A Voting System for Internet Based Demo ra y Stefan Dirnstorfer Thetaris GmbH stefanthetaris.de Abstra t. A lot of resear h has been spent on the pro ess of de ision making in large groups. While we easily nd widely a epted solutions within groups of friends, the pro ess somehow does not s ale up to larger and more distant groups of individuals. Some of problems have been attributed to the multiple dilemmas known in voting theory. This paper introdu es an intera tive voting game that avoids the loss of pareto optimallity and the inuen e of voting ta ti s that is based on non-publi ballots ast by peer voters. At its down side it might be subje t to non onverging voting behaviour and not produ e a nal result within a onned time frame. 1 The So ial Dilemma The so ial dilemma, often des ribed as the prisoners dilemma, is said to be the fundamental problem of so ial ooperation. If the so ial dilemma is present, the group de ision will not be pareto optimal, due to the individuals pursuing their self interests. This se tion explains that the so ial dilemma is present in a when voting on a number of inter onne ted issues in a separate vote. Any voting system that is not pareto optimal is not suitable to solve the so ial dilemma. Minor disputes between group members must be set aside in favor of a ommon goal. The examples below show that separate voting on separate de isions is not suitable for resolving so ial dilemmas that require the arbitration over multiple initiatives. 1.1 Example 1 Mu h thought has been spent on the pro edures of a single vote on mutually ex lusive hoi es, su h as the ele tion of a representative or the ele tion of a politi al party [1, 4℄. As soon as multiple hoi es have to be made on separate issues demo rati voting does a very bad job in satisfying the involved voters, as will be shown by the following example. Figure 1 shows a ballot paper that ould be used to ondu t a vote on two separate initiatives. As it turns out, a lassi al voting system is often unable to produ e the most satisfying result. Assume that two people parti ipate in this vote. For simpli ity they will be alled Left and Right. Left is a fanati supporter of initiative A and slightly biased against initiative B. Right is a radi al fan of initiative B and somewhat biased against initiative A. Sin e both initiatives don't ex lude ea h other, it is obvious that they ould both be implemented with satisfying results for Left and Right. However, that is not the result of a demo rati vote. The fundamental idea behind a demo rati vote is that ea h parti ipant expresses their personal preferen es in a selsh fashion and yet, the optimal Fig. 1. Ballot paper with two non-ex lusive hoi es Initiative A Initiative B Yes Yes No No de ision ould be derived through a mathemati al ounting s heme. Under su h an obje tive Left would vote (Yes,No) and Right would vote (No,Yes). Without ooperation neither initiative an a hieve the ne essary majority and the result would be (No,No), although both parti ipants agree that (Yes,Yes) would have been a better option. This result is known in game theory as the Prisoners Dilemma [5℄. Figure 2 shows the dilemma in matrix form. Fig. 2. So ial dilemma in a vote on two independent initiatives: Left would be happiest with (Yes,No), while Right favors (No,Yes). Both ould be satised with a (Yes,Yes) result, but unarbitrated voting leads to a suboptimal (No,No) de ision. Right: (No, Yes) Right: (Yes,Yes) Left: (Yes, No) Result: (No, No) Result: (Yes, No) Left: (Yes, Yes) Result: (No,Yes) Result: (Yes, Yes) In pra ti al politi s the optimal solution requires a benevolent member of parliament who arbitrates between Left and Right. This job an only be done through a small number of delegated representatives who an build su ient trust in ea h other. Internet based a tivist groups usually do not have the re- sour es to sustain su h arbitration eorts, or it even ontrasts self sele ted grass- roots demo rati prin iples. With an in reasing number of de isions that have to be made, more and more arbitration opportunities will be lost and group satisfa tion de reases. Looking at our simple example one might rightfully obje t that there must have been someone to ome up with a voting s heme that eli its the interde- penden e between the two initiatives. Many su h s hemes are subsumed under the Condor et voting system whi h requires the parti ipants to assign ordinal preferen e numbers to the four mutually ex lusive hoi es (No,No), (No,Yes), (Yes,No), and (Yes,Yes) as shown in gure 3. All ommon variations of the