Sequential Deliberation for Social Choice
Brandon Fain, Ashish Goel, Kamesh Munagala, Sukolsak Sakshuwong
SSequential Deliberation for Social Choice
Brandon Fain ∗ Ashish Goel † Kamesh Munagala ‡ Sukolsak Sakshuwong § Abstract
In large scale collective decision making, social choice is a normative study of how one oughtto design a protocol for reaching consensus. However, in instances where the underlying decisionspace is too large or complex for ordinal voting, standard voting methods of social choice may beimpractical. How then can we design a mechanism - preferably decentralized, simple, scalable,and not requiring any special knowledge of the decision space - to reach consensus? We proposesequential deliberation as a natural solution to this problem. In this iterative method, successivepairs of agents bargain over the decision space using the previous decision as a disagreementalternative. We describe the general method and analyze the quality of its outcome when thespace of preferences define a median graph. We show that sequential deliberation finds a 1.208-approximation to the optimal social cost on such graphs, coming very close to this value withonly a small constant number of agents sampled from the population. We also show lowerbounds on simpler classes of mechanisms to justify our design choices. We further show thatsequential deliberation is ex-post Pareto efficient and has truthful reporting as an equilibriumof the induced extensive form game. We finally show that for general metric spaces, the secondmoment of of the distribution of social cost of the outcomes produced by sequential deliberationis also bounded.
Suppose a university administrator plans to spend millions of dollars to update her campus, andshe wants to elicit the input of students, staff, and faculty. In a typical social choice setting, shecould first elicit the bliss points of the students, say “new gym,” “new library,” and “new studentcenter.” However, voting on these options need not find the social optimum, because it is not clearthat the social optimum is even on the ballot. In such a setting, deliberation between individualswould find entirely new alternatives, for example “replace gym equipment plus remodeling campusdining plus money for scholarship.” This leads to finding a social optimum over a wider space ofsemi-structured outcomes that the system/mechanism designer was not originally aware of, andthe participants had not initially articulated.We therefore start with the following premise: The mechanism designer may not be able toenumerate the outcomes in the decision space or know their structure, and this decision space maybe too big for most ordinal voting schemes. (For instance, ordinal voting is difficult to implement ∗ Department of Computer Science, Duke University, 308 Research Drive, Durham, NC 27708. [email protected] . Supported by NSF grants CCF-1408784, CCF-1637397, and IIS-1447554. † Supported by the Army Research Office Grant No. 116388, the Office of Naval Research Grant No. 11904718,and by the Stanford Cyber Initiative. Author’s Address: Management Science and Engineering Department, StanfordUniversity, Stanford CA 94305. Email: [email protected] ‡ Department of Computer Science, Duke University, Durham NC 27708-0129. [email protected] . Supportedby NSF grants CCF-1408784, CCF-1637397, and IIS-1447554. § Management Science and Engineering Department, Stanford University, 353 Serra Mall, Stanford, CA 94305. [email protected] a r X i v : . [ c s . G T ] O c t n complex combinatorial spaces [24] or in continuous spaces [14].) However, we assume that agentscan still reason about their preferences and small groups of agents can negotiate over this spaceand collaboratively propose outcomes that appeal to all of them. Our goal is to design protocolsbased on such a primitive by which small group negotiation can lead to an aggregation of societalpreferences without a need to formally articulate the entire decision space and without every agenthaving to report ordinal rankings over this space.The need for small groups is motivated by a practical consideration as well as a theoretical one.On the practical side, there is no online platform, to the best of our knowledge, that has a successfulhistory of large scale deliberation and decision making on complex issues; in fact, large online forumstypically degenerate into vitriol and name calling when there is substantive disagreement amongthe participants. Thus, if we are to develop practical tools for decision making at scale, a sequenceof small group deliberations appears to be the most plausible path. On the theoretical side, weunderstand the connections between sequential protocols for deliberation and axiomatic theories ofbargaining for small groups, e.g. for pairs [33, 8], but not for large groups, and we seek to bridgethis gap. Summary of Contributions.
Our main contributions in this paper are two-fold: • A simple and practical sequential protocol that only requires agents to negotiate in pairs andcollaboratively propose outcomes that appeal to both of them. • A canonical analytic model in which we can precisely state properties of this protocol in termsof approximation of the social optimum, Pareto-efficiency, and incentive-compatibility, as wellas compare it with simpler protocols.
Before proceeding further, we review bargaining, the classical framework for two-player ne-gotiation in Economics. Two-person bargaining, as framed in [28], is a game wherein there is adisagreement outcome and two agents must cooperate to reach a decision; failure to cooperateresults in the adoption of the disagreement outcome. Nash postulated four axioms that the bar-gaining solution ought to satisfy assuming a convex space of alternatives: Pareto optimality (agentsfind an outcome that cannot be simultaneously improved for both of them), symmetry betweenagents, invariance with respect to affine transformations of utility (scalar multiplication or additivetranslation of any agent’s utility should not change the outcome), and independence of irrelevantalternatives (informally that the presence of a feasible outcome that agents do not select does notinfluence their decision). Nash proved that the solution maximizing the Nash product (that wedescribe later) is the unique solution satisfying these axioms. To provide some explanation of howtwo agents might find such a solution, [33] shows that Nash’s solution is the subgame perfect equi-librium of a simple repeated game on the two agents, where the agents take turns making offers,and at each round, there is an exogenous probability of the process terminating with no agreement.The two-person bargaining model is therefore clean and easy to reason about. As a conse-quence, it has been extensively studied. In fact, there are other models and solutions to two-personbargaining, each with a slightly different axiomatization [20, 21, 27], as well as several experimentalstudies [32, 29, 7]. In a social choice setting, there are typically many more than two agents, eachagent having their own complex preferences. Though bargaining can be generalized to n agentswith similar axiomatization and solution structure, such a generalization is considered impractical.This is because in reality it is difficult to get a large number of individuals to negotiate coherently;2omplexities come with the formation of coalitions and power structures [18, 23]. Any model forsimultaneous bargaining, even with three players [6], needs to take these messy aspects into account. In this paper, we take a middle path, avoiding both the complexity of explicitly specifyingpreferences in a large decision space that any individual agent may not even fully know (fullycentralized voting), and that of simultaneous n -person bargaining (a fully decentralized cooperativegame). We term this approach sequential deliberation . We use 2-person bargaining as a basicprimitive, and view deliberation as a sequence of pairwise interactions that refine good alternativesinto better ones as time goes by.More formally, there is a decision space S of feasible alternatives (these may be projects, setsof projects, or continuous allocations) and a set N of agents. We assume each agent has a hiddencardinal utility for each alternative. We encapsulate deliberation as a sequential process. Theframework that we analyze in the rest of the paper is captured in Figure 1.1. In each round t = 1 , , . . . , T :(a) A pair of agents u t and v t are chosen independently and uniformly at random withreplacement.(b) These agents are presented with a disagreement alternative a t , and perform bargaining,which is encoded as a function B ( u, v, a ) as described below.(c) Agents u t and v t are asked to output a consensus alternative; if they fail to reach aconsensus then the alternative a t is output.(d) Let o t denote the alternative that is output in round t . We set a t +1 = o t , where weassume a is the bliss point of an arbitrary agent.2. The final social choice is a T . Note that this is equivalent to drawing an outcome at randomfrom the distribution generated by repeating this protocol several times.Figure 1: A framework for sequential pairwise deliberation.Our framework is simple with low cognitive overhead, and is easy to implement and reasonabout. Though we don’t analyze other variants in this paper, we note that the framework is flexible.For instance, the bargaining step can be replaced with any function B ( u, v, a ) that corresponds toan interaction between u and v using a as the disagreement outcome; we assume that this functionmaximizes the Nash product, that is, it corresponds to the Nash bargaining solution. Similarly,the last step of social choice could be implemented by a central planner based on the distributionof outcomes produced. The framework in Figure 1 is well-defined and practical irrespective of an analytical model.However, we provide a simple analytical model for specifying the preferences of the agents in whichwe can precisely quantify the behavior of this framework as justification.
Median Graphs.
