Strategic Behavior is Bliss: Iterative Voting Improves Social Welfare

1 Introduction . Voting is one of the most popular methods for a group of agents to make a collective decision based on their preferences . Whether a decision is for a high-stakes presidential election or a routine luncheon , agents submit their preferences and a voting rule is applied to select a winning alternative . One critical flaw of voting is its susceptibility to strategic manipulations . That is , agents may have an incentive to misreport their preferences ( i.e . votes ) to obtain a more favorable outcome . Unfortunately , manipulation is inevitable under any non-dictatorial single-round voting systems when there are three or more alternatives , as recognized by the celebrated Gibbard-Satterthwaite theorem [ Gibbard , 1973 , Satterthwaite , 1975 ] . Consequently , decades of research sought to deter manipulation , especially by high computational barriers [ Bartholdi et al. , 1989 , Faliszewski et al. , 2010 , Faliszewski and Procaccia , 2010 ] ; see [ Conitzer and Walsh , 2016 ] for a recent survey of the field . While there is a large body of literature on manipulation of single-round voting systems , sequential and iterative voting procedures are less understood . Indeed , these procedures occur in a variety of applications , such as Doodle or presidential election polls , where people finalize their votes after previewing others ’ responses [ Meir et al. , 2010 , Desmedt and Elkind , 2010 , Xia and Conitzer , 2010 , Reijngoud and Endriss , 2012 , Zou et al. , 2015 ] . Our key question is : What is the effect of strategic behavior in sequential and iterative voting ? A series of work initiated by Meir et al . [ 2010 ] characterizes the dynamics and equilibria of iterative voting , where agents sequentially and myopically improve their reported preferences based on other agents ’ reports [ Reyhani and Wilson , 2012 , Lev and Rosenschein , 2012 , Brânzei et al. , 2013 , Grandi et al. , 2013 , Obraztsova et al. , 2013 , Meir et al. , 2014 , Rabinovich et al. , 2015 , Obraztsova et al. , 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . 2015 , Endriss et al. , 2016 , Meir , 2016 , Tsang and Larson , 2016 , Koolyk et al. , 2017 ] . While the convergence of iterative voting has been investigated for many commonly studied voting rules , the effect of strategic behavior , in terms of aggregate social welfare , remains largely unclear . A notable exception is Brânzei et al . [ 2013 ] ’ s work that introduced and characterized the additive dynamic price of anarchy ( ADPoA ) of iterative voting with respect to the plurality , veto , and Borda social choice functions . The ( additive ) DPoA measures the social welfare ( difference ) ratio between the truthful winner and an iterative policy ’ s equilibrium winners when an adversary minimizes aggregate social welfare by controlling both the order in which agents make their strategic manipulations and agents ’ truthful preferences altogether . In particular , Brânzei et al . [ 2013 ] proved that under iterative plurality , the number of agents whose top preference is an equilibrium winner is at most one less than that of the truthful plurality winner . Therefore , strategic behavior does not have a significant negative impact on the social welfare measured by the sum plurality score of the winner . Nevertheless , it is unclear whether this observation holds for other notions of social welfare . 1.1 Our Contributions . We address the key question discussed above in the iterative voting framework , first proposed by Meir et al . [ 2010 ] , by characterizing Brânzei et al . [ 2013 ] ’ s ADPoA under plurality dynamics and rank-based utility functions that differ from the iteration method . Given m ≥ 3 alternatives , a ranked-based utility function is characterized by a utility vector ~u such that each agent receives ui utility if their i-th ranked alternative wins , although this alternative may differ for each agent . We study iterative plurality due to its simplicity and popularity in practice . Moreover , our results absolve the need for the mechanism ’ s center to know ~u exactly , thus conserving agents ’ privacy . Still , we assume this is constant for all agents . Our first main result ( Theorem 1 ) states that , unfortunately , for any fixed m ≥ 3 and utility vector ~u , the ADPoA is Θ ( n ) for n agents . Therefore , the positive result achieved by Brânzei et al . [ 2013 ] is not upheld if ~u differs from plurality utility under the iterative plurality mechanism . To overcome this negative worst-case result , we introduce the notion of expected additive dynamic price of anarchy ( EADPoA ) , which presumes agents ’ truthful preferences to be generated from a probability distribution . Our second main result ( Theorem 2 ) is positive and surprises us : for any fixed m ≥ 3 and utility vector ~u , the EADPoA is −Ω ( 1 ) when agents ’ preferences are i.i.d . uniformly at random , known as Impartial Culture ( IC ) in social choice . In particular , our result suggests that strategic behavior is bliss because iterative voting helps agents choose an alternative with higher expected social welfare , regardless of the order of agents ’ strategic manipulations . Techniques . We compute the EADPoA by partitioning the ( randomly generated ) profiles according to their potential winners – the alternatives that can be made to win by incrementing their plurality scores by at most one . Conditioned on profiles with two potential winners , we show that iterative plurality returns the alternative that beats the other in a head-to-head competition ( Lemma 1 ) . This type of “ self selection ” improves the expected social welfare over truthful plurality winner by Ω ( 1 ) ( Lemma 2 ) . When there are three or more potential winners , we further show that the expected welfare loss is o ( 1 ) ( Lemmas 3 and 4 ) . Since the likelihood of k-way ties is exponentially small ( in fact , Θ ( n− k−1 2 ) [ Xia , 2021 ] ) , the overall social welfare is improved in expectation . We provide an experimental justification of our second main result in Appendix B . 1.2 Related Work and Discussions . Sequential and iterative voting . Since iterative voting ’ s inception in 2010 , many researchers have studied its convergence properties under differing assumptions and iteration protocols . Meir et al . [ 2010 ] first established the convergence of iterative plurality with deterministic tie-breaking from any initial preference profile or with randomized tie-breaking from the truthful profile . However , this result appears quite sensitive to its assumptions , since the authors found counter-examples when allowing agents to manipulate simultaneously , using better- instead of best-replies , or weighing agents ’ votes unequally . Lev and Rosenschein [ 2012 ] and Reyhani and Wilson [ 2012 ] independently showed that no other scoring rule besides veto necessarily converges , while Koolyk et al . [ 2017 ] demonstrated the same for common non-scoring rules , such as Maximin , Copeland , and Bucklin . Similarly , Obraztsova et al . [ 2013 , 2015 ] and Rabinovich et al . [ 2015 ] each analyze the conditions for Nash equilibrium for iterative voting rules and their truth-biased or lazy voting counterparts [ Thompson et al. , 2013 ] . Rabinovich et al . [ 2015 ] conclude that determining whether a given profile is a reachable Nash equilibrium is NP-complete . Most iterative voting rules require agents to have full information about each others ’ votes in order to compute their best-responses . To relax this strong assumption , Reijngoud and Endriss [ 2012 ] and Endriss et al . [ 2016 ] , inspired by Chopra et al . [ 2004 ] , introduce communication graphs and poll information functions that restricts the amount information each agent receives and better respects voter privacy . The researchers subsequently provide susceptibility , immunity , and convergence results according to different voting rules . Tsang and Larson [ 2016 ] use these concepts to simulate iterative plurality on a social network and allow agents to infer their