Politonomics: A Meta-Theory Encompassing Political and Economic Decision Making

By John Lawrence July 14, 2011

Abstract

In “Social Choice and Individual Values,” Kenneth Arrow said ¹, “In a capitalist democracy there are essentially two methods by which social choices can be made: voting, typically used to make ‘political’ decisions, and the market mechanism, typically used to make ‘economic’ decisions.” This paper resolves that dichotomy by developing a meta-theory from which can be derived methods for both political and economic decision making. This theory overcomes Arrow’s Impossibility Theorem in which he postulates that social choice is impossible and compensates for strategic voting, an undesirable aspect of decision making according to Gibbard and Satterthwaite ^{2, 3}. Thus the politonomics meta-theory spawns both political and economic systems which are indeed possible and which cannot be gamed. In a typical voting system the outcome of an election among several candidates results in one realized outcome – the winner of the election - which applies to all voters. In a typical economic system, a consumer may choose among a variety of possible baskets of consumer items and work programs with the result that multiple realized outcomes are possible with a unique or quasi-unique outcome for each worker/consumer. As the number of possible realized outcomes of a political-economic decision making process increases, the process becomes more economic and less political in nature and vice versa. We show that as the number of possible realized outcomes increases, voter/consumer/worker satisfaction or utility increases both individually and collectively.

Introduction

A possible realized outcome is an alternative or candidate that can apply to a voter or worker/consumer after the voting or selection process occurs. For example, in traditional voting in a single member constituency there may be several alternatives or candidates, but, after the voting process occurs, there can be only one possible realized outcome and that is the winner of the election. This outcome applies to all voters. Now imagine a hypothetical political system in which there are two possible realized outcomes. Let’s say that after the voting process among several candidates occurs, A and B are both winners and that all those who prefer A to B are governed by A and all those who prefer B to A are governed by B. It’s as if there has been a split into two jurisdictions and two constituencies with A governing in one and B governing in the other. Please note that geography need not have anything to do with the two jurisdictions and constituencies. From an economic viewpoint, this process could be conceived of as a group of consumers selecting, out of several possibilities, one or the other of two baskets – A or B – of consumer items. Alternatively, in the political case for multiple possible realized outcomes, the winners could all be considered to be members of a parliament so that each voter would be represented to a greater or lesser degree by various members of parliament depending on how closely each member correlated with that voter’s expressed preferences. We take a utilitarian approach but avoid the controversy about interpersonal comparisons⁴ by assuming that, even for a welfare economy, voting methods are used, and hence each individual chooser is allocated the power of one vote thus equalizing all interpersonal comparisons. By the same token, utilitarian economic methods are used for voting rather than rank order preferences. Hillinger has made the case for utilitarian voting. Lehtinen⁶ asserts: “One reason why one indivdual has one vote under most rules is that each individual’s voting choice is considered equally important, and each individual’s utility is taken to carry at least roughly equal weight in the welfare function.” All individual choosers can express preference intensity but those preference intensities are considered to have equal weight.We assume without loss of generality a method of indicating preferences and intensities consisting of a rating scale with the lowest rating being zero and the highest rating being one. A real number between zero and one indicates preference as well as intensity. By using ratings instead of rankings or, if you prefer, a cardinal rather than an ordinal approach, Arrow’s impossibility theorem is rendered moot. So if candidate A is rated highly on voter 1’s preference rating scale and candidate B is rated quite low, we could say that the member of Congress or Parliament that best represents voter 1’s interests is candidate A with high intensity. If A is preferred to B, but only slightly, we could say the preference has low intensity.

Range/Approval Hybrid Voting

We know that for only one possible realized outcome, there is an optimal voting process called the range/approval hybrid ^7,8. Sincere range voting⁹ in which each voter expresses his or her preferences by rating the candidates with a real number between 0 and 1 will tend to maximize social utility although Lehtinen⁵ has shown that strategic voting can actually improve collective utility. Strategic considerations lead to applying a formula for each individual to his or her sincere preference ratings to maximize the outcome. If no prior information is known regarding other voters’ preferences, then all ratings greater than the individual’s average are changed to “1” and all ratings less than the average are changed to “0”. This changes the preference rating to an approval¹⁰ style vote which is the best one can do in terms of strategy in the absence of information regarding the other voters. This is equivalent to placing a threshold just below the mean of the preference ratings and adjusting the ratings of every sincere preference above the threshold to “1” and every rating below the threshold to “0”.