We assume that the set S of alternatives are vertices of a median graph . Amedian graph has the property that for each triplet of vertices u, v, w , there is a unique point that3s common to the three sets of shortest paths (since there may be multiple pairwise shortest paths),those between u, v , between v, w , and between u, w . This point is the unique median of u, v, w . Weassume each agent u has a bliss point p u ∈ S , and his disutility for an alternative a ∈ S is simply d ( p u , a ), where d ( · ) is the shortest path distance function on the median graph. (Note that thisdisutility can have an agent-dependent scale factor.) Several natural graphs are median graphs,including trees, points on the line, hypercubes, and grid graphs in arbitrary dimensions [22]. As wediscuss in Section 1.5, because of their analytic tractability and special properties, median graphshave been extensively studied as structured models for spatial preferences in voting theory. Someof our results generalize to metric spaces beyond median graphs; see Section 5 and Appendix B. Nash Bargaining.
The model for two-person bargaining is simply the classical
Nash bargaining solution described before. Given a disagreement alternative a , agents u and v choose that alternative o ∈ S that maximizes:Nash product = ( d ( p u , a ) − d ( p u , o )) × ( d ( p v , a ) − d ( p v , o ))subject to individual rationality, that is, d ( p v , o ) ≤ d ( p v , a ) and d ( p u , o ) ≤ d ( p u , a ). The Nashproduct maximizer need not be unique; in the case of ties we postulate that agents select theoutcome that is closest to the disagreement outcome. As mentioned before, the Nash product is awidely studied axiomatic notion of pairwise interactions, and is therefore a natural solution conceptin our framework. Social Cost and Distortion.
The social cost of an alternative a ∈ S is given by SC ( a ) = (cid:80) u ∈N d ( p u , a ). Let a ∗ ∈ S be the minimizer of social cost, i.e. , the generalized median . Wemeasure the Distortion of outcome a asDistortion( a ) = SC ( a ) SC ( a ∗ ) (1)where we use the expected social cost if a is the outcome of a randomized algorithm.Note that our model is fairly general. First, the bliss points of the agents in N form anarbitrary subset of S . Further, the alternative chosen by bargaining need not correspond to anybliss point, so that pairs of agents are exploring the entire space of alternatives when they bargain,instead of just bliss points. Assuming that disutility is some metric over the space follows recentliterature [3, 2, 9, 10, 16], and our tightest results are for median graphs specifically. Before presenting our results, we re-emphasize that while we present analytical results for se-quential deliberation in specific decision spaces, the framework in Figure 1 is well defined regardlessof the underlying decision space and the mediator’s understanding of the space. At a high level,this flexibility and generality in practice are its key advantages.
Bargaining and Medians.
We first show in Section 2 that on a median graph, Nash bargainingbetween agents u and v using disagreement outcome a outputs the median of p u , p v , a . Therefore, B ( u, v, a ) = Median( p u , p v , a ). On a general metric space, we show in Section 5.1 that the NashBargaining outcome would lie on the shortest path between the agents, and the distance from anagent is proportional to its distance to the disagreement outcome. In a sense, agents only need toexplore options on the shortest path between them.4 ounding Distortion. Our main result in Section 3 shows that for sequential Nash bargainingon a median graph, the expected Distortion of outcome a T has an upper bound approaching 1 . T → ∞ . Surprisingly, we show that in T = log (cid:15) + 2 .
575 steps, the expected Distortion isat most 1 .
208 + (cid:15) , independent of the number of agents, the size of the median space, and theinitial disagreement point a . For instance, the Distortion falls below 1 .
22 in at most 9 steps ofdeliberation, which only requires a random sample of at most 20 agents from the population toimplement.In Section 3.2, we ask:
How good is our numerical bound?
We present a sequence of lowerbounds for social choice mechanisms that are allowed to use increasingly richer information aboutthe space of alternatives on the median graph. This also leads us to make qualitative statementsabout our deliberation scheme. • We show that any social choice mechanism that is restricted to choosing the bliss point ofsome agent cannot have Distortion better than 2. More generally, it was recently shown [17]that even eliciting the top k alternatives for each agent does not improve the bound of 2 formedian graphs unless k = Ω( |S| ). In effect, we show that forcing the agents to reason aboutcardinal utilities via deliberation leads to new alternatives that are more powerful at reducingDistortion than simply eliciting and aggregating reasonably detailed ordinal rankings. • Next consider mechanisms that choose, for some triplet ( u, v, w ) of agents with bliss points p u , p v , p w , the median outcome m uvw = B ( u, v, p w ). We show this has Distortion at least1 . o where a is the bliss point of some agent. • Finally, for every pair of agents ( u, v ), consider the set of alternatives on a shortest pathbetween p u and p v . This encodes all deliberation schemes where B finds a Pareto-efficientalternative for some 2 agents at each step. We show that any such mechanisms has Distortionratio at least 9 / . Properties of Sequential Deliberation.
We next show that sequential deliberation has severalnatural desiderata on median graphs in Section 4. In particular: • Under mild assumptions, the limiting distribution over outcomes of sequential deliberation is unique . • For every T ≥
1, the outcome o T of sequential deliberation is ex-post Pareto-efficient , meaningthat there is no other alternative that has at most that social cost for all agents and strictlybetter cost for one agent. This is not a priori obvious, since the outcome at any one roundonly uses inputs from two agents; though it is Pareto-efficient for those two agents, it couldvery well be sub-optimal for the other agents. • Interpreted as a mechanism, truthful play is a sub-game perfect Nash equilibrium of sequentialdeliberation. More precisely, we consider a different view of the function B ( u, v, a ) thatencodes Nash Bargaining. Suppose agents u and v report their bliss points, and the platformimplements the function B that computes the median of p u , p v , and a . In a sequential setting,would any agent have incentive to misreport their bliss point so that they gain an advantage(in terms of expected distance to the final social choice) in the induced extensive form game?We show that the answer is negative – on a median graph, truthfully reporting bliss pointsis a sub-game perfect Nash equilibrium of the induced extensive form game.5 eyond Median Graphs. In Section 5, we consider general metric spaces. We show that theDistortion of sequential deliberation is always at most a factor of 3. More surprisingly, we showthat sequential deliberation has constant distortion even for the second moment of the distributionof social cost of the outcomes, i.e. , the latter is at most a constant factor worse than the optimumsquared social cost. This has the following practical implication: A policy designer can look at thedistribution of outcomes produced by deliberation, and know that the standard deviation in socialcost is comparable to its expected value, which means deliberation eliminates outlier alternativesand concentrates probability mass on more central alternatives. We also show that such a claimcannot be made for random dictatorship, whose distortion on squared social cost is unbounded.
While the real world complexities of the model are beyond the analytic confines of this work,deliberation as an important component of collective decision making and democracy is studied inpolitical science. For examples (by no means exhaustive), see [13, 36]. There is ongoing relatedwork on Distortion of voting for simple analytical models like points in R [12], and in general metricspaces [3, 2, 9, 10, 16]. This work focuses on optimally aggregating ordinal preferences, say thetop k preferences of a voter [17]. In contrast, our scheme elicits alternatives as the outcome ofbargaining rounds that require agents to reason about cardinal preferences.. As mentioned before,we essentially show that for median graphs, unless k is very large, such deliberation has provablylower distortion than social choice schemes that elicit purely ordinal rankings.Median graphs and their ordinal generalization, median spaces, have been extensively studiedin the context of social choice. The special cases of trees and grids have been studied as structuredmodels for voter preferences [35, 5]. For general median spaces, it is known that the Condorcetwinner (that is an alternative that pairwise beats any other alternative in terms of voter preferences)is strongly related to the generalized median [4, 34, 38] – if the former exists, it coincides with thelatter. Nehring and Puppe [30] shows that any single-peaked domain which admits a non-dictatorialand neutral strategy-proof social choice function is a median space. Clearwater et al. [11] alsoshowed that any set of voters and alternatives on a median graph will have a Condorcet winner. Ina sense, these are the largest class of structured and spatial preferences where ordinal voting overthe entire space of alternatives leads to a “clear winner” even by pairwise comparisons. Our workstems from the assumption that this space may not be fully known to the mechanism designer orall agents.Our paper is inspired by the triadic consensus results of Goel and Lee [15]. In that work, theauthors focus on small group interactions with the goal of reaching consensus. In their model,three people deliberate at the same time, and they choose a fourth individual to whom they granttheir votes. This individual takes these votes and participates in future rounds, until all votesaccumulate with one individual, who is the consensus outcome. The analysis proceeds througha median graph, on which the authors show that the Distortion of the consensus approaches 1.However, the protocol crucially assumes individuals know the positions of other individuals, andrequires the space of alternatives to coincide with the space of individuals. We make neither ofthese assumptions – in our case, the space of alternatives can be much larger than the number ofagents, and further, individuals interact with others only via bargaining. This makes our protocolmore practical, but at the same time, restricts our Distortion to be bounded away from 1.The notion of democratic equilibrium [19, 14] considers social choice mechanisms in continuousspaces where individual agents with complex utility functions perform update steps inspired by See also recent work by [37] that considers minimizing the variance of randomized truthful mechanisms. iterative voting where the current alternative is put to vote against one proposed bydifferent random agent chosen each step [1, 25, 31], or other related schemes [26]. In contrast withour work, these protocols are not deliberative and require voting among several agents each step;furthermore, the analysis focuses on convergence to an equilibrium instead of welfare or efficiencyguarantees.
