II. Moral externalities – external and internal sanctions ^*

Ethical theory in general maintains that moral institutions and the virtues supported by these institutions cannot be assessed on merely instrumental grounds. Even those theoreticians who contrary to this prevailing opinion argue that moral institutions must be justified with respect to their instrumental value still regard it as a matter of choice whether or not the individual addressee adopts proposed norms. As a rational being in the economic sense of that term he will do so if and only if the norms and the institutions based upon them are instrumental to the pursuit of his own aims, ends, or values. In such a case no externality exists. The norm was accepted freely within an autonomous act of choice.

Recently James Buchanan has suggested (in oral communication) to go one step further in a somewhat Nietzschean direction. The basic idea is a very simple one: From an economic point of view individuals should be expected to invest into activities that will impose certain constraints on other individuals' behavior if the imposition of the constraints furthers the aims, ends, or values of the investors. The investors use morals as an instrument of the pursuit of their own ends. They exert a moral externality on other individuals because this is helpful for reaching investors' aims, ends, goals etc. -- regardless of whether the imposed moral code is in the sense of any enlightened moral system "good" or "bad", "right" or "wrong".

The concept of a "moral externality", as it will be used subsequently, indicates that an individual Or group of individuals have the ability to influence the preferences over some risky prospects of another individual or group of individuals without the consent of the latter. The individuals who exert the influence may do so either by threatening to impose in a regular manner sanctions on certain forms of conduct (external moral sanctions) or by modifying the internal choice processes -- the emotions and expectations stirred by mental representations of choice situations -- of the choosing individuals (internal moral sanctions). In the first case individuals in the role of enforcers will be guided by some norm to impose sanctions on other individuals should they fail to act in certain ways. This is a conventional procedure in any norm guided system that incorporates punishments. In the second case individuals who adopt the role of enforcers will train other individuals to feel and choose differently. Though the training process also may involve the infliction of evils or punishments these sanctions are used differently than within the external sanctioning system. It is not the expectation of the sanction but rather the past experience of sanctions that guides choices in a more or less indirect way.

Subsequently it is assumed that both ways of social control, those using internally and [hose using externally effective moral externalities, are technically viable. Under this premise I will analyze some criteria for the choice of internal and external techniques of "moral control" in a standard two by two prisoner's dilemma (henceforth "PID"). First, some thresholds are determined beyond which internal as well as external sanctions become behaviorally effective. Secondly, it is shown formally that in principle the same behavioral effect may be reached with both methods of moral control. This leads, thirdly, to the question of optimally choosing between the two methods of exerting an externality of equal behavioral force on another individual by taking into account the costs of the two techniques.

1                    A model of external sanctioning in prisoners´ dilemmas

Imagine a standard prisoner´s dilemma with two players, A, B who can each perform two strategies. Let C_i denote the cooperative strategies of i=A, B while D_i indicates deviant or defection strategies of the two players A, B.

From this we get the conventional PD matrix

with tⁱ>rⁱ>pⁱ>sⁱ, i=A, B; i. e. Temptation>Reward>Punishment>Sucker holds good among the payoffs.

Assume now that the PD is only a subgame of a larger game that allows for additional actions of player A. A can choose either to play the original PD or to play a new game -- henceforth NG -- in which a sanction is imposed on B for playing the uncooperative strategy D_B.

B has full control only over his own choice of actions but not over the result that emerges from the choices of both players. Therefore B can be held responsible only for his action choice. A rational individual B can be motivated only by a sanction attached to this choice of an action. Let us assume further that the sanction is a negative one or a punishment (positive sanctions would do also) and that A -- for the time being -- does not incur costs of inflicting the punishment on B. Then the n. g. matrix becomes:

It should be noted first that A has a dominant strategy in NG as well as in PD regardless of the strength of ∂. For, A´s payoff is not affected by imposing a sanction on B. A is still facing the PD payoffs as far as his own payoff structure is concerned. A´s choice, as before, will be the dominant alternative D_A. Therefore any equilibrium of the new game will be in the second row. If this is common knowledge the crucial relationship is the one between s^B and (p^B-∂). A will get the temptation payoff only if

s^{B ≥} (p^B-∂).

To reach this result A will want to make sure that the strict inequality holds good

s^{B >} (p^B-∂).

In this case rational play of B will bring about A´s most favored outcome. B is threatened to be punished by A with an utility loss of ∂ should B ever choose to perform an uncooperative action. Therefore, if sanctioning is costless for A he should choose ∂ such that

s^B - p^B > - ∂ or

∂ > p^B - s^B.

If sanctioning is not costless then A must check whether in terms of his own utility function the cost of imposing sanction ∂, c^A(∂), fulfils

c^A(∂) < t^A - p^A

which is equivalent to

t^A - c^A(∂) > p^A.

Imposing the sanction is worthwhile for A if the expected payoff t^A of the new game NG minus the costs c^A(∂) of bringing into effect the new "rules" -- i.e. t^A - c^A(∂) -- is higher than the expected payoff p^A of the old game -- both according to the predicted equilibria. In this case A should choose to impose an externality and thus choose to play NG

The extensive form of such choice may be sketched in the following way

It may finally be noted that an individual A who wants "to play save" may be interested in reaching an equilibrium that is not only an iterated dominant equilibrium but fulfils the stronger criterion of a dominant strategy equilibrium. In that case A must make sure that choosing the cooperative action  becomes a dominant strategy for B. Then A must see to it that not only ∂ > p^B - s^B holds good but rather that two inequalities are fulfilled

∂ > p^B - s^B

and

∂ > t^B - r^B.

In that case additional investment into punishment activities is necessary if the difference between temptation and reward exceeds the one between punishment and sucker’s payoff i.e. if t^B - r^B > p^B - s^B. A will invest the additional amount only if c( ∂>t^B - r^B  ) < t^A - p^A. Including punishment costs -- infliction and suffering -- the matrices of subgames of  NG become for c(∂)<t^A - p^A:               

while if c(∂) is too large the original PD of Matrix 1 emerges. In the latter case the exertion of an external moral externality is too costly to be worthwhile and NG will not be chosen by A.

2                    A model of internal sanctioning in prisoners´ dilemmas

Explicitly introducing utility functions of the v. Neumann-Morgenstern-Ramsey-type and indexing them with i=A, B we get an equivalent statement of the payoff structure of the PD that reminds us of the underlying strategy combinations of the two players

u> u> u> u

and

u> u> u> u.

The conventional utility representation of preferences over lotteries is all inclusive in that it shows what the individual chooser all things considered prefers. It takes into account all value dimensions that the choosing subject applies to the world (including attitude towards risk, hedonistic, aesthetic, moral values ...) and in a sense "aggregates" them to one numerical scale representing the order among alternatives that eventually  emerges from the deliberations of the subject. If we violate this premise of the standard model and assume that the world has two different value dimensions a natural one -- indicated by an upper index "n" attached to the utility index -- and a moral one -- indicated by an upper index "m" -- we get two different representations. (Of course one could split the analysis even further but this would not add any insights to the arguments presently pursued -- though Arrow-Condorcet-like cycling might occur with evaluation along more than two dimensions in the intra-personal aggregation process.)

Assume now that the natural ("naked") preferences of the two individuals have a PD structure of the following form

u> u> u> u

and

u> u> u> u.

Assume further that the two individuals are such (their internal technology of decision making is such) that they can adopt a moral order besides the natural one. They can look at the world through two different windows. For the sake of the present argument it is suggested that the moral order is a broadly speaking Kantian one (assurance game preferences as "moral" preferences would form another less radical and thus perhaps more plausible consequentialistic candidate) in which each actor focuses exclusively on whether or not he himself  has performed his duty (i.e. cooperated). We get

u= u> u= u

and

u= u> u= u.

Evidently the assumption that there are two orders influencing the behavior of an individual actor may be interpreted as modelling the classical tension between "duty" and "desire". The mental struggle between "desire" and "duty", between aiming at naturally preferred end states and striving to perform the morally right act (regardless of consequences) will eventually lead to an effective ordering. All things (or both dimensions) considered one of the two influences on the individual will prevail. The emerging effective, overall or aggregate ordering may be denoted by:

ufor i=A, B.

From the natural and the moral order we do not know yet what the effective ordering is. We have to know the relative weight of the two orderings and how they can be combined to an overall order. Following Rainer Hegselmann (cf. 1988) a quite natural way to form an effective ordering seems to result from a simple convex combination of the natural and the moral utility function:

u:= µ_i u+ (1-µ_i ) umit  0 ≤µ_i ≤ 1,  i=A, B.

If the original natural and moral utility functions conform to the requirements for forming expected utilities then this function will too. In this sense the convex wheighing process seems to be adequate. The utility functions being unique up to positive monotonic linear transformations only it is obviously necessary to find a non-arbitrary way to choose some linear transformations and values for µ_i, i=A, B, such that the aggregation is adequate in an interpretative sense. Let us assume for the sake of argument that this problem of intrasubjective utility comparisons can be solved in a reasonable way. Then the factors µ_i, i=A, B show the weight that moral preferences get in the individuals´ determination of their effective preference orders.

Following Rainer Hegselmann µ_i may be dubbed the morality coefficient of the individuals A and B respectively. This coefficient indicates how strongly moral convictions or internal moral sanctions count in the decision process of the chooser. Obviously, 1-µ_i shows the weight of the non-moral or natural preferences.

If µ_i = 1, i=A, B, then the individuals are perfect moral actors in the Kantian sense. If at least one actor has a morality coefficient less than "1", then natural preferences can at least potentially have some influence on the effective preferences and thus on how the new game NG is eventually played. On the other hand, should µ_i = 0 hold good for i=A, B then the game is in any case played as a natural PD

A numerical example again borrowed from Hegselmann (1988) may illustrate these considerations. Let the natural payoff matrix:             

The matrix of the moral point of view of the two Kantian players might be: 

With µ_A = 0,3 and µ_B = 0,4  we get the payoff matrix of the effective game:

Here the moral point of view prevails over the natural point of view. Cooperative strategies become dominant. However, and quite trivially so, this will not necessarily be the case. All depends on whether or not the morality coefficient is higher or lower than certain thresholds.

To analyse the problem of thresholds in a general manner we can again use variables for the utilities (natural, moral, effective). Then we get:                       

with T>R>P>S for the game under natural payoffs, while the game played under moral payoffs is                   

with H>N, whereas

is the payoff matrix of the new game after convex combination or aggregation of the natural and moral utilities to an effective utility measure. The latter matrix may be written as:

For convenience it was again assumed that the values T, R, P, S, H, N  are identical for both individuals. This assumption could be given up easily -- utilities are unique only up to positive monotonic linear transformation anyway and, in any case, the morality coefficients may be different across individuals and thus effective orderings may differ  between individuals. (Both modelling assumptions, the identity of the payoffs and the inequality between the moral coefficients, seem to be helpful in isolating the influence of the disposition to act according to moral considerations.)

Thresholds of the morality coefficients such that the original PD in natural payoffs will be eliminated in NG are of great interest.. As is known and was illustrated already by the numerical example the original dilemma is eliminated if

r_A > t_A and  s_A > p_A

and

r_B > t_B and s_B > p_B;

for, then the equilibrium of the effective game is characterized by (C_A, C_B). Cooperative behavior becomes the dominant strategy equilibrium of the game. Weaker conditions that also may lead to deviations from the original uncooperative dominant strategy equilibrium under natural payoffs can easily be formulated. Using some simple algebra Hegselmann has shown that two equations which characterize relevant thresholds a, b are of central importance here:

    (i.)           

  (ii.)           

Using these equations one can easily that cooperation becomes a dominant strategy for player i=A, B iff

(1)   µ_i ≥ max {a, b}, for a ≠ b

or

µ_i > a & µ_i > b, for a =b .

The effective game is a PD still iff

(2)   µ_A < min {a, b} and µ_B < min {a, b}.

mutual cooperation is an equilibrium in the effective game iff 

(3)   µ_A ≥ a  and µ_B ≥ a.

Further, in terms of their original natural preferences players who have not yet adopted moral norms will anticipate the truth of the two following propositions:

(4)   If a cooperative equilibrium exists (µ_A ≥ a  and µ_B ≥ a) both players are as well off by playing their cooperative strategies as if playing any other equilibrium strategy ("mixing" included).

(5)   If even "µ_A > a  and µ_B > a" holds good, then both players are strictly better off if both are playing their cooperative strategies.

([ar_A + b] > [at_A + b] iff r_A > t_A -- for  a > 0 -- therefore the propositions about thresholds are invariant with respect to linear transformations and thus in this sense meaningful.)

The propositions (1) to (5) --  and in particular (4) and (5) -- characterize why rational players might think of the adoption of a morality as a useful device to further their own ends in terms of their original or natural or status quo preferences. From these considerations the players know how it would pay off for them if they could change the payoffs of the game. Or, to put it slightly different, they know how they should change that part of the rules of a PD that is related to the preference or payoff structure if they can.

Now, from the point of view of every day language it is perhaps a bit strange to talk about preference changes as changes of the rules of a game. Usually we think of what has technically been described as the "game form" -- i.e. rules of the game except for the preferences (or allowing for any preference orders whatsoever or at least classes of preference orders) -- if we imagine the "rules of the game". However, in the usage of standard game theory the rules of the game comprise anything that is beyond the strategic intervention of the players. And this, in general, includes the preferences. As game theory focuses on strategic interaction this is a technically sound approach as long as preferences are taken as unalterable. If as in the present discussion one assumes that preferences may be changed by strategic intervention then it is obvious for the same technical reasons that they no longer can be subsumed under the rubric of the "rules of the game".

Again, any form of analysis that is not designed to impose externally preferences on the players of a game -- i.e. the analyst is not willing "to play God" -- has to take some of the players´ preferences as given. The evaluation of outcomes of strategic interaction has to start from something after all. Natural preferences of a time period that is taken as the starting point of the game may serve here as a natural point of reference. (If we want to avoid the somewhat doubtful notion of "naked" preferences then the status quo preferences may serve as the natural preferences. This is also a good reason why the status quo is of special importance within an economic analysis. For otherwise the analyst must impose his own or at least externally determined values. Status quo is not special for the players but rather for the analyst!).

If preference changes can be induced by actions within the play of an ongoing game then we should model these strategic possibilities explicitly. One not altogether implausible way to do this may arise from pursuing the idea of an effective utility function that is the outcome of the convex combination (or weighted sum) of a natural utility function (over end states) and a moral utility (derived from performing the right acts) somewhat further.

Taking up the ideas of section one this can be accomplished if we assume that µ_A is a function that may be influenced by actions of B and that vice versa µ_B is a function of certain measures A can take. We may again take into account the costs j incurs in modifying µ_i, i.e. c^j(µ_i), i≠j, j, i=A, B. The analysis proceeds in two steps. First, it is assumed that only one of the players can modify the payoffs of the other one. The potential for exerting a moral externality via influencing the morality coefficient of another person is unilaterally distributed. This way of exerting a moral externality is related to the conventional external punishment behavior then and it is shown that both ways are -- at least in a sense -- formally equivalent. In a second step it is indicated that a bilateral potential to exert moral externalities may give rise to "moral war". Though moral disarmament may lead to a Pareto superior state this state may be in disequilibrium.

3                    Equivalence of internal and external moral externalities

If only one of the players has the option to modify the payoffs of the other one then all depends on whether the costs for eliminating the dominance of the deviant act of the other player are less than the gains to the modifying player (or the difference between the temptation and the punishment payoff of the latter one). It is not necessary to repeat the simple analysis from above here. If the relevant threshold has been determined for the internal sanctioning process and if both the external and the internal methods of sanctioning are viable then the rational chooser has to check which of the two alternative methods is the "cheaper" one. He can indeed do this all the time: Provided that internal and external mechanisms may be modelled in the simple ways suggested in the present note it can be shown that for any external sanction mechanism there exists an internal one of equal strength.

Assume that individual B is the one who can exert a moral externality either by triggering the internal mechanism of A or by imposing external sanctions on A. A cannot influence B´s preferences but B can exert a moral externality on A in that B can change A´s value of µ_A. Consider

(In this matrix B´s payoffs are printed in bold letters. These payoffs cannot be changed by A because B´s utility function is beyond A´s influence.)

For i=A and 0<µ_A<1 (disregarding the extreme cases for the time being) define the following positive monotone linear transformation :

[αx_A + β] =: x'_A for x_A= r_A, s_A, t_A, p_A  with

α:= > 0

β:= H .

Applying this transformation to the original values x_A= r_A, s_A, t_A, p_A  i. e. to the expression given in "{}" in the next formula below we immediately get:

r'_A= [α r_A + β] = [ {( 1-µ_A ) R + µ_AH} + H ] = R

s'_A= [α s_A + β] = [ {( 1-µ_A ) S + µ_AH} + H ] = S

t'_A= [α t_A + β] =  [ {( 1-µ_A ) T + µ_AN} + H ] = T -

p'_A= [α p_A+ β] =  [ {( 1-µ_A ) P + µ_AN} + H ] = P -

This simple result has an interesting interpretative implication. One should observe that by assumption H>N. Therefore we get

 > 0 or  - < 0 .

The effective utility function that takes the values t'_A,  p'_A after transformation is equivalent (represents exactly the same preference order over lotteries) to the effective utility function that adopts the values t_A, p_A respectively. Both utility functions represent the outcome of a process of internal  sanctioning which was modelled by the convex weighing of natural and moral utility. The argument shows that an equivalent effective utility function could be reached also if the original natural utility function is transformed by subtracting a constant -- namely  -- from t_A and p_A.

Now, an external  sanction applied to the performance of uncooperative acts amounts to the same thing as substracting a constant value g from the natural utility values of the player whenever he should perform that act.

Let  g:=  , then one gets a very simple expression for the effective game which is played after B has influenced A´s internal decision procedure such that A gives some positive weight µ_A to moral considerations and thus values cooperation as an end in itself

Thus, instead of influencing µ_A player B could have threatened to punish A after performing an uncooperative act by an external sanction that A would evaluate at utility -g. This shows that -- under the present assumptions -- for a given effect on ordering there exist equivalent external and internal sanctions. This is a simple formal justification for the widely shared view that internal and external forms of punishment are somehow in the same category. Further, for any effect beyond the relevant thresholds costs of external and internal sanctioning can be compared. The player who can exert the moral externality can choose between alternative means to reach his end of constraining choices of the other player. Knowing that he will be indifferent between the two effects of his investment he can compare the costs and even determine the optimal mix of exerting a moral externality. Marginal rates of transformation could be calculated, etc. -- 1 will not pursue any further these more conventional economic questions here but rather make a final remark on what is going to happen if both players can exert a moral externality with both techniques. As indicated before, this might lead to what be called moral war.

4                    Bilateral moral externalities

The unconstrained exertion of externalities is a kind of war. In this sense people may be at war with each other if they engage in exerting moral externalities on each other.

In a typical prisoner´s dilemma game the outcome of war as evaluated in terms of natural preferences may be desirable because of the fact that the dominant strategy equilibrium is Pareto dominated by the payoff combination from cooperative play of both players. The result of the process of exerting moral externalities is                       

From preceding considerations it is obvious that this matrix is equivalent to the matrix of mutual influence on internal sanction systems

 (C_A, C_B) may become an equilibrium outcome here. This is desirable from the point of view of the natural preferences of the two players provided that the investment for reaching this result is not too costly for both players.

However, moral war like other kinds of war may lead to Pareto inferior results. And, as in other kinds of war disarmament may be desirable but at the same time hard to achieve.

If for instance a game like the battle of the sexes is played then typically new problems will emerge on a higher level with respect to moral disarmament. Consider the following matrix of the “battle of the sexes”

with a>b>0.

Taking into account additional ways of influencing each other under conditions of moral war one can derive a new payoff matrix. In terms of his own utility A incurs investment costs of c(k) for exerting the externality of amount k on B -- where k is evaluated according to B´s utility function. B is punished for not complying with A´s preference for the (X, X)-equilibrium. At the same time B incurs investment costs of c(h) for exerting the externality h on A. He punishes A for not complying with B´s preference for the (Y, Y)-equilibrium and thus for not playing X_A.

If we assume again that the costs of influencing each other are basically set up costs then the matrix becomes

Now, evidently the c(h) and c(k) terms do not affect the ordering of alternatives. They are simply a dead weight loss. We can neglect them without affecting the structure of the game. After substituting the expressions in squares by the original ones we get -- as far as the incentive structure of forward looking choices is concerned -- an equivalent representation of the new game

If both punishment activities are successful beyond the relevant thresholds such that compliance with the other player´s wishes becomes a dominant strategy then we must have a-h<0 and a-k<0 -- and thus h>a>b and k>a>b. In that case the dominant strategy equilibrium will be (Y_A, X_B). This outcome is worth for both players than any of the equilibria of the old game.

Taking into account the sunk costs or the investment into what may be called moral rent seeking things get even worse. The equilibrium will yield an actual payoff of (0-c(k), 0-c(h)). Of course, the same argument holds good too as far as the other payoffs are concerned because they are diminished by a constant too.

It is obvious that both players have a common interest in avoiding moral war in the battle of sexes game. On the other hand, both players know that the best state of affairs for them might be to go to moral war unilaterally as long as the investment costs c(k), c(h) are both less than the difference a-b as evaluated in their individual subjective utilities. In such a situation the decision structure becomes quite complicated if as necessary all subgames of the overall game are studied carefully and the information structure is taken into account. (As in any other arms race much depends on whether or not the players can immediately observe the "moves" or investments of their adversaries. Of course, the above calculations for an internalised sanction system with a morality coefficient could also be applied again.)

5                    Concluding remark

Though moral war is a bad from the point of view of the original or natural preferences, from the point of view of effective preferences the utility loss is possibly not "felt" anymore. This leads to a new class of philosophical issues about the interpretation of utility functions. One might wonder whether it is still meaningful to talk about the natural preferences if they do not exist anymore. Because what exists are effective preferences only. If the internal decision situation of the chooser is neglected and only observed choice matters then evidently the new preferences would be sufficient to give a full account of the situation. But from an internal point of view of the decision maker there may be several utility functions representing different value dimensions. There may be desire and a feeling of duty opposed to it. There may be a tension that is felt and not only an order of states of affairs. As analysts we must decide whether the classical hedonistic connotations of "feeling utility" in terms of pleasure and pain shall be involved or whether we want to use a purely representative utility notion in analysing the moral games of life.

Some related topics will be addressed in the next paper discussing the specific example of a work ethic which as most of the themes discussed in the present paper were taken up recently by James Buchanan too. When addressing the theme of work ethic it may be helpful to keep in mind the somewhat sketchy remarks on moral externalities that were made so far. They can serve as a kind of background against which the problems of work ethic may be seen somewhat clearer. On the one hand, there are dangers of moral war. On the other hand, the equivalence of internal and external sanctions should remind us that ethical indoctrination with a work ethic is morally problematic.

<< References >>

*	Kliemt, Hartmut: Papers on Buchanan and related Subjects. – Munich (Accedo Verlagsgesellschaft), 1990: p. 37 – 60. (Studies in Economics and Social Science (SESS), Vol. 1 (1990))

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