Bounding the inefficiency of compromise in opinion formation
Ioannis Caragiannis, Panagiotis Kanellopoulos, Alexandros A. Voudouris
aa r X i v : . [ c s . G T ] F e b Bounding the inefficiency of compromise in opinion formation ∗ Ioannis Caragiannis [email protected]
University of Patras
Panagiotis Kanellopoulos [email protected]
University of Patras & CTI “Diophantus”
Alexandros A. Voudouris [email protected]
University of Patras
Abstract
Social networks on the Internet have seen an enormous growth recently and play a crucialrole in different aspects of today’s life. They have facilitated information dissemination in waysthat have been beneficial for their users but they are often used strategically in order to spreadinformation that only serves the objectives of particular users. These properties have inspired arevision of classical opinion formation models from sociology using game-theoretic notions andtools. We follow the same modeling approach, focusing on scenarios where the opinion expressedby each user is a compromise between her internal belief and the opinions of a small number ofneighbors among her social acquaintances. We formulate simple games that capture this behaviorand quantify the inefficiency of equilibria using the well-known notion of the price of anarchy. Ourresults indicate that compromise comes at a cost that strongly depends on the neighborhood size.
1. Introduction
Opinion formation has been the subject of much research in sociology, economics, physics, andepidemiology. Due to the widespread adoption of the Internet and the subsequent blossoming ofsocial networks, it has recently attracted the interest of researchers in AI (e.g., see Auletta et al.[2016], Schwind et al. [2015], Tsang and Larson [2014]) and CS at large (e.g., see Bindel et al.[2015], Mossel and Tamuz [2014], Olshevsky and Tsitsiklis [2009]) as well.An influential model that captures the adoption of opinions in a social context has been pro-posed by Friedkin and Johnsen [1990]. According to this, each individual has an internal belief onan issue and publicly expresses a (possibly different) opinion; internal beliefs and public opinionsare modeled as real numbers. In particular, the opinion that an individual expresses follows by aver-aging between her internal belief and the opinions expressed by her social acquaintances. Recently,Bindel et al. [2015] show that this behavior can be explained through a game-theoretic lens: aver-aging between the internal belief of an individual and the opinions in her social circle is simply a strategy that minimizes an implicit cost for the individual.Bindel et al. [2015] use a quadratic function to define this cost. Specifically, this function isequal to the total squared distance of the opinion that the individual expresses from her belief andthe opinions expressed in her social circle. In a sense, this behavior leads to opinions that followthe majority of her social acquaintances. Bindel et al. [2015] consider a static snapshot of the social ∗ . A preliminary version of this paper entitled “Bounding the inefficiency of compromise” appeared in Proceedingsof the 26th International Joint Conference on Artificial Intelligence (IJCAI) , pages 142-148, 2017. This work waspartially supported by COST Action CA 16228 “European Network for Game Theory” and by a PhD scholarshipfrom the Onassis Foundation. neighbors . So, in their model, opinionformation co-evolves with the neighborhood for each individual: her neighborhood consists of thosewho have opinions similar to her belief. Then, the opinion expressed is assumed to minimize thesame cost function used by Bindel et al. [2015], taking into account the neighborhood instead of thewhole social circle.We follow the co-evolutionary model of Bhawalkar et al. [2013], but we deviate from their costdefinition and instead consider individuals that seek to compromise with their neighbors. Hence, weassume that each individual aims to minimize the maximum distance of her expressed opinion fromher belief and each of her neighbors’ opinion. Like Bhawalkar et al. [2013], we assume that opinionformation co-evolves with the social network. Each individual’s neighborhood consists of the k other individuals with the closest opinions to her belief. Naturally, these modeling decisions leadto the definition of strategic games, which we call k - compromising opinion formation (or, simply, k -COF) games. Each individual is a (cost-minimizing) player with the opinion expressed as herstrategy. We study questions related to the existence, computational complexity, and quality of equilibria in k -COF games. We begin by proving several properties about the geometric structure of opinionsand beliefs at pure Nash equilibria, i.e., in states of the game where each player minimizes herindividual cost assuming that the remaining players will not change their opinions, and, thus, hasno incentive to deviate by expressing a different opinion.Using these structural properties we show that there exist simple k -COF games that do notadmit pure Nash equilibria. Furthermore, we prove that even in games where equilibria do exist,their quality may be suboptimal in terms of the social cost , i.e., the total cost experienced by allplayers. To quantify this inefficiency, we show that the optimistic measure known as the price ofstability (introduced by Anshelevich et al. [2008]) which, informally, is defined as the ratio of theminimum social cost achieved at any pure Nash equilibrium to the minimum social cost at anypossible state of the game, grows linearly with k .For the special case of -COF games, we show that each such game admits a representation asa directed acyclic graph, in which every pure Nash equilibrium corresponds to a path between twodesignated nodes. Hence, the problems of computing the best or worst (in terms of the social cost)pure Nash equilibrium (or even of computing whether such an equilibrium exists) are equivalent tosimple path computations that can be performed in polynomial time.For general k -COF games, we quantify the inefficiency of the worst-case pure Nash equi-libria by bounding the pessimistic measure known as the price of anarchy (introduced byKoutsoupias and Papadimitriou [1999]) which, informally, is defined as the ratio of the maximumsocial cost achieved at any pure Nash equilibrium to the minimum possible social cost at any stateof the game. Specifically, we present upper and lower bounds on the price of anarchy of k -COFgames (with respect to both pure and mixed Nash equilibria) that suggest a linear dependence on k .Our upper bound on the price of anarchy exploits, in a non-trivial way, linear programming dualityin order to lower-bound the optimal social cost. For the fundamental case of -COF games, weobtain a tight bound of using a particular charging scheme in the analysis. Our contribution issummarized in Table 1. 2 PoA MPoA PoS Existence/Complexity (Thms 19, 20) ≥ ≥ / PNE may not exist for any k (Thm. 6)(Thm. 21) (Thm. 8) Best and worst PNE is in P (Thm. 10) ≤ (Thm. 12) ≥ / ≥ / Open question: Is computing a PNE ≥ / (Thm. 22) (Thm. 23) (Thm. 9) in P when k ≥ ? ≥ ≤ k + 1) (Thm. 12) ≥ k + 2 ≥ k +13 ≥ k + 1 (Thm. 22) (Thm. 23) (Thm. 7) Table 1: Summary of our results for k -COF games. The table presents our bounds on the priceof anarchy over pure Nash equilibria (PoA) and mixed Nash equilibria (MPoA), on the price ofstability (PoS) as well as the existence and complexity of pure Nash equilibria (PNE). Clearly, anyupper bound on PoA is also an upper bound on PoS. DeGroot [1974] proposed a framework that models the opinion formation process, whereeach individual updates her opinion based on a weighted averaging procedure. Subsequently,Friedkin and Johnsen [1990] refined the model by assuming that each individual has a private be-lief and expresses a (possibly different) public opinion that depends on her belief and the opinionsof people to whom she has social ties. More recently, Bindel et al. [2015] studied this model andproved that, for the setting where beliefs and opinions are in [0 , , the repeated averaging pro-cess leads to an opinion vector that can be thought of as the unique equilibrium in a correspondingopinion formation game.Deviating from the assumption that opinions depend on the whole social circle, Bhawalkar et al.[2013] consider co-evolutionary opinion formation games, where as opinions evolve so does theneighborhood of each person. This model is conceptually similar to previous ones that have beenstudied by Hegselmann and Krause [2002], and Holme and Newman [2006]. Both Bindel et al.[2015] and Bhawalkar et al. [2013] show constant bounds on the price of anarchy of the gamesthat they study. In contrast, the modified cost function we use in order to model compromise yieldsconsiderably higher price of anarchy.A series of recent papers from the EconCS community consider discrete models with binaryopinions. Chierichetti et al. [2018] consider discrete preference games, where beliefs and opinionsare binary and study questions related to the price of stability. For these games, Auletta et al. [2015,2017b] characterize the social networks where the belief of the minority can emerge as the opinionof the majority, while Auletta et al. [2017a] examine the robustness of such results to variants of themodel. Auletta et al. [2016] generalize discrete preference games so that players are not only inter-ested in agreeing with their neighbors and more complex constraints can be used to represent theplayers’ preferences. Bil`o et al. [2016] extend co-evolutionary formation games to the discrete set-ting. Other models assume that opinion updates depend on the entire social circle of each individual,who consults a small random subset of social acquaintances; see the recent paper by Fotakis et al.[2016] and the survey of Mossel and Tamuz [2014].When there are more than one issues to be discussed, Jia et al. [2015] propose and analyze theDeGroot-Friedkin model for the evolution of an influence network between individuals who formopinions on a sequence of issues, while Xu et al. [2015] introduce a modification to the DeGroot-3riedkin model so that each individual may recalculate the weight given to her opinion, i.e., herself-confidence, after the discussion of each issue.Another line of research considers how fast a system converges to a stable state. In this spirit,Etesami and Basar [2015] consider the dynamics of the Hegselmann-Krause model [2002], whereopinions and neighborhoods co-evolve, and study the termination time in finite dimensions underdifferent settings. Similarly, Ferraioli et al. [2016] study the speed of convergence of decentralizeddynamics in finite opinion games, where players have only a finite number of opinions available.Ferraioli and Ventre [2017] consider the role of social pressure towards consensus in opinion gamesand provide tight bounds on the speed of convergence for the important special case where the socialnetwork is a clique.Das et al. [2014] perform a set of online user studies and argue that widely studied theoreticalmodels do not completely explain the experimental results obtained. Hence, they introduce ananalytical model for opinion formation and present preliminary theoretical and simulation resultson the convergence and structure of opinions when users iteratively update their respective opinionsaccording to the new model.Chazelle [2012] analyzes influence systems, where each individual observes the location of herneighbors and moves accordingly, and presents an algorithmic calculus for studying such systems.Kempe et al. [2016] present a novel model of cultural dynamics and study the interplay between se-lection and influence. Their results include an almost complete characterization of stable outcomesand guaranteed convergence from all starting states. Gomez-Rodriguez et al. [2012] consider net-work diffusion and contagion propagation. Their goal is to infer an unknown network over whichcontagion propagated, tracing paths of diffusion and influence. Finally, Kempe et al. [2015] studythe optimization problem for influence maximization in a social networks, where each individualmay decide to adopt an idea or an innovation depending on how many of her neighbors already do.The goal is to select an initial seed set of early adopters so that the number of adopters is maximized.In spite of the extensive related literature on opinion formation in many different disciplines,our model introduces a novel cost function that, we believe, captures the tendency of individuals tocompromise more accurately. We begin with preliminary definitions and notation in Section 2. Then, in Section 3 we presentseveral structural properties of pure Nash equilibria, while Section 4 is devoted to the existenceand the price of stability of these equilibria. In Section 5, we present an algorithm that determineswhether pure Nash equilibria exist in a -COF game, and, in addition, computes the best and worstsuch equilibria, when they exist. In Sections 6 and 7 we prove upper bounds on the price of anarchyof k -COF and -COF games, respectively, while Section 8 contains our lower bounds. We concludein Section 9 with a discussion of open problems and possible extensions of our work.
2. Definitions and notation
A compromising opinion formation game defined by the k nearest neighbors (henceforth, called k -COF game) is played by a set of n players whose beliefs lie on the line of real numbers. Let s = ( s , s , . . . , s n ) ∈ R n be the vector containing the players’ beliefs such that s i ≤ s i +1 for each i ∈ [ n − . Let z = ( z , z , . . . , z n ) ∈ R n be a vector containing the (deterministic or randomized)opinions expressed by the players; these opinions define a state of the game. We denote by z − i the4pinion vector obtained by removing z i from z . In an attempt to simplify notation, we omit k fromall relevant definitions.Given vector z (or a realization of it in case z contains randomized opinions), we define theneighborhood N i ( z , s ) of player i to be the set of k players whose opinions are the closest to thebelief of player i breaking ties arbitrarily (but consistently). For each player i , we define I i ( z , s ) asthe shortest interval of the real line that includes the following points: the belief s i , the opinion z i ,and the opinion z j for each player j ∈ N i ( z , s ) . Furthermore, let ℓ i ( z , s ) and r i ( z , s ) be the playerswith the leftmost and rightmost point in I i ( z , s ) , respectively. For example, ℓ i ( z , s ) can be equalto either player i or some player j ∈ N i ( z , s ) , depending on whether the leftmost point of I i ( z , s ) is s i , z i , or z j . To further simplify notation, we will frequently use ℓ ( i ) and r ( i ) instead of ℓ i ( z , s ) and r i ( z , s ) when z and s are clear from the context. In the following, we present the relevantdefinitions for the case of possibly randomized opinion vectors; clearly, these can be simplifiedwhenever z consists entirely of deterministic opinions.Given a k -COF game with belief vector s , the cost that player i experiences at the state of thegame defined by an opinion vector z is E [cost i ( z , s )] = E (cid:20) max j ∈ N i ( z , s ) (cid:26) | z i − s i | , | z j − z i | (cid:27)(cid:21) = E (cid:20) max (cid:26) | z i − s i | , | z r i ( z , s ) − z i | , | z i − z ℓ i ( z , s ) | (cid:27)(cid:21) . (1)For the special case of -COF games, we denote by σ i ( z , s ) (or σ ( i ) when z and s are clear fromthe context) the player (other than i ) whose opinion is closest to the belief s i of player i ; notice that σ ( i ) is the only member of N i ( z , s ) . In this case, the cost of player i can be simplified as E [cost i ( z , s )] = E (cid:20) max (cid:26) | z i − s i | , | z σ i ( z , s ) − z i | (cid:27)(cid:21) . (2)We say that an opinion vector z consisting entirely of deterministic opinions is a pure Nashequilibrium if no player i has an incentive to unilaterally deviate to a deterministic opinion z ′ i inorder to decrease her cost, i.e., cost i ( z , s ) ≤ cost i (( z ′ i , z − i ) , s ) , where by ( z ′ i , z − i ) we denote the opinion vector in which player i chooses the opinion z ′ i and allother players choose the opinions they have according to vector z . Similarly, a possibly randomizedopinion vector z is a mixed Nash equilibrium if for any player i and any deviating deterministicopinion z ′ i we have E [cost i ( z , s )] ≤ E z − i [cost i (( z ′ i , z − i ) , s )] . Let PNE ( s ) and MNE ( s ) denote the sets of pure and mixed Nash equilibria, respectively, of the k -COF game with belief vector s .The social cost of an opinion vector z is the total cost experienced by all players, i.e., E [ SC ( z , s )] = n X i =1 E [cost i ( z , s )] . z ∗ ( s ) be a deterministic opinion vector that minimizes the social cost for the given k -COF gamewith belief vector s ; we will refer to it as an optimal opinion vector for s .The price of anarchy (PoA) over pure Nash equilibria of a particular k -COF game with beliefvector s is defined as the ratio between the social cost of its worst (in terms of the social cost) pureNash equilibrium and the optimal social cost, i.e.,PoA ( s ) = sup z ∈ PNE ( s ) SC ( z , s ) SC ( z ∗ ( s ) , s ) . The price of stability (PoS) over pure Nash equilibria of the k -COF game with belief vector s isdefined as the ratio between the social cost of the best pure Nash equilibrium (in terms of socialcost) and the optimal social cost, i.e.,PoS ( s ) = inf z ∈ PNE ( s ) SC ( z , s ) SC ( z ∗ ( s ) , s ) . Similarly, the price of anarchy and the price of stability over mixed Nash equilibria of a k -COFgame with belief vector s are defined asMPoA ( s ) = sup z ∈ MNE ( s ) E [ SC ( z , s )] SC ( z ∗ ( s ) , s ) and MPoS ( s ) = inf z ∈ MNE ( s ) E [ SC ( z , s )] SC ( z ∗ ( s ) , s ) , respectively.Then, the price of anarchy and the price of stability of k -COF games, for a fixed k , are definedas the supremum of PoA ( s ) and PoS ( s ) over all belief vectors s , respectively.We conclude this section with an example. Example 1.
Consider the -COF game with three players and belief vector s = ( − , , whichis depicted in Figure 1(a). For simplicity, we will refer to the players as left ( ℓ ), middle ( m ), andright ( r ).Let us examine the opinion vector z = ( − , − , which is depicted in Figure 1(b). We havethat σ ( ℓ ) = m since the opinion z m = − of the middle player is closer to the belief s ℓ = − ofthe left player than the opinion z r = 4 of the right player. Therefore, the cost of the left player is cost ℓ ( z , s ) = max {| −
10 + 10 | , | −
10 + 5 |} = 5 . Similarly, the neighbors of the middle and rightplayers are σ ( m ) = r and σ ( r ) = m , while their costs are cost m ( z , s ) = max { , } = 9 and cost r ( z , s ) = max { − , } = 9 , respectively. The social cost is SC ( z , s ) = 23 .Now, consider the alternative pure Nash equilibrium opinion vector z ′ = ( − . , , which isdepicted in Figure 1(c). Observe that even though z ′ = z , each player has the same neighbor as in z and no player has an incentive to deviate in order to decrease her cost. Indeed, let us focus on themiddle player for whom it is σ ( m ) = r . Her opinion is in the middle of the interval defined by herbelief s m = 3 and the opinion z ′ r = 5 of the right player. Hence, this opinion minimizes her cost byminimizing the maximum between the distance from her belief and the distance from the opinionof the right player. It is easy to verify that the same holds for the left and right players. The playercosts are now . , , and , respectively, yielding a social cost of . .6 (a) −
10 2 5 − (b) −
10 2 5 − . (c) Figure 1: The game examined in Example 1. (a) Illustration of the belief vector s = ( − , , . Theblack squares correspond to player beliefs. The notation [ x ] is used to denote the number of playersthat have the same beliefs; here we have only one player per belief. (b) Illustration of the opinionvector z = ( − , − , . The dots correspond to player opinions and each arrow connects the beliefof a player to her opinion. (c) Illustration of the equilibrium opinion vector z ′ = ( − . , , .
3. Some properties about equilibria
We devote this section to proving several some interesting properties of pure Nash equilibria; thesewill be useful in the following. The first one is obvious due to the definition of the cost function.
Lemma 1.
In any pure Nash equilibrium z of a k -COF game with belief vector s , the opinion ofany player i lies in the middle of the interval I i ( z , s ) . The next lemma allows us to argue about the order of player opinions in a pure Nash equilibrium z . Lemma 2.
In any pure Nash equilibrium z of a k -COF game with belief vector s , it holds that z i ≤ z i +1 for any i ∈ [ n − such that s i < s i +1 .Proof. For the sake of contradiction, let us assume that z i +1 < z i for a pair of players i and i +1 with s i < s i +1 . Then, it cannot be the case that the leftmost endpoint of the interval I i ( z , s ) of player i isat the left of (or coincides with) the leftmost endpoint of interval I i +1 ( z , s ) of player i + 1 and therightmost endpoint of I i ( z , s ) is at the left of (or coincides with) the rightmost endpoint of I i +1 ( z , s ) .In other words, it cannot be the case that min { s i , z ℓ ( i ) } ≤ min { s i +1 , z ℓ ( i +1) } and max { s i , z r ( i ) } ≤ max { s i +1 , z r ( i +1) } hold simultaneously. Since, by Lemma 1, points z i and z i +1 lie in the middleof the corresponding intervals, we would have z i ≤ z i +1 , contradicting our assumption.So, at least one of the two inequalities between the interval endpoints above must not hold. In thefollowing, we assume that min { s i , z ℓ ( i ) } > min { s i +1 , z ℓ ( i +1) } (the case where max { s i , z r ( i ) } > max { s i +1 , z r ( i +1) } is symmetric). This assumption implies that z ℓ ( i +1) < s i < s i +1 (i.e., min { s i +1 , z ℓ ( i +1) } = z ℓ ( i +1) ), and, subsequently, that z ℓ ( i +1) < z ℓ ( i ) . In words, player ℓ ( i + 1) does not belong to interval I i ( z , s ) . Furthermore, since z ℓ ( i +1) < s i +1 , and as (by Lemma 1) z i +1 lies in the middle of I i +1 ( z , s ) , we also have that the leftmost endpoint of interval I i +1 ( z , s ) cannot7elong to player i + 1 , i.e., ℓ ( i + 1) = i + 1 . An example of the relative ordering of points (beliefsand opinions), after assuming that z i +1 < z i and min { s i , z ℓ ( i ) } > min { s i +1 , z ℓ ( i +1) } is depicted inFigure 2. z ℓ ( i +1) z ℓ ( i ) s i z i +1 z i s i +1 I i +1 ( z , s ) · · · I i ( z , s ) · · · Figure 2: An example of the argument used in the proof of Lemma 2.Since ℓ ( i + 1) does not belong to I i ( z , s ) , there are at least k players different than ℓ ( i + 1) and i that have opinions at distance at most s i − z ℓ ( i +1) from belief s i . Since s i < s i +1 and z ℓ ( i +1) < z ℓ ( i ) ,all these players are also at distance strictly less than s i +1 − z ℓ ( i +1) from belief s i +1 . This contradictsthe fact that the opinion of player ℓ ( i + 1) is among the k closest opinions to s i +1 .In the following, in any pure Nash equibrium z , we assume that z i ≤ z i +1 for any i ∈ [ n − .This follows by Lemma 2 when s i < s i +1 and by a convention for the identities of players withidentical belief.In addition to the ordering of opinions in a pure Nash equilibrium, we can also specify the rangeof neighborhoods (in Lemma 3) and opinions (in Lemma 4). Lemma 3.
Let z be a pure Nash equilibrium of a k -COF game with belief vector s . Then, for eachplayer i , there exists j with i − k ≤ j ≤ i such that I i ( z , s ) is the shortest interval that contains theopinions z j , z j +1 , ..., z j + k and belief s i .Proof. If I i ( z , s ) consists of a single point, the lemma follows trivially by the definition of theneighborhood and Lemma 2 since at least k +1 consecutive players including i should have opinionsin I i ( z , s ) . Otherwise, by Lemma 2, the lemma is true if there is at most one opinion in each ofthe left and the right boundary of I i ( z , s ) ; in this case, there are exactly k + 1 consecutive playersincluding player i with opinions in I i ( z , s ) .In the following, we handle the subtleties that may arise due to tie-breaking at the boundariesof I i ( z , s ) . Let Y ℓ and Y r be the set of players with opinions at the leftmost and the rightmost pointof I i ( z , s ) , respectively. From Lemma 1, player i belongs neither to Y ℓ nor to Y r . Now consider thefollowing set of players: the | Y ℓ ∩ N i ( z , s ) | players with highest indices from Y ℓ , the | Y r ∩ N i ( z , s ) | players with lowest indices from Y r and all players with opinions that lie strictly in I i ( z , s ) . Due tothe definition of N i ( z , s ) and by Lemma 2, there are k + 1 players in this set, including player i ,with consecutive indices.In the following, irrespectively of how ties are actually resolved, we assume that N i ( z , s ) ∪ { i } consists of k + 1 players with consecutive indices. This does not affect the cost of player i atequilibrium in the proofs of our upper bounds (since, by Lemma 3, the interval defined is exactlythe same), while our lower bound constructions are defined carefully so that the results hold nomatter how ties are actually resolved. 8 emma 4. Let z be a pure Nash equilibrium of a k -COF game with belief vector s . Then, for eachplayer i , it holds that s ℓ ( i ) ≤ z i ≤ s r ( i ) .Proof. Since N i ( z , s ) ∪ { i } consists of k + 1 players with consecutive indices, we have that s ℓ ( i ) ≤ s i ≤ s r ( i ) . For the sake of contradiction, let us assume that s ℓ ( i ) ≤ s r ( i ) < z i for some player i (the case where z i lies at the left of s ℓ ( i ) is symmetric). Since s r ( i ) < s i and as z i is at the middleof I i ( z , s ) , it holds that z r ( i ) > z i (i.e., r ( i ) = i ). Also, since z r ( i ) > z i > s r ( i ) , and because z r ( i ) is in the middle of I r ( i ) ( z , s ) , it holds that z r ( r ( i )) > z r ( i ) and, by Lemma 2, r ( r ( i )) > r ( i ) ; seeFigure 3 for an example of the relative ordering of points (beliefs and opinions) when assuming that s r ( i ) < z i . s ℓ ( i ) s i s r ( i ) z i z r ( i ) z r ( r ( i )) I r ( i ) ( z , s ) · · · Figure 3: An example of the argument used in the proof of Lemma 4.We now claim that ℓ ( i ) / ∈ N r ( i ) ( z , s ) . Assume otherwise that ℓ ( i ) ∈ N r ( i ) ( z , s ) . By definition, r ( r ( i )) ∈ N r ( i ) ( z , s ) . Then, Lemma 2 implies that any player j , different than r ( i ) , with ℓ ( i ) < j
4. Existence and quality of equilibria
Our first technical contribution is a negative statement: pure Nash equilibria may not exist for any k (Theorem 6). Then, we show that even in k -COF games that admit pure Nash equilibria, the bestequilibrium may be inefficient; in other words, the price of stability is strictly greater than for anyvalue of k , and, actually, depends linearly on k . These results appear in Theorems 7, 8, and 9. We begin with a technical lemma. The lemma essentially presents necessary conditions so that aparticular set of neighborhoods, and corresponding intervals, may coexist in a pure Nash equilib-rium. 9 emma 5.
Consider a k -COF game and any three players a, b, c with beliefs s a ≤ s b ≤ s c ,respectively. For any pure Nash equilibrium z where I a ( z , s ) = [ s a , z b ] , I b ( z , s ) = [ s b , z c ] and I c ( z , s ) = [ z b , s c ] , it must hold that s b ≥ s a +5 s c , while for any pure Nash equilibrium z where I a ( z , s ) = [ s a , z b ] , I b ( z , s ) = [ z a , s b ] and I c ( z , s ) = [ z b , s c ] , it must hold that s b ≤ s a +3 s c .Proof. It suffices to prove the first case; the second case is symmetric. Since I b ( z , s ) = [ s b , z c ] and I c ( z , s ) = [ z b , s c ] , by Lemma 1 it holds that z b = ( s b + z c ) / and z c = ( z b + s c ) / which yieldthat z b = s b + s c − s b and z c = s b + s c − s b )3 . Hence, we obtain that z c − s b = 2( s c − s b )3 . (3)Similarly, since I a ( z , s ) = [ s a , z b ] , it holds that z a = s a + z b = s a +2 s b + s c and, therefore, we obtainthat s b − z a = − s a + 4 s b − s c . (4)Since I b ( z , s ) = [ s b , z c ] , we have that a / ∈ N b ( z , s ) and, subsequently, that z c − s b ≤ s b − z a which,together with (3) and (4), yields that s b ≥ s a +5 s c as desired.The proof of the next theorem is inspired by a construction of Bhawalkar et al. [2013] andexploits Lemma 5. Theorem 6.
For any k , there exists a k -COF game with no pure Nash equilibria.Proof. Consider a k -COF game with k + 1 players partitioned into three sets called L , M , and R ,where L and R each contain k players, while M = { m } is a singleton. We set s i = 0 for each i ∈ L , s i = 2 for each i ∈ R , while s m = 1 − ǫ , where ǫ < / is an arbitrarily small positiveconstant.Let us assume that there exists a pure Nash equilibrium z . Then, clearly, for any i ∈ L it musthold that N i ( z , s ) = L \ { i } ∪ { m } , and, therefore, I i ( z , s ) = [0 , z m ] . Similarly, for any i ∈ R we have N i ( z , s ) = R \ { i } ∪ { m } , and I i ( z , s ) = [ z m , . Now, concerning player m , if all herneighbors are in L , then, it holds that I m ( z , s ) = [ z i , s m ] for some i ∈ L . But then, observe thateven though the intervals defined above exhibit the structure described in Lemma 5, the belief vector s does not satisfy the corresponding necessary conditions of that lemma as − ǫ > / ; hence, z isnot a pure Nash equilibrium. The same reasoning applies in case all of m ’s neighbors are in R .It remains to consider the case where m has at least one neighbor in each of L and R . By thedefinition of I i ( z , s ) for i ∈ L ∪ R , as stated above, Lemma 1 implies that z i = z m / for any i ∈ L , while z i = 1 + z m / for any i ∈ R . Then, Lemma 4 implies that z m / ≤ s m = 1 − ǫ and z m / ≥ s m , and, consequently, I m ( z , s ) = [ z m / , z m / . Again, by Lemma 1 we havethat z m = z m / z m / , i.e., z m = 1 . But then, we obtain z i = 1 / for any i ∈ L and z i = 3 / forany i ∈ R , which implies that all k players in L are strictly closer to s m than any player in R ; thiscontradicts the assumption that m has neighbors in both L and R .An example of the construction used in the proof of Theorem 6 is presented in Figure 4.10 [ k ] 1 − ǫ [1] 2[ k ] (a) − ǫ x/ x x/ (b) Figure 4: (a) The k -COF game considered in the proof of Theorem 6 where the k players of set L have belief , player m has s m = 1 − ǫ and the k players of set R have belief . (b) Lemma 5implies that there is no pure Nash equilibrium where m has neighbors in strictly one of L , R . In theremaining case, it must hold that x = 1 , but then all players in L are strictly closer to s m than anyplayer in R . We will now prove that the price of stability of k -COF games is strictly higher than , i.e., thereexist games without any efficient pure Nash equilibria (even when they exist). In particular, for anyvalue of k we show that there exist rather simple games with price of stability in Ω( k ) . Theorem 7.
The price of stability of k -COF games, for k ≥ , is at least ( k + 1) / .Proof. Consider a k -COF game with k + 1 players, where k of them have belief , while theremaining one has belief . Let ˜ z be the opinion vector where each player has opinion . Clearly,SC (˜ z , s ) = 1 , and, hence the optimal social cost is at most .Now, consider any pure Nash equilibrium z . Since, there are k + 1 players, the neighborhood ofeach player includes all remaining ones. Let x be the opinion that the player with belief expressesat z . By Lemma 4, we have that x ∈ [0 , , and by Lemma 1, we have that all remaining playersmust have opinion x/ . Therefore, again by Lemma 1, x must satisfy the equation x = (1+ x/ / ,i.e., x = 2 / . Therefore, there exists a single pure Nash equilibrium z where all players with belief have opinion / and the single player with belief has opinion / , and we obtain SC ( z ) =( k + 1) / which implies the theorem.Clearly, the above result states the inefficiency of the best pure Nash equilibrium only when k ≥ . For the remaining cases where k ∈ { , } we present slightly more complicated instances,where the proofs rely on Lemma 5. Recall that, for -COF games, σ ( i ) denotes the single neighborof player i . Theorem 8.
The price of stability of -COF games is at least / .Proof. We use the following -COF game with six players and belief vector s = (0 , − λ, , ,
18 + 3 λ, , where λ ∈ (0 , / .Consider the opinion vector ˜ z = (3 − λ, − λ, − λ,
16 + 6 λ,
17 + 2 λ,
20 + λ ) .
11t can be easily seen that it has social cost SC (˜ z , s ) = 10 + 12 λ . So, clearly, SC ( z ∗ , s ) ≤
10 + 12 λ for any optimal opinion vector z ∗ .Now, consider the opinion vector z = (cid:18) − λ , − λ , , ,
59 + 6 λ ,
64 + 3 λ (cid:19) with social cost SC ( z , s ) = 34 / − λ . It is not hard to verify (by showing, as Lemma 1 requires,that each opinion lies in the middle of its player’s interval) that z is a pure Nash equilibrium; weargue that this equilibrium is unique.We claim that, by Lemma 5, there cannot be a pure Nash equilibrium where both σ ( j −
1) = j and σ ( j + 1) = j for any j ∈ { , } . To see this, assume otherwise and note that the correspondingintervals satisfy the conditions of the lemma. However, by observing the belief vector s , it holds that s j − +3 s j +1 < s j < s j − +5 s j +1 , for j ∈ { , } , i.e., s does not satisfy the conditions of Lemma 5;this contradicts our original assumption.The above observation, together with Lemma 2, implies that σ (1) = 2 , σ (3) = 4 , σ (4) = 3 and σ (6) = 5 in any equilibrium. This leaves only σ (2) ∈ { , } and σ (5) ∈ { , } undefined.Consider an equilibrium z ′ with σ (2) = 3 ; the case σ (5) = 4 is symmetric. Since σ (3) = 4 ,Lemma 4 implies that z ′ > s = 8 and, hence z ′ − s > λ. (5)Since σ (1) = 2 , σ (2) = 3 and z ′ = s + z ′ , Lemma 4 implies that z ′ > s and we obtain that z ′ > − λ and, hence, s − z ′ < − λ . (6)By inequalities (5) and (6), we get z ′ − s > s − z ′ , which contradicts our assumption that σ (2) = 3 . So, it must hold that σ (2) = 1 (and, respectively, σ (5) = 6 ) which implies that z is theunique pure Nash equilibrium.We conclude that the price of stability is lower-bounded bySC ( z , s ) SC ( z ∗ , s ) = 34 / − λ
10 + 12 λ , and the theorem follows by taking λ to be arbitrarily close to . Theorem 9.
The price of stability of -COF games is at least / .Proof. Consider a -COF game with four players a , b , c , and d , with belief vector s = (0 , , , .Let ˜ z = (1 , , , / be an opinion vector and observe that SC (˜ z , s ) = 3 / ; note that ˜ z is not apure Nash equilibrium as player a has an incentive to deviate. Clearly, the optimal social cost is atmost / .Now consider any pure Nash equilibrium z . By the structural properties of equilibria, N a ( z , s ) = N d ( z , s ) = { b, c } , while b ∈ N c ( z , s ) and c ∈ N b ( z , s ) . It remains to argue aboutthe second neighbor of b and c . We distinguish between two cases depending on whether b and c have a common second neighbor in { a, d } or not.12n the first case, let a be the common neighbor; the case where d is that neighbor is symmetric.By Lemma 1, we have that z b = z c = (1 + z a ) / . Then, we have that I a ( z , s ) = [0 , z b ] , I b ( z , s ) =[ z a , , and I d ( z , s ) = [ z b , . Note that by applying Lemma 5 on players a , b , and d , we obtain acontradiction to the fact that z is a pure Nash equilibrium.In the second case, without of loss of generality, let N b ( z , s ) = { a, c } and N c ( z , s ) = { b, d } which, by Lemma 4, imply that z b ∈ [0 , and z c ∈ [1 , . Then, Lemma 1 yields z a = z c / , z b = ( z a + z c ) / , z c = ( z b + z d ) / , and z d = 1 + z b / . By solving this system of equations, weobtain that z = (4 / , / , / , / and, hence, SC ( z ) = 12 / .
5. Complexity of equilibria
In this section we focus entirely on -COF games. We present a polynomial-time algorithm thatdetermines whether such a game admits pure Nash equilibria, and, in case it does, allows us tocompute the best and worst pure Nash equilibrium with respect to the social cost. We do so byestablishing a correspondence between pure Nash equilibria and source-sink paths in a suitablydefined directed acyclic graph. See Example 2 below for an instance execution of the followingprocedure.Assume that we are given neighborhood information according to which each player i has eitherplayer i − or player i + 1 as neighbor. From Lemma 3, such a neighborhood structure is necessaryin a pure Nash equilibrium. We claim that this information is enough in order to decide whetherthere is a consistent opinion vector that is a pure Nash equilibrium or not. All we have to do is touse Lemma 1 and obtain n equations that relate the opinion of each player to her belief and herneighbor’s opinion. These equations have a unique solution which can then be verified whether itindeed satisfies the neighborhood conditions or not. So, the main idea of our algorithm is to cleverlysearch among all possible neighborhood structures that are not excluded by Lemma 3 for one thatdefines a pure Nash equilibrium.For integers ≤ a ≤ b < c ≤ n , let us define the segment C ( a, b, c ) to be the set of players { a, a + 1 , ..., c } together with the following neighborhood information for them: σ ( p ) = p + 1 for p = a, ..., b and σ ( p ) = p − for p = b + 1 , ..., c . It can be easily seen that the neighborhoodinformation for all players at a pure Nash equilibrium can always be decomposed into disjointsegments. Importantly, given the neighborhood information in segment C ( a, b, c ) and the beliefs ofits players, the opinions they could have in any pure Nash equilibrium that contains this segmentare uniquely defined using Lemma 1. In particular, the opinions of the players within a segment C ( a, b, c ) are computed as follows. First, we set z b = s b + s b +1 − s b and z b +1 = s b + s b +1 − s b )3 .Then, we set z p = s p + z p +1 if a ≤ p < b , and z p = s p + z p − if b < p ≤ c .We remark that the opinion vector implied by a segment is not necessarily consistent to thegiven neighborhood structure. So, we call segment C ( a, b, c ) legit if a = 2 , c = n − (so that it canbe part of a decomposition) and the uniquely defined opinions are consistent to the neighborhoodinformation of the segment, i.e., if | z σ ( p ) − s p | ≤ | z p ′ − s p | for any pair of players p, p ′ (with p = p ′ )in C ( a, b, c ) . This process appears in Algorithm 1.A decomposition of neighborhood information for all players will consist of consecutive seg-ments C ( a , b , c ) , C ( a , b , c ) , ..., C ( a t , b t , c t ) so that a = 1 , c t = n , a ℓ = c ℓ − + 1 for ℓ = 2 , ..., t . Such a decomposition will yield a pure Nash equilibrium if it consists of legit segmentsand, furthermore, the uniquely defined opinions of players in consecutive segments are consistentto the neighborhood information. 13n particular, consider the directed graph G that has two special nodes designated as the sourceand the sink, and a node for each legit segment C ( a, b, c ) . Note that G has O ( n ) nodes. The sourcenode is connected to all segment nodes C (1 , b, c ) while all segment nodes C ( a, b, n ) are connectedto the sink. An edge from segment node C ( a, b, c ) to segment node C ( a ′ , b ′ , c ′ ) exists if a ′ = c + 1 and the uniquely defined opinions of players in the two segments are consistent to the neighborhoodinformation in both of them. This consistency test has to check1. whether the leftmost opinion z a ′ in segment C ( a ′ , b ′ , c ′ ) is indeed further away from the belief s c of player c than the opinion z c − of the designated neighbor of c in segment C ( a, b, c ) , i.e., | z c − − s c | ≤ | z a ′ − s c | , and whether2. whether the rightmost opinion z c in segment C ( a, b, c ) is further away from the belief s a ′ ofplayer a ′ than the opinion z a ′ +1 of the designated neighbor of a ′ in segment C ( a ′ , b ′ , c ′ ) , i.e., | z a ′ +1 − s a ′ | ≤ | z c − s a ′ | .By the definition of segments and of its edges, G is acyclic. This process appears in Algorithm 2.Based on the discussion above, there is a bijection between pure Nash equilibria and source-sinkpaths in G . In addition, we can assign a weight to each node of G that is equal to the total cost ofthe players in the corresponding segment, i.e., weight( C ( a, b, c )) = X a ≤ p ≤ c | z p − s p | . Then, the total weight of a source-sink path P is equal to the social cost of the corresponding pureNash equilibrium, i.e, SC ( z , s ) = X C ( a,b,c ) ∈P weight( C ( a, b, c )) . Hence, standard algorithms for computing shortest or longest paths in directed acyclic graphscan be used not only to detect whether a pure Nash equilibrium exists, but also to compute theequilibrium of best or worst social cost.
Theorem 10.
Given a -COF game, deciding whether a pure Nash equilibrium exists can be donein polynomial time. Furthermore, computing a pure Nash equilibrium of highest or lowest socialcost can be done in polynomial time as well. Example 2.
Consider a -COF game with four players with belief vector s = (0 , , , . Accord-ing to the discussion above, there are segments of the form C ( a, b, c ) with ≤ a ≤ b < c ≤ ,but it can be shown that only of them are legit; these are C (1 , , , C (3 , , (see Figure 5a),and C (1 , , (see Figure 5b). For example, segment C (1 , , , in which σ (1) = 2 , σ (2) = 1 , σ (3) = 2 , and σ (4) = 3 , corresponds to the opinion vector (3 , , , . This is not consistent tothe neighborhood information σ (2) = 1 in the segment, as the belief of player coincides with theopinion of player , while the opinion of player is further away. The resulting directed acyclicgraph G (see Figure 5c) implies that there exist two pure Nash equilibria for this -COF game,namely the opinion vectors (3 , , , and (5 , , , .14 lgorithm 1: Segment
Input: belief vector s = ( s , ..., s n ) , parameters a , b , and c such that a ≤ b < c Output: opinion vector z a : c = ( z a , ..., z c ) , segment weight, legit indicatorlegit := 0 if a = 2 or c = n − then legit := 1 z b := s b + ( s b +1 − s b ) z b +1 := s b + ( s b +1 − s b ) for p := b − downto a do z p := ( s p + z p +1 ) endfor p := b + 2 to c do z p := ( s p + z p − ) endfor p := a + 1 to b doif | z p − − s p | < | z p +1 − s p | then legit := 0 endendfor p := b + 1 to c − doif | z p +1 − s p | < | z p − − s p | then legit := 0 endend weight := 0 for p := a to c do weight := weight + | z p − s p | endend return [ z a : c , weight, legit] 15 lgorithm 2: ConstructGraph
Input: belief vector s = ( s , ..., s n ) Output: a node-weighted directed acyclic graph GV := ∅ for a := 1 to n − dofor b := a to n − dofor c := b + 1 to n do [ z a : c , weight, legit] := Segment ( s , a, b, c ) if legit = 1 then C.a = a , C.b = b , C.c = c , C. z a : c = z a : c , C.weight = weightV := V ∪ C endendendend V := V ∪ { source , sink } E := ∅ for C ∈ V doif C.a = 1 then E := E ∪ ( source , C ) else if C.c = n then E := E ∪ ( C, sink ) endendfor all segment pairs ( C, D ) such that D.a = C.c + 1 doif | C.z c − − s C.c | ≤ |
D.z a − s C.c | and | D.z a +1 − s D.a | ≤ |
C.z c − s D.a | then E := E ∪ ( C, D ) endend return G = ( V, E ) (a) (b)source sink C (1 , , C (3 , , C (1 , , (c) Figure 5: The -COF game considered in Example 2. (a) The legit segments C (1 , , and C (3 , , which imply the opinion vector (3 , , , . (b) The legit segment C (1 , , whichimplies the opinion vector (5 , , , . (c) The directed acyclic graph G which shows that thereexist two pure Nash equilibria in the game.
6. Upper bounds on the price of anarchy
In this section we prove upper bounds on the price of anarchy of k -COF games. In our proof, werelate the social cost of any deterministic opinion vector, including optimal ones, to a quantity thatdepends only on the beliefs of the players and can be thought of as the cost of the truthful opinionvector (in which the opinion of every player is equal to her belief). In particular, we prove a lowerbound on the optimal social cost (in Lemmas 11) and an upper bound on the social cost of any pureNash equilibrium, both expressed in terms of this quantity. The bound on the price of anarchy thenfollows by these relations; see the proof of Theorem 12.Consider an n -player k -COF game with belief vector s = ( s , ..., s n ) . For player i , we denote by ℓ ∗ ( i ) and r ∗ ( i ) the integers in [ n ] such that ℓ ∗ ( i ) ≤ i ≤ r ∗ ( i ) , r ∗ ( i ) − ℓ ∗ ( i ) = k , and | s r ∗ ( i ) − s ℓ ∗ ( i ) | is minimized. The proof of the next lemma exploits linear programming and duality. Lemma 11.
Consider a k -COF game with belief vector s = ( s , ..., s n ) and let z be any determin-istic opinion vector. Then, SC ( z , s ) ≥ k + 1) n X i =1 | s r ∗ ( i ) − s ℓ ∗ ( i ) | . Proof.
Consider any deterministic opinion vector z and let π be a permutation of the players so that z π ( j ) ≤ z π ( j +1) for each j ∈ [ n − . We refer to player π ( j ) as the player with rank j . Foreach player i , we will identify an effective neighborhood F i ( z , s ) that consists of k + 1 players withconsecutive ranks and includes player i . Define ˜ ℓ ( i ) and ˜ r ( i ) to be the players in F i ( z , s ) with the
1. Note that we have proved monotonicity of opinions for pure Nash equilibria only (Lemma 2) and it is not clearwhether such a monotonicity property holds for opinion vectors of minimum social cost. In addition, the statementof Lemma 11 refers to any opinion vector. This clearly includes non-monotonic ones, so we need to rank players interms of opinions in the proof. F i ( z , s ) have thesame belief, we let ˜ ℓ ( i ) and ˜ r ( i ) be the players with the lowest and highest ranks, respectively. Theeffective neighborhood will be defined in such a way that it satisfies the properties cost i ( z , s ) ≥ z ˜ r ( i ) − z i and cost i ( z , s ) ≥ z i − z ˜ ℓ ( i ) .Let N i ( z , s ) denote the neighborhood of player i , i.e., the set of players (not including i ) withthe k closest opinions to the belief s i of player i . Let J i ( z , s ) be the smallest contiguous intervalcontaining all opinions of players in N i ( z , s ) ∪{ i } and let D i ( z , s ) be the set of players with opinionsin J i ( z , s ) . Clearly, | D i ( z , s ) | ≥ k + 1 . We define F i ( z , s ) to be a subset of D i ( z , s ) that consistsof k + 1 players with consecutive ranks including player i . See Figure 6 for an illustrative exampleof all quantities defined above. s z s z z s z s z s s z Figure 6: An example of the quantities used in the proof of Lemma 11. Let k = 2 and i = 4 . Then,the neighborhood of player is N ( z , s ) = { , } , the smallest contiguous interval containing theopinions of players in N ( z , s ) ∪ { } is J ( z , s ) = [ z , z ] , the set of players with opinions in J ( z , s ) is D ( z , s ) = { , , , , } , the effective neighborhood is F ( z , s ) = { , , } , and, hence, ˜ ℓ (4) = 1 , and ˜ r (4) = 4 .Let ℓ ′ ( i ) and r ′ ( i ) be the players in N i ( z , s ) with the leftmost and rightmost opinion. In orderto show that the definition of F i ( z , s ) satisfies the two desired properties, we distinguish betweenthree different cases depending on the location of opinion z i among the players in N i ( z , s ) ∪ { i } . • Case I:
Player i has neither the leftmost nor the rightmost opinion in N i ( z , s ) ∪ { i } , i.e., z ℓ ′ ( i ) < z i < z r ′ ( i ) . In this case, J i ( z , s ) = [ z ℓ ′ ( i ) , z r ′ ( i ) ] . Then, the definition of N i ( z , s ) implies that cost i ( z , s ) ≥ z r ′ ( i ) − z i and cost i ( z , s ) ≥ z i − z ℓ ′ ( i ) . Hence, cost i ( z , s ) ≥ | z j − z i | for every z j ∈ J i ( z , s ) or, equivalently, j ∈ D i ( z , s ) and, subsequently, for each j ∈ F i ( z , s ) .This implies the two desired properties cost i ( z , s ) ≥ z ˜ r ( i ) − z i and cost i ( z , s ) ≥ z i − z ˜ ℓ ( i ) . • Case II:
Player i has the leftmost opinion in N i ( z , s ) ∪ { i } , i.e., z i ≤ z ℓ ′ ( i ) . Then, J i ( z , s ) =[ z i , z r ′ ( i ) ] . Now, the definition of N i ( z , s ) implies that cost i ( z , s ) ≥ z r ′ ( i ) − z i and, hence, cost i ( z , s ) ≥ | z j − z i | for every z j ∈ J i ( z , s ) or, equivalently, j ∈ D i ( z , s ) and, subsequently,for each j ∈ F i ( z , s ) . Again, this implies the two desired properties. • Case III:
Player i has the rightmost opinion in N i ( z , s ) ∪{ i } , i.e., z i ≥ z r ′ ( i ) . Then, J i ( z , s ) =[ z ℓ ′ ( i ) , z i ] . Now, the definition of N i ( z , s ) implies that cost i ( z , s ) ≥ z i − z ℓ ′ ( i ) and, hence, cost i ( z , s ) ≥ | z j − z i | for every z j ∈ J i ( z , s ) or, equivalently, j ∈ D i ( z , s ) and, subsequently,for every j ∈ F i ( z , s ) . Again, the two desired properties follow.By setting the variable t i equal to cost i ( z , s ) for i ∈ [ n ] , the discussion above and the fact that cost i ( z , s ) ≥ | s i − z i | imply that the opinion vector z together with t = ( t , . . . , t n ) is a feasible
2. Case I cannot appear when k = 1 . X i ∈ [ n ] t i subject to t i + z i ≥ s i , ∀ i ∈ [ n ] t i − z i ≥ − s i , ∀ i ∈ [ n ] t i + z i − z ˜ r ( i ) ≥ , ∀ i ∈ [ n ] such that ˜ r ( i ) = it i + z ˜ ℓ ( i ) − z i ≥ , ∀ i ∈ [ n ] such that ˜ ℓ ( i ) = it i , z i ≥ , ∀ i ∈ [ n ] Using the dual variables α i , β i , γ i , and δ i associated with the four constraints of the above LP,we obtain its dual LP:maximize X i ∈ [ n ] s i α i − X i ∈ [ n ] s i β i subject to α i + β i + γ i · { ˜ r ( i ) = i } + δ i · { ˜ ℓ ( i ) = i } ≤ , ∀ i ∈ [ n ] α i − β i + γ i · { ˜ r ( i ) = i } − δ ( i ) · { ˜ ℓ ( i ) = i } − X j = i :˜ r ( j )= i γ j + X j = i :˜ ℓ ( j )= i δ j ≤ , ∀ i ∈ [ n ] α i , β i , γ i , δ i ≥ The indicator { X } is equal to when the condition X is true, and otherwise. We will show thatthe solution defined as α i = |{ j ∈ [ n ] : ˜ r ( j ) = i }| k + 1) ,β i = |{ j ∈ [ n ] : ˜ ℓ ( j ) = i }| k + 1) ,γ i = δ i = 12( k + 1) , is a feasible dual solution. Indeed, to see why the first dual constraint is satisfied, first observe thatplayer i belongs to at most k + 1 different effective neighborhoods. Hence, player i can have thelowest or highest belief among the players in the effective neighborhood of at most k + 1 players(implying that α i + β i ≤ − k +1) ) when ˜ r ( i ) = i or ˜ ℓ ( i ) = i and of at most k players (implyingthat α i + β i ≤ − k +1 ) when ˜ r ( i ) = i and ˜ ℓ ( i ) = i . The first constraint follows.It remains to show that the second constraint is satisfied as well (with equality). We do so bydistinguishing between three cases: • When ˜ r ( i ) = i and ˜ ℓ ( i ) = i , the dual solution guarantees that α i = P j = i :˜ r ( j )= i γ j andthe term α i in the left-hand side of the second constraint cancels out with the sum of γ ’s.Similarly, β i = P j = i :˜ ℓ ( j )= i δ j and the term β i cancels out with the sum of δ ’s. Also, theterms γ i and δ i are both equal to k +1) and cancel out as well.19 When ˜ r ( i ) = i (then, clearly, ˜ ℓ ( i ) = i ), we have that α i = δ i · { ˜ ℓ ( i ) = i } + P j = i :˜ r ( j )= i γ j (cancelling out the first, fourth and fifth terms) and β i = P j = i :˜ ℓ ( j )= i δ j (cancelling out thesecond and sixth terms), and the second constraint is satisfied with equality as the third termis zero. • Finally, when ˜ ℓ ( i ) = i (now, it is ˜ r ( i ) = i ), we have that α i = P j = i :˜ r ( j )= i γ j (cancelling outthe first and fifth terms) and β i = γ i · { ˜ r ( i ) = i } + P j = i :˜ ℓ ( j )= i δ j (cancelling out the second,third and sixth terms), and the second constraint is satisfied with equality as the fourth termis zero.So, the social cost of the solution z is lower-bounded by the objective value of the primal LPwhich, by duality, is lower-bounded by the objective value of the dual LP. HenceSC ( z , s ) ≥ X i ∈ [ n ] s i α i − X i ∈ [ n ] s i β i = 12( k + 1) X i ∈ [ n ] |{ j ∈ [ n ] : ˜ r ( j ) = i }| s i − X i ∈ [ n ] |{ j ∈ [ n ] : ˜ ℓ ( j ) = i }| s i = 12( k + 1) X i ∈ [ n ] ( s ˜ r ( i ) − s ˜ ℓ ( i ) )= 12( k + 1) X i ∈ [ n ] | s ˜ r ( i ) − s ˜ ℓ ( i ) | . The last equality follows since s ˜ r ( i ) ≥ s ˜ ℓ ( i ) , by the definition of ˜ r ( i ) and ˜ ℓ ( i ) .Note that for each player i , there are at least k + 1 beliefs of different players with values in [ s ˜ ℓ ( i ) , s ˜ r ( i ) ] , including player i . By the definition of ℓ ∗ ( i ) and r ∗ ( i ) for each player i , the aboveinequality yields SC ( z , s ) ≥ k + 1) X i ∈ [ n ] | s r ∗ ( i ) − s ℓ ∗ ( i ) | , as desired.We are now ready to prove our upper bound on the price of anarchy for k -COF games. In ourproof, we exploit the mononicity of opinions in a pure Nash equilibrium and we associate the costof each player in the equilibrium to the same quantity used in the statement of Lemma 11. Theorem 12.
The price of anarchy of k -COF games over pure Nash equilibria is at most k + 1) .Proof. Consider a k -COF game with belief vector s = ( s , . . . , s n ) , and let z ∗ = ( z ∗ , . . . , z ∗ n ) beany opinion vector that minimizes the social cost. By Lemma 11, we haveSC ( z ∗ , s ) ≥ k + 1) n X i =1 | s r ∗ ( i ) − s ℓ ∗ ( i ) | . (7)20ow, consider any pure Nash equilibrium z of the game. We will show thatSC ( z , s ) ≤ n X i =1 | s r ∗ ( i ) − s ℓ ∗ ( i ) | , (8)and the theorem will then follow by inequalities (7) and (8).The rest of this proof is, therefore, devoted to showing inequality (8). To this end, we will showthat, for any player i , we have cost i ( z , s ) ≤ s r ∗ ( i ) − s ℓ ∗ ( i ) ) . Then, inequality (8) will follow bysumming over all players.Consider an arbitrary player i and, without loss of generality, let us assume that z i ≥ s i (the case z i ≤ s i is symmetric). Recall that ℓ ( i ) and r ( i ) denote the players in N i ( z , s ) ∪ { i } with the leftmostand rightmost point, respectively, in I i ( z , s ) and note that r ( i ) − ℓ ( i ) = k . First, observe that if z r ( i ) = z i , the assumption z i ≥ s i implies that all players in N i ( z , s ) ∪{ i } have opinions at s i (since,by Lemma 1, z i is in the middle of interval I i ( z , s ) at equilibrium). In this case, cost i ( z , s ) = 0 andthe desired inequality holds trivially. So, in the following, we assume that r ( i ) > i and z r ( i ) > z i ,i.e., z r ( i ) is at the right of z i which in turn is at the right of (or coincides with) s i .Recall that, for player i , ℓ ∗ ( i ) and r ∗ ( i ) denote the integers in [ n ] such that ℓ ∗ ( i ) ≤ i ≤ r ∗ ( i ) , r ∗ ( i ) − ℓ ∗ ( i ) = k , and | s r ∗ ( i ) − s ℓ ∗ ( i ) | is minimized. Since r ( i ) − ℓ ( i ) = r ∗ ( i ) − ℓ ∗ ( i ) = k , wedistinguish between two main cases depending on the relative order of r ( i ) and r ∗ ( i ) . Case I r ( i ) > r ∗ ( i ) and ℓ ( i ) > ℓ ∗ ( i ) . Since z r ( i ) is at the right of s i and ℓ ∗ ( i ) does not belong tothe neighborhood of player i (while player r ( i ) does so by definition), z ℓ ∗ ( i ) is at the left of s i and,furthermore, z r ( i ) − s i ≤ s i − z ℓ ∗ ( i ) or, equivalently, z r ( i ) ≤ s i − z ℓ ∗ ( i ) . (9)This yields cost i ( z , s ) = z r ( i ) − z i ≤ s i − z ℓ ∗ ( i ) − z i . (10)These inequalities will be useful in several places of the proof for this case below.If z ℓ ∗ ( i ) ≥ s ℓ ∗ ( i ) then, since r ∗ ( i ) ≥ i and z i ≥ s i , inequality (10) becomes cost i ( z , s ) ≤ s i − s ℓ ∗ ( i ) ≤ s r ∗ ( i ) − s ℓ ∗ ( i ) and the desired inequality follows. So, in the following, we assume that z ℓ ∗ ( i ) < s ℓ ∗ ( i ) i.e., z ℓ ∗ ( i ) is (strictly) at the left of s ℓ ∗ ( i ) . Hence, ℓ ∗ ( i ) has her leftmost neighbor with z ℓ ( ℓ ∗ ( i )) < z ℓ ∗ ( i ) and, by Lemma 1, z ℓ ∗ ( i ) = z ℓ ( ℓ ∗ ( i )) + max { s ℓ ∗ ( i ) , z r ( ℓ ∗ ( i )) } . (11)Since r ∗ ( i ) − ℓ ∗ ( i ) = k and ℓ ( ℓ ∗ ( i )) < ℓ ∗ ( i ) , we have r ∗ ( i ) − ℓ ( ℓ ∗ ( i )) > k , and, therefore, r ∗ ( i ) does not belong to the neighborhood of ℓ ∗ ( i ) . Hence, s ℓ ∗ ( i ) − z ℓ ( ℓ ∗ ( i )) ≤ z r ∗ ( i ) − s ℓ ∗ ( i ) or,equivalently z ℓ ( ℓ ∗ ( i )) ≥ s ℓ ∗ ( i ) − z r ∗ ( i ) ≥ s ℓ ∗ ( i ) − s i + z ℓ ∗ ( i ) , (12)where the second inequality follows by our case assumption z r ∗ ( i ) ≤ z r ( i ) and inequality (9).We now further distinguish between two cases, depending on whether player i belongs to theneighborhood of player ℓ ∗ ( i ) or not. 21 ase I.1 i ∈ N ℓ ∗ ( i ) ( z , s ) ; see also Figure 7a for an example of this case. Then, we have z i ≤ z r ( ℓ ∗ ( i )) and, subsequently, max { s ℓ ∗ ( i ) , z r ( ℓ ∗ ( i )) } ≥ z r ( ℓ ∗ ( i )) ≥ z i . (13)Using inequalities (12) and (13), (11) yields z ℓ ∗ ( i ) ≥ s ℓ ∗ ( i ) − s i + z ℓ ∗ ( i ) z i , which implies that z ℓ ∗ ( i ) ≥ s ℓ ∗ ( i ) − s i + z i . Now, inequality (10) becomes cost i ( z , s ) ≤ s i − s ℓ ∗ ( i ) − z i ≤ s i − s ℓ ∗ ( i ) ≤ s r ∗ ( i ) − s ℓ ∗ ( i ) ) as desired. The second inequality follows since z i ≥ s i and the last one follows since r ∗ ( i ) ≥ i . Case I.2 i N ℓ ∗ ( i ) ( z , s ) ; see also Figure 7b for an example. Then, we have s ℓ ∗ ( i ) − z ℓ ( ℓ ∗ ( i )) ≤ z i − s ℓ ∗ ( i ) , which implies that z ℓ ( ℓ ∗ ( i )) ≥ s ℓ ∗ ( i ) − z i . Using this inequality together with the factthat max { s ℓ ∗ ( i ) , z r ( ℓ ∗ ( i )) } ≥ s ℓ ∗ ( i ) , (11) yields z ℓ ∗ ( i ) ≥ s ℓ ∗ ( i ) − z i and inequality (10) becomes cost i ( z , s ) ≤ s i − s ℓ ∗ ( i ) − z i ≤ s i − s ℓ ∗ ( i ) ≤ s r ∗ ( i ) − s ℓ ∗ ( i ) ) , as desired. The second last inequality follows since z i ≥ s i and the last one follows since r ∗ ( i ) ≥ i . Case II r ( i ) ≤ r ∗ ( i ) and ℓ ( i ) ≤ ℓ ∗ ( i ) . Since z i is in the middle of the interval I i ( z , s ) and z r ( i ) isthe rightmost opinion in I i ( z , s ) , we have z i = min { s i , z ℓ ( i ) } + z r ( i ) ≤ z ℓ ( i ) + z r ( i ) ≤ z ℓ ∗ ( i ) + z r ∗ ( i ) . Since s i ≤ z i , the last inequality yields z ℓ ∗ ( i ) ≥ s i − z r ∗ ( i ) . (14)We also have cost i ( z , s ) = z r ( i ) − z i ≤ z r ∗ ( i ) − z i . (15)If z r ∗ ( i ) ≤ s r ∗ ( i ) then, since s ℓ ∗ ( i ) ≤ s i ≤ z i , inequality (15) yields cost i ( z , s ) ≤ s r ∗ ( i ) − s i ≤ s r ∗ ( i ) − s ℓ ∗ ( i ) , which is even stronger than the desired inequality. So, in the following we assumethat z r ∗ ( i ) > s r ∗ ( i ) , i.e., z r ∗ ( i ) is at the right of s r ∗ ( i ) . Since z r ∗ ( i ) is in the middle of the interval I r ∗ ( i ) ( z , s ) , we have that r ( r ∗ ( i )) > r ∗ ( i ) and, therefore, z r ∗ ( i ) = min { s r ∗ ( i ) , z ℓ ( r ∗ ( i )) } + z r ( r ∗ ( i )) . (16)Moreover, since r ( r ∗ ( i )) − ℓ ∗ ( i ) > r ∗ ( i ) − ℓ ∗ ( i ) = k , player ℓ ∗ ( i ) does not belong to the neigh-borhood of player r ∗ ( i ) . Hence, z r ( r ∗ ( i )) − s r ∗ ( i ) ≤ s r ∗ ( i ) − z ℓ ∗ ( i ) which, together with inequality(14), yields that z r ( r ∗ ( i )) ≤ s r ∗ ( i ) − z ℓ ∗ ( i ) ≤ s r ∗ ( i ) − s i + z r ∗ ( i ) . (17)We now further distinguish between two cases, depending on whether player i belongs to theneighborhood of player r ∗ ( i ) or not. 22 ase II.1 i ∈ N r ∗ ( i ) ( z , s ) ; see also Figure 7c for an example. Then, using the fact that min { s r ∗ ( i ) , z ℓ ( r ∗ ( i )) } ≤ z ℓ ( r ∗ ( i )) ≤ z i and inequality (17), equation (16) becomes z r ∗ ( i ) ≤ z i + 2 s r ∗ ( i ) − s i + z r ∗ ( i ) and, equivalently, z r ∗ ( i ) ≤ z i + 2 s r ∗ ( i ) − s i . Hence, inequality (15) yields cost i ( z , s ) ≤ s r ∗ ( i ) − s i ≤ s r ∗ ( i ) − s ℓ ∗ ( i ) ) , as desired. The last inequality follows since ℓ ∗ ( i ) ≤ i . Case II.2 i N r ∗ ( i ) ( z , s ) ; see Figure 7d for an example. Since i does not belong to the neigh-borhood of player r ∗ ( i ) but player r ( r ∗ ( i )) does, we have that z r ( r ∗ ( i )) − s r ∗ ( i ) ≤ s r ∗ ( i ) − z i or,equivalently, z r ( r ∗ ( i )) ≤ s r ∗ ( i ) − z i . Then, using also the fact that min { s r ∗ ( i ) , z ℓ ( r ∗ ( i )) } ≤ s r ∗ ( i ) ,equation (16) becomes z r ∗ ( i ) ≤ s r ∗ ( i ) − z i and (15) yields cost i ( z , s ) ≤
32 ( s r ∗ ( i ) − z i ) ≤
32 ( s r ∗ ( i ) − s ℓ ∗ ( i ) ) , which is even stronger than the desired inequality. The last inequality follows since z i ≥ s i and ℓ ∗ ( i ) ≤ i .So, we have shown that in the pure Nash equilibrium z and for any player i , we have that cost i ( z , s ) ≤ s r ∗ ( i ) − s ℓ ∗ ( i ) ) . By summing over all players, we obtain inequality (8) and thetheorem follows.
7. An improved bound on the price of anarchy for -COF games For the case of -COF games we can prove an even stronger statement following a similar proofroadmap as in the previous section, but using simpler (and shorter) arguments. We denote by η ( i ) the player (other than i ) that minimizes the distance | s i − s η ( i ) | ; note that η ( i ) ∈ { i − , i + 1 } . Theproof of the next lemma (which can be thought of as a stronger version of Lemma 11 for 1-COFgames) relies on a particular charging scheme that allows us to lower-bound the cost of each playerin any deterministic opinion vector. Lemma 13.
Consider a -COF game with belief vector s = ( s , . . . , s n ) and let z be any deter-ministic opinion vector. Then, SC ( z , s ) ≥ n X i =1 | s i − s η ( i ) | . Proof.
We begin by classifying the players into groups and, subsequently, we show how the costs ofdifferent groups can be combined so that the lemma holds. We call a player i with z i / ∈ [ s i − , s i +1 ] a kangaroo player and associate the quantity excess i with her. If z i ∈ [ s j , s j +1 ] for some j > i , wesay that the players in the set C i = { i +1 , ..., j } are covered by player i and define excess i = z i − s j .23 ℓ ( ℓ ∗ ( i )) z ℓ ∗ ( i ) s ℓ ∗ ( i ) s ℓ ( i ) s i z i s r ∗ ( i ) s r ( i ) I ℓ ∗ ( i ) ( z , s ) · · · (a) z ℓ ( ℓ ∗ ( i )) z ℓ ∗ ( i ) s ℓ ∗ ( i ) s ℓ ( i ) s i z i s r ∗ ( i ) s r ( i ) I ℓ ∗ ( i ) ( z , s ) (b) s ℓ ( i ) s ℓ ∗ ( i ) s i z i s r ( i ) s r ∗ ( i ) z r ∗ ( i ) z r ( r ∗ ( i )) I r ∗ ( i ) ( z , s ) · · · (c) s ℓ ( i ) s ℓ ∗ ( i ) s i z i s r ( i ) s r ∗ ( i ) z r ∗ ( i ) z r ( r ∗ ( i )) I r ∗ ( i ) ( z , s ) (d) Figure 7: Indicative examples of the different cases in the proof of Theorem 12. Subfigures (a) and(b) concern Case I, as r ( i ) > r ∗ ( i ) and ℓ ( i ) > ℓ ∗ ( i ) , while subfigures (c) and (d) fall under Case II,as r ( i ) ≤ r ∗ ( i ) and ℓ ( i ) ≤ ℓ ∗ ( i ) .Otherwise, if z i ∈ [ s j − , s j ] for some j < i , we say that the players in the set C i = { j, ..., i − } are covered by player i and define excess i = s j − z i .Let K be the set of kangaroo players and C the set of players that are covered by a kangaroo;these need not be disjoint. We now partition the players not in K ∪ C into the set L of large playerssuch that, for any i ∈ L , it holds cost i ( z , s ) ≥ ( | s i − s η ( i ) | ) , and the set S that contains theremaining players who we call small . See also Figure 8 for an example of these sets. s s z z s z z s z s Figure 8: An example with kangaroos, covered, large, and small players. In particular, ∈ K as z / ∈ [ s , s ] , ∈ K ∩ C as she is covered by player and, in addition, z / ∈ [ s , s ] . Similarly, ∈ C as she is covered by player , while and are neither kangaroo nor covered. Since cost ( z , s ) < ( s − s ) , it is ∈ S , while, since cost ( z , s ) ≥ ( s − s ) , we have ∈ L .24e proceed to prove five useful properties (Claims 14–18); recall that σ ( i ) denotes the singleneighbor of player i . Claim 14.
Let i ∈ K . Then, cost i ( z , s ) − excess i ≥ ( | s i − s η ( i ) | + P j ∈ C i | s j − s η ( j ) | ) .Proof. We assume that z i > s i (the other case is symmetric). Let ℓ be the player with the rightmostbelief that is covered by i . Then, excess i = z i − s ℓ . We have cost i ( z , s ) − excess i = max {| s i − z i | , | z i − z σ ( i ) |} − ( z i − s ℓ ) ≥ s ℓ − s i = ℓ − X j = i ( s j +1 − s j ) ≥
13 ( | s i − s η ( i ) | + X j ∈ C i | s j − s η ( j ) | ) as desired. Claim 15.
Let i ∈ S such that σ ( i ) ∈ K . Then, cost i ( z , s ) + excess σ ( i ) ≥ | s i − s η ( i ) | .Proof. We assume that σ ( i ) > i (the other case is symmetric). If z σ ( i ) > s σ ( i ) , then cost i ( z , s ) = max {| s i − z i | , | z i − z σ ( i ) |}≥
12 ( z σ ( i ) − s i ) >
12 ( s σ ( i ) − s i ) ≥ | s i − s η ( i ) | , which contradicts the fact that i is a small player. Hence, z σ ( i ) ∈ [ s i , s σ ( i ) ] , otherwise player i would be covered. Let j be the player with the leftmost belief that is covered by player σ ( i ) . Then, excess σ ( i ) = s j − z σ ( i ) . We have cost i ( z , s ) + excess σ ( i ) = max {| s i − z i | , | z i − z σ ( i ) |} + s j − z σ ( i ) ≥
12 ( z σ ( i ) − s i ) + 12 ( s j − z σ ( i ) ) = 12 ( s j − s i ) ≥ | s i − s η ( i ) | as desired. Claim 16.
Let i ∈ S such that σ ( i ) ∈ L or σ ( i ) ∈ C \ K . Then, cost i ( z , s ) + cost σ ( i ) ( z , s ) ≥ ( | s i − s η ( i ) | + | s σ ( i ) − s η ( σ ( i )) | ) .Proof. We assume that σ ( i ) > i (the other case is symmetric). If z σ ( i ) > s σ ( i ) , then cost i ( z , s ) = max {| s i − z i | , | z i − z σ ( i ) |}≥
12 ( z σ ( i ) − s i ) >
12 ( s σ ( i ) − s i ) ≥ | s i − s η ( i ) | , i is a small player. Hence, z σ ( i ) ∈ [ s i , s σ ( i ) ] , otherwise player i wouldbe covered. Then, cost i ( z , s ) + cost σ ( i ) ( z , s ) = max {| s i − z i | , | z i − z σ ( i ) |} + max {| s σ ( i ) − z σ ( i ) | , | z σ ( i ) − z σ ( σ ( i )) |}≥ z σ ( i ) − z i + s σ ( i ) − z σ ( i ) = s σ ( i ) − z i . Since i is small, we have z i < s i + ( s σ ( i ) − s i ) and we get cost i ( z , s ) + cost σ ( i ) ( z , s ) ≥
23 ( s σ ( i ) − s i ) ≥ | s i − s η ( i ) | + 13 | s σ ( i ) − s η ( σ ( i )) | as desired.Let N ( S ) denote the set of players j that are neighbors of players in S (i.e., j ∈ N ( S ) when σ ( i ) = j for some player i ∈ S ). Claim 17. N ( S ) does not contain small players.Proof. Assume otherwise that for some player i ∈ S , σ ( i ) also belongs to S . Without loss ofgenerality σ ( i ) > i . If z σ ( i ) ≥ s σ ( i ) , then cost i ( z , s ) ≥ | z σ ( i ) − s i | ≥ | s σ ( i ) − s i | ≥ | s i − s η ( i ) | contradicting the fact that i ∈ S . So, z σ ( i ) < s σ ( i ) . Also, z σ ( i ) ≥ s i (since neither i is coverednor σ ( i ) is kangaroo). Since σ ( i ) is small, s σ ( i ) − z σ ( i ) < | s σ ( i ) − s η ( σ ( i )) | ≤ ( s σ ( i ) − s i ) , i.e., z σ ( i ) > s σ ( i ) + s i . Hence, cost i ( z , s ) ≥
12 ( z σ ( i ) − s i ) >
13 ( s σ ( i ) − s i ) , which contradicts i ∈ S . Claim 18.
For every two players i, i ′ ∈ S , σ ( i ) = σ ( i ′ ) .Proof. Assume otherwise and let σ ( i ) = σ ( i ′ ) = j with i < i ′ . If z j [ s i , s i ′ ] , then the costof either i or i ′ is at least ( s i ′ − s i ) , contradicting the fact that both players are small. Hence, z j ∈ [ s i , s i ′ ] . Notice that s j ∈ [ s i , s i ′ ] as well, otherwise either i or i ′ would be covered by j . Nowthe fact that i and i ′ are small implies that cost i ( z , s ) + cost i ′ ( z , s ) < | s i − s η ( i ) | + 13 | s i ′ − s η ( i ′ ) | ≤
13 ( s j − s i ) + 13 ( s i ′ − s j ) = 13 ( s i ′ − s i ) . On the other hand, cost i ( z , s ) + cost i ′ ( z , s ) ≥
12 ( z j − s i ) + 12 ( s i ′ − z j ) = 12 ( s i ′ − s i ) , a contradiction. 26e now consider the social cost of z due to players of different groups and exploit the claimsabove so that we obtain the lemma. In particular, we haveSC ( z , s ) = n X i =1 cost i ( z , s ) ≥ X i ∈ S : σ ( i ) ∈K (cid:0) cost i ( z , s ) + excess σ ( i ) (cid:1) + X i ∈ S : σ ( i ) ∈ L ∪ ( C\K ) (cid:0) cost i ( z , s ) + cost σ ( i ) ( z , s ) (cid:1) + X i ∈K (cost i ( z , s ) − excess i ) + X i ∈ L \ N ( S ) cost i ( z , s ) ≥ X i ∈ S : σ ( i ) ∈K | s i − s η ( i ) | + 13 X i ∈ S : σ ( i ) ∈ L ∪ ( C\K ) (cid:0) | s i − s η ( i ) | + | s σ ( i ) − s η ( σ ( i )) | (cid:1) + 13 X i ∈K | s i − s η ( i ) | + X j ∈ C i | s j − s η ( j ) | + 13 X i ∈ L \ N ( S ) | s i − s η ( i ) |≥ n X i =1 | s i − s η ( i ) | , as desired. The first inequality follows by the classification of the players and due to Claims 17 and18. The second one follows by Claims 15, 16, and 14, and by the definition of large players. Thelast one follows since the players enumerated in the first two sums at its left cover the whole set S (by Claim 17).We are ready to present our upper bound on the price of anarchy for -COF games. Theorem 19.
The price of anarchy of -COF games over pure Nash equilibria is at most .Proof. Let us consider a -COF game with n players and belief vector s . Let z ∗ be an optimalopinion vector and recall that η ( i ) is the player that minimizes the distance | s i − s η ( i ) | . By Lemma13, we have SC ( z ∗ , s ) ≥ n X i =1 | s i − s η ( i ) | . (18)Now, consider any pure Nash equilibrium z of the game. We will show thatSC ( z , s ) ≤ n X i =1 | s i − s η ( i ) | . (19)The theorem then follows by (18) and (19).In particular, we will show that cost i ( z , s ) ≤ | s i − s η ( i ) | for each player i . Let us assume that η ( i ) = i − ; the case η ( i ) = i + 1 is symmetric. Recall that σ ( i ) is the neighbor of player i in thepure Nash equilibrium z . We distinguish between four cases.27 Case I: σ ( i ) = i − . By Lemma 4, we have s i − ≤ z i ≤ s i . Then, clearly, cost i ( z , s ) = | s i − z i | ≤ | s i − s i − | as desired. • Case II: σ ( i ) = i + 1 and σ ( i −
1) = i . By Lemmas 2 and 4, we have s i − ≤ z i − ≤ s i ≤ z i .Since player i has player i + 1 as neighbor, we have | z i +1 − s i | ≤ | s i − z i − | . Hence, cost i ( z , s ) = | z i − s i | ≤ | z i +1 − s i | ≤ | s i − z i − | ≤ | s i − s i − | . • Case III: σ ( i ) = i + 1 , σ ( i −
1) = i − , and cost i ( z , s ) ≤ cost i − ( z , s ) . By the definitionof σ ( · ) and Lemma 2, we have z i − ≤ z i − ≤ s i − ≤ s i ≤ z i ≤ z i +1 . We have cost i ( z , s ) ≤ i − ( z , s ) − cost i ( z , s )= | s i − − z i − | − | z i − s i |≤ | z i − s i − | − | z i − s i | = | s i − s i − | . The second inequality follows since player i − (instead of i ) is the neighbor of player i − . • Case IV: σ ( i ) = i + 1 , σ ( i −
1) = i − , and cost i ( z , s ) > cost i − ( z , s ) . cost i ( z , s ) < i ( z , s ) − cost i − ( z , s )= | z i +1 − s i | − | s i − − z i − |≤ | s i − z i − | − | s i − − z i − | = | s i − s i − | . The second inequality follows since player i + 1 (instead of i − ) is the neighbor of player i .This completes the proof.
8. Lower bounds on the price of anarchy
This section contains our lower bounds on the price of anarchy. We begin by considering thesimpler case of -COF games, for which we present a tight lower bound of for pure Nash equilibria(Theorem 20) and a lower bound of for mixed Nash equilibria (Theorem 21). We remark that,for -COF games, this implies that mixed Nash equilibria are strictly worse than pure ones. Then,we study the general case of k -COF games and we show lower bounds for pure and mixed Nashequilibria (Theorems 22 and 23, respectively) that grow linearly with k . -COF games We now present our lower bounds for the case of -COF games; both results rely on the same, andrather simple, instance. Theorem 20.
The price of anarchy of -COF games over pure Nash equilibria is at least .
3. We remark that our lower bounds on the price of stability in Section 4 are also lower bounds on the price of anarchy.However, the lower bounds presented in this section are much stronger. − λ [2] − − λ [1] 2 + λ [1] 10 + λ [2] (a) − − λ − − λ λ
10 + λ − − λ λ (b) − − λ − − λ λ
10 + λ − − λ λ (c) Figure 9: (a) The -COF game considered in the proofs of Theorems 20 and 21. (b) The pure Nashequilibrium vector z (see the proof of Theorem 20) with social cost . (c) The opinion vector ˜ z withsocial cost λ . Proof.
Let λ ∈ (0 , and consider a -COF game with six players and belief vector s = ( − − λ, − − λ, − − λ, λ,
10 + λ,
10 + λ ) . This game is depicted in Figure 9a. We can show thatthe opinion vector (see Figure 9b) z = ( − − λ, − − λ, − − λ, λ,
10 + λ,
10 + λ ) is a pure Nash equilibrium with social cost SC ( z , s ) = 8 . The first two players suffer zero cost asthey follow each other and their opinions coincide with their beliefs; the same holds also for the lasttwo players. For the third player, it is σ (3) ∈ { , } since | z − s | = | z − s | = 8 < | z − s | =8 + 2 λ and z is in the middle of the interval [ − − λ, − − λ ] ; hence, cost ( z , s ) = 4 . Similarly,we have σ (4) ∈ { , } , z lies in the middle of the interval [2 + λ,
10 + λ ] and cost ( z , s ) = 4 .Hence, z is indeed a pure Nash equilibrium.Now, consider the opinion vector (see Figure 9c) ˜ z = (cid:18) − − λ, − − λ, − − λ , λ ,
10 + λ,
10 + λ (cid:19) which yields a social cost of SC (˜ z , s ) = λ ; here, again, the first and last two players have zerocost, but players and now each have cost λ since they follow each other. The optimal socialcost is upper bounded by SC (˜ z ) and, hence, the price of anarchy is at leastSC ( z , s ) SC (˜ z , s ) = 31 + λ/ , and the theorem follows by setting λ arbitrarily close to .Our next theorem gives a lower bound on the price of anarchy over mixed Nash equilibria for -COF games; we remark that this lower bound is greater than the upper bound of Theorem 19 forthe price of anarchy over pure Nash equilibria. 29 heorem 21. The price of anarchy of -COF games over mixed Nash equilibria is at least .Proof. Consider again the -COF game depicted in Figure 9a with six players and belief vector s = ( − − λ, − − λ, − − λ, λ,
10 + λ,
10 + λ ) , where λ ∈ (0 , . To simplify the followingdiscussion, we will refer to the first two players as the L players, the third player as player ℓ , thefourth player as player r , and the last two players as the R players.Let z be a randomized opinion vector according to which z i = s i for every i ∈ L ∪ R , z ℓ ischosen equiprobably from {− − λ, − λ } , and z r is chosen equiprobably from { λ, − λ } .Observe that σ ( ℓ ) ∈ L whenever z r = 6 + λ , and σ ( ℓ ) = r whenever z r = 6 − λ ; each of theseevents occurs with probability / . Hence, we obtain E [cost ℓ ( z , s )] = E [cost r ( z , s )] = 12 (cid:18)
42 + 4 + 4 λ (cid:19) + 12 (cid:18) − λ − λ (cid:19) = 8 − λ, and, thus, E [ SC ( z , s )] = 16 − λ . In the following, we will prove that z is a mixed Nash equilibrium.First, observe that all players in sets L and R have no incentive to deviate since they follow eachother and have zero cost. We will now argue about player ℓ ; due to symmetry, our findings willapply to player r as well.Consider a deterministic deviating opinion y for player ℓ . We will show that E [cost ℓ ( z , s )] ≤ E z − ℓ [cost ℓ ( y, z − ℓ ) , s ] for any y , which implies that player ℓ has no incentive to deviate from therandomized opinion z ℓ . Indeed, we have that E z − ℓ [cost ℓ (( y, z − ℓ ) , s )]= 12 max {| − − λ − y | , | y + 10 + λ |} + 12 max {| − − λ − y | , | − λ − y |}≥
12 ( y + 10 + λ ) + 12 (6 − λ − y )= 8 − λ, where the inequality holds since max {| a | , | b |} ≥ a for any a and b . Hence, player ℓ has no incentiveto deviate from her strategy in z , and neither has player r due to symmetry. Therefore, z is a mixedNash equilibrium.Now, consider the opinion vector ˜ z = (cid:18) − − λ, − − λ, − − λ , λ ,
10 + λ,
10 + λ (cid:19) which, as in Theorem 20, yields a social cost of SC (˜ z , s ) = λ . Hence, the optimal social cost isupper bounded by SC (˜ z , s ) , and the price of anarchy over mixed equilibria is at least E [ SC ( z , s )] SC (˜ z , s ) = 3 16 − λ λ , and the theorem follows by setting λ arbitrarily close to . k -COF games with k ≥ We will now present lower bounds on the price of anarchy for k -COF games, with k ≥ . We startwith the case of pure Nash equilibria and continue with the more general case of mixed equilibria.As in the case of -COF games, a particular game will be used in order to derive the lower boundsboth for pure and mixed Nash equilibria. 30 − λ [ k + 1] − − λ [1] 0[ k −
1] 4 + λ [1] 16 + 2 λ [ k + 1] (a) − − λ − − λ λ
16 + 2 λ − − λ λ (b) − − λ − − λ λ
16 + 2 λ (c) − − λ − − λ λ
16 + 2 λ − − λ λ (d) Figure 10: (a) The k -COF game considered in the proofs of Theorems 22 and 23, for k ≥ . (b)The pure Nash equilibrium opinion vector z (see the proof of Theorem 22). (c) The optimal opinionvector ˜ z for k ≥ . (d) The optimal opinion vector ˜ z for k = 2 . Observe that the optimal opinionvector changes at k = 2 due to the neighborhood size. Theorem 22.
The price of anarchy of k -COF games over pure Nash equilibria is at least k + 1 for k ≥ , and at least / for k = 2 .Proof. Let λ ∈ (0 , and consider a k -COF game with k +3 players, for k ≥ , that are partitionedinto the following five sets. The first set L consists of k + 1 players with s i = − − λ for any i ∈ L , the second set consists of a single player ℓ with s ℓ = − − λ , the third set M has k − players with s i = 0 for any i ∈ M , the fourth set is a single player r with s r = 4 + λ , and thelast set R consists of k + 1 players with s i = 16 + 2 λ for any i ∈ R . This instance is depicted inFigure 10a.Let z be the following opinion vector: z i = − − λ for any i ∈ R , z ℓ = − − λ , z i = 0 for any i ∈ M , z r = 8 + λ , and z i = 16 + 2 λ for any i ∈ R ; see Figure 10b. It is not hard to verify that thisopinion vector is a pure Nash equilibrium with social cost SC ( z , s ) = (8 + λ )( k + 1) . First, observethat all players in sets L and R have zero cost, and, hence, have no incentive to deviate to anotheropinion. Furthermore, no player i ∈ M has an incentive to deviate either since z i lies in the middleof the interval [ − − λ, λ ] which is defined by the opinions of players ℓ and r who, together withthe remaining players of M , constitute the neighborhood N i ( z , s ) of player i . The cost experiencedby such a player i is λ . Finally, the neighborhood N ℓ ( z , s ) of player ℓ consists of all playersin M (who have opinions that are closest to s ℓ ) and some player i ∈ L ; note that player r does notbelong to N ℓ ( z , s ) since z r − s ℓ = 12 + 2 λ > − λ = s ℓ − z i for all i ∈ L . Hence, player ℓ has noincentive to deviate to another opinion since z ℓ lies in the middle of the interval [ − − λ, and31he experiences cost equal to λ . Due to symmetry, player r does not have incentive to deviateas well. Hence, z is indeed a pure Nash equilibrium with SC ( z , s ) = (8 + λ )( k + 1) .We now present an opinion vector ˜ z with social cost SC (˜ z , s ) = 8 + 2 λ for k ≥ and cost(˜ z , s ) = (4 + λ ) for k = 2 . In particular, for k ≥ , ˜ z is defined as follows: ˜ z i = − − λ for any i ∈ L , ˜ z ℓ = ˜ z i = ˜ z r = 0 for any i ∈ M , and ˜ z i = 16 + 2 λ for any i ∈ R ; see Figure 10c.Observe that all players in L , M , and R have zero cost, while players ℓ and r have cost equal to λ each. For k = 2 , ˜ z is defined as follows: ˜ z i = − − λ for any i ∈ L , ˜ z ℓ = − (4 + λ ) , ˜ z i = 0 for any i ∈ M , ˜ z r = (4 + λ ) , and ˜ z i = 16 + 2 λ for any i ∈ R ; see Figure 10d. Again, allplayers in L and R have zero cost. However, players ℓ and r now each have cost (4 + λ ) and theunique player in M has cost (4 + λ ) .Clearly, since SC (˜ z , s ) is an upper bound on the optimal social cost, we conclude that the priceof anarchy over pure Nash equilibria is at least (8+ λ )( k +1)8+2 λ for k ≥ and λ )5(4+ λ ) for k = 2 , and thetheorem follows by setting λ arbitrarily close to .We now consider the case of mixed Nash equilibria; we remark that, in this case, our lowerbounds for k ≥ are smaller than the corresponding upper bounds for pure Nash equilibria. Theorem 23.
The price of anarchy of k -COF games over mixed Nash equilibria is at least k + 2 for k ≥ , and at least / for k = 2 .Proof. As in the proof of Theorem 22, let λ ∈ (0 , and consider the k -COF game depicted inFigure 10a with k + 3 players that form sets. Again, the first set L consists of k + 1 playerswhere s i = − − λ for all i ∈ L , the second set consists of a single player ℓ with s ℓ = − − λ ,the third set M has k − players with s i = 0 for all i ∈ M , the fourth set is a single player r with s r = 4 + λ , and the last set R consists of k + 1 players with s i = 16 + 2 λ for all i ∈ R .Consider the following (randomized) opinion vector z : z i = s i for every i ∈ L ∪ M ∪ R , while z ℓ is chosen uniformly at random among {− − λ, − λ } and z r is chosen uniformly at randomamong { − λ, λ } . We will show that the opinion vector z is a mixed Nash equilibrium with E [ SC ( z , s )] = 8 k + 16 − λ .First, observe that the players in sets L and the R constitute local neighborhoods, that is, N i ( z , s ) = L \ { i } for any player i ∈ L , and N i ( z , s ) = R \ { i } for any player i ∈ R . Hence, allthese players have zero cost and no incentive to deviate.Next, let us focus on a player i ∈ M . Clearly, the neighborhood of player i consists of theremaining k − players in M as well as players ℓ and r . The expected cost of player i in z is E [cost i ( z , s )] = (8 + λ ) + (8 − λ ) = 8 since at least one of players ℓ and r is at distance λ with probability / and both of them are at distance − λ with probability / . Hence, these k − players contribute k − to the expected social cost of z . We now argue that if player i ∈ M deviates to a deterministic opinion y , her expected cost does not decrease. Clearly, if y ≥ λ , thenthis trivially holds as the expected cost of i is at least y − z ℓ which is at least y + 8 − λ ; the casewhere y ≤ − λ is symmetric. Hence, it suffices to consider the case where | y | < λ . The expected32ost of i when deviating to y is E z − i [cost i (( y, z − i ) , s )]= 14 max { λ − y, y + 8 + λ } + 14 max { λ − y, y + 8 − λ } + 14 max { − λ − y, y + 8 + λ } + 14 max { − λ − y, y + 8 − λ }≥
14 (8 + λ − y ) + 14 (8 + λ − y ) + 14 ( y + 8 + λ ) + 14 ( y + 8 − λ )= 8 , where the inequality holds since max { a, b } ≥ a for any a and b .Now, let us examine player r ; the case of player ℓ is symmetric. Observe that the k − playersin M always belong to the neighborhood N r ( z , s ) of player r and it remains to argue about theidentity of the last player in N r ( z , s ) . Whenever z ℓ = − λ , then ℓ ∈ N r ( z , s ) , otherwