessary to negotiate, form coalitions, or influence the policy agenda. The actors no longer feel the need to change their strategies. The set Γ of all these stable utility vectors is called the *core* of the game^{26}. This can also be formulated in an abstract manner. Namely, suppose that there is a game with characteristic form (K, V), then the core Γ consists of the vectors __u__ in V(K), which can not be blocked by any coalition C. A coalition will block a vector __u__, as soon as a vector __w__ in V(C) exists, which satisfies u_{k} < w_{k}, with k in C ^{27}. In this case it is said, that __w__ dominates over __u__ ^{28}.

As an illustration of these definitions, consider again the game in the table 1. Without cooperation the actors select (30, 30, 40). The assumption is, that mixing strategies is not allowed for the coalition with K=3. In other words, suppose that the function V, applied to the *grand* coalition {1, 2, 3}, simply reproduces this table^{29}. Then the coalition {1, 2, 3} will also select (30, 30, 40). But it has just been shown, that V({1, 2}) contains vectors w_{k} > 30 for k=1, 2. Therefore this coalition will block the outcome (30, 30, 40). However, the coalition reduces the outcome for actor 3 to w_{3} < 40. Therefore actor 3 will not cooperate, and his strategy is undetermined. The vector __w__ is certainly not in V(K). Apparently in this case the core is empty. It is worth mentioning, that (30, 30, 40) does form the core, when the outcome for the strategy combination (s_{2}(1), s_{2}(2), s_{2}(3)) in the table 1 is replaced by __u__ = (20, 10, 70) ^{30}.

This illustration shows, that a game without core is unstable. The component 3 of the vector __w__ is undetermined. The game of the table 1 gives an interesting illustration of this^{31}. Suppose that the utility value u_{3} remains valid, even after the coalition C = {1, 2} is formed and it applies the mixed strategy of the formula 1. Then it is rational for the actor 3 to prefer s_{1}(3). However, then the actor 1 gets a preference for the pure strategy s_{2}(C) = (s_{2}(1), s_{1}(2)). And the actor 2 now prefers the pure strategy s_{3}(C) = (s_{2}(1), s_{2}(2)). Subsequently, the actor 3 will change his view because of the new choice s_{2}(1), and get a preference for the strategy s_{2}(3). Etcetera. In practice this means, that the 3 actors are engaged in a continuous negotiation^{32}.

Unfortunately games *with* a core are sometimes also not determined completely. In some games the core can be entered by means of various, mutually different, coalitions. Therefore, despite the existence of the core, it can not predicted which coalition will be formed. Only the outcomes __u__ can be calculated in advance. Policy analysts believe that this is a serious defect of this type of game theories, because in practice usually only the coalitions are observable^{33}. It is difficult to measure outcomes. For instance, concessions and obligations sometimes only have effects in the long term.

β_{1}=−,−,− | β_{2}=+,−,− | β_{3}=−,+,− | β_{4}=−,−,+ | β_{5}=+,+,− | β_{6}=+,−,+ | β_{7}=−,+,+ | β_{8}=+,+,+ | |
---|---|---|---|---|---|---|---|---|

actor 1 | 0 | 5 | 1 | -2 | 6 | 3 | -1 | 4 |

actor 2 | 0 | 6 | -2 | -1 | 4 | 5 | -4 | 3 |

actor 3 | 0 | -2 | 4 | 1 | 2 | -1 | 5 | 3 |

As an illustration the table 2 shows the utility values __u__ of K=3 actors, who must vote with regard to 3 policy proposals^{34}. This is the game in its normal form. A policy analyst wants to know which coalition is formed, and which policy is selected. Both the coalition and the policy are determined in negotiations^{35}. There are 8 possible outcomes β_{n} (n = 1, ..., 8) of the vote. Each β_{n} indicates which proposals have been accepted (+) or rejected (−). Obviously a coalition of 2 actoren always wins the vote^{36}. The coalition {1, 2} has as Pareto optimal outcomes β_{2} and β_{5}. For {1, 3} they are β_{3}, β_{5} and β_{8}, and for {2, 3} they are β_{2}, β_{5}, β_{6} and β_{8}. Apparently only β_{5} is Pareto optimal for all coalitions. The corresponding outcome __u__ = (6, 4, 2) is not dominated by any vector. Apparently this __u__ is the core Γ. In this situation the question of the policy analyst is not answered completely, because the core can be entered by different coalitions.

Furthermore is must be remarked, that the phenomenon of the core is only poorly confirmed by experiments. Ordeshook mentions a laboratory study with test persons, where merely 47% of the experiments entered the core as the result of the negotiations^{37}. In the concerned study the reason was, that each actor k mainly tried to agree on the outcome with his highest u_{k} (in table 2: β_{5} for actor 1, β_{2} for actor 2, and β_{7} for actor 3). Thus the attempt to enter the core fails. Ordeshook calls this strategy a heuristic. The aim of the heuristic is to limit the transaction costs. It may evidently be hoped, that for really important proposals the actors will act more rationally. Nevertheless, Ordeshook concludes, that institutions such as voting procedures, agenda's and committees can significantly affect the outcome and therefore the policy^{38}.

