Sequential Deliberation for Social Choice

10/02/2017 ∙ by Brandon Fain, et al. ∙ Duke University Stanford University 0

In large scale collective decision making, social choice is a normative study of how one ought to design a protocol for reaching consensus. However, in instances where the underlying decision space is too large or complex for ordinal voting, standard voting methods of social choice may be impractical. 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 propose sequential deliberation as a natural solution to this problem. In this iterative method, successive pairs of agents bargain over the decision space using the previous decision as a disagreement alternative. We describe the general method and analyze the quality of its outcome when the space 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 with only a small constant number of agents sampled from the population. We also show lower bounds on simpler classes of mechanisms to justify our design choices. We further show that sequential deliberation is ex-post Pareto efficient and has truthful reporting as an equilibrium of the induced extensive form game. We finally show that for general metric spaces, the second moment of of the distribution of social cost of the outcomes produced by sequential deliberation is also bounded.

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1 Introduction

Suppose a university administrator plans to spend millions of dollars to update her campus, and she wants to elicit the input of students, staff, and faculty. In a typical social choice setting, she could first elicit the bliss points of the students, say “new gym,” “new library,” and “new student center.” However, voting on these options need not find the social optimum, because it is not clear that the social optimum is even on the ballot. In such a setting, deliberation between individuals would find entirely new alternatives, for example “replace gym equipment plus remodeling campus dining plus money for scholarship.” This leads to finding a social optimum over a wider space of semi-structured outcomes that the system/mechanism designer was not originally aware of, and the participants had not initially articulated.

We therefore start with the following premise: The mechanism designer may not be able to enumerate the outcomes in the decision space or know their structure, and this decision space may be too big for most ordinal voting schemes. (For instance, ordinal voting is difficult to implement in complex combinatorial spaces [24] or in continuous spaces [14].) However, we assume that agents can still reason about their preferences and small groups of agents can negotiate over this space and collaboratively propose outcomes that appeal to all of them. Our goal is to design protocols based on such a primitive by which small group negotiation can lead to an aggregation of societal preferences without a need to formally articulate the entire decision space and without every agent having 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 successful history of large scale deliberation and decision making on complex issues; in fact, large online forums typically degenerate into vitriol and name calling when there is substantive disagreement among the participants. Thus, if we are to develop practical tools for decision making at scale, a sequence of small group deliberations appears to be the most plausible path. On the theoretical side, we understand the connections between sequential protocols for deliberation and axiomatic theories of bargaining for small groups, e.g. for pairs [33, 8], but not for large groups, and we seek to bridge this 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 and collaboratively propose outcomes that appeal to both of them.

  • A canonical analytic model in which we can precisely state properties of this protocol in terms of approximation of the social optimum, Pareto-efficiency, and incentive-compatibility, as well as compare it with simpler protocols.

1.1 Background: Bargaining Theory

Before proceeding further, we review bargaining, the classical framework for two-player negotiation in Economics. Two-person bargaining, as framed in [28], is a game wherein there is a disagreement outcome and two agents must cooperate to reach a decision; failure to cooperate results in the adoption of the disagreement outcome. Nash postulated four axioms that the bargaining solution ought to satisfy assuming a convex space of alternatives: Pareto optimality (agents find an outcome that cannot be simultaneously improved for both of them), symmetry between agents, invariance with respect to affine transformations of utility (scalar multiplication or additive translation of any agent’s utility should not change the outcome), and independence of irrelevant alternatives (informally that the presence of a feasible outcome that agents do not select does not influence their decision). Nash proved that the solution maximizing the Nash product (that we describe later) is the unique solution satisfying these axioms. To provide some explanation of how two agents might find such a solution,  [33]

shows that Nash’s solution is the subgame perfect equilibrium 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 consequence, it has been extensively studied. In fact, there are other models and solutions to two-person bargaining, each with a slightly different axiomatization [20, 21, 27], as well as several experimental studies [32, 29, 7]. In a social choice setting, there are typically many more than two agents, each agent having their own complex preferences. Though bargaining can be generalized to agents with 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; complexities come with the formation of coalitions and power structures [18, 23]. Any model for simultaneous bargaining, even with three players [6], needs to take these messy aspects into account.