Finally, the approval style votes for each candidate are summed over all the voters and the candidate with the most votes is declared the winner. We will only consider the case in which individual strategy occurs without knowledge of information regarding other voters. The more general case in which knowledge of the sincere preferences of other voters is taken into account before strategizing is beyond the purview of this paper.

This process can be undertaken by individuals on a haphazard basis or it can be done on a universal basis by the social decision function itself just prior to the summation of utilities process. If done in this way, it negates the advantage of strategizing by individuals and equalizes the benefits of strategizing for all voters. There may be a loss or a gain of collective or social utility when strategy is used but this is either the price to be paid for eliminating strategic advantages for some voters or represents a serendipitous bonus in social utility. Therefore, since we know the optimal strategy for each individual, we can take it out of the hands of individuals and make it a part of the social decision function. By doing this two things are accomplished: 1) there is nothing to be gained by an individual voter in further strategizing so that each voter has an incentive to vote sincerely and 2) each voter will be assured that he will gain the advantages of an optimal strategy. Thus as Dowding and Van Hees¹¹ have stated: “Rather than being concerned about the strategic or gaming nature of voting systems, we should celebrate those aspects … [because] … the possibility of manipulation is often a virtue rather than a vice.”

Increasing the Number of Possible Realized Outcomes

Now if we imagine that there are a number of possible realized outcomes and not just one, we might ask how that affects strategic considerations? Obviously, if there is a very large number of possible realized outcomes compared to the number of voters, there is no need for strategy. Each voter can vote sincerely in the knowledge that he will get his first choice or close to his first choice as an outcome. But let us consider the case of two possible realized outcomes. Obviously, the threshold can be raised from just below the average of all the individual voter’s ratings since the voter has two chances to get an outcome closer to his most preferred outcome.

Let’s examine an individual citizen’s vote which represents a specification of utilities over the candidates with each utility corresponding to a position on the preference rating scale between 0 and 1. The greater the indicated utility, the higher the probability that a particular candidate will be elected due to that individual alone. We assume a variable number of possible realized outcomes or alternatives - m. For each m we place the threshold such that the expected individual utility using strategy is a maximum. The expected utility has two components – one is the utility gained from the fact that there are multiple possible realized outcomes. The second is due to the fact that all alternatives above threshold are strategically raised to 1, and all alternatives below threshold are strategically lowered to 0. As m increases, the individual voter under consideration has an increased chance of winning in the sense that one of his above threshold candidates becomes one of the possible realized outcomes. The average expected utility associated with a voter’s gaining one of his above threshold candidates as a possible realized outcome equals the probability associated with that voter’s above threshold candidates times the average of the above threshold utilities divided by the number of above threshold candidates. There is an additional expected utility associated with the increase in probability when the alternatives above threshold are strategically raised to one and the alternatives below threshold are strategically lowered to 0. Therefore, there are two components to the expected utility, and these two components are independent and additive. We seek to place the threshold in such a way as to maximize the sum of these two components.

Again the winners are chosen by summing the votes for each individual candidate across all voters. When there are m possible realized outcomes, the top m vote getters are declared winners, and an individual voter is assumed to have picked a winner if at least one of his above threshold candidates is one of the realized outcomes. For example, if there are m possible realized outcomes of baskets of consumer items, an individual will choose that basket which has the highest utility for him out of the m winners. If none of the m winners was an above threshold choice for that individual, he would just choose that basket among the m realized outcomes with highest utility for him. Although the analysis proceeds considering one individual voter, the social decision function can compute thresholds for all voters and apply the optimal strategy for each before summation of votes.

Maximizing Expected Utility for m = 1

Let’s consider the expected utility for a particular voter as a function of the number of possible realized outcomes and also due to strategy. Let n be the number of candidates and u_i be a utility (1 £ i £ n) associated with the i^th candidate. Let the components of probability under an individual’s control in the election be p_i. p_i = u_i for sincere voting (0 £ p_i = u_i £ 1). Let there be n_a candidates above threshold and n_b below threshold.