Con- dor et vote an solve this simple problem [4℄. Su h overar hing voting s hemes 2 1. THE SOCIAL DILEMMA are feasible for a small number of learly interlinked issues, but eort grows exponentially with the number of involved de isions. Fig. 3. A Condor et vote ould solve the so ial dilemma for two hoi es, but be omes impra - ti al for a larger number of seemingly independent de isions. Choi es for initiatives A, B Left's preferen es Right's preferen es (No, No) 3rd 3rd (No, Yes) 4th 1st (Yes, No) 1st 4th (Yes, Yes) 2nd 2nd 1.2 Example 2 This se ond example will again demonstrate that demo rati voting does not automati ally rea h the pareto optimal solution for the group, if it is applied separately on seemingly un onne ted issues. This time initiative A and B are not te hni ally ex luding ea h other, but they ompete for some sort of limited resour e that makes in impra ti al to implement both, A and B. Fig. 4. A possible vote distribution for two hoi es that are not stri tly ex lusive but unduly strain group resour es, when ombined. pro A ontra A pro B 10% 45% ontra B 45% 0% Suppose two initiatives, A and B, are ompletely independent in terms of phenomenologi al ee t, but both deplete the group's limited nan ial or natu- ral resour es. Only 10% of the ele torate think that it makes sense to implement both measures, while 90% think that only one of the suggested measures an be implemented in a sustainable fashion. Yet, under these tight onditions sepa- rated demo rati voting an lead to exa tly this highly unsatisfa tory out ome for the group. Figure 4 shows the distribution of voters and gure 5 shows the results. Fig. 5. Separated voting favors both initiatives. Only 10% are satised, while 90% omplain about wasted resour es. pro ontra Initiative A 55% 45% Initiative B 55% 45% 3 If su h a demo rati system with independent votes on independent politi al issues governs a politi al state at least a nan ial solution an be found easily: Charge future generation. However, for small groups as well as for prot oriented businesses this is not an option. That is why grassroots demo rati prin iples are never found in the industry on any signi ant s ale and it is the reason, why grassroots organizations always break apart qui kly. 2 The Internet Voting System The internet voting system is a voting system primarily designed to help solving the so ial dilemma. It is to be played intera tively and requires all votes to be publi ly viewable during the voting phase. Sin e oni ting positions an only be resolved by arbitrations over multiple initiatives, the voting system must onsider all open de isions simultaneously. Thus, a huge number of potential out omes have to be evaluated, whi h is only possible by utilizing the unique features of an intera tive omputational platform without the restri tions of a physi al ballot sheet. Hen e, the internet voting system is spe i ally designed for the internet and is not appli able to paper based voting. The internet voting system is more akin to ooperative and intera tive game play than to lassi al voting. Many strategy games prove that it is possible to take simultaneous and ne grained judgments in a huge number of politi al elds. This potential must be exploited to nd widely agreed preferen e orders. The results will be more fuzzy in terms of absolute valuations, but more agree- able and onsistent in terms of its result. At its downside it does not enfor e a nal winner if the voters preferen e orders are ir ular. 2.1 Vote Counting The internet vote ounting s heme follows the physi al model of me hani al for es. Dierent spa ial dire tions orrespond to politi al initiatives that an be supported or opposed. All users of the system an pull with equal for e into any dire tion that orresponds to their mix of politi al onvi tions. Just as in physi s transverse for es an el ea h other, while orrelated for es add to ea h other. We an represent the voting behavior as a matrix V ∈ Rm×n . The entries vpi refer to the voting weight assigned by parti ipant p to initiative i. The weights vpi ∈ R an be positive to express support and negative to express opposition. The total voting weight assigned by a parti ipant is onned to length one of the resulting for e ve tor vp in Eu lidean spa e. Hen e, the summed squared voting weights are onned to one. n (1) X 2 kvp k = (vpi )2 = 1 i=1 The resulting group preferen e R(i) for an initiative i an be obtained as the sum of all dire tional for es applied in the i-th omponent of all parti ipant's 4 2. THE INTERNET VOTING SYSTEM voting ve tors. m (2) X