best-responses based on their neighbors ’ reports . The authors demonstrate how correlating agents ’ preferences affects the PoA and DPoA of strategic outcomes . Sequential but non-iterative voting games have also been investigated in the literature . Desmedt and Elkind [ 2010 ] characterized the subgame perfect Nash equilibrium of a voting game where agents vote sequentially and are allowed to abstain from voting . Xia and Conitzer [ 2010 ] characterized the subgame perfect Nash equilibrium of a similar voting game where agents are not allowed to abstain , and proved that the equilibrium winner is highly unfavorable in the worst case , which can be viewed as an ordinal PoA . Our paper focuses on iterative voting setting proposed by Meir et al . [ 2010 ] and therefore differs from these works . Best-response mechanisms . A separate line of research from iterative voting has studied the convergence and acyclicity properties of best-response mechanisms . Monderer and Shapley [ 1996 ] first introduced the finite improvement property applied to games with agents that sequentially change their actions . Apt and Simon [ 2015 ] and Fabrikant et al . [ 2010 ] subsequently characterized betterresponse dynamics in weakly acyclic games , which encapsulate potential and dominance-solvable games , and demonstrate bounds on finding their Nash equilibrium . The relationship between iterative voting and best-response mechanisms was explored by Meir et al . [ 2014 ] and Meir [ 2016 ] , who fully characterized the acyclicity and local dominance properties of iterative voting rules . Implications of our main results . Our results provide completeness and explanatory power to the empirical studies of Grandi et al . [ 2013 ] and Tsang and Larson [ 2016 ] . The former work shows an increase in additive social welfare using the Borda welfare vector due to plurality dynamics when agents have restricted manipulations and independent or correlated preferences [ Berg , 1985 ] . The latter work shows a similar gain when agents have single-peaked preferences , are embedded on a social network , and make their manipulations based on estimates of their neighbors ’ reports . Put together , iterative voting provides a social welfare benefit that serves as an additional defense of strategic manipulation to those presented by Dowding and Hees [ 2008 ] . We believe our results will benefit a further study of non-strategyproof mechanisms in other social choice domains , such multi-issue voting [ Bowman et al. , 2014 , Grandi et al. , 2020 ] . 2 Preliminaries . Basic setting . Let A = [ m ] , { 1 , . . . , m } denote the set of m ≥ 3 alternatives and n ∈ N denote the number of agents . We denote by L ( A ) the set of all strict linear orders over A and use Rj ∈ L ( A ) , j ≤ n to represent agents ’ preference rankings . Preferences are aggregated into profiles P = ( R1 , . . . , Rn ) , and we use top ( Rj ) ∈ A to denote agent j ’ s top preferred alternative . For any pair of alternatives a , b ∈ A , we use P [ a b ] to denote the number of agents that prefer a to b in P . Integer positional scoring rules . An ( integer ) positional scoring rule r~s is characterized by an integer scoring vector ~s = ( s1 , . . . , sm ) ∈ Zm≥0 with s1 ≥ s2 ≥ · · · ≥ sm ≥ 0 and s1 > sm . For example , plurality uses the vector ~splu = ( 1 , 0 , . . . , 0 ) , veto uses ( 1 , . . . , 1 , 0 ) , and Borda uses ( m− 1 , m− 2 , . . . , 0 ) . In this work we focus on the plurality rule rplu = r~splu and define the score of a ∈ A according to profile P as sP ( a ) = ∑ R∈P 1 { top ( R ) = a } . We use the resolute function rplu ( P ) = arg maxa∈A sP ( a ) to select a single winning alternative , breaking ties lexicographically and favoring that with the smallest index . Assume r = rplu unless stated otherwise . Rank-based utility and additive social welfare . We assume that agents have additive utilities characterized by a rank-based utility vector ~u = ( u1 , . . . , um ) ∈ Rm≥0 with u1 ≥ . . . ≥ um ≥ 0 and u1 > um . Like the scoring rule , each agent j gets ~u ( Rj , a ) = ui