It has just been remarked, that each actor must make an estimate of the utility of certain policy decisions. In principle the individual utility function u_{k}(__x__) of actor k can assume any form. Here the vector __x__ refers to the quantities x_{n}(k) of the good n, which is owned by k. However, it is more clarifying to assume a simple form. In game theory the form with transferable utility is popular:

(2) u_{k}(x_{1}(k), x_{2}(k), ..., x_{N}(k)) = x_{1}(k) + ω_{k}(x_{2}(k), ..., x_{N}(k))

In the formula 2, ω_{k} is a utility function, which no longer contains x_{1}(k). Apparently the utility u_{k} changes in a linear manner with x_{1}(k). The function is *quasi-*linear. This has the advantage, that the set of utility possibilities assumes a simple form^{39}. Namely, sum the formula 2 for all K:

(3) Σ_{k=1}^{K} u_{k}(__x__(k)) = x_{1} + Σ_{K=1}^{K} ω_{k}(x_{2}(k), ..., x_{N}(k))

Since only similar quantities can be added, the formula 2 assumes a cardinal utility. The formulas 2 and 3 imply, that the actors can mutually exchange utility unis, simply by transfering a unit of the good n=1. Therefore this is called a *transferable* utility. The good n=1 is called the *numéraire*, as a bearer of utility.

In the right-hand side of the formula 3, x_{1}(k) is no longer explicitly present. It can be interpreted as a constant X, when one is only interested in the variation of __u__ due to the distribution of x_{1}. Now the equation Σ_{K=1}^{K} u_{k} = X is a hyperplane in the K-dimensional __u__ space, with as perpendicular the vector __η__ = (1, 1, ..., 1) ^{40}. This notably also holds for the boundary of the set U of utility possibilities. Then the characteristic function V of C (with c participants) can be represented by the mathematical expression:

(4) V(C) = {__u__ in R^{c}: Σ_{k in C} u_{k} ≤ ν(C)}

In the formula 4, ν(C) is called the value of the coalition^{41}. This shows the advantage of the assumption of transferable utility: the function V, which generates a utility space, is replaced by a function ν(C), which generates a value. Then the game in its characteristic form is (K, ν). And now the core Γ is simply the set of vectors __u__ with Σ_{k in C} u_{k} ≥ ν(C) for all C ^{42}. The exchange good n=1 is naturally clearly distinguished from the other benefits n = 2, ..., N, which affect utility in a more complex manner. Usually n=1 is interpreted as money. This is to say, the K actors can compensate each other by means of payments for the possible immaterial concessions, which are made during the negotiations about the policy decision^{43}.

The situation in the game changes, when the utility is transferable. For, then the actors can compensate each other for unfavourable results of votes, by means of a payment with the numéraire good (n=1)^{44}. For instance, consider the table 2, now with the assumption of transferable utility. Then the coalition {1, 2} would be preferred above the outcome β_{2}, when the actor 2 pays a quantity x_{1} with value in the interval [1, 2] to the actor 1. It may be questioned, whether such transactions are realistic. According to Ordeshook it is strange, that the policy is partly determined by mutual payments between the actors^{45}. When the actors are politicians, then Ordeshook calls this a *graft*, and therefore objectionable. This may be true for politicians. But when one considers actors in a policy network, then mutual payments may be acceptable.

Compensation could evidently be immaterial. For instance, it could be assumed, that the actors consist of politicians, who promise electoral favours to each other. But then the quasi-linear assumption is dubious. In some cases the application is conceivable, for instance when a coalition distributes the posts of ministers in a cabinet^{46}. Then x_{1} is the number of posts. So the merit of models with transferable utility is more the theoretical transparency than the similarity with reality^{47}.

When a game has a core, then the actors usually will select their decision here. For, no single actor can improve his outcome by blocking this decision. So this choice has the fairnes of the unavoidable. However, other criteria have also been proposed for calling a decision just. Such claims of justice are naturally normative. They illustrate the institutional approach of policy formation, which downplays rationality and power. Perhaps the most famous proposal for a just decision has been done by L. Shapley. The outcome must be the *Shapley value* __ζ__ (so actually a vector). The Shapley value is mainly interesting under the assumption of a transferable utility, and therefore this is the starting poiny of the present paragraph^{48}. Shapley makes a proposal for the distribution within the coalition C, but does not want to intervene in the status quo (utility possibilities, available strategies, and the like)^{49}.

Shapley derives his just value from four properties. This is called the axiomatic approach. They are efficiency (Σ_{k in C} ζ_{k} = ν(C)), symmetry (the outcome for actor k does not change under permutations of actors), linearity in ν(C), and the dummy axiom (ζ_{k}=0 for an actor k, who does not add value by himself)^{50}. These axioms are less self-evident than for instance those in the bargaining model of Nash^{51}. Nash derives from these four axioms, that the Shapley value is given by:^{52}

(5) ζ_{k}(K, ν) = Σ_{C in K} ((K − c)! × (c − 1)! / K!) × (ν(C) − ν(C\{k}))

In the formula 5, k is an element in the set C, which includes c actors in total. The notation C\{k} represents the set, which is obtained by removing k from C. The k! notation represents the mathematical faculty k × (k-1) × ... × 1. One defines 0!=1. The term ν(C) − ν(C\{k}) expresses the value, which the actor k adds by joining the coalition. This is the marginal contribution of k to C, as it were. The faculties are a weighing factor corresponding to the number of ways (by means of permutations) to form this C. Thus ζ_{k} in the formula 5 is an average of all marginal contributions of k.