R(i) = vpi p=1 Fig. 6. Comparison of possible vote distributions in dierent voting systems. A marker shows an example for voting weights that ould be allo ated in a vote on two initiatives. a) Independent b) Cumulative ) The internet voting voting voting system 1 (1, 1) 1 1 (0.67, 0.33) (0.89, 0.45) 0 0 0 -1 -1 -1 -1 0 1 -1 0 1 -1 0 1 Figure 6 ompares the possible vote distributions of dierent voting systems in a vote with two initiatives. The voting weight assigned to the rst initiative is plotted on the x-axis, while the se ond weight is plotted against the y-axis. All a hievable vote ombinations are shown in the graph. a) If parti ipants an vote on both initiatives independently an asso iated voting weight between -1 and 1 an o ur on both axes. If su h a voting system is applied to non-politi al topi s, su h as ustomer satisfa tion, it is usually pos- sible to assign fra tional weights. In a politi al struggle, however, hardly anyone would deliberately limit their potential inuen e. Hen e, slight onvi tions for any side of the debate leads to a full vote assigned in the orresponding dire tion. b) In a umulative voting system parti ipants an distribute a limited total voting weight onto dierent initiatives. Under su h a voting system it an make sense to split a vote onto dierent initiatives. If, however, one initiative is seen as onsiderable more important than the other, parti ipants would hardly split votes at all. Why give up a fra tion of a vote in a major issue, when one an just gain the same fra tion in a minor issue. ) The internet voting system uses a limited radial voting weight. The system allows maximum total inuen e, if the vote is spread evenly on both initiatives. At the same time, the system allows higher inuen e on one initiative, if more weight is assigned to it. As dis ussed in more detail below, this is the only voting system that makes it optimal to orre tly reveal relative preferen es. Figure 7 shows a possible voting strategy. Four parti ipants have ontrary positions on two initiatives. P favors (Yes,Yes), Q (No,Yes), R (No,No), and S wants (Yes,No). Due to dierent priorities assigned by ea h parti ipant the resulting sum has a slight tenden y towards (no,no). As will be dis ussed in more detail, any two parti ipants an over ome the so ial dilemma and ooperate to 5 Fig. 7. Unde idable vote with four parti ipants and two initiatives. Cooperation an in rease inuen e. In the right graph, P and S express their ooperative instead of their individual preferen es. Un ooperative ase P and S ooperate (no,yes) (yes,yes) (no,yes) (yes,yes) Q P Q result P,S result R R S (no,no) (yes,no) (no,no) (yes,no) in rease their inuen e on the result. While ooperating ea h group member votes a ording to group preferen es. Here, R and S push the result slightly towards a (Yes,No). 2.2 Vote Delegation Vote delegation is often seen as a way to hand over responsibility to someone who is trusted to make the most informed de isions. The internet voting sys- tem allows vote delegation to a number of dierent delegates, whose onsenting positions are expressed as the delegators vote, while dissenting positions are abstained from. Furthermore, ir ular votes delegations are allowed. Hen e, del- egated votes are not ne essarily passed one way upstream, but the an ir ulate within a group and ensure a ertain degree of ooperative voting. A Parti ipant p an delegate a portion dpq of her voting weight to parti ipant q and hange her vote from vp to an average with vq . The vote delegation me hanism then omputes a transformed vote ve tor ṽp whi h is expressed as the new vote ve tor and whi h is ounted in the evaluation of the total result. ! (3) X ṽp = n vp + dpq ∗ ṽq q Fun tion n is a ve tor normalization, that ensures full voting a tivity, even if delegates disagree on any of the initiatives. v n(v) = (4) kvk Te hni ally speaking vote delegation ex hanges ones own vote ve tor with a weighted ombination of delegate vote ve tors. From a gaming perspe tive this has signi ant onsequen es. Delegations are publi expressions of ooperative intentions, but they are no proofs that ooperation in maintained. In fa t, the ooperative behavior an not be observed and ea h group member an defe t into voting along her initial preferen es. This leads to the so ial dilemma as shown below, with the spe ial feature that ooperative opportunities exist between almost any two parti ipants. 