utility for the alternative a ∈ A ranked ith in Rj . Unlike prior work , however , we do not presume that ~u is the same as the scoring vector ~s . We define the additive social welfare of a according to P as SW~u ( P , a ) = ∑n j=1 ~u ( Rj , a ) . Iterative plurality voting . Given agents ’ truthful preferences P , we consider an iterative process of profiles ( P t ) t≥0 that describe agents ’ reported preferences ( i.e . votes ) ( Rt1 , . . . , R t n ) t≥0 over time . For each round t , one agent j is chosen by a scheduler φ to make a myopic improvement step , denoted by Rtj j−→ R′j , to their report . This step is called a better-response if j prefers the new outcome r ( R′j , R t −j ) to the prior one r ( R t j , R t −j ) , whereas it is a best-response ( BR ) if , additionally , j could not have achieved a more preferred outcome than manipulating to R′j from P t. 1 Following Brânzei et al . [ 2013 ] , we limit our discussion to strategic manipulations beginning from the truthful profile P 0 = P . This guarantees that all improvement steps R′j j−→ R′′j from profile P ′ to P ′′ are best-responses that change the iterative winner : r ( P ′ ) 6= top ( R′j ) ∧ r ( P ′′ ) = top ( R′′j ) . 2 As a result , any sequence of BR steps converges inO ( nm ) rounds [ Reyhani and Wilson , 2012 ] . The profiles { P ∗ } with no further improvement steps are therefore Nash equilibrium ( NE ) with respect to P . We define EW ( P ) as the set of equilibrium winning alternatives corresponding to all NE reachable from P via some BR sequence . That is , EW ( P ) = { r ( P ∗ ) : ∃ a BR sequence from P leading to the NE profile P ∗ } Lastly , we ’ ll define the set of potential winning alternatives of any profile P as those who could become a winner if their plurality score were to increment by one , including the current winner . That is , some agent could make these alternatives win by taking a BR step that increases their plurality score , if the agent ’ s ranking permits . Following [ Rabinovich et al. , 2015 ] , we have : PW ( P ) = { a ∈ A : { sP ( a ) = sP ( r ( P ) ) − 1 , a is ordered before r ( P ) sP ( a ) = sP ( r ( P ) ) , a is ordered after r ( P ) } ∪ { r ( P ) } where the ordering is lexicographical for tie-breaking . Reyhani and Wilson [ 2012 ] proved that the potential winning set is monotonic in t : ∀t ≥ 0 , PW ( P t+1 ) ⊆ PW ( P t ) , which implies EW ( P ) ⊆ PW ( P 0 ) . As a result , iterative plurality voting acts like a sequential tie-breaking mechanism whose outcome follows from the scheduler φ . The following example demonstrates this section ’ s concepts . Example 1 . Let n = 9 , m = 3 , and consider the truthful profile P defined with R1 = R2 = R3 = [ 1 3 2 ] , R4 = R5 = [ 2 3 1 ] , R6 = [ 2 1 3 ] , and R7 = R8 = R9 = [ 3 2 1 ] . We observe from the plurality scores ( sP ( 1 ) , sP ( 2 ) , sP ( 3 ) ) = ( 3 , 3 , 3 ) that r ( P ) = 1 and PW ( P ) = { 1 , 2 , 3 } , representing a three-way tie . Next , Figure 1 describes the five BR sequences from P : We therefore conclude EW ( P ) = { 2 , 3 } . Moreover , consider the utility vector ~u = ( u1 , u2 , u3 ) . Then the social welfare for each alternative is ( SW~u ( P , 1 ) , SW~u ( P , 2 ) , SW~u ( P , 3 ) ) = ( 3u1 + 1u2 + 5u3 , 3u1 + 3u2 + 3u3 , 3u1 + 5u2 + 1u3 ) . 2 1Note that φ must select a BR step if one exists [ Apt and Simon , 2015 ] . 2These are characterized as Type 1 or direct best replies in the literature [ Meir , 2016 ] . Conversely , Type 3 best-responses ( r ( P ′ ) = top ( R′j ) ∧ r ( P ′′ ) = top ( R′′j ) ) do not occur in improvement sequences from the truthful profile . Note that no BR step is of Type 2 ( r ( P ′ ) = top ( R′j ) ∧ r ( P ′′ ) 6= top ( R′′j ) ) .

Summary: The authors study the welfare of Nash equilibria reached via best-response dynamics from the truthful profile. For worst-case profiles, these equilibria can have linearly less welfare than the truthful outcome. Assuming impartial culture, however, the expected welfare of these equilibria is higher than the welfare for the truthful outcome, assuming rank-based utilities.