The Shapley value can be clarified with the following game, for K=3 ^{53}. The actors 1 and 2 are agents (U) of policy, and the actor 3 is a developer (O) of policy. According to the figure 1, a design must be followed by its execution, so that only the combination {O, U} has some value, say 1. Suppose that the policy requires just a single agent. Thus one finds the values ν({k}) = ν({1, 2}) = 0, and ν({1, 3}) = ν({2, 3}) = ν({1, 2, 3}) = 1. Thanks to the formula 5 the Shapley value can be computed, namely __ζ__ = (^{1}/_{6}, ^{1}/_{6}, ^{2}/_{3}). Note that the actor 3 is a monopoly, which can play the actors 1 and 2 off against each other^{54}. Therefore the core of the game consists of Γ = {(0, 0, 1)}. Here apparently __ζ__ is not in the core. It is just to such an extent, that it somewhat levels the steep inequality of Γ.

In the parliamentary system, parties form a coalition, which supports its government. The government programme consists of compromises between the parties about the most important subjects. This can be interpreted as an exchange of votes, where the actors grant each other something. In this case, in the parliamentary sessions the parties do not vote according to their true preferences regarding the subjects. The exchange of votes is also useful, when a collective programme is absent, and the members of parliament determine their actions in isolation. Namely, then each politician wants to further decisions, which favour his rank-and-file. Also in this situation the policitians mutually conclude agreements in order to coordinate their votes. They temporarily engage in coalitions. This is called *log-rolling*, or *vote trading*. Another conceivable application is the policy formation in a policy network, which decides by means of voting.

β_{1}=−,− | β_{2}=+,− | β_{3}=−,+ | β_{4}=+,+ | |
---|---|---|---|---|

actor 1 | 0 | -2 | -2 | -4 |

actor 2 | 0 | 5 | -2 | 3 |

actor 3 | 0 | -2 | 5 | 3 |

with

As an illustration the table 3 shows the utility values __u__ of K=3 actors, who must vote about 2 policy proposals^{55}. The interpretation of the table is, that a proposal yields a utility of 7 for a single actor, whereas the costs of 6 are borne by all 3 actors (so u=-2). Each actor promotes a minority interest (otherwise a coalition would be unnecessary). The actor is rent seeking, where he shifts the costs of his preference to the majority^{56}. When the actors vote according to their preference, then both proposals are rejected, so that the outcome is β_{1}. However, the outcome β_{4} is better for the actors 2 and 3. Therefore it pays for these actors to engage in a voting coalition, where they no longer follow their own preferences. This is an exchange of votes, which is beneficial thanks to the intense preference of the actors 2 and 3 for, respectively, the first and second policy proposal.

The exchange is even socially desirable. Suppose for instance, that the social welfare function is given by W(__u__) = Σ_{k=1}^{3} u_{k}. Then the outcome β_{4} increases welfare, more than β_{1}, β_{2} or β_{3} ^{57}. Unfortunately logrolling can also lead to undesirable policies^{58}. For, suppose that in the table 3 each proposal has a utility of 5 for the favoured actor (instead of 7). This is to say, u_{2}=3 for the outcome β_{2} etcetera. Then one has W = -1, -1 and -2 for respectively β_{2}, β_{3} and β_{4}. All in all the social costs are larger than the benefits. The proposal unnecessarily increases the state ratio, because actually it must be rejected. Apparently the other actors in the coalition are misled by the high utility (5), compared to the low costs (-2) per actor^{59}. It can only be hoped, that the competition in the bargaining between the actors keeps them alert, and moderates this danger.

Furthermore, note that at least theoretically situations with exchanges of votes are never stable^{60}. This is illustrated in the table 3, and is shown in the figure 4. First the actors 2 and 3 decide to form a coalition in order to realize β_{4}. But the actor 1 will oppose this by offering a more attractive coalition, say {1, 2} with outcome β_{2}. But next the actor 3 will offer {1, 3} to the actor 1, with outcome β_{1}. And in this situation the coalition {2, 3} is obviously again attractive. Such a sequence is called a *voting cycle*^{61}. This can be moderated when the actors engage in predictions. For instance, the actor 2 can reject the offer of actor 1, and stick to the outcome β_{4}.

An instability also emerges, when the bundle of proposals is not put to the vote at once, but each proposal is discussed separately. Then it is tempting to defect from the agreed voting behaviour. The actor votes for his own proposal, but against the other ones. Yet in the parliamentary practice it turns out that decisions are fairly stable. Thanks to durable relations the members of parliament have built up trust. They realize, that they need each other also in the future^{62}.

First it must be emphasized, that the concept of utility is essential for the presented social choice theory. One can only decide, when preferences exist. But utility is an elusive and dynamic variable. According as an actor disposes of more means, more can be exchanged, and then the set of utility possibilities of the other actors increases. An actor can derive utility from immaterial variables, such as status, trust, knowledge, or even ideology. And the effort, which is needed in order to acquire the good, again reduces its utility. When all these factors are taken into account, then the social choice theory would be universally applicable. But such a calculation is naturally impossible. Therefore one can only expect realistic results from the social choice theory, when a single cause of utility dominates. Often this utility has a material origin, for instance money.

Furthermore, the mentioned models make assumptions, which are controversial. The actors must dispose of complete information. Moreover, in the previous paragraphs it has usually been assumed, that the actors in a coalition honour their agreements. The enforcement of the "contract" would be automatic. In reality this is evidently all not true, and information can be used as a source of power. The Gazette has already often discussed this theme, usually in the form of the *principal-agent* problem. Incidentally, the public choice theory offers solutions for this. In specific situations self-enforcing contracts are sought. In coproduction of policy the ownership of the various means and tasks must be distributed wisely among the actors in the network^{63}.

It is worthwhile to integrate the social choice theory in the decision theory of the science of public administration. They can compensate each other's weaknesses. The present paragraph wants to analyze this possibility.

The social choice theory uses a limited number of standard models with a wide applicability, such as the theory of rent seeking. Methodologically it builds on two well-tried and sound approaches, namely game theory and the theory of social exchange^{64}. The social choice theory describes the strategies, which various actors use in their policy network. Their motivation is defined clearly, namely as an (enlightened) self interest. The formation of coalitions can even be modeled in complex situations, such as for decisions about bundles of policy proposals (*package deal*). Here compromises are concluded, so that the actors vote against their own self interest for some proposals. Situations can also be described, where the actors compensate each other for unfavourable decisions. And finally the morals and institutions are included thanks to the rules of the game and inventions such as the Shapley value^{65}.

The primacy of the self interest fits nicely with societies, consisting of well educated citizens, who embrace individualism. The distribution of power is reflected in the inviolable rules of the game. A previous column has already described the proposal of Buchanan and Tullock to externally impose collective rights in the constitution. According to the social choice theory the policy formulation is devoid of goals, within these legal boundaries^{66}. Only the utilities count, and thus guarantee the support for the policy. So the theory is deductive, and this makes it easily applicable. Notably, games without core show the essence of administrative concepts such as muddling through and the garbage can policy. Then the normal form in game theory is a suited instrument for the analysis of the policy network.

The assumption, that the behaviour of actors is determined by their utility, is a powerful abstraction. But it also conceils certain aspects. For, various factors contribute to the utility, and many of these contributions are difficult to measure. First, the utility of an actor is partly determined by his morals and by the social rules of the game. The social choice theory assumes, that the rules of the game are given in advance. Therefore the institutions and the social morals are an exogenous factor in the models. The morals and institutions become only visible in the utility of the actor, as far as they give him a feeling of satisfaction or displeasure. This reduces morals to a preference. On the other hand, this has the advantage, that the social choice theory is ideologically neutral. The instruments for the execution of the policy also remain undetermined. They are rules of the game and contribute to the utility values of actors, in the form of means and resources.

The actor can only weigh the costs and benefits of a decision, when he can develop rational expectations. Therefore the various probabilities must be known. In reality this is often not the case, because the policy always has unintended side effects^{67}. Furthermore, the compromises in the decision sometimes result in vague formulations of the policy goals. Then the policy only takes shape during the execution^{68}. Also remember the garbage can model. Policy networks try to reduce this uncertainty by mutually sharing information.

Game theory and the social exchange theory do not study the specific nature of the means of power, which are used by the actors^{69}. The struggle to control the agenda is ignored. The social choice theory does study the various institutional forms of policy agenda's^{70}. The social choice theory hardly elaborates on the way, in which networks are coordinated, except for the rules of the game^{71}. It also remains silent about the effect of communication on policies. And finally, boundedly rational behaviour, such as heuristics and roles, actually fit poorly with the actor model of the homo economicus. Then it must be assumed, that a heuristic or role is selected, because it has less transaction costs than rational behaviour.

The present column has made clear, that the social choice models are not capable of accurately describing reality. In fact, this has never been the intention. They merely want to give *insight* in the general processes, which underlie policy formation. As actor-model the homo economicus is used. This model has proved its value in economics, sociology and social psychology. Incidentally, the model of the homo economicus is usually applied in such a manner, that he acts within the framework of the institutions.

On the other hand, policy analysis often uses narrative models, also in decision theory. Then all factors can be addressed, which are relevant for policies. But this approach also has disadvantages. The analyst may get bogged down in a swamp of details. And the researchers do not have the support of tested standard models. Each researcher develops his own model, so that the debate about the truth is fragmented. This increases the probability, that scientists may succumb to the temptation of propagating morals or an ideology. Here the concept of utility of the social choice theory works like a remedy. because it can often reveal the weak sides of such doctrines.

- See chapter 3 in
*Beleid in beweging*(2012, Boom Lemma uitgevers) by V. Bekkers. This distinction of four categories is interesting, but not common in the science of public administration. (back) - Incidentally, the bargaining behind the veil of ignorance, which has been discussed in a previous column, can be seen as a form of power-free communication. Furthermore, communication is inherent in processes of power, which are regularly discussed in the Gazette. Due to the relation between communication and power your columnist questions the cultural approach as a separate policy model. (back)
- See p.20 in
*Beleid in beweging*, and more extensively on p.58. The struggle for the agenda is described in paragraph 5.4. The policy cycle is also presented on p.153-157 in*Politiques publiques*(1989, Presses Universitaires de France) by Y. Meny and J.-C. Thoenig. They discuss the struggle for the agenda on p.166-173, and 183-187. A peculiarity of this latter book is, that the neo-marxist current in the public administration is still discussed, on p.80-96. At the time the Leninist block in Eastern Europe still exists. Furthermore, note that each administrative layer has its own policy cycles. In new public management (NPM) there is a purposive distinction between the formulation and execution of policies. Then the political policy cycle consists mainly of*controlling*, where the*Ist*- and*Soll*-values of the policy theme are compared. Politics assigns means to the execution and the programme, but does not interfere with the precise allocation of these means. See paragraph 7.2 in*New public management*(2011, Haupt Verlag, German language) by K. Schedler and I. Proeller, notably the cycle on p.186. (back) - See p.164 in
*The public administration theory primer*(2003, Westview Press) by H.G. Frederickson and K.B. Smith, p.54 and 202 in*Beleid in beweging*, p.77 and 207-211 in*Politiques publiques*, or p.116 in*The Sage handbook of public administration*(2008, Sage Publications), edited by B. Guy Peters en J. Pierre. (back) - See p.166 in
*The public administration theory primer*, p.79 in*Beleid in beweging*, or p.132 in*The Sage handbook of public administration*. (back) - See p.167 in
*The public administration theory primer*, or p.79 in*Politiques publiques*. On p.38 in this second book it is remarked, that because of incrementalism a change in the political ideology of the government has only limited effects. Here reference can also be made to the stability of the constitution, which must be the starting point for all policies. See p.64-70 in*Political economy in macroeconomics*(2000, Princeton paperbacks) by A. Drazen. (back) - See p.177 in
*The public administration theory primer*, p.71 and 212-226 in*Politiques publiques*, or p.28 and especially p.100 in*New public management*. Besides the collapse of the Leninist system the disappointments with the American*Great Society*have also contributed to the rise of decentralization. Your columnist plans to analyze the Great Society in a future column. (back) - See chapter 8 in
*The public administration theory primer*. There are naturally also policy analysts, who (implicitly) take the rational choice paradigm as their starting point. See the text of L.J. O'Toole, published as chapter 18 in*The Sage handbook of public administration*. For a long time your columnist has hoped that the social choice theory could be omitted in the Gazette, but it is really indispensable. (back) - On p.162 in
*Beleid in beweging*a summary is presented of the means, which are a possible source of power. The sociologist J.S. Coleman relates in*Foundations of social theory*(1994, First Harvard University Press) the power of an actor to his available means. See for instance p.381. A large quantity of means allows the actor to increase the utility possibilities of other actors. This can even be done via an intermediairy, in the indirect exchange. Coleman has a broad interpretation of the concept of utility, where he also includes the effects of morals (p.386). Nevertheless, some circles of policy analysts see the social choice and public choice theory as threats. The public morals, notably the attention for the general interest, would be considered insufficiently. And the preference for free markets, or at least competition, implies that individuals are seen relatively more as customers, and less as participating citizens. Thus one reads on p.202 in*The public administration theory primer*: "Students, teachers, and scholars of public administration are left with an identity crisis". This refers to new public management (NPM), which borrows many insights from public choice. It is true, that the rational choice paradigm takes as its starting point the critical citizens, and calculating politicians, and assertive bureaucrats. But this obviously does not totally exclude the moral motives. On p.146 in*The Sage handbook of public administration*Knott and Hammond rightly state: "There is nothing about the enterprise of formal theory (or rational choice theory) that is generically either conservative or liberal". (back) - See chapter 9 in
*The public administration theory primer*. On p.107 in*Beleid in beweging*it is called*new*public governance. According to Bekkers the new public management (NPM) has even transformed into governance. Your columnist believes that the identification of NPM with governance is confusing. NPM is mainly a plea for competition in order to realize effective and efficient policies. Many public services can be supplied by the markets. See chapter 2 in*New public management*. In*Graaiers of redders?*(2011, Uitgeverij Atlas) W. Dicke, B. Steenhuisen and W. Veeneman present a cautiously positive judgement about the Dutch experiences with NPM. They conclude, that free markets in public services also require an approach of*muddling through*. Your columnist has learned to appreciate this book more and more. (back) - See paragraph 6.3 in
*New public management*. (back) - On p.219 in
*The public administration theory primer*it is stated: "Most visions of [governance] recognize the fundamental difference between the public and private sectors and that corporatizing the latter has broad implications for the underpinnings of a democratic polity". Thus governance is indeed separated from new public management. According to the same page, governance is more process-oriented (value-rationality) than goal-directed (instrumental rationality). But networks benefit from social entrepreneurship. See p.145 in*Beleid in beweging*, and p.171-173 in*Politiques publiques*. According to p.179-180 in this latter book such entrepreneurs use symbols. This is studied in the cultural model of the science of public administration. (back) - See p.98-99 in
*Beleid in beweging*or p.222 in*The public administration theory primer*. Bekkers compares polycentric management with the christian sovereignty in the personal circle. The public-private partnership is a special form of polycentric management, because it partly employs markets. See paragraph 8.3 in*New public management*. (back) - The participation of groups is discussed in detail on p.184-187 in
*Beleid in beweging*, p.304-305 in*The Sage handbook of public administration*, and p.55 in*New public management*. (back) - See p.97-106 in
*Politiques publiques*. According to p.42, tight networks are formed, consisting of the sector, policy agents, and related politicians. These are called*iron triangles*. A well-known example is agriculture. In (neo-)corporatism the execution actually dictates the contents of policies (p.267). According to p.237 in*The Sage handbook of public administration*this can lead to an undesirable rigidity, because outsiders are kept out of the network. (back) - See p.63, paragraph 6.2.2 and especially paragraph 6.4.2 van
*Beleid in beweging*, or p.111 and further in*Politiques publiques*. So the science of public administration acknowledges the relevance of game theory, although applications remain rare. (back) - The commensurability of material and immaterial benefits (empathy, pride, sense of duty, and the like) is also propagated by H. Gintis on p.206 in
*The bounds of reason*(2014, Princeton University Press). Gintis advocates the use of game theory in the political sciences and policy analysis. (back) - In the science of public administration the evaluation based on utility is called a cost-benefit analysis. See p.150, 155 and 174 in
*Beleid in beweging*. On p.278 it is remarked, that qualitative costs and benefits are difficult to measure. See furthermore p.304-305 in*Politiques publiques*. (back) - Note, that the outer boundary has a bulging form. Then the set of utility possibilities is convex. The convex form is caused by the law of diminishing marginal utility. According as an actor k receives more benefits x
_{n}(k), his utility per added unit of this good will decrease. At the same time, giving up such a unit becomes more painful for the other actors. See p.310 in*Game theory and political theory*(1993, Cambridge University Press) by P.C. Ordeshook for the definition of U, and p.309 for a figure with K=2. On p.675 in the excellent and now classic reference book*Microeconomic theory*(1995, Oxford University Press) by A. Mas-Colell, M.D. Whinston, and J.R. Green a similar analysis is presented for K=3. Here the exchange of goods between three persons is surprisingly interpreted as a cooperation, and not as competition. In this case the cooperation consists of the collective decision to engage in exchanges. In a situation without cooperation the initial distribution of the goods determines the utility of each actor. According as more actors participate in the exchange, the utility possibilities expand. The best outcomes are obtained by the*grand*coalition {1, 2, 3}.

A policy problem with three actors is not exceptional. The introduction of this column mentioned the workers, entrepreneurs and the state in (neo-)corporatism. (back) - In paragraph 5.1 of
*Public choice III*(2009, Cambridge University Press) by D.C. Mueller it is remarked, that a policy decision always implies a redistribution of the existing means. Care must be taken, that this redistribution does not end in an exploitation of minorities by the majority. Moreover, redistribution by means of taxes commonly leads to welfare losses, due to the dead-weight loss. (back) - This example has been copied from p.305 in
*Game theory and political theory*. It is repeated on p.341, in a slightly modified form. (back) - Apparently the game yields more than the status quo, because otherwise the actors would not be prepared to play. It could be assumed, that a strategy s
_{j}(k) can be simply remaining passive. But the formation of coalitions is a concept, which suggests action. The consulted literature about social choice addresses this question merely in passing. On p.162-164 and 283 in*Beleid in beweging*a reference to network analysis is found. Here the mutual dependency is explained as a relation of power. For, the dependency is determined partly by the available alternatives. On p.267-284 in*Politiques publiques*networks are also analyzed. (back) - The argument is as follows. Consider the combination of strategies (s
_{1}(1), s_{1}(2)). When here the actor 3 chooses s_{1}(3), then the outcome for the coalition is (57, 50). When the actor 3 chooses s_{2}(3), then its outcome is (55, 20). So actor 1 may rest assured of the outcome 55, and actor 2 certainly obtains 20. In the same manner the guaranteed outcomes of the other coalitions are determined. (back) - See p.675 in
*Microeconomic theory*. On p.308 and 312 in*Game theory and political theory*the same matter is discussed, albeit in slightly different wordings. The characteristic function V(C) is mentioned on p.676 in*Microeconomic theory*, and on p.308 in*Game theory and political theory*. Apparently in this example V(C) is defined in such a manner, that for coalitions with 2 actors the mixed strategy is also allowed. When your columnist understands the example well, then the function V does not allow the mixed strategy for the coalition {1, 2, 3}. For, when this would be allowed, then the actors could mix various strategies, with a better outcome than (30, 30, 40). A promising candidate is the mixed strategy α × (s_{2}(1), s_{1}(2), s_{1}(3)) + (1−α) × (s_{2}(1), s_{1}(2), s_{2}(3)), with 0.55 < α < 0.8. But Ordeshook states on p.305: "Although all three persons [in {1, 2, 3} EB] could move to any outcome in the game, no outcome Pareto-dominates (30, 30, 40), so unanimous agreement to move elsewhere is impossible". (back) - One is inclined to think, that for the coalition C = {1, 2} the actor 3 in isolation will prefer his highest utility value u
_{3}in the table 1. But it is more advantageous to threaten each actor in the coalition with a punishment. For, in practice the utility distributions are dynamic. For instance, it is true that the actor 2 gets more than 30 thanks to his coalition with 1 and a convenient p mixing factor. But when subsequently the actor 3 chooses s_{2}(3), then the actor 2 will yet receive less than when the actor 3 would choose s_{1}(3). See p.344 in*Game theory and political theory*. When an attempt is made to calculate this bargaining process, then the risk aversion of the actors 1 and 2 must also be taken into account. Furthermore, your columnist can imagine, that the utility value u_{3}changes due to the formation of the coalition, perhaps because it affects the reputation of actor 3, or because it damages the mutual trust between 1, 2, and 3. (back) - By far the most well-known core is the contract curve in the exchange of goods. In chapter 25 of
*Foundations of social theory*Coleman applies this model to a broad exchange, where for instance also status or information are included. This social model is theoretically extremely powerful. But on p.686 he warns, that such a social exchange differs significantly from reality. For instance, the transaction costs are ignored. (back) - This definition originates from p.678 in
*Microeconomic theory*. (back) - This can be found on p.340-341 in
*Game theory and political theory*. (back) - Once more (see a previous footnote): on p.305 in
*Game theory and political theory*it is stated: "Although all three persons [in {1, 2, 3} EB] could move to any outcome in the game, no outcome Pareto-dominates (30, 30, 40), so unanimous agreement to move elsewhere is impossible". It can be implicitly deduced from this, that mixed strategies are not allowed, and according to your columnist only for this case the argument is indeed valid. (back) - Namely, in this case V({1, 2}) in the figure 3 shrinks significantly. The point μ
_{4}shifts downwards to (0, 10). Therefore the Nash equilibrium E lies above the boundary of V({1, 2}). The coalition is no longer attractive. The reader can check this for himself, or read it on p.341-344 in*Game theory and political theory*. It is curious that in this example Ordeshook also somewhat changes the outcomes of (s_{1}(1), s_{1}(2), s_{1}(3)) and (s_{2}(1), s_{2}(2), s_{1}(3)) in comparison with the table 1, which according to your columnist is unnecessary. (back) - The now following argument can be found on p.382 in
*Game theory and political theory*. (back) - On p.40 in
*Just playing*(1998, The MIT Press) K. Binmore remarks, that the concept of the core assumes actors, which are myopic with regard to the future. Many games do not have a core, so that the actors are indeed inclined to, for instance, reject a certain outcome u in favour of another outcome w. But next w will probably again be rejected by another actor. A rational actor is supposed to take this into account, and will probably yet accept the outcome u. On p.477 Binmore remarks, that policy decisions often evolve gradually. Compare the incrementalism of Lindblom. The policy cycle is completed many times before the selected solution becomes definitive. The policy problem can not be solved in one cycle. On p.65 and 204 in*Beleid in beweging*Bekkers even states, that*muddling through*and incrementalism can not be reconciled with the (sequential) policy cycle. On p.189-202 in*Politiques publiques*cases are described, where the policy is gradually developed, without a concrete political decision. On p.248 it is stated, that even during the execution the policies can still change. (back) - See p.408 in
*Game theory and political theory*. (back) - This example has been copied from p.347 in
*Game theory and political theory*. (back) - Here the bargaining model of Nash is not useful, because this example has more than two actors. On p.78 in
*Playing fair*(1994, The MIT Press) and p.496 in*Just playing*K. Binmore deliberately limits his argument to K=2, because the analysis of coalitions is so complex. (back) - This supposes, that the acceptance of the proposal requires a simple majority of more than 50%. The threshold could also be chosen lower or higher. See chapter 4 in
*Public choice III*. (back) - See p.376-377 in
*Game theory and political theory*. (back) - In the same way K. Binmore states on p.77-78 in
*Playing fair*, that usually actors reject some possible coalitions in advance. It is conceivable, that this will also eliminate the core. On p.404-420 in*Foundations of social theory*the effect of heuristics on policy decisions is studied with an abstract-sociological perspective. Here the various voting procedures are analyzed. When it is assumed, that actors apply heuristics without thinking, then in fact the actor-model of the homo sociologicus is applied. See*Handeln und Strukturen*(2016, Beltz Juventa) by U. Schimank. (back) - See for the following argument p.317 and 325-326 in
*Microeconomic theory*. (back) - More generally, the formula of the inner product
__η__·__u__= X represents a hyperplane with perpendicular__η__. Each__u__vector with its endpoint in the plane can be split into a component perpendicular to__η__, and a parallel component. Only the parallel component is relevant for the inner product. Thus all vectors__u__with endpoint in the hyperplane have the same projected distance on__η__with respect to the origin.