6 3. UTILITARIAN CONSIDERATIONS 3 Utilitarian Considerations The utilitarian approa h provides a mathemati al framework for the behavior of rational individuals [2℄, whi h allows us to derive a number of strategi properties of the dis ussed voting system. We onsider a utility fun tion up that measures the satisfa tion that parti ipant p gets from the results. Naturally, p wants to maximize her utility and, assuming rational behavior, votes a ordingly. vp = argmax up R(1), R(2), · · · , R(n) (5)  In the following, it must be assumed that the utility fun tion up is smooth and without lo al extrema. Smoothness an be derived from the fa t that no sudden or dis ontinuous de isions an be taken from the system. All results are either of advisory nature, or are run through a slow and smooth approval pro ess. The la k of lo al extrema implies that all parti ipants always want to hange the politi al lands ape into any dire tion and are never fully satised with the status quo. 3.1 Optimal Voting Weights Correspond to Real Preferen es The rst property tea hes us that it is optimal to allo ate voting weights a - ording to real preferen es. Hen e, the ratios of optimal voting weights equal the ratios of real subje tive preferen es. These an be expressed as the utility gradient ∇up that points in the dire tion steepest as ent, i.e. the dire tion with qui kest gain in utility. It will be shown that the voting ve tor vp points in the same dire tion, but with a normalized length. Hen e, a vote vp reveals a rst order approximation of p's utility fun tion. ∇up vp = n(∇up ) = (6) k∇up k Let xr be the total result of the vote xr = q vq and x0 be the result P Proof: prior to the parti ipation of p, i.e. x0 = xr − vp . The ontribution of p an adjust the result within the onstraints provided by the voting system φ = kxr −x0 k−1 where φ(xr ) = 0. Now, p wants to maximize her utility with subje t to above onstraints. By the method of Lagrange multipliers we an on lude that utility maximization is a hieved when ∃λ : ∇u = λ∇φ, whereas ∇φ = 2(xr − x0 ) = 2vp and nally λ = k∇uk/2. While few people will be able to spe ify their marginal utility with su h an a ura y it is never the less important that utility an not be in reased by voting against ones personal preferen es. Se ond, a urate preferen e ve tors an be derived from averages over a number of delegated group members. 3.2 The First Marginal Vote has Zero Cost The se ond property onrms that the rst marginal voting weight omes for free. Hen e, it is just rational to assign some voting weight to an initiative at ones 7 slightest onvi tion. It does not require noti eable redu tion of voting power on other initiatives. This property also has strong impli ations for ooperative voting strategies. Any topi you feel almost indierent about an serve as a valuable asset in a vote swapping deal. Su h deals are extremely important in the quest for solving the so ial dilemma. Naturally, you would prefer to enter su h a ooperative deal with someone who has mat hing ore prin iples. Assuming parti ipant p intends to allo ate some voting weight to a new initiative i. Due to the limited overall voting weight p has to withdraw some votes from any other initiative. Here, she de ides to ut ba k on j . The question now is how mu h weight must be subtra ted from j in order to in rease her vote on i by one marginal unit. As indi ated previously the answer is zero. dvpj lim =0 (7) vpi →0 dvpi Proof: The votingPweight is onned by kvp k2 − 1 = 0. Building the total dierential yields 2vpi dvpi = 0. Division by dpi yields dvpj /dvpi = −vpi/vpj . The high ex hange value of the rst marginal vote is onsistent with real world observations. Few se onds spent on signing a petition are met with hours of ampaigning eorts, while a full time ommitment to a politi al movement hardly pays the bills. Dedi ated interest groups also prot from the large status gained by representing their members in spe ialized issues. 3.3 Any Two Parti ipants Can Cooperate The third property on erns the ability to in rease inuen e through ooperative voting. Joint voting ta ti s are known to prevail in all demo rati voting systems [3℄. Often it is not easy to nd su h opportunities, sin e they might o ur only between ertain parti ipants with a suitable preferen e mat h. In this respe t the internet voting system is mu h more demo rati . Any two parti ipants an ooperate, almost regardless of their politi al preferen es. The internet voting system is designed as an intera tive game, where votes an be viewed and reassigned. Ta ti al voting is an intended part of the system. The internet voting system does not try to resolve pathologi al ases with ir- ular and twisted preferen e orders. It