1.2 A Practical Compromise: Sequential Pairwise Deliberation

In this paper, we take a middle path, avoiding both the complexity of explicitly specifying preferences in a large decision space that any individual agent may not even fully know (fully centralized voting), and that of simultaneous -person bargaining (a fully decentralized cooperative game). We term this approach sequential deliberation. We use 2-person bargaining as a basic primitive, and view deliberation as a sequence of pairwise interactions that refine good alternatives into better ones as time goes by.

More formally, there is a decision space of feasible alternatives (these may be projects, sets of projects, or continuous allocations) and a set of agents. We assume each agent has a hidden cardinal utility for each alternative. We encapsulate deliberation as a sequential process. The framework that we analyze in the rest of the paper is captured in Figure 1.

In each round : A pair of agents and are chosen independently and uniformly at random with replacement. These agents are presented with a disagreement alternative , and perform bargaining, which is encoded as a function as described below. Agents and are asked to output a consensus alternative; if they fail to reach a consensus then the alternative is output. Let denote the alternative that is output in round . We set , where we assume is the bliss point of an arbitrary agent. The final social choice is . Note that this is equivalent to drawing an outcome at random from 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 reason about. 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 that corresponds to an interaction between and using as the disagreement outcome; we assume that this function maximizes 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 distribution of outcomes produced.

1.3 Analytical Model: Median Graphs and Sequential Nash Bargaining

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 which we can precisely quantify the behavior of this framework as justification.

Median Graphs.

We assume that the set of alternatives are vertices of a median graph. A median graph has the property that for each triplet of vertices , there is a unique point that is common to the three sets of shortest paths (since there may be multiple pairwise shortest paths), those between , between , and between . This point is the unique median of . We assume each agent has a bliss point , and his disutility for an alternative is simply , where is the shortest path distance function on the median graph. (Note that this disutility 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 we discuss in Section 1.5, because of their analytic tractability and special properties, median graphs have been extensively studied as structured models for spatial preferences in voting theory. Some of 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 , agents and choose that alternative that maximizes:

subject to individual rationality, that is, and . The Nash product maximizer need not be unique; in the case of ties we postulate that agents select the outcome that is closest to the disagreement outcome. As mentioned before, the Nash product is a widely studied axiomatic notion of pairwise interactions, and is therefore a natural solution concept in our framework.

Social Cost and Distortion.

The social cost of an alternative is given by . Let be the minimizer of social cost, i.e., the generalized median. We measure the Distortion of outcome as

(1)

where we use the expected social cost if is the outcome of a randomized algorithm.

Note that our model is fairly general. First, the bliss points of the agents in form an arbitrary subset of . Further, the alternative chosen by bargaining need not correspond to any bliss 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 recent literature [3, 2, 9, 10, 16], and our tightest results are for median graphs specifically.

1.4 Our Results

Before presenting our results, we re-emphasize that while we present analytical results for sequential deliberation in specific decision spaces, the framework in Figure 1 is well defined regardless of 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 bargaining between agents and using disagreement outcome outputs the median of . Therefore, . On a general metric space, we show in Section 5.1 that the Nash Bargaining outcome would lie on the shortest path between the agents, and the distance from an agent is proportional to its distance to the disagreement outcome. In a sense, agents only need to explore options on the shortest path between them.

Bounding Distortion.

Our main result in Section 3 shows that for sequential Nash bargaining on a median graph, the expected Distortion of outcome has an upper bound approaching as . Surprisingly, we show that in steps, the expected Distortion is at most , independent of the number of agents, the size of the median space, and the initial disagreement point . For instance, the Distortion falls below in at most steps of deliberation, which only requires a random sample of at most agents from the population to implement.

In Section 3.2, we ask: How good is our numerical bound? We present a sequence of lower bounds for social choice mechanisms that are allowed to use increasingly richer information about the space of alternatives on the median graph. This also leads us to make qualitative statements about our deliberation scheme.

  • We show that any social choice mechanism that is restricted to choosing the bliss point of some agent cannot have Distortion better than . More generally, it was recently shown [17] that even eliciting the top alternatives for each agent does not improve the bound of for median graphs unless . In effect, we show that forcing the agents to reason about cardinal utilities via deliberation leads to new alternatives that are more powerful at reducing Distortion than simply eliciting and aggregating reasonably detailed ordinal rankings.

  • Next consider mechanisms that choose, for some triplet of agents with bliss points , the median outcome . We show this has Distortion at least , which means that sequential deliberation is superior to one-shot deliberation that outputs where is the bliss point of some agent.

