Consequentialism -X- (Pareto Efficiency) -X- Deontology

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Let us check the Polity to begin with:

1. N is the set of all individuals in society.

And that which their politics concerns – the state of society.

2. S is the set of all possible information contained within society, so that a set s ∈ 2S (2S being the set of all possible subsets of S) contains all extant information about a particular iteration of society and will be called the state of society. S is an arbitrary topological space.

And the means by which individuals make judgements about that which their politics concerns. Their preferences over the information contained within the state of society.

3. Each individual i ∈ N has a complete and transitive preference relation ≽i defined over a set of preference-information Si ⊂ S such that si ≽ s′i can be read “individual i prefers preference information si at least as much as preference-information s′i”.

Any particular set of preference-information si ⊂ Si can be thought of as the state of society as viewed by individual i. The set of preference-information for individual i is a subset of the information contained within a particular iteration of society, so si ⊂ s ⊂ S.

A particular state of society s is a Pareto efficient if there is no other state of society s′ for which one individual strictly prefers their preference-information s′i ⊂ s′ to that particular state si ⊂ s, and the preference-information s′j ⊂ s′ in the other state s′ is at least as preferred by every other individual j ≠ i.

4. A state s ∈ S is said to be Pareto efficient iff ∄ s′ ∈ 2S & i ∈ N : s′i ≻ si & s′j ≽ sj ∀ j ≠ i ∈ N.

To put it crudely, a particular state of society is Pareto efficient if no individual can be made “better off” without making another individual “worse off”. A dynamic concept which mirrors this is the concept of a Pareto improvement – whereby a change in the state of society leaves everyone at least indifferent, and at least one individual in a preferable situation.

5. A movement between two states of society, s → s′ is called a Pareto improvement iff ∃ i ∈ N : s′i ≻ si & s′j ≽ sj ∀ j ≠ i ∈ N .

Note that this does not imply that s′ is a Pareto efficient state, because the same could potentially be said of a movement s′ → s′′. The state s′ is only a Pareto efficient state if we cannot find yet another state for which the movement to that state is a Pareto improvement. The following Theorem, demonstrates this distinction and gives an alternative definition of Pareto efficiency.

Theorem: A state s ∈ 2S is Pareto efficient iff there is no other state s′ for which the movement s → s′ is a Pareto improvement.

If one adheres to a consequentialist political doctrine (such as classical utilitarianism) rather than a deontological doctrine (such as liberalism) in which action is guided by some categorical imperative other than consequentialism, the guide offered by Pareto improvement is the least controversial, and least politically committal criterion to decision-making one can find. Indeed if we restrict political statements to those which concern the assignation of losses, it is a-political. It makes a value judgement only about who ought gain (whosoever stands to).

Unless one holds a strict deontological doctrine in the style, say, of Robert Nozick’s Anarchy state and Utopia (in which the maintenance of individual freedom is the categorical imperative), or John Rawls’ A Theory of Justice (in which again individual freedom is the primary categorical imperative and the betterment of the “poorest” the second categorical imperative), it is more difficult to argue against implementing some decision which will cause a change of society which all individuals in society will be at worst indifferent to. Than arguing for some decision rule which will induce a change of society which some individual will find less preferable. To the rationalisitic economist it seems almost petty, certainly irrational to argue against this criterion, like those individuals who demand “fairness” in the famous “dictator” experiment rather than accept someone else becoming “better off”, and themselves no “worse off”.

Greed

In greedy exchange, when two individuals meet, the richer person takes one unit of capital from the poorer person, as represented by the reaction scheme (j, k) → (j + 1, k − 1) for j ≥ k. In the rate equation approximation, the densities ck(t) now evolve according to

dck/dt = ck-1j=1k-1cj + ck+1j=k+1cj – ckN – c2k —– (1)

The first two terms account for the gain in ck(t) due to the interaction between pairs of individuals of capitals (j, k−1), with j k, respectively. The last two terms correspondingly account for the loss of ck(t). One can check that the wealth density M1 ≡ ∑k=1 k ck(t) is conserved, and that the population density obeys

dN/dt = -c1N —– (2)

Equation (1) are conceptually similar to the Smoluchowski equations for aggregation with a constant reaction rate. Mathematically, however, they appear to be more complex and we have been unable to solve them analytically. Fortunately, equation (1) is amenable to a scaling solution. For this purpose, we first re-write equation (1) as

dck/dt = -ck(ck + ck+1) + N(ck-1 – ck) + (ck+1 – ck-1)∑j=kcj —– (3)

Taking the continuum limit and substituting the scaling ansatz,

ck(t) ≅ N2C(x), with x = kN —– (4)

transforms equations (2) and (3) to

dN/dt = -C(0)N3 —– (5)

and

C(0)[2C + xC’] = 2C2 + C'[1 – 2∫xdyC(y)] —– (6)

where C ′ = dC/dx. Note also that the scaling function must obey the integral relations

xdxC(x) = 1 and ∫xdxxC(x) = 1 —– (7)

The former follows from the definition of density, N = ∑ck(t) ≅ N∫dx C(x), while the latter follows if we set, without loss of generality, the conserved wealth density equal to unity, ∑kkck(t) = 1.

Introducing B(x) = ∫0x dyC(y) recasts equation (6) into C(0)[2B′ + xB′′] = 2B′2 + B′′[2B − 1]. Integrating twice gives [C(0)x − B][B − 1] = 0, with solution B(x) = C(0)x for x < xf and B(x) = 1 for x ≥ xf, from which we conclude that the scaled wealth distribution C(x) = B′(x) coincides with the zero-temperature Fermi distribution;

C(x) = C(0), for x < xf

= 0, for x ≥ xf —– (8)

Hence the scaled profile has a sharp front at x = xf, with xf = 1/C(0) found by matching the two branches of the solution for B(x). Making use of the second integral relation, equation (7), gives C(0) = 1/2 and thereby closes the solution. Thus, the unscaled wealth distribution ck(t) reads,

ck(t) = 1/(2t), for k < 2√t

= 0, for k ≥ 2√t —– (9)

and the total density is N(t) = t-1/2

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Figure: Simulation results for the wealth distribution in greedy additive exchange based on 2500 configurations for 106 traders. Shown are the scaled distributions C(x) versus x = kN for t = 1.5n, with n = 18, 24, 30, and 36; these steepen with increasing time. Each data set has been av- eraged over a range of ≈ 3% of the data points to reduce fluctuations.

These predictions by numerical simulations are shown in the figure. In the simulation, two individuals are randomly chosen to undergo greedy exchange and this process is repeated. When an individual reaches zero capital he is eliminated from the system, and the number of active traders is reduced by one. After each reaction, the time is incremented by the inverse of the number of active traders. While the mean-field predictions are substantially corroborated, the scaled wealth distribution for finite time actually resembles a finite-temperature Fermi distribution. As time increases, the wealth distribution becomes sharper and approaches equation (9). In analogy with the Fermi distribution, the relative width of the front may be viewed as an effective temperature. Thus the wealth distribution is characterized by two scales; one of order √t characterizes the typical wealth of active traders and a second, smaller scale which characterizes the width of the front.

To quantify the spreading of the front, let us include the next corrections in the continuum limit of the rate equations, equation (3). This gives,

∂c/∂t = 2∂/∂k [c∫kdjc(j)] – c∂c/∂k – N∂c/∂k + N/2 ∂2c/∂k2 —– (10)

Here, the second and fourth terms on the RHS denote the second corrections. since, the convective third term determines the location of the front to be at kf = 2√t, it is natural to expect that the diffusive fourth term describes the spreading of the front. the term c∂c/∂k  turns out to be negligible in comparison to the diffusive spreading term and is henceforth neglected. The dominant convective term can be removed by transforming to a frame of reference which moves with the front namely, k → K = k − 2√t. among the remaining terms in the transformed rate equation, the width of the front region W can now be determined by demanding that the diffusion term has the same order of magnitude as the reactive terms, i.e. N ∂2c/∂k∼ c2. This implies W ∼ √(N/c). Combining this with N = t−1/2 and c ∼ t−1 gives W ∼ t1/4, or a relative width w = W/kf ∼ t−1/4. This suggests the appropriate scaling ansatz for the front region is

ck(t) = 1/t X(ξ), ξ = (k – 2√t)/ t1/4 —– (11)

Substituting this ansatz into equation (10) gives a non-linear single variable integro-differential equation for the scaling function X(ξ). Together with the appropriate boundary conditions, this represents, in principle, a more complete solution to the wealth distribution. However, the essential scaling behavior of the finite-time spreading of the front is already described by equation (11), so that solving for X(ξ) itself does not provide additional scaling information. Analysis gives w ∼ t−α with α ≅ 1/5. We attribute this discrepancy to the fact that w is obtained by differentiating C(x), an operation which generally leads to an increase in numerical errors.