just shows the presen e of su h in onsis- ten ies as a sequen e of non- onverging voting results. If y li voting behavior was to be resolved, more negotiations have to take pla e and new initiatives must be added to serve as a potential in entive for arbitration. In the most simple ooperative situation between two parti ipants p and q ea h side delegates a positive voting weight ǫ. The symmetri delegation is ertainly not the only ooperative option. There might be a number of reasons why the other side should delegate a larger voting fra tion. Anyway, for the following dis ussion p delegates to q with dpq = ǫ and vi e versa dqp = ǫ. Following equation (3) new voting ve tors ṽp and ṽq an be derived. vp + ǫvq vq + ǫvp ṽp = , ṽq = (8) kvp + ǫvq k kvq + ǫvp k 8 3. UTILITARIAN CONSIDERATIONS This mutual delegation is almost always bene ial for both sides. There are only two ex eptions. First, p and q already have equal opinions and equal preferen es on all initiatives, i.e. vp = vq . Maximum ooperation is a hieved. It is not possible to ooperate further. Se ond, p and q have exa tly opposing preferen es vp = −vq . Both sides disagree on all initiatives with equal preferen e weight. Ex luding these two spe ial ases there always exists a delegation weight ǫ for whi h both parti ipants an in rease their utility. ∃ ǫ > 0 : up (ṽp + ṽq ) > up (vp + vq ) ∧ (9) uq (ṽp + ṽq ) > uq (vp + vq ) Proof: We start p with building the derivativepof the normalizing denominator: dǫ kvp +ǫvq k = (vp + ǫvq ) · (vp + ǫvq ) = 1/2 k · k(vq ·(vp +ǫvq )+(vp +ǫvq )·vq . d Inserting an initial ǫ=0 yields vp · vq No we an ontinue to onstru t the derivative of uP at ǫ = 0. dǫd up (n(vp + ǫvq )+ n(vq + ǫvp )) = (∇up )·(vq − vp (vp ·vq )+ vp − vq (vp ·vq )). Inserting ∇up = vp and vp · vp = 1 yields 1 − (vp · vq )2 , whi h is positive for vp 6= ±vq . Joining a ooperative formation group members do no longer vote along their personal preferen es. Instead, they must strive to optimize group utilities with n(∇ũp ) = ṽp and n(∇ũq ) = ṽq . If one side fails to do so ooperation breaks. Be ause the global existen e of ũp and ũq is not guaranteed, building onsistent and sustainable ooperation remains a hallenge. It is important to realize that temporary ooperation an o ur even between parti ipants who disagree on all issues. As soon as absolute preferen es deviate ea h side an gain support in an important area, in ex hange for giving up what is per eived as a less important issue. Now it is time to revisit our initial example from se tion 1.2. Two large fra tions disagree on all two initiatives with (Yes,No) against (No,Yes). In a dis onne ted vote it is up to a small minority to de ide for an arbitrary out ome. In the suggested voting system this is not the ase. The two large groups an ooperate temporarily and favor a (No,No) over a (Yes,Yes), just by putting a little more preferen e on opposing the other initiative than on supporting the own. Assuming minimal ooperative intentions the minority group is only left with joining either of the larger fra tions. It an not open up a third (Yes,Yes) path. 3.4 Cooperation Leads to the So ial Dilemma The fourth property reminds us that the suggested voting system does not automati ally resolve the so ial dilemma. It an only help doing so and support the required negotiations. The ooperative state is not a Nash equilibrium. It still takes mutual so ial links and a fundamental will to work together. The politi al onsequen es of a failed ooperative state are often severe. Two parties rst agree to ooperate and start implementing a joint set of initiatives. But then, ooperation breaks and ea h side denies ne essary amendments to initiatives they originally supported. The results are in omplete and in onsistent laws that don't mat h the intended spirit. 9 The internet voting system bears a similar dilemma. Despite the fa t that voting ve tors are permanently visible, the ooperative status an not be ob- je tively determined. This is due to an abundan e of non-defe tive reasons why the voting ve tor an hange. With a large number of ex uses at hand one does not need to fear immediate retaliation when defe ting in a ooperative vote. Delegation weights an publi ly announ e ooperative intentions, but neither their existen e nor their absen e bears any proof. The ee t ould as well have been repli ated manually, or undone with the remaining voting weights. The observation of the voting ve tors an also be misleading. If a parti ipant restores her voting ve tor to a pre ooperative state it an have many reasons. For one, her preferen es might have hanged to a more extreme position and she now votes what she believes to be the fair ompromise. Another reason might be that she entered a ooperation with an even more extreme parti ipant. Her ooperative inuen e is then visible in the third parti ipant's vote. Even an un hanged voting ve tor does not ne essarily proof un hanged o- operative intentions. Maybe the parti ipant displayed an exaggerated position prior to the ooperative deal and would have voted for what she pretends to be a ooperative on ession anyway. Figure 8 shows an example of a ooperative dilemma. Both parti ipants agree on the rst initiative, but disagree on the se ond. Sin e their voting power on the se ond initiative an els out anyway, they ould as well fo us entirely on the rst initiative. Whi hever side manages to get away with a defe tive vote an get the best results. Fig. 8. So ial dilemma in ooperative voting. P has a utility of up = R(1) + R(2) while Q's utility is uq = R(1) − R(2). In the ooperative state both an gain a utility of up = uq = 2.0. If one side defe ts its utility in reases to 2.41 while the other side is left with a mere 1.0. Q: (0.71, -0.71) Q: (1.0, 0.0) P: (0.71, 0.71) (1.41, 0.0) (1.71, 0.71) P: (1.0, 0.0) (1.71, -0.71) (2.0, 0.0) The only known solution to the so ial dilemma exists in a multiperiod re- peated play. Therefore the internet voting system is more assistive in solving the so ial dilemma. Bene ial ooperative states an be upheld for a long period without the need to renegotiate after any hanges in the politi al lands ape. The internet voting system awards the ability to ooperate even in the light of politi al battles with an in reased inuen e on the nal result. No matter what the out ome of the vote, these people are the ones that should have the most power. Only they an guarantee sustainable and stable poli ies. Given a group with low entry and exit barriers, it is fair to assume that the fra tion whi h has higher ooperative tenden ies is more adept to represent the group. Maybe it's even fair to say: they are the group. 10 4. CONCLUSION 4 Con lusion A voting system that grants full voting weight on ea h initiative does appeal to the politi al hardliners. Abstention or any indi ation of low preferen e must be seen as a weakness. The vote ould have been swapped or sold for a politi al favor in an other area. The so ial and politi al impli ations are severe. Politi ians must falsely display strong preferen es on any issue, whi h then ompli ates arbitration eorts. This do ument suggests a xed voting system that an be used by groups of a tivists with good governan e as a entral goal. The system awards a bal- an ed expression of politi al views with higher voting weight in key areas and it supports the formation of ooperating groups of voters that an span many de isions regardless of a tual preferen es. Thus, the voting system puts its fo us on arbitration and on the solution of the so ial dilemma. Harvesting the intera tive features of the internet makes the suggested vot- ing system the rst of its kind. By ombining elements of online games and so ial networks it enables a new dimension of problem solving apabilities. Ea h par- ti ipant is represented by a voting ve tor. As soon as mutual onsent is a hieved, the ve tors start pointing into the same dire tion. This does not automate the pro ess of solving the so ial dilemma, but provides some metri s of well you did. Demo ra y is often expe ted to provide an automated problem solving ma- hine that turns personal preferen es into the publi good. In many Western demo rati states this fatal mis on eption is so strong that it even survives mil- itary defeat and the obvious inability to solve the sustainability problem. Many fanati s are in power, who sti k to a mantra of individualisti thinking leading to the publi good. This mantra is re ited in the awake of de ay and rises on a global s ale. If this do ument and the herein suggested voting system an play a role in a ooperative politi al revolution then it has served its purpose. Referen es 1. Mi hael A. Jones. The Geometry behind paradoxes of voting power. Mont lair State Uni- versity, 2006. 2. Alfred Marshall. Prin iples of E onomi s. London: Ma millan, 1920. 3. Philip J. Reny. Arrow's Theorem and the Gibbard-Satterthwaite Theorem: A Unied Ap- proa h. University of Chi ago, 2000. 4. Warren D. Smith. Des riptions of single-winner voting systems. 2006. 5. John von Neumann and Os ar Morgenstern. Theory of Games and E onomi Behavior. Prin eton University Press, 1944. 11