In this section we will use the notation N for a set of agents, S for the space of feasiblealternatives, and H for a distribution over S . Most of our results are for the analytic model givenearlier wherein the set S of alternatives are vertices of a median graph ; see Figure 2 for someexamples. Definition 1.
A median graph G ( S , E ) is an unweighted and undirected graph with the followingproperty: For each triplet of vertices u, v, w ∈ S × S × S , there is a unique point that is common tothe shortest paths (which need not be unique between a given pair) between u, v , between v, w , andbetween u, w . This point is the unique median of u, v, w . Figure 2: Examples of Median GraphsIn the framework of Figure 1, we assume that at every step, two agents perform Nash bargainingwith a disagreement alternative. The first results characterize Nash bargaining on a median graph.In particular, we show that Nash bargaining at each step will select the median of bliss points ofthe two agents and the disagreement alternative. After that, we show that we can analyze theDistortion of sequential deliberation on a median graph by looking at the embedding of that graphonto the hypercube.
Lemma 1.
For any median graph G = ( S , E ) , any two agents u, v with bliss points p u , p v ∈ S , andany disagreement outcome a ∈ S , let M be the median. Then M maximizes the Nash product of u and v given a , and is the maximizer closest to a .Proof. Since G is a median graph, M exists and is unique; it must by definition be the intersection ofthe three shortest paths between ( p u , p v ) , ( p u , a ) , ( p v , a ). Note that we can therefore write d ( p u , a ) = d ( p u , M ) + d ( M, a ) and similarly for d ( p v , a ). Let α = d ( p u , M ); β = d ( p v , M ); and γ = d ( a, M ).Suppose Nash bargaining finds an outcome o ∗ ∈ S . Let x = d ( o ∗ , p u ) and y = d ( o ∗ , p v ). Observingthat M lies on the shortest path between p u and p v , and using the triangle inequality, we obtainthat x + y ≥ α + β = ⇒ β ≤ x + y − α .Noting that d ( p u , a ) = α + γ and d ( p v , a ) = β + γ , the Nash product of the point o ∗ is:( α + γ − x ) × ( β + γ − y ) ≤ ( α + γ − x )( γ − ( α − x )) = γ − ( α − x ) x = α and y = β . One possible maximizer is therefore the point o ∗ = M .Suppose d ( o ∗ , a ) < γ , then by the triangle inequality, d ( p u , o ∗ ) > α , and similarly d ( p v , o ∗ ) > β .Therefore, there cannot be a closer maximizer of the Nash product to a than the point M . Hypercube Embeddings.
For any median graph G = ( S , E ), there is an isometric embedding φ : G → Q of G into a hypercube Q [22]. This embedding maps vertices S into a subset of verticesof Q so that all pairwise distances between vertices in S are preserved by the embedding. A simpleexample of this embedding for a tree is shown in Figure 3. We use this embedding to show thefollowing result, in order to simplify subsequent analysis.Figure 3: The hypercube embedding of a 4-vertex star graph Lemma 2.
Let G ( S , E ) be a median graph, and let φ be its isometric embedding into hypercube Q ( V, E (cid:48) ) . For any three points t, u, v ∈ S , let M G be the median of vertices t, u, v and let M Q bethe median of vertices φ ( t ) , φ ( u ) , φ ( v ) ∈ V . Then φ ( M G ) = M Q .Proof. By definition, since φ is an isometric embedding [22], d ( x, y ) = d ( φ ( x ) , φ ( y )) for all x, y ∈ S (2)Since G is a median graph, M G is the unique median of t, u, v ∈ S , which by definition satisfies theequalities: d ( t, u ) = d ( t, M G ) + d ( M G , u ) d ( t, v ) = d ( t, M G ) + d ( M G , v ) d ( u, v ) = d ( u, M G ) + d ( M G , v ) Q ( V, E (cid:48) ) is a hypercube, and is thus also a median graph, so M Q is the unique median of φ ( t ) , φ ( u ) , φ ( v ) ∈ V , which by definition satisfies the equalitiesI d ( φ ( t ) , φ ( u )) = d ( φ ( t ) , M Q ) + d ( M Q , φ ( u ))II d ( φ ( t ) , φ ( v )) = d ( φ ( t ) , M Q ) + d ( M Q , φ ( v ))III d ( φ ( u ) , φ ( v )) = d ( φ ( u ) , M Q ) + d ( M Q , φ ( v ))Applying Equation (2) to the first set of equalities shows that φ ( M G ) satisfies equalities I,II, andIII respectively. But φ ( M G ) ∈ V and M Q is the unique vertex in V satisfying equalities I,II, andIII. Therefore, φ ( M G ) = M Q . In this section, we show that the Distortion of sequential deliberation is at most 1 . .1 Upper Bounding Distortion Recall the framework for sequential deliberation in Figure 1 and the definition of Distortion inEquation (1). We first map the problem into a problem on hypercubes using Lemma 2.
Corollary 1.
Let G = ( S , E ) be a median graph, let φ : G → Q be an isometric embedding of G onto a hypercube Q ( V, E (cid:48) ) , and let N be a set of agents such that each agent u has a bliss point p u ∈ S . Then the Distortion of sequential deliberation on G is at most the Distortion of sequentialdeliberation on φ ( G ) where each agent’s bliss point is φ ( p u ) .Proof. Fix an initial disagreement outcome a ∈ S and an arbitrary list of T pairs of agents ( u , v ),( u , v ), ..., ( u T , v T ). In round 1 bargaining on G , Lemma 1 implies that sequential deliberationwill select o = median( a , p u , p v ). Furthermore, Lemma 2 implies that if we had considered φ ( G )and bargaining on φ ( a ) , φ ( p u ) , φ ( p v ) instead, sequential deliberation would have selected φ ( o ).Suppose at some round t that we have a disagreement outcome a t . Then the same argument yieldsthat if o t is the bargaining outcome on G , φ ( o t ) would have been the bargaining outcome on φ ( G ).Thus, by induction, we have that if the list of outcomes on G is o , ..., o T then the list of outcomeson φ ( G ) is φ ( o ) , ..., φ ( o T ). But recall that φ ( · ) is an isometric embedding and the social cost ofan alternative (as defined in Section 1.3) is just its sum of distances to all points in N , so a T and φ ( a T ) have the same social cost.Furthermore, let a ∗ ∈ S denote the generalized median of N . Then, φ ( a ∗ ) has the same socialcost as a ∗ . This means the median of the embedding of N into Q has at most this social cost, whichin turn means that the Distortion of sequential deliberation in the embedding cannot decrease.Our main result in this section shows that as t → ∞ , the Distortion of sequential deliberationapproaches 1 . t and independent of thenumber of agents |N | , the size of the median space |S| , and the initial disagreement point a . Inparticular, the Distortion is at most 1 .
22 in at most 9 steps of deliberation, which is indeed a verysmall number of steps.
Theorem 1.
Sequential deliberation among a set N of agents, where the decision space S is amedian graph, yields E [ Distortion ( a t )] ≤ .
208 + t .Proof. By Corollary 1, we can assume the decision space is a D -dimensional hypercube Q so thatdecision points (and thus bliss points) are vectors in { , } D . For every dimension k , let f k be thefraction of agents whose bliss point has a 1 in the k th dimension, and let p u,k be the 0 or 1 bit inthe k th dimension of the bliss point p u for agent u . Let a ∗ ∈ { , } D be the minimum social costdecision point, i.e., a ∗ := argmin a ∈ Q SC ( a ). Clearly, a ∗ k is 1 if f k > / a ∗ k = 0 for f k = 1 / k , the total distance to a ∗ k , summed over N is: (cid:88) u ∈N | a ∗ k − p u,k | = |N | min { f k , − f k } Now, note that sequential deliberation defines a Markov chain on Q . The state in a given stepis just a t , and the randomness is in the random draw of the two agents. Let H ∗ be the stationarydistribution of the Markov chain. Then we can writelim t →∞ E [Distortion( a t )] = E a ∈H ∗ [Distortion( a )]To write down the transition probabilities, we assume this random draw is two independentuniform random draws from N , with replacement. We also note that Lemma 1 implies that on9 , sequential deliberation will pick the median in every step, i.e. , given a disagreement outcome a t and two randomly drawn agents with bliss points p u , p v , the new decision point will be o t =median( p u , p v , a t ). On a hypercube, the median of three points is just the dimension-wise majority.Thus, we get a 2-state Markov chain in each dimension k , with transition probabilitiesPr[ o tk = 1 | a tk = 1] = f k + 2 f k (1 − f k ) and Pr[ o tk = 1 | a tk = 0] = f k Let P ∗ k = lim t →∞ Pr[ o tk = 1], and let H ∗ k denote this stationary distribution for the corresponding2-state Markov chain. Then, P ∗ k = (cid:0) f k − f k (cid:1) P ∗ k + (cid:0) f k (cid:1) (1 − P ∗ k ) ⇒ P ∗ k = f k f k − f k By linearity of expectation, the total expected distance for every dimension k , summed over u ∈ N to the final outcome is given by E a k ∈H ∗ k (cid:34) (cid:88) u ∈N | a k − p u,k | (cid:35) = |N | (cid:18)(cid:18) f k f k + (1 − f k ) (cid:19) (1 − f k ) + (cid:18) − f k f k + (1 − f k ) (cid:19) f k (cid:19) = |N | (cid:18) f k (1 − f k ) f k + (1 − f k ) (cid:19) Without loss of generality, let f k ∈ [0 , /
2] so that for dimension k , the total distance to a ∗ k is |N | f k .Then the ratio of the expected total distance to H ∗ k to the total distance to a ∗ k is at most: E a k ∈H ∗ k (cid:2)(cid:80) u ∈N | p u,k − a k | (cid:3)(cid:80) u ∈N | p u,k − a ∗ k | ≤ max f k ∈ [0 , / − f k f k + (1 − f k ) ≤ . E a ∈H ∗ [Distortion( a )] = (cid:80) Dk =1 E a k ∈H ∗ k (cid:2)(cid:80) u ∈N | p u,k − a k | (cid:3)(cid:80) Dk =1 (cid:80) u ∈N | p u,k − a ∗ k | ≤ . Convergence Rate.
Now that we have bounded the Distortion of the stationary distribution,we need to consider the convergence rate. We will not bound the mixing time of the overall Markovchain. Rather, note that in the preceding analysis, we only used the marginal probabilities H ∗ k for every dimension k . Furthermore, the Markov chain defined by sequential deliberation need notwalk along edges on Q , so we can consider separately the convergence of the chain to the stationarymarginal in each dimension.After t steps, let P kt = Pr[ o tk = 1] and let H tk denote this distribution. Assume f k ∈ [0 , / between H tk and H ∗ k is (cid:15) f k f k +(1 − f k ) , then it is easy to check that theexpected total distance to H tk is within a (1+ (cid:15) ) factor of the expected distance to H ∗ k , which impliesa Distortion of at most 1 .
208 + (cid:15) in that dimension. We therefore bound how many steps it takesto achieve total variation distance (cid:15) f k f k +(1 − f k ) in any dimension k ; if this bound holds uniformlyfor all dimensions k , this would imply the overall Distortion is at most 1 .
208 + (cid:15) , completing theproof. total variation distance is particularly simple for these distributions with support of just two points: T V D ( H tk , H ∗ k ) = | P kt − P ∗ k | k , two executions of the 2-state Markov chain along that dimension couple ifthe agents picked in a time step have the same value in that dimension. At any step, this happenswith probability at least (cid:0) f k + (1 − f k ) (cid:1) . Therefore, the probability that the chains have notcoupled in t steps is at most (cid:0) − (cid:0) f k + (1 − f k ) (cid:1)(cid:1) t = (2 f k (1 − f k )) t We therefore need T large enough so that(2 f k (1 − f k )) T ≤ (cid:15) f k f k + (1 − f k ) ⇒ T = max f k ∈ [0 , / log (cid:15) log f k (1 − f k ) + log f k + log( f k + (1 − f k ) )log f k (1 − f k ) ⇒ T ≤ log (cid:15) + 2 . T holds uniformly for all dimensions, this directly implies the theorem. We will now show that the Distortion bounds of sequential deliberation are significant, meaningthat mechanisms from simpler classes are constrained to have higher Distortion values. We presenta sequence of lower bounds for social choice mechanisms that use increasingly rich informationabout the space of alternatives on a median graph G = ( S , E ) with a set of agents N with blisspoints V N ⊆ S .We first consider mechanisms that are constrained to choose outcomes in V N . For instance,this captures the random dictatorship that chooses the bliss point of a random agent as the finaloutcome. It shows that the compromise alternatives found by deliberation do play a role in reducingDistortion. Lemma 3.
Any mechanism constrained to choose outcomes in V N has Distortion at least 2.Proof. It is easy to see that the k -star graph (the graph with a central vertex connected to k othervertices none of which have edges between themselves) is a median graph. Consider an |N | -stargraph where V N are the non central vertices; that is, each and every agent has a unique blisspoint on the periphery of the star. Then any mechanism constrained to choose outcomes in V N must choose one of these vertices on the periphery of the star. The social cost of such a point is( |N |− ×
2, whereas the social cost of the optimal central vertex is clearly just |N | . The Distortiongoes to 2 as |N | grows large.We draw more contrasts between sequential deliberation and random dictatorship in appendix A.In particular, we show that sequential deliberation dominates random dictatorship on every instancefor median graphs, and converges to a Distortion of one for nearly unanimous instances, unlike ran-dom dictatorship. We next consider mechanisms that are restricted to choosing the median of thebliss points of some three agents in N . In particular, this captures sequential deliberation run for T = 1 steps, as well as mechanisms that generalize dictatorship to an oligarchy composed of atmost 3 agents. This shows that iteratively refining the bargaining outcome has better Distortionthan performing only one iteration. 11 emma 4. Any mechanism constrained to choose outcomes in V N or a median of three points in V N must have Distortion at least 1.316.Proof. Let G be a median graph; in particular let G be the D -dimensional hypercube { , } D .For every dimension, an agent has a 1 in that dimension of their bliss point independently withprobability p . In expectation p |N | agents’ bliss points have a 1 in any given dimension. We assume0 < p < / a ∗ being the all 0’s vector, E [ SC ( a ∗ )] = D |N | p , wherethe randomness is in the construction. Now, suppose D = polylog( |N | ). Then, for any β <
1, withprobability at least 1 − poly ( |N | ) every three points in V N has at least β (3 p − p ) D ones. Byunion bounds, for some α ∈ ( β, V N is at least: E [ SC ( a )] ≥ D |N | α (cid:2) (3 p − p )(1 − p ) + p (1 − p + 2 p ) (cid:3) = D |N | α (cid:0) p − p + 3 p + p (cid:1) (3)where again, the randomness is in the construction. Then there is nonzero probability that E [ SC ( a )] E [ SC ( a ∗ )] ≥ α (cid:0) p − p + 3 p + 1 (cid:1) (4)If we choose the argmax of Equation 4, we get nonzero probability over the construction that theDistortion is at least (1 . α . Letting α grow close to 1 and noting that the nonzero probabilityover the construction implies the existence of one such instance completes the argument.We finally consider a class of mechanisms that includes sequential deliberation as a special case.We show that any mechanism in this class cannot have Distortion arbitrarily close to 1. This alsoshows that sequential deliberation is close to best possible in this class. Lemma 5.
Any mechanism constrained to choose outcomes on shortest paths between pairs ofoutcomes in V N must have Distortion at least / . .Proof. The construction of the lower bound essentially mimics that of Lemma 4. In this casehowever, we get that each point on a shortest path between two agents has at least α ( p × (1 − p ) + (1 − p ) × p ) D ones, so E [ SC ( a )] ≥ D |N | α (cid:2) p × (1 − p ) + (1 − p ) × p (cid:3) = D |N | α (cid:0) − p + p + p (cid:1) So there is nonzero probability over the construction that E [ SC ( a )] E [ SC ( a ∗ )] ≥ α (cid:0) − p + p + 1 (cid:1) where the maximum of − p + p + 1 over p < / / p = 1 /
4. The rest of the argumentfollows as in Lemma 4.The significance of the lower bound in Lemma 5 should be emphasized: though there is alwaysa Condorcet winner in median graphs, it need not be any agent’s bliss point, nor does it need tobe Pareto optimal for any pair of agents. The somewhat surprising implication is that any localmechanism (in the sense that the mechanism chooses locally Pareto optimal points) is constrainedaway from finding the Condorcet winner. 12
Properties of Sequential Deliberation
In this section, we study some natural desirable properties for our mechanism: uniqueness ofthe stationary distribution of the Markov chain, ex-post Pareto-efficiency of the final outcome, andsubgame perfect Nash equilibrium.
Uniqueness of the Stationary Distribution
We first show that the Markov chain correspond-ing to sequential deliberation converges to a unique stationary distribution on the actual mediangraph, rather than just showing that the marginals and thus the expected distances from theperspectives of the agents converge.To prove uniqueness, it will be helpful to note that the Markov chain defined by sequentialbargaining on G by N only puts nonzero probability mass on points in the median closure M ( V N )of V N (see Definition 2 and Figure 4 for an example). This is the state space of the Markov chain,and there is a directed edge (i.e., nonzero transition probability) from x to v if there exist u, w ∈ V N such that v = M ( x, u, w ) (where M ( x, u, w ) is the median of x, u, w by a slight abuse of notation). Definition 2.
Let V N ⊆ S be the set of bliss points of agents in N . A point v is in M ( V N ) if v ∈ V N or if there exists some sequence ( x , y ) , ( x , y ) , . . . , ( x t , y t ) such that every point in everypair in the sequence is in V N and there is some z ∈ V N s.t. v = M ( x t , y t , M ( x t − , y t − , M ( . . . M ( x , y , z ) . . . ))) ∈ V N = M ( x , y , z ) / ∈ M ( V N )Figure 4: The median closure of the red points is given by the red and blue points. Theorem 2.
The Markov chain defined in Theorem 1 has a unique stationary distribution.Proof.
Let G = ( S , E ) be a median graph, let N be a set of agents, and let V N ⊆ S be the set ofbliss points of the agents in N . The Markov chain will have a unique stationary distribution if it isaperiodic and irreducible. To see that the chain is aperiodic, note that for any state o t = v ∈ M ( V N )of the Markov chain at time t , there is a nonzero probability that o t +1 = v . This is obvious if v ∈ V N ,as the agent corresponding to that bliss point might be drawn twice in round t +1 (remember, agentsare drawn independently with replacement from N ). If instead v / ∈ V N , we know by definition of M ( V N ) that there exist u, w ∈ V N and x ∈ M ( V N ) such that v = median( u, w, x ). But then v = median( u, w, v ). Clearly we can write d ( u, v ) = d ( u, v ) + d ( v, v ) and d ( w, v ) = d ( w, v ) + d ( v, v ),then the fact that v = median( u, w, x ) implies that d ( u, w ) = d ( u, v ) + d ( v, w ). Taken together,these equalities imply that v is the median chosen in round t + 1. So in either case, there is someprobability that o t +1 = v . The period of every state is 1, and the chain is aperiodic.To argue that the chain is irreducible, suppose for a contradiction that there exist t, v ∈ M ( V N )such that there is no path from t to v . Then v / ∈ V N , since all nodes in V N clearly have an To interpret this sequence, note that z represents the initial disagreement alternative drawn as the bliss pointof a random agent. ( x t , y t ) are the bliss points of the random agents drawn to bargain in every round. v thereforerepresents a feasible outcome of sequential deliberation. M ( V N ). Then by definition there exists some sequence( x , y ) , ( x , y ) , . . . , ( x t , y t ) ∈ V N and some z ∈ V N such that v = M ( x t , y t , M ( x t − , y t − , M ( . . . M ( x , y , z ) . . . ))) . Since z ∈ V N , z must have an incoming edge from t . But then there is a path from t to v . Thisis a contradiction; the chain must be irreducible as well. Both properties together show that theMarkov chain has a unique stationary distribution. Pareto-Efficiency
The outcome of sequential deliberation is ex-post Pareto-efficient on a mediangraph. In other words, in any realization of the random process, suppose the final outcome is o ;then there is no other alternative a such that d ( a, v ) ≤ d ( o, v ) for every v ∈ N , with at least oneinequality being strict. This is a weak notion of efficiency, but it is not trivial to show; while it is easyto see that a one shot bargaining mechanism using only bliss points is Pareto efficient by virtue ofthe Pareto efficiency of bargaining, sequential deliberation defines a potentially complicated Markovchain for which many of the outcomes need not be bliss points themselves. Theorem 3.
Sequential deliberation among a set N of agents, where the decision space S is amedian graph and the initial disagreement point a is the bliss point of some agent, yields an ex-post Pareto Efficient alternative.Proof. Let G = ( S , E ) be a median graph, let N be a set of agents, and let V N ⊆ S be the set of blisspoints of the agents in N . It follows from the proof of Corollary 1 that without loss of generality wecan suppose S is a hypercube embedding. Consider some realization ( x , y ) , ( x , y ) , ..., ( x T , y T ) ofsequential bargaining, where ( x t , y t ) ∈ S × S are the bliss points of the agents drawn to bargain instep t . Let o T denote the final outcome. For the sake of contradiction assume there is an alternative z that Pareto-dominates o T , i.e., d ( z, v ) ≤ d ( o T , v ) for each v ∈ N , with at least one inequalitybeing strict. V N t x t y t a t ! o t K t ! ! ! ∅ ! Figure 5: An example of sequential deliberation with K t labeled. The dimensions are numbered1 , , . . . , K t be the set of dimensions of the hypercube S that are “de-cided” by the agents in round t in the sense that these agents agree on that dimension (andcan thus ignore the outside alternative in that dimension) and all future agents disagree onthat dimension (and thus keep the value decided by bargaining in round t ). Formally, K t = (cid:110) k : x tk = y tk and ∀ t (cid:48) > t, x t (cid:48) k (cid:54) = y t (cid:48) k (cid:111) . Then by the majority property of the median on the hyper-cube, for any dimension k such that k ∈ K t for some t ∈ { , . . . , T } , it must be that o Tk = x tk . Anexample is shown in Figure 5. 14onsider the final round T . It must be that ∀ k ∈ K T , z k = o Tk . If this were not the case, z would not be pairwise efficient (i.e., on a shortest path from x T to y T ), whereas o T is pairwiseefficient by definition of the median, so one of the agents in round T would strictly prefer o T to z ,violating the dominance of z over o T .Next, consider round T −
1. Partition the dimensions of S into K T , K T − and all others. Supposefor a contradiction that ∃ k ∈ K T − such that z k (cid:54) = o Tk , where x T − k = y T − = o Tk by definition.Then the agents in round T − o T to z on the dimensions in K T − . But for k ∈ K T , we know that z k = o Tk , so the agents are indifferent between z and o T on the dimensionsin K T . Furthermore, for k / ∈ K T ∪ K T − , x T − k (cid:54) = y T − k , so at least one of the two agents at leastweakly prefers o T to z on the remaining dimensions. But then at least one agent must strictlyprefer o T to z , contradicting the dominance of z over o T .Repeating this argument yields that for all k ∈ K ∪ K ∪ . . . ∪ K T , z k = o Tk . For all otherdimensions, o Tk takes on the value a , which is the bliss point of some agent. Since that agent mustweakly prefer z to o T , z must also take the value of her bliss point on these remaining dimensions.But then z = o T , so z does not Pareto dominate o T , a contradiction. Truthfulness of Extensive Forms
Finally, we show that sequential deliberation has truth-telling as a sub-game perfect Nash equilibrium in its induced extensive form game. Towards thisend, we formalize a given round of bargaining as a 2-person non-cooperative game between twoplayers who can choose as a strategy to report any point v on a median graph; the resulting outcomeis the median of the two strategy points chosen by the players and the disagreement alternativepresented. The payoffs to the players are just the utilities already defined; i.e., the player wishesto minimize the distance from their true bliss point to the outcome point. Call this game thenon-cooperative bargaining game (NCBG).The extensive form game tree defined by non-cooperative bargaining consists of 2 T alternatinglevels: Nature draws two agents at random, then the two agents play NCBG and the outcomebecomes the disagreement alternative for the next NCBG. The leaves of the tree are a set of pointsin the median graph; agents want to minimize their expected distance to the final outcome. Theorem 4.
Sequential NCBG on a median graph has a sub-game perfect Nash equilibrium whereevery agent truthfully reports their bliss point at all rounds of bargaining.Proof.
The proof is by backward induction. Let G = ( S , E ) be a median graph. In the base case,consider the final round of bargaining between agents u and v with bliss points p u and p v anddisagreement alternative a . The claim is that u playing p u and v playing p v is a Nash equilibrium.By Lemma 2, we can embed G isometrically into a hypercube Q as φ : G → Q and consider thebargaining on this embedding. Then for any point z that agent u plays d ( p u , M ( z, p v , a )) = d ( φ ( p u ) , M ( φ ( z ) , φ ( p v ) , φ ( a )))The median on the hypercube is just the bitwise majority, so if u plays some z where for somedimension φ k ( z ) (cid:54) = φ k ( p u ), it can only increase u ’s distance to the median. So playing p u is a bestresponse.For the inductive step, suppose u is at an arbitrary subgame in the game tree with t roundsleft, including the current bargain in which u must report a point, and assuming truthful play in allsubsequent rounds. Let { ( x , y ) , ( x , y ) , ..., ( x t , y t ) } represent ( x , y ) as the outside alternativeand other agent bliss point against which u must bargain, ( x , y ) as the bliss points of the agentsdrawn in the next round, and so on. We want to show that it is a best response for agent u tochoose p u , i.e., to truthfully represent her bliss point. Define M tu := M ( x t , y t , M ( x t − , y t − , M ( . . . M ( x , y , p u ) . . . )))15here M ( · ) indicates the median, guaranteed to exist and be unique on the median graph. Also,for any point z , similarly define M tz := M ( x t , y t , M ( x t − , y t − , M ( . . . M ( x , y , z ) . . . )))Suppose by contradiction that p u is not a best response for agent u , then there must exist z (cid:54) = p u and some r > d ( p u , M ru ) > d ( p u , M rz ). We embed G isometrically into a hypercube Q as φ : G → Q . Then by the isometry property, d ( φ ( p u ) , φ ( M ru )) > d ( φ ( p u ) , φ ( M rz )). By the proofof Corollary 1, we can pretend the process occurs on the hypercube.Consider some dimension k . If φ k ( x t ) = φ k ( y t ) for some t ≤ r , then this point becomes themedian in that dimension, so the median becomes independent of φ k ( w ), where w is the initialreport of agent u . Till that time, the bargaining outcome in that dimension is the same as φ k ( w ).In either case, for all times t ≤ r and all k , we have: | φ k ( p u ) − φ k ( M tu ) | ≤ | φ k ( p u ) − φ k ( M tz ) | Summing this up over all dimensions, d ( φ ( p u ) , φ ( M tu )) ≤ d ( φ ( p u ) , φ ( M tz )), which is a contradiction.Therefore, p u was a best response for agent u . Therefore, every agent truthfully reporting theirbliss points at all rounds is a subgame perfect Nash equilibrium of Sequential NCBG. We now work in the very general setting that the set S of alternatives are points in a finitemetric space equipped with a distance function d ( · ) that is a metric. As before, we assume eachagent u ∈ N has a bliss point p u ∈ S . An agent’s disutility for an alternative a ∈ S is simply d ( p u , a ). We first present results for the Distortion, and subsequently define the second moment, orSquared-Distortion. For both measures, we show that the upper bound for sequential deliberationis at most a constant regardless of the metric space. Theorem 5.
The Distortion of sequential deliberation is at most 3 when the space of alternativesand bliss points lies in some metric, and this bound is tight.Proof.
Each agent u ∈ N has a bliss point p u ∈ S . An agent’s disutility for an alternative a ∈ S is simply d ( p u , a ). Let a ∗ ∈ S be the social cost minimizer, i.e., the generalized median. Forconvenience, let Z a = d ( a, a ∗ ) for a ∈ S . By a slight abuse of notation, let Z i = d ( p i , a ∗ ) for i ∈ N ,i.e., the distance from agent i ’s bliss point to a ∗ .We will only use the assumption that B ( u, v, a ) finds a Pareto efficient point for u and v , sorather than taking an expectation over the choice of the disagreement alternative, we take the worstcase. Let a ∗ be the social cost minimizer with social cost OP T . We can write then write expected It is important to note that we are not assuming that agent u will not bargain again in the subgame; there areno restrictions on the values of { ( x , y ) , ( x , y ) , ..., ( x t , y t ) } . (cid:88) i ∈N (cid:88) j ∈N |N | max a ∈S (cid:88) k ∈N d ( B ( i, j, a ) , p k ) ≤ (cid:88) i,j ∈N |N | max a ∈S (cid:88) k ∈N d ( B ( i, j, a ) , a ∗ ) + Z k = OP T + (cid:88) i,j ∈N |N | max a ∈S ( d ( B ( i, j, a ) , a ∗ )) (cid:2) Triangle inequality (cid:3) ≤ OP T + (cid:88) i,j ∈N |N | max a ∈S ( d ( B ( i, j, a ) , p i ) + Z i + d ( B ( i, j, a ) , p j ) + Z j ) (cid:2) Triangle inequality (cid:3) ≤ OP T + (cid:88) i,j ∈N |N | (2 Z i + 2 Z j ) (cid:2) Pareto efficiency of B ( · ) (cid:3) = 3 OP T
Since this holds for the worst case choice of disagreement alternative, it holds over the wholesequential process. The tight example is a weighted graph that can be constructed as follows: startwith a star with a single agent’s bliss point located at each leaf of the star (i.e., an | N | -star) whereevery edge in the star has weight (or distance) 1. Now, for every pair of agents, add a vertexconnected to the bliss points of both of the agents with weight 1 − (cid:15) for (cid:15) >
0. Then as | N | → ∞ ,every round of Nash bargaining selects one of these pairwise vertices. But the central vertex ofthe star has social cost | N | whereas the pairwise vertices all have social cost approaching 3 | N | as (cid:15) → / We now show that for any metric space, sequential deliberation has a crucial advantage interms of the distribution of outcomes it produces. For this, we consider the second moment, orthe expected squared social cost. Recall that the social cost of an alternative a ∈ S is given by SC ( a ) = (cid:80) u ∈N d ( p u , a ). Let a ∗ ∈ S be the minimizer of social cost, i.e. , the generalized median .Then define: Squared-Distortion = E [( SC ( a )) ]( SC ( a ∗ )) where the expectation is over the set of outcomes a produced by Sequential Deliberation. We will show that sequential deliberation has Squared-Distortion upper bounded by a constant.This means the standard deviation in social cost of the distribution of outcomes is comparable tothe optimal social cost. This has a practical implication: A policy designer can run sequentialdeliberation for a few steps, and be sure that the probability of observing an outcome that has γ times the optimal social cost is at most O (1 /γ ). In contrast, we show that Random Dictatorship The motivation for considering Squared-Distortion instead of the standard deviation is that the latter mightprefer a more deterministic mechanism with a worse social cost, a problem that the Squared-Distortion avoids. Z a = d ( a, a ∗ ) for a ∈ S . By a slight abuse of notation, let Z i = d ( p i , a ∗ )for i ∈ N , i.e., the distance from agent i ’s bliss point to a ∗ . We will need the following technicallemma bounding these distances. The lemma addresses the following question: given an arbitraryagent u , how far away can the outcome of a bargaining round with agents i and j and disagreementalternative a be from p u ? The answer is that it cannot be much further than the values of Z u andthe smaller of Z i , Z j , Z a . The two min functions in the bound serve to eliminate outliers, and thisis crucial for bounding the Squared-Distortion. Lemma 6.
For all i, j, u ∈ N and a ∈ S we have that d ( B ( i, j, a ) , p u ) ≤ Z u + 2 min ( Z i , Z j ) + min ( Z a , max ( Z i , Z j )) Proof.
Assume w.l.o.g. that Z i ≤ Z j . Then the lemma statement reduces to d ( B ( i, j, a ) , p u ) ≤ Z u + 2 Z i + min ( Z a , Z j ) (5)Recall that Nash bargaining asserts that given a disagreement alternative a , agents i and j choosethat alternative o ∈ S that maximizes:Nash product = ( d ( p i , a ) − d ( p i , o )) × ( d ( p j , a ) − d ( p j , o ))Maximizing this on a general metric yields that d ( B ( i, j, a ) will be chosen on the p i to p j shortestpath (that is, the Pareto efficient frontier) at a distance of d ( p i ,p j )2 + d ( p i ,a ) − d ( p j ,a )2 from p i . Therefore,we have that: d ( B ( i, j, a ) , p i ) = d ( p i , p j )2 + d ( p i , a ) − d ( p j , a )2 (6)Now we can show equation 5 by repeatedly using the triangle inequality. We have d ( B ( i, j, a ) , p u ) ≤ d ( p i , p u ) + d ( B ( i, j, a ) , p i )= d ( p i , p u ) + d ( p i , p j )2 + d ( p i , a ) − d ( p j , a )2 (cid:2) equation 6 (cid:3) ≤ d ( p i , a ∗ ) + d ( p u , a ∗ ) + d ( p i , a ∗ ) + d ( p j , a ∗ ) + d ( p i , a ∗ ) + d ( a, a ∗ )2 − d ( p j , a )2= Z u + 2 Z i + Z j + Z a − d ( p j , a )2Note that we can apply the triangle inequality again to get two bounds, since Z j ≤ d ( p j , a ) + Z a and Z a ≤ d ( p j , a ) + Z j . Since both bounds must hold, we have that d ( B ( i, j, a ) , p u ) ≤ Z u + 2 Z i +min ( Z a , Z j ).It is not hard to see that this bound is tight in the worst case. Suppose agents p i , p j , p u formthe leaves of a 3-star (with weight 1 edges) and a is the center of the star. Now suppose thereis a point o connected to p i and to p j each by edges of weight 1 − (cid:15) for (cid:15) > o will clearlybe the bargaining outcome, so d ( B ( i, j, a ) , p u ) = 3 − (cid:15) . But the right hand side of the inequality, Z u + 2 min ( Z i , Z j ) + min ( Z a , max ( Z i , Z j )) = 3, so the inequality becomes tight as (cid:15) → heorem 6. The Squared-Distortion of sequential deliberation for T ≥ is at most when thespace of alternatives and bliss points lies in some metric. Furthermore, the Squared-Distortion ofrandom dictatorship is unbounded.Proof. Let
OP T be the squared social cost of a ∗ . Note that OPT does not depend on T . Inparticular, OP T = (cid:0)(cid:80) i ∈N Z i (cid:1) = (cid:80) i,j ∈N Z i Z j .For arbitrary T , the disagreement alternative in the final step of bargaining generated by se-quential deliberation. We will therefore rely on the first moment bound given in Theorem 5 forsuch an outcome. Let ALG be the expected squared social cost of sequential deliberation with T steps, where the disagreement alternative a ∈ S is used in the final round of bargaining. i and j are the last two agents to bargain, and p k is the bliss point of an arbitrary agent. We can write ALG as ALG = (cid:88) i ∈N (cid:88) j ∈N |N | (cid:32) (cid:88) k ∈N d ( B ( i, j, a ) , p k ) (cid:33) ≤ |N | (cid:88) i,j ∈N (cid:32) (cid:88) k ∈N Z k + 2 min ( Z i , Z j ) + min ( Z a , max ( Z i , Z j )) (cid:33) (cid:2) By lemma 6 (cid:3) = 1 |N | (cid:88) i,j ∈N (cid:32) |N | min ( Z i , Z j ) + |N | min ( Z a , max ( Z i , Z j )) + (cid:32) (cid:88) k ∈N Z k (cid:33)(cid:33) Now we expand the square and analyze term by term, using the facts that min ( x, y ) ≤ x × y and max ( x, y ) ≤ x + y .= 1 |N | (cid:88) i,j ∈N (cid:32) |N | min ( Z i , Z j ) + |N | min ( Z a , max ( Z i , Z j )) + (cid:32) (cid:88) k ∈N Z k (cid:33) + 4 |N | min ( Z i , Z j ) min ( Z a , max ( Z i , Z j )) + 4 |N | min ( Z i , Z j ) (cid:32) (cid:88) k ∈N Z k (cid:33) + 2 |N | min ( Z a , max ( Z i , Z j )) (cid:32) (cid:88) k ∈N Z k (cid:33) (cid:33) ≤ |N | (cid:88) i,j ∈N (cid:32) |N | Z i Z j + |N | min ( Z a , max ( Z i , Z j )) + OP T + 4 |N | Z i Z a + 4 |N | Z i (cid:32) (cid:88) k ∈N Z k (cid:33) + 2 |N | Z a (cid:32) (cid:88) k ∈N Z k (cid:33) (cid:33) We can trivially sum out the terms not involving a , leaving us with inequality 7. ALG ≤ OP T + (cid:88) i,j ∈N (cid:32) Z a ( Z i + Z j ) + 4 Z i Z a + 2 |N | Z a (cid:32) (cid:88) k ∈N Z k (cid:33) (cid:33) (7)Note that the triangle inequality implies that for any i ∈ N , Z a ≤ d ( a, p i ) + Z i . Applying this19epeatedly in inequality 7 and simplifying yields: ALG ≤ OP T + (cid:88) i,j ∈N (cid:32) Z i Z j + 5 d ( a, p j ) Z i + d ( a, p i ) Z j + 2 |N | ( d ( a, p i ) + Z i ) (cid:32) (cid:88) k ∈N Z k (cid:33)(cid:33) = 17 OP T + (cid:88) i ∈N d ( a, p i ) (cid:32) (cid:88) k ∈N Z k (cid:33) + (cid:88) j ∈N d ( a, p j ) Z i + d ( a, p i ) Z j Recall that Theorem 5 implies that (cid:80) i ∈N d ( a, p i ) ≤ (cid:80) i ∈N Z i . ALG ≤ OP T + 6
OP T + 15
OP T + 3
OP T = 41
OP T
It is easy to see that Random Dictatorship has an unbounded Squared-Distortion. Recall thatRandom Dictatorship chooses the bliss point of an agent chosen uniformly at random. Considerthe simple graph with two nodes, a fraction f of the agents on one node and 1 − f on the other. Let f < /
2. Then the expected squared social cost of Random Dictatorship is just f (1 − f ) +(1 − f ) f whereas the optimal solution has squared social cost f , so Random Dictatorship has Squared-Distortion (1 − f ) /f + (1 − f ), which is unbounded as f → Our work is the first step to developing a theory around practical deliberation schemes. Wesuggest several future directions. First, we do not have a general characterization of the Distortionof sequential deliberation for metric spaces. We have shown that for general metric spaces there is asmall but pessimistic bound on the Distortion of 3, but that for specific metric spaces the Distortionmay be much lower. We do not have a complete characterization of what separates these good andbad regimes.More broadly, an interesting question is extending our work to take opinion dynamics into ac-count, i.e. , proving stronger guarantees if we assume that when two agents deliberate, each agent’sopinion moves slightly towards the other agent’s opinion and the outside alternative. Furthermore,though we have shown that all agents deliberating at the same time does not improve on dictator-ship, it is not clear how to extend our results to more than two agents negotiating at the same time.This runs into the challenges in understanding and modeling multiplayer bargaining [18, 23, 6].Finally, it would be interesting to conduct experiments to measure the efficacy of our frameworkon complex, real world social choice scenarios. There are several practical hurdles that need to beovercome before such a system can be feasibly deployed. 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A Random Dictatorship and N -Person Bargaining on Median Graphs A simple baseline algorithm for our problem is Random Dictatorship: Output the bliss point p u of a random agent u ∈ N . Such an algorithm also satisfies the desiderata that we presented inSection 4: The outcome is trivially Pareto-efficient and the mechanism is truthful. The Distortionof Random Dictatorship on social cost is at most 2, and Theorem 3 shows this bound is tight. Wenow show two ways that sequential deliberation is superior to Random Dictatorship despite theirseeming similarity. These statements are stronger than what the worst-case Distortion boundsimply. Theorem 7.
For any set of agents N and outcomes S that define a median graph, the expectedsocial cost of sequential deliberation is at most that of Random Dictatorship.Proof. As in the proof of Theorem 1, we consider the hypercube embedding, and decompose thesocial cost along each dimension k . Let f k denote the fraction of agents mapping to 1 alongdimension k . Then, the previous proof showed that the social cost of sequential deliberation isSocial Cost of Sequential Deliberation = |N | (cid:18) f k (1 − f k )1 + 2 f k − f k (cid:19) Random Dictatorship would choose an agent mapping to 1 with probability f k , in which case afraction (1 − f k ) of agents pay cost 1, and vice versa. Therefore its social cost isSocial Cost of Random Dictatorship = |N | (2 f k (1 − f k ))It is an easy exercise to show that for all f k ∈ [0 , efinition 3. A set of agents N is said to be (cid:15) - unanimous on a set of outcomes S if all but an (cid:15) fraction of the agents have the same bliss point in S . Corollary 2.
Sequential deliberation with a (cid:15) -unanimous set of agents on a median graph hasDistortion at most (1 + (cid:15) ) . In contrast, Random Dictatorship has Distortion − (cid:15) .Proof. As in Theorem 1, we can analyze sequential deliberation on the hypercube. As before, let f k be the fraction of agents with a 1 in dimension k and assume w.l.o.g. that f k ∈ [0 , /
2] so that theoptimal point is the all 0 vector. In the proof of Theorem 1, we derived the following as a boundfor the Distortion of sequential deliberation.max f k ∈ [0 , / − f k f k − f k ≤ .
208 (8)If (cid:15) ≥ .
208 then the result is immediate. Otherwise, note that under (cid:15) -unanimity, it must in factbe the case that f k ∈ [0 , (cid:15) ], so the maximization is over a more restricted domain. Furthermore, for (cid:15) < . (cid:15) . It is easy to see this by taking the derivative. Thus, for (cid:15) < . f k ∈ [0 ,(cid:15) ] − f k f k − f k = 1 − (cid:15) (cid:15) − (cid:15) ≤ − (cid:15) (cid:15) − (cid:15) − (cid:15) (cid:15) = 1 + (cid:15) In contrast it is easy to see that even for the simple median graph consisting of two points (theone dimensional hypercube), Random Dictatorship can have Distortion 2 even as the agents are (cid:15) -unanimous. On such a graph, the optimal social cost is just (cid:15) |N | but Random Dictatorship hasexpected social cost (1 − (cid:15) ) (cid:15) |N | + (cid:15) (1 − (cid:15) ) |N | = 2 (cid:15) (1 − (cid:15) ) |N | . The ratio goes to 2 as (cid:15) goes to 0.This shows Random Dictatorship can actually obtain its worst case Distortion even on the simplestpossible instances. N -Person Nash Bargaining. Finally, we present a canonical instance of bargaining that de-volves into Random Dictatorship by allowing all agents to bargain in a single round. Consider aline metric. Suppose n > n = N − n > o is any point x ∈ [0 , N agents simultaneously bargainto maximize the Nash product of the increase in their utility with respect to o . It is easy to showthe following result: Theorem 8. N -person Nash bargaining for N agents whose bliss points lie at and on a line,with the disagreement outcome x ∈ [0 , , outputs the same point x as the final outcome. This follows because any deviation makes at least one agent strictly less happy, and this agentwill stick to disagreeing. Suppose x corresponds to the bliss point of a random agent, then thismechanism coincides with Random Dictatorship and has Distortion 2. This shows that the Distor-tion of N -person Nash bargaining even with a random bliss point as disagreement outcome is worsethan pairwise deliberation. Note that the result only needs enough agents so that there is one agentat 0 and the other at 1. This confirms (on the line) the intuitive observation that deliberation inlarge groups tends to break down and produce random outcomes.24 Distortion of Deliberation on the Unit Simplex
We have shown that for general metric spaces there is a small but pessimistic bound on theDistortion of 3, but that for median graphs, the Distortion is much lower. This begs the question:
Are there spaces beyond median graphs for which sequential deliberation has Distortion much lessthan ? Though we leave the characterization of such spaces as an open question, we show thatour results do extend beyond median graphs. Consider the following special case motivated bybudgeting applications. The outcome space S is the d -dimensional standard simplex. Agents arelocated at the vertices of this simplex, and the distances are (cid:96) . It is clear that this is not a medianspace; in fact, the shortest paths between pairs of vertices from a triplet do not intersect. We canview the vertices as items of unit size, and the restriction to a simplex as a unit budget constraint.Agents’ bliss points correspond to single items, while the outcome space is all possible fractionalallocations. We show that the Distortion of sequential deliberation in this setting is 4 /
3, whilethe corresponding bound for random dictatorship is again 2. Though this setting is stylized in thesense that approval voting on items coincides with the social optimum, our scheme itself is generaland not tied to the designer knowing that utilities lie in this specific space.The space S is the d -dimensional standard simplex: S = (cid:126)x ∈ (cid:60) d | d (cid:88) j =1 x j = 1; x j ≥ ∀ j ∈ { , , . . . , d } The are N agents whose bliss points are its vertices. We denote the vertex with x j = 1 as v j .Suppose the probability mass of agents located at vertex j is p j , so that (cid:80) dj =1 p j = 1.Consider a disagreement alternative (cid:126)a ∈ S , and Nash bargaining between agents at v i and v j with j (cid:54) = i . It is easy to check that this produces the outcome with o i = (1 + a i − a j ) /
2, and o j = (1 + a j − a i ) /
2. When i = j , we have o i = 1.Let π σ denote the distribution over alternatives σ ∈ S in steady state. Define the expectedvalue of coordinate i as: s i = (cid:88) σ π σ σ i ∀ i ∈ { , , . . . , d } The steady state conditions imply the set of equations: (cid:88) σ π σ σ i = p i + (cid:88) σ π σ (cid:88) j (cid:54) = i p i p j (cid:18) σ i − σ j (cid:19) The LHS is the expected value of σ i according to distribution π . The RHS is how the value evolvesin one step, which should yield the same quantity. By definition of s i , we therefore have: s i = p i + (cid:88) j (cid:54) = i p i p j (cid:18) s i − s j (cid:19) ∀ i ∈ { , , . . . , d } and (cid:80) di =1 s i = 1. Solving for this system, we obtain: s i = p i − p i × (cid:80) dj =1 p j − p j Note next that the optimal social cost is
OP T = min i (1 − p i ). Without loss of generality,assume this is 1 − p . Let α = p , so that OP T = 1 − α . The social cost with respect to the steady25tate distribution is given by ALG = (cid:88) σ π σ (cid:88) i p i (1 − σ i ) = (cid:88) i p i (1 − s i ) = 1 − (cid:88) i p i s i = 1 − (cid:80) i p i − p i (cid:80) j p j − p j = α + (cid:80) j> p jα − α + (cid:80) j> p j − p j = 1 α − α + (cid:80) j> p j − p j ≤ α − α + (cid:80) j> p j = 1 α − α + 1 − α = 1 − α − α + α Therefore, the Distortion of sequential deliberation is at mostmax α ∈ [0 , − α + α = 43Note that on this instance, the distortion of random dictatorship is α (1 − α ) + (cid:80) i> p i (1 − p i )1 − α ≤ α ≤≤