For m = 1, the probability of a particular candidate being elected in the absence of a priori information is 1/n. The probability of at least one of the candidates being elected in the set that is above threshold for the individual under consideration is n_a /n. Therefore, the component of average expected utility per candidate associated with those utilities above threshold is

E_m=1(u) = {n_a /n}S{p_i u_i}/n_a

^{all utilities above threshold}

The increase in expected utility due to the individual’s use of strategy, E_s(u), comes about from increasing those utilities above threshold to 1 and decreasing those utilities below threshold to 0.

Each utility represents a vote for a particular candidate expressed as a real number between zero and one. Strategically increasing a particular candidate’s utility increases the probability that that candidate will be elected, and strategically decreasing an opponent’s utility also increases the probability that that candidate will be elected. Suppose a voter has u_i > u_k. Candidate i has utility u_i and increasing u_i on the voter’s utility rating scale increases the probability that i will be elected. Decreasing opponent k’s utility rating, u_k, decreases the probability that k will be elected. If we increase candidate i’s utility rating strategically from u_i to 1, the increase in i’s probability of election due to a particular voter is proportional to 1 – u_i. If we decrease candidate k’s utility rating strategically from u_k to 0, the decrease in probability of k’s election due to a particular voter is proportional to u_k.

We consider the average increase in utility per candidate above threshold and the average decrease in utility per candidate below threshold so we divide each term by n_a and n_b respectively. The increase in expected utility due to strategy is, therefore,

E_s(u) = {1/n_a}S(1 - p_i) u_i + {1/n_b}S p_i u_i

^{all utilities above threshold all utilities below threshold}

The total average expected utility is then

E_total(u) = E_m=1(u) + E_s(u)

We set the threshold so as to maximize E_total(u). This gives the maximum strategic boost to the group of candidates located above threshold while minimizing the chances of that group of candidates located below threshold and takes into account the number of possible realized outcomes.

Increasing Number of Possible Realized Outomes

Now let’s consider the situation in which there are two or more possible realized outcomes. If one of the candidates above threshold for an individual voter is selected for one of the possible realized outcomes, this is considered a winning candidate for that voter. Therefore, in order to lose, all of the candidates filling the winning slots would have to be below threshold candidates. If there are n_b candidates below threshold and m possible realized outcomes, the number of ways that n_b candidates can fill m slots is the number of ways n_b things can be taken m at a time which is

n_b!/m!( n_b-m)!

For example, if m=2 and n_b = 4, we would have 4!/2!2! = 6

The total number of ways m slots can be filled is

n!/m!(n-m)!

So the probability that an individual’s winning candidates would all be below threshold is

{n_b!/m!( n_b-m)!}/{n!/m!(n-m)!}

The probability that at least one winner would be above threshold, P(win), would be

1 - {n_b!/m!( n_b-m)!}/{n!/m!(n-m)!}

= {n!/(n-m)! - n_b!/( n_b-m)!}/n!/(n-m)!

If m=1 and in the absence of prior information, P(win/m=1) = (n - n_b)/n = n_a/n which agrees with our previous calculation for E_m(u).

It is also true that n!/ (n-m)! < n^m Therefore, P(win) < 1 – n_b^m/ n^m. It can be seen that the probability of a winner goes up exponentially as the number of possible realized outcomes increases and approaches 1 asymptotically.

Therefore, the average expected utility for the component of utility, E_m(u), is

E_m(u) = [P(win)]S{p_i u_i}/n_a

^{all utilities above threshold}

Let S{p_i u_i }/ n_a = u_a

^{all utilities above threshold}

The average expected utility due to multiple winners is then

E_m(u) = [P(win)] u_a

The increase in utility due to an increase in the value of m is

[P(win)/m>1][u_a] – [P(win/m=1)][u_a]

Example: m=2, n=4, n_b =2

P(win) = {12 – 2}/12 = 5/6

P(win/m=1) = ½

So there has been a gain in utility by adding possible realized outcomes.

Examples for different data sets and various values of m are worked out in the appendix. The algorithm for setting the threshold is very simple: compute E_total(u) for each possible threshold and choose that threshold that maximizes E_total(u). The examples show that an individual voter’s total expected utility increases as the number of possible realized outcomes, m, increases. If the social decision function optimizes each individual vote for strategy, voters will have an incentive to vote sincerely knowing that the system will provide the strategy for them. The application of strategy by the social decision function itself couldprovide an increase in collective social utility.

If everyone votes sincerely, the only gains in expected utility would come from increasing the value of m, the number of possible realized outcomes. Therefore, the possible increase in social utility due to applying the optimal strategy can be computed as the difference between expected utility taking into count both strategy and the number of possible realized outcomes and the expected utility without using strategy. If the optimal strategy is applied to every voter by the social decision function itself, then the possible gain in collective expected utility would be the number of voters times the individual gain in expected utility due to strategy.

Therefore, if v = the number of voters and , the gain in collective utility from using strategy is potentially

v{E_total(u) - E_s(u)} = vE_m(u)

Conclusions

We have shown that both political and economic utility or satisfaction increase as the number of possible realized outcomes - either of an election process or of consumer/worker choices - increases. Range/Approval Hybrid voting has been shown to be an optimal strategy for maximizing individual utility when there is one possible realized outcome; it also transcends Arrow’s Impossibility Theorem because it rates rather than ranks preferences. A threshold is chosen in such a way that utilities that exceed the threshold are strategically maximized while those that are less than the threshold are strategically minimized. That optimal strategy can be incorporated into the social decision function and applied to every voter/chooser. The result is that the individual has nothing to gain by voting insincerely and the benefits of strategy are applied to everyone which could lead to an increase in collective social utility over the case in which all voters voted sincerely as Lehtinen⁵ suggests the possibility of. Since the optimal strategy for each individual is known, strategy can be taken out of the hands of individuals and placed in the social decision function which applies the optimal strategy for each individual before his or her vote is added to the tally. Any gains from strategizing are thus distributed on an equal basis throughout the electorate. This makes it impossible for an individual to gain anything by voting insincerely. As the number of possible realized outcomes increases, the efficacy of strategy decreases since the individual chooser is more likely to get an outcome he or she prefers regardless of strategy. Thus Arrow’s Impossibility Theorem as well as Gibbard and Satterthwaite’s concerns about manipulation have been averted. This theory represents a meta-theory from which both political and economic solutions can be derived and unifies the split in social choice theory between political and economic decision making.

Appendix

Let us consider the following example. Let m equal the number of possible realized outcomes.

Example 1:

u₁ = 1, u₂ = .8, u₃ = .6, u₄ = .4, u₅ = .35, u₆ = .3, u₇ = .2, u₈ = 0.

m = 1, n = 8.

E_m

(all utilities above threshold)

p_i = u_i n_a/n p_i u_i (1/ n_a) S p_i u_i (n_a/n)S( p_i u_i)/ n_a Threshold under u_i

1 .125 1 1 .125 1

.8 .25 .64 .9 .225 .8

.6 .375 .36 .8 .3 .6

.4 .5 .16 .7 .35 .4

.35 .625 .1225 .63 .39 .35

.3 .75 .09 .58 .435 .3

.2 .875 .04 .52 .455 .2

0 1 0 .45 .45 0

E_s

Above Below Threshold

Threshold Threshold Under

u_i = p_i p_i u_i 1-p_i (1-p_i )u_i {S(1- p_i) u_i}/n_a {Sp_i u_i}/n_b u_i E_s(u)

1 1 0 0 0 .20 1 .20

.8 .64 .2 .16 .08 .129 .8 .209

.6 .36 .4 .24 .13 .08 .6 .21

.4 .16 .6 .24 .16 .063 .4 .223

.35 .1225 .65 .2275 .1735 .043 .35 .2165

.3 .09 .7 .21 .18 .02 .3 .20

.2 .04 .8 .16 .177 0 .2 .177

0 0 1 0 .155 0 0 .155

Threshold under u_i E_m E_s E_total(u) = E_m(u)+ E_s(u)

1 .125 .20 .325

.8 .225 .209 .434

.6 .3 .21 .51

.41 .35 .223 .573

.351 .39 .2165 .6065

.31 .435 .20 .635

.22 .455 .177 .632

0 .45 .155 .605

Therefore, the threshold is placed just under u_i = .31 because u_i reaches a maximum of .635 there.

Example 2:

u₁ = 1, u₂ = .8, u₃ = .6, u₄ = .4, u₅ = .35, u₆ = .3, u₇ = .2, u₈ = 0.

m = 2, n = 8

P(win) = 1 - {n_b!/m!( n_b-m)!}/{n!/m!(n-m)!}

= 1 - n_b! (n-m)! /( n_b-m)!n!

= 1 - n_b!6!/( n_b-2)!8!

= 1 - n_b!/56(n_b-2)!

= {n!/m!(n-m)! - n_b!/m!( n_b-m)!}/ {n!/m!(n-m)!}

= {n!/(n-m)! - n_b!/( n_b-m)!} /n!/(n-m)!

= {(n-m)!}{n!/(n-m)! - n_b!/( n_b-m)!}/n!

= 6!{8!/6! - n_b!/( n_b-2)!}/8!

= {56 - n_b!/( n_b-2)!}56

= 1 - n_b!/( n_b-2)!56

E_m

(all utilities above threshold)

p_i = u_i P(win) n_a u_a E_r Threshold under u_i

1 .25 1 1 .25 1

.8 .464 2 .9 .42 .8

.6 .643 3 .8 .51 .6

.4 .786 4 .7 .55 .4

.35 .893 5 .63 .563 .35

.3 .964 6 .575 .554 .3

.2 .982 7 .52 .51 .2

0 1 8 .46 .46 0

E_s

Above Below Threshold

Threshold Threshold Under

u_i = p_i p_i u_i 1-p_i (1-p_i )u_i {S(1- p_i) u_i}/ n_a {Sp_i u_i}/ n_b u_i E_s(u)

1 1 0 0 0 .202 1 .202

.8 .64 .2 .16 .08 .129 .8 .209

.6 .36 .4 .24 .13 .083 .6 .213

.4 .16 .6 .24 .16 .063 .4 .223

.35 .1225 .65 .2275 .1735 .043 .35 .1995

.3 .09 .7 .21 .1795 .02 .3 .1865

.2 .04 .8 .16 .177 0 .2 .177

0 0 1 0 .155 0 0 .155

Threshold under u_i E_m E_s E_total(u) = E_m(u)+ E_s(u)

1 .25 .202 .452

.8 .42 .209 .629

.6 .51 .213 .723

.4 .55 .223 .773

.35 .563 .1995 .7625

.3 .554 .1865 .741

.2 .51 .177 .687

0 .46 .155 .615

Therefore, threshold is placed under u_i = .4 because E_total(u) reaches a maximum there. This shows that the threshold has been raised as m has increased from 1 (in Example 1) to 2 in Example 2 for the same data set.

Example 3

u₁ = 1, u₂ = .9, u₃ = .85, u₄ = .35, u₅ = .3, u₆ = .25, u₇ = .2, u₈ = 0.

m = 2, n = 8

P(win) = 1 - n_b!/( n_b-2)!/56

E_m

(all utilities above threshold)

p_i = u_i P(win) n_b u_a E_r Threshold under u_i

1 .25 7 1 .25 1

.9 .464 6 .95 .44 .9

.85 .643 5 .92 .59 .85

.35 .786 4 .775 .60 .35

.3 .893 3 .68 .61 .3

.25 .964 2 .61 .59 .25

.2 .982 1 .55 .54 .2

0 1 0 .48 .48 0

E_s

Above Below Threshold

Threshold Threshold Under

u_i = p_i p_i u_i 1-p_i (1-p_i )u_i {S(1- p_i) u_i}/n_a {Sp_i u_i}/n_b u_i E_s(u)

1 1 0 0 0 .264 1 .264

.9 .81 .1 .09 .045 .1725 .9 .2175

.85 .72 .15 .1275 .0725 .063 .85 .1355

.35 .1225 .65 .2275 .1113 .048 .35 .1593

.3 .09 .7 .21 .131 .034 .3 .165

.25 .0625 .75 .1875 .140 .02 .25 .160

.2 .04 .8 .16 .143 0 .2 .143

0 0 1 0 - - - -

Threshold under u_i E_m E_s E_total(u) = E_m(u)+ E_s(u)

1 .25 .264 .514

.9 .44 .2175 .6575

.85 .59 .1355 .7255

.35 .60 .1593 .7593

.3 .61 .165 .775

.25 .59 .160 .75

.2 .54 .143 .683

0 - - -

Therefore the threshold is placed just under u_i = .3

Example 4

u₁ = 1, u₂ = .9, u₃ = .85, u₄ = .35, u₅ = .3, u₆ = .25, u₇ = .2, u₈ = 0.

m = 4, n = 8

P(win) = 1 - {n_b!/m!( n_b-m)!}/{n!/m!(n-m)!}

= 1 - n_b!(n-m)! /( n_b-m)!n!

= 1 - n_b!(4)! /8!(n_b-4)!

= 1 - n_b!/8.7.6.5(n_b-4)!

= 1 - n_b!/1680(n_b-4)!

E_m

(all utilities above threshold)

p_i = u_i P(win) n_b u_a E_r Threshold under u_i

1 .5 7 1 .5 1

.9 .786 6 .95 .747 .9

.85 .929 5 .92 .855 .85

.35 .986 4 .775 .764 .35

.3 .997 3 .68 .678 .3

.25 .999 2 .61 .61 .25

.2 .9995 1 .55 .55 .2

0 1 0 .48 .48 0

E_s

Above Below Threshold

Threshold Threshold Under

u_i = p_i p_i u_i 1-p_i (1-p_i )u_i {S(1- p_i) u_i}/n_a {Sp_i u_i}/n_b u_i E_s(u)

1 1 0 0 0 .264 1 .264

.9 .81 .1 .09 .045 .1725 .9 .2175

.85 .72 .15 .1275 .0725 .063 .85 .1355

.35 .1225 .65 .2275 .1113 .048 .35 .1593

.3 .09 .7 .21 .131 .034 .3 .165

.25 .0625 .75 .1875 .140 .02 .25 .160

.2 .04 .8 .16 .143 0 .2 .143

0 0 1 0 - - - -

Threshold under u_i E_m E_s E_total(u) = E_m(u)+ E_s(u)

1 .5 .264 .764

.9 .747 .2175 .9645

.85 .855 .1355 .9905

.35 .764 .1593 .9233

.3 .678 .165 .843

.25 .61 .160 .77

.2 .55 .143 .693

0 - - -

Therefore, the threshold is placed just under u_i = .85. The threshold has been raised as m has increased from 2 (in Example 3) to 4 in Example 4 for the same data set.

References

1. Kenneth J. Arrow, Social Choice and Individual Values, Yale University Press, 1951, p. 1.
2. Allan Gibbard, "Manipulation of voting schemes: a general result", Econometrica, Vol. 41, No. 4 (1973), pp. 587–601.
3. Mark A. Satterthwaite, "Strategy-proofness and Arrow's Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions", Journal of Economic Theory 10 (April 1975), 187–217.
4. Camacho, A Societies and Social Decision Functions. Dordrecht: Reidel
5. Hillinger, “The Case for Utilitarian Voting”, Homo Oeconomicus, Vol.22, No. 3 (2005)
6. Lehtinen, Aki (2011) A Welfarist Critique of Social Choice Theory. Journal of Theoretical Politics 23(3): 359-381.
7. Smith, Warren, http://www.rangevoting.org/RVstrat3.html#conc
8. John C. Lawrence, http://www.socialchoiceandbeyond.com/strategicrangevoting.rtf
9. See: http://groups.yahoo.com/group/RangeVoting/
10. 10. See: http://rangevoting.org/Approval.html
11. Keith Dowding and Martin Van Hees, “In Praise of Manipulation, Cambridge University Press, 38, 2007, 1-15.