This model can be presented graphically for the case K=3. Since the boundary of the set of utility possibilities is a hyperplane, the boundary cuts out a triangle with identical sides in the R^{3}-space. Then the analysis can be focused on this triangular plane. See also the column about the inequality of incomes, notably the discussion of the budget plane. The triangle is called a simplex. See p.169 and 677-678 in*Microeconomic theory*, or p.128-129 and 319 in*Game theory and political theory*. (back) - See p.676 in
*Microeconomic theory*. A similar formula can be found on p.318 in*Game theory and political theory*. (back) - See p.678 in
*Microeconomic theory*or p.350 in*Game theory and political theory*. (back) - On p.301 in
*Just playing*Binmore mentions a theoretical objection, namely that the transfer of units of utility assumes a neutral attitude towards risk. However, actors are almost always risk averse. According to p.676 in*Microeconomic theory*it is often possible to generalize models, which have been initially developed with the assumption of transferable utility, to the case, where this transfer is not allowed. (back) - See p.318 in
*Game theory and political theory*. (back) - See p.320-321 in
*Game theory and political theory*. (back) - See p.321 in
*Game theory and political theory*. (back) - On p.174-175 in
*Beleid in beweging*compensations are mentioned as a real option. Here it is called a*package deal*. See also the paragraph about exchanging votes. On p.279 it is stated, that*side payments*to some actors can be unavoidable in order to obtain support for the policy. (back) - According to p.463 in
*Game theory and political theory*the Shapley value is difficult to interpret in games without transferable utility. Moreover, then various fair solutions can exist. On the other hand, in the main text it has already been remarked that, according to Ordeshook, unfortunately the transferable utility is rare in the public administration. Binmore also is not convinced by the Shapley value, on p.175 in*Just playing*. Your columnist still discusses the Shapley value in order to show, that morals can affect a policy decision. For instance, the Shapley value could be dictated institutionally by society. (back) - On p.466 in
*Game theory and political theory*and p.204 in*Rational-Choice-Theorie*(2011, Juventa Verlag) by N. Braun and T. Gautschi it is remarked, that therefore the Shapley value is yet also an index of power. (back) - See p.682 in
*Microeconomic theory*, or p.205 in*Rational-Choice-Theorie*. In this latter book the linearity is called*addability*. (back) - On p.463 in
*Game theory and political theory*Ordeshook believes that especially the axiom of linearity with regard to ν(C) is controversial, because ν is a*slippery concept*. (back) - See p.682 in
*Microeconomic theory*, p.465 in*Game theory and political theory*, or p.205 in*Rational-Choice-Theorie*. None of these sources derives the formula 5, although according to*Microeconomic theory*this is quite simple. (back) - See p.681-682 in
*Microeconomic theory*. There it is called the glove game. Each player has a single worthless glove. The coalition can form pairs of these, which do have value. (back) - Already four years ago your columnist described in a column various situations of bargaining between a monopoly and a duopoly. (back)
- See paragraph 5.9 in
*Public choice III*. (back) - According to paragraph 5.1 and p.106 in
*Public choice III*a decision via a (qualified) majority is also a redistribution of utilities. (back) - See p.107 in
*Public choice III*. (back) - See p.106 in
*Public choice III*, or p.327-331 in*Political economy in macroeconomics*. On p.84 and 87 Drazen notes, that in a system with logrolling by politicians the electorat must vote strategically, so deviate from its true preference. (back) - See p.327 in
*Political economy in macroeconomics*. Drazen also suggests, that the costs are often underestimated. The benefits can be large, for instance in a proposal for regional infrastructure, which also creates employment in the same region. (back) - See p.118 in
*Public choice III*, or in its general form p.108. (back) - The voting cycle must evidently not be identified with the policy cycle. (back)
- According to p.118 in
*Public choice III*especially the chairmen of the parliamentary groups supervise the honouring of deals with other parliamentary groups. See also the column about the repeated prisoner's dilemma game. (back) - On p.161 and further in
*Beleid in beweging*the ownership of policy problems is addressed. In chapter 4 of*Foundations of social theory*Coleman shows, that actors benefit from transfering the right to act to a skilled representative or agent. (back) - Game theory and the theory of social choice (rational choice paradigm) partly overlap each other, which is also apparent from the models in the present column. (back)
- The institutional frame of the strategies of the actors is described on p.112-116 in
*Politiques publiques*. (back) - See also p.170 in
*Beleid in beweging*, and p.212 and further in*Politiques publiques*. According to Meny and Thoenig (p.247) the social and public choice models in essence describe the*bottom-up*policy formation. The network can yet formulate policy morals. But the social choice theory distrusts dictated goals, and prefers methodological individualism. (back) - See p.138-139 in
*Politiques publiques*or paragraph 7.1 in*Handeln und Strukturen*. (back) - See p.297-298 in
*Politiques publiques*, or p.302 in*The Sage handbook of public administration*. (back) - See paragraph 5.3 of
*Beleid in beweging*for a survey of means of power. (back) - See paragraph 5.12 in
*Public choice III*. Also*Game theory and political theory*analyzes various agenda's, but your columnist has not yet studied this text. (back) - See p.234-238 in
*Beleid in beweging*, or p.240 in*The Sage handbook of public administration*for forms of coordination. (back)