  • Finally, for every pair of agents , consider the set of alternatives on a shortest path between and . This encodes all deliberation schemes where finds a Pareto-efficient alternative for some agents at each step. We show that any such mechanisms has Distortion ratio at least . This space of mechanisms captures sequential deliberation, and shows that sequential deliberation is close to best possible within this space.

Properties of Sequential Deliberation.

We next show that sequential deliberation has several natural desiderata on median graphs in Section 4. In particular:

  • Under mild assumptions, the limiting distribution over outcomes of sequential deliberation is unique.

  • For every , the outcome of sequential deliberation is ex-post Pareto-efficient, meaning that there is no other alternative that has at most that social cost for all agents and strictly better cost for one agent. This is not a priori obvious, since the outcome at any one round only uses inputs from two agents; though it is Pareto-efficient for those two agents, it could very well be sub-optimal for the other agents.

  • Interpreted as a mechanism, truthful play is a sub-game perfect Nash equilibrium of sequential deliberation. More precisely, we consider a different view of the function that encodes Nash Bargaining. Suppose agents and report their bliss points, and the platform implements the function that computes the median of , and . 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 points is a sub-game perfect Nash equilibrium of the induced extensive form game.

Beyond Median Graphs.

In Section 5, we consider general metric spaces. We show that the Distortion of sequential deliberation is always at most a factor of . More surprisingly, we show that sequential deliberation has constant distortion even for the second moment of the distribution of social cost of the outcomes, i.e.

, the latter is at most a constant factor worse than the optimum squared social cost. This has the following practical implication: A policy designer can look at the distribution of outcomes produced by deliberation, and know that the standard deviation in social cost is comparable to its expected value, which means deliberation eliminates outlier alternatives and concentrates probability mass on more central alternatives.

111See also recent work by [37]

that considers minimizing the variance of randomized truthful mechanisms.

We also show that such a claim cannot be made for random dictatorship, whose distortion on squared social cost is unbounded.

1.5 Related Work

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 in political science. For examples (by no means exhaustive), see [13, 36]. There is ongoing related work on Distortion of voting for simple analytical models like points in [12], and in general metric spaces [3, 2, 9, 10, 16]. This work focuses on optimally aggregating ordinal preferences, say the top k preferences of a voter [17]. In contrast, our scheme elicits alternatives as the outcome of bargaining rounds that require agents to reason about cardinal preferences.. As mentioned before, we essentially show that for median graphs, unless is very large, such deliberation has provably lower distortion than social choice schemes that elicit purely ordinal rankings.

Median graphs and their ordinal generalization, median spaces, have been extensively studied in the context of social choice. The special cases of trees and grids have been studied as structured models for voter preferences [35, 5]. For general median spaces, it is known that the Condorcet winner (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 the latter. Nehring and Puppe [30] shows that any single-peaked domain which admits a non-dictatorial and neutral strategy-proof social choice function is a median space. Clearwater et al. [11] also showed that any set of voters and alternatives on a median graph will have a Condorcet winner. In a sense, these are the largest class of structured and spatial preferences where ordinal voting over the entire space of alternatives leads to a “clear winner” even by pairwise comparisons. Our work stems from the assumption that this space may not be fully known to the mechanism designer or all agents.

Our paper is inspired by the triadic consensus results of Goel and Lee [15]. In that work, the authors 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 grant their votes. This individual takes these votes and participates in future rounds, until all votes accumulate with one individual, who is the consensus outcome. The analysis proceeds through a median graph, on which the authors show that the Distortion of the consensus approaches . However, the protocol crucially assumes individuals know the positions of other individuals, and requires the space of alternatives to coincide with the space of individuals. We make neither of these assumptions – in our case, the space of alternatives can be much larger than the number of agents, and further, individuals interact with others only via bargaining. This makes our protocol more practical, but at the same time, restricts our Distortion to be bounded away from .

The notion of democratic equilibrium [19, 14] considers social choice mechanisms in continuous spaces where individual agents with complex utility functions perform update steps inspired by gradient descent, instead of ordinal voting on the entire space. However, these schemes do not involve deliberation between agents and have little formal analysis of convergence. Several works have considered iterative voting where the current alternative is put to vote against one proposed by different random agent chosen each step [1, 25, 31], or other related schemes [26]. In contrast with our 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 efficiency guarantees.

2 Median Graphs and Nash Bargaining

In this section we will use the notation for a set of agents, for the space of feasible alternatives, and for a distribution over . Most of our results are for the analytic model given earlier wherein the set of alternatives are vertices of a median graph; see Figure 2 for some examples.

Definition 1.

A median graph is an unweighted and undirected graph with the following property: For each triplet of vertices , there is a unique point that is common to the shortest paths (which need not be unique between a given pair) between , between , and between . This point is the unique median of .

Figure 2: Examples of Median Graphs

In the framework of Figure 1, we assume that at every step, two agents perform Nash bargaining with 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 of the two agents and the disagreement alternative. After that, we show that we can analyze the Distortion of sequential deliberation on a median graph by looking at the embedding of that graph onto the hypercube.

Lemma 1.

For any median graph , any two agents with bliss points , and any disagreement outcome , let be the median. Then maximizes the Nash product of and given , and is the maximizer closest to .

Proof.

Since is a median graph, exists and is unique; it must by definition be the intersection of the three shortest paths between . Note that we can therefore write and similarly for . Let ; ; and . Suppose Nash bargaining finds an outcome . Let and . Observing that lies on the shortest path between and , and using the triangle inequality, we obtain that .

Noting that and , the Nash product of the point is:

This is maximized when and . One possible maximizer is therefore the point . Suppose , then by the triangle inequality, , and similarly . Therefore, there cannot be a closer maximizer of the Nash product to than the point .

Hypercube Embeddings.

For any median graph , there is an isometric embedding of into a hypercube  [22]. This embedding maps vertices into a subset of vertices of so that all pairwise distances between vertices in are preserved by the embedding. A simple example of this embedding for a tree is shown in Figure 3. We use this embedding to show the following result, in order to simplify subsequent analysis.

Figure 3: The hypercube embedding of a 4-vertex star graph
Lemma 2.

Let be a median graph, and let be its isometric embedding into hypercube . For any three points , let be the median of vertices and let be the median of vertices . Then .

Proof.

By definition, since is an isometric embedding [22],

(2)

Since is a median graph, is the unique median of , which by definition satisfies the equalities:

is a hypercube, and is thus also a median graph, so is the unique median of , which by definition satisfies the equalities

Applying Equation (2) to the first set of equalities shows that satisfies equalities I,II, and III respectively. But and is the unique vertex in satisfying equalities I,II, and III. Therefore, . ∎

3 The Efficiency of Sequential Deliberation

In this section, we show that the Distortion of sequential deliberation is at most . We then show that this bound is significant, meaning that mechanisms from simpler classes are necessarily constrained to have higher Distortion values.

3.1 Upper Bounding Distortion

Recall the framework for sequential deliberation in Figure 1 and the definition of Distortion in Equation (1). We first map the problem into a problem on hypercubes using Lemma 2.

Corollary 1.

Let be a median graph, let be an isometric embedding of onto a hypercube , and let be a set of agents such that each agent has a bliss point . Then the Distortion of sequential deliberation on is at most the Distortion of sequential deliberation on where each agent’s bliss point is .

Proof.

Fix an initial disagreement outcome and an arbitrary list of pairs of agents , , …, . In round 1 bargaining on , Lemma 1 implies that sequential deliberation will select . Furthermore, Lemma 2 implies that if we had considered and bargaining on instead, sequential deliberation would have selected . Suppose at some round that we have a disagreement outcome . Then the same argument yields that if is the bargaining outcome on , would have been the bargaining outcome on . Thus, by induction, we have that if the list of outcomes on is then the list of outcomes on is . But recall that is an isometric embedding and the social cost of an alternative (as defined in Section 1.3) is just its sum of distances to all points in , so and have the same social cost.

Furthermore, let denote the generalized median of . Then, has the same social cost as . This means the median of the embedding of into has at most this social cost, which in turn means that the Distortion of sequential deliberation in the embedding cannot decrease. ∎

Our main result in this section shows that as , the Distortion of sequential deliberation approaches , with the convergence rate being exponentially fast in and independent of the number of agents , the size of the median space , and the initial disagreement point . In particular, the Distortion is at most in at most steps of deliberation, which is indeed a very small number of steps.

Theorem 1.

Sequential deliberation among a set of agents, where the decision space is a median graph, yields .

Proof.

By Corollary 1, we can assume the decision space is a -dimensional hypercube

so that decision points (and thus bliss points) are vectors in

. For every dimension , let be the fraction of agents whose bliss point has a 1 in the th dimension, and let be the 0 or 1 bit in the th dimension of the bliss point for agent . Let be the minimum social cost decision point, i.e., . Clearly, is 1 if and 0 otherwise [assume w.l.o.g. that for ], so for every dimension , the total distance to , summed over is:

Now, note that sequential deliberation defines a Markov chain on

. The state in a given step is just , and the randomness is in the random draw of the two agents. Let be the stationary distribution of the Markov chain. Then we can write

To write down the transition probabilities, we assume this random draw is two independent uniform random draws from , with replacement. We also note that Lemma 1 implies that on , sequential deliberation will pick the median in every step, i.e., given a disagreement outcome and two randomly drawn agents with bliss points , the new decision point will be . On a hypercube, the median of three points is just the dimension-wise majority. Thus, we get a -state Markov chain in each dimension , with transition probabilities

Let , and let denote this stationary distribution for the corresponding -state Markov chain. Then,

By linearity of expectation, the total expected distance for every dimension , summed over to the final outcome is given by

Without loss of generality, let so that for dimension , the total distance to is . Then the ratio of the expected total distance to to the total distance to is at most:

Since the above bound holds in each dimension of the hypercube, we can combine them as:

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 Markov chain. Rather, note that in the preceding analysis, we only used the marginal probabilities for every dimension . Furthermore, the Markov chain defined by sequential deliberation need not walk along edges on , so we can consider separately the convergence of the chain to the stationary marginal in each dimension.

After steps, let and let denote this distribution. Assume . If the total variation distance222total variation distance is particularly simple for these distributions with support of just two points: between and is , then it is easy to check that the expected total distance to is within a factor of the expected distance to , which implies a Distortion of at most in that dimension. We therefore bound how many steps it takes to achieve total variation distance in any dimension ; if this bound holds uniformly for all dimensions , this would imply the overall Distortion is at most , completing the proof.

For any dimension , two executions of the -state Markov chain along that dimension couple if the agents picked in a time step have the same value in that dimension. At any step, this happens with probability at least . Therefore, the probability that the chains have not coupled in steps is at most

We therefore need large enough so that

Since this bound of holds uniformly for all dimensions, this directly implies the theorem. ∎

3.2 Lower Bounds on Distortion

We will now show that the Distortion bounds of sequential deliberation are significant, meaning that mechanisms from simpler classes are constrained to have higher Distortion values. We present a sequence of lower bounds for social choice mechanisms that use increasingly rich information about the space of alternatives on a median graph with a set of agents with bliss points .

We first consider mechanisms that are constrained to choose outcomes in . For instance, this captures the random dictatorship that chooses the bliss point of a random agent as the final outcome. It shows that the compromise alternatives found by deliberation do play a role in reducing Distortion.

Lemma 3.

Any mechanism constrained to choose outcomes in has Distortion at least 2.

Proof.

It is easy to see that the -star graph (the graph with a central vertex connected to other vertices none of which have edges between themselves) is a median graph. Consider an -star graph where are the non central vertices; that is, each and every agent has a unique bliss point on the periphery of the star. Then any mechanism constrained to choose outcomes in must choose one of these vertices on the periphery of the star. The social cost of such a point is , whereas the social cost of the optimal central vertex is clearly just . The Distortion goes to as 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 instance for median graphs, and converges to a Distortion of one for nearly unanimous instances, unlike random dictatorship. We next consider mechanisms that are restricted to choosing the median of the bliss points of some three agents in . In particular, this captures sequential deliberation run for steps, as well as mechanisms that generalize dictatorship to an oligarchy composed of at most agents. This shows that iteratively refining the bargaining outcome has better Distortion than performing only one iteration.

Lemma 4.

Any mechanism constrained to choose outcomes in or a median of three points in must have Distortion at least 1.316.

Proof.

Let be a median graph; in particular let be the -dimensional hypercube . For every dimension, an agent has a 1 in that dimension of their bliss point independently with probability . In expectation agents’ bliss points have a 1 in any given dimension. We assume is an absolute constant. For being the all ’s vector, , where the randomness is in the construction. Now, suppose . Then, for any , with probability at least every three points in has at least ones. By union bounds, for some , the social cost of any median of three points in is at least:

(3)

where again, the randomness is in the construction. Then there is nonzero probability that

(4)

If we choose the argmax of Equation 4, we get nonzero probability over the construction that the Distortion is at least . Letting grow close to and noting that the nonzero probability over 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 . This also shows that sequential deliberation is close to best possible in this class.

Lemma 5.

Any mechanism constrained to choose outcomes on shortest paths between pairs of outcomes in must have Distortion at least .

Proof.

The construction of the lower bound essentially mimics that of Lemma 4. In this case however, we get that each point on a shortest path between two agents has at least ones, so

So there is nonzero probability over the construction that

where the maximum of over is when . The rest of the argument follows as in Lemma 4. ∎

The significance of the lower bound in Lemma 5 should be emphasized: though there is always a Condorcet winner in median graphs, it need not be any agent’s bliss point, nor does it need to be Pareto optimal for any pair of agents. The somewhat surprising implication is that any local mechanism (in the sense that the mechanism chooses locally Pareto optimal points) is constrained away from finding the Condorcet winner.

4 Properties of Sequential Deliberation

In this section, we study some natural desirable properties for our mechanism: uniqueness of the stationary distribution of the Markov chain, ex-post Pareto-efficiency of the final outcome, and subgame perfect Nash equilibrium.

Uniqueness of the Stationary Distribution

We first show that the Markov chain corresponding to sequential deliberation converges to a unique stationary distribution on the actual median graph, rather than just showing that the marginals and thus the expected distances from the perspectives of the agents converge.

To prove uniqueness, it will be helpful to note that the Markov chain defined by sequential bargaining on by only puts nonzero probability mass on points in the median closure of (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 to if there exist such that (where is the median of by a slight abuse of notation).

Definition 2.

Let be the set of bliss points of agents in . A point is in if or if there exists some sequence such that every point in every pair in the sequence is in and there is some s.t.

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 be a median graph, let be a set of agents, and let be the set of bliss points of the agents in . The Markov chain will have a unique stationary distribution if it is aperiodic and irreducible. To see that the chain is aperiodic, note that for any state of the Markov chain at time , there is a nonzero probability that . This is obvious if , as the agent corresponding to that bliss point might be drawn twice in round (remember, agents are drawn independently with replacement from ). If instead , we know by definition of that there exist and such that . But then . Clearly we can write and , then the fact that implies that . Taken together, these equalities imply that is the median chosen in round . So in either case, there is some probability that . 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 such that there is no path from to . Then , since all nodes in clearly have an incoming edge from every other node in . Then by definition there exists some sequence and some such that

Since , must have an incoming edge from . But then there is a path from to . This is a contradiction; the chain must be irreducible as well. Both properties together show that the Markov chain has a unique stationary distribution. ∎

Pareto-Efficiency

The outcome of sequential deliberation is ex-post Pareto-efficient on a median graph. In other words, in any realization of the random process, suppose the final outcome is ; then there is no other alternative such that for every , with at least one inequality being strict. This is a weak notion of efficiency, but it is not trivial to show; while it is easy to see that a one shot bargaining mechanism using only bliss points is Pareto efficient by virtue of the Pareto efficiency of bargaining, sequential deliberation defines a potentially complicated Markov chain for which many of the outcomes need not be bliss points themselves.

Theorem 3.

Sequential deliberation among a set of agents, where the decision space is a median graph and the initial disagreement point is the bliss point of some agent, yields an ex-post Pareto Efficient alternative.

Proof.

Let be a median graph, let be a set of agents, and let be the set of bliss points of the agents in . It follows from the proof of Corollary 1 that without loss of generality we can suppose is a hypercube embedding. Consider some realization of sequential bargaining, where are the bliss points of the agents drawn to bargain in step . Let denote the final outcome. For the sake of contradiction assume there is an alternative that Pareto-dominates , i.e., for each , with at least one inequality being strict.

Figure 5: An example of sequential deliberation with labeled. The dimensions are numbered .

Recall that on the hypercube, the median of three points has the particularly simple form of the dimension wise majority. Let be the set of dimensions of the hypercube that are “decided” by the agents in round in the sense that these agents agree on that dimension (and can thus ignore the outside alternative in that dimension) and all future agents disagree on that dimension (and thus keep the value decided by bargaining in round ). Formally, . Then by the majority property of the median on the hypercube, for any dimension such that for some , it must be that . An example is shown in Figure 5.

Consider the final round . It must be that . If this were not the case, would not be pairwise efficient (i.e., on a shortest path from to ), whereas is pairwise efficient by definition of the median, so one of the agents in round would strictly prefer to , violating the dominance of over .

Next, consider round . Partition the dimensions of into and all others. Suppose for a contradiction that such that , where by definition. Then the agents in round must strictly prefer to on the dimensions in . But for , we know that , so the agents are indifferent between and on the dimensions in . Furthermore, for , so at least one of the two agents at least weakly prefers to on the remaining dimensions. But then at least one agent must strictly prefer to , contradicting the dominance of over .

Repeating this argument yields that for all , . For all other dimensions, takes on the value , which is the bliss point of some agent. Since that agent must weakly prefer to , must also take the value of her bliss point on these remaining dimensions. But then , so does not Pareto dominate , 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 this end, we formalize a given round of bargaining as a 2-person non-cooperative game between two players who can choose as a strategy to report any point on a median graph; the resulting outcome is the median of the two strategy points chosen by the players and the disagreement alternative presented. The payoffs to the players are just the utilities already defined; i.e., the player wishes to minimize the distance from their true bliss point to the outcome point. Call this game the non-cooperative bargaining game (NCBG).

The extensive form game tree defined by non-cooperative bargaining consists of alternating levels: Nature draws two agents at random, then the two agents play NCBG and the outcome becomes the disagreement alternative for the next NCBG. The leaves of the tree are a set of points in 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 where every agent truthfully reports their bliss point at all rounds of bargaining.

Proof.

The proof is by backward induction. Let be a median graph. In the base case, consider the final round of bargaining between agents and with bliss points and and disagreement alternative . The claim is that playing and playing is a Nash equilibrium. By Lemma 2, we can embed isometrically into a hypercube as and consider the bargaining on this embedding. Then for any point that agent plays

The median on the hypercube is just the bitwise majority, so if plays some where for some dimension , it can only increase ’s distance to the median. So playing is a best response.

For the inductive step, suppose is at an arbitrary subgame in the game tree with rounds left, including the current bargain in which must report a point, and assuming truthful play in all subsequent rounds. Let represent as the outside alternative and other agent bliss point against which must bargain, as the bliss points of the agents drawn in the next round, and so on. We want to show that it is a best response for agent to choose , i.e., to truthfully represent her bliss point. Define

where indicates the median, guaranteed to exist and be unique on the median graph. Also, for any point , similarly define

Suppose by contradiction that is not a best response for agent , then there must exist and some such that . We embed isometrically into a hypercube as . Then by the isometry property, . By the proof of Corollary 1, we can pretend the process occurs on the hypercube.

Consider some dimension . If for some , then this point becomes the median in that dimension, so the median becomes independent of , where is the initial report of agent . Till that time, the bargaining outcome in that dimension is the same as . In either case, for all times and all , we have:

Summing this up over all dimensions, , which is a contradiction. Therefore, was a best response for agent 444It is important to note that we are not assuming that agent will not bargain again in the subgame; there are no restrictions on the values of .. Therefore, every agent truthfully reporting their bliss points at all rounds is a subgame perfect Nash equilibrium of Sequential NCBG. ∎

5 General Metric Spaces

We now work in the very general setting that the set of alternatives are points in a finite metric space equipped with a distance function that is a metric. As before, we assume each agent has a bliss point . An agent’s disutility for an alternative is simply . We first present results for the Distortion, and subsequently define the second moment, or Squared-Distortion. For both measures, we show that the upper bound for sequential deliberation is at most a constant regardless of the metric space.

Theorem 5.

The Distortion of sequential deliberation is at most 3 when the space of alternatives and bliss points lies in some metric, and this bound is tight.

Proof.

Each agent has a bliss point . An agent’s disutility for an alternative is simply . Let be the social cost minimizer, i.e., the generalized median. For convenience, let for . By a slight abuse of notation, let for , i.e., the distance from agent ’s bliss point to .

We will only use the assumption that finds a Pareto efficient point for and , so rather than taking an expectation over the choice of the disagreement alternative, we take the worst case. Let be the social cost minimizer with social cost . We can write then write expected worst case social cost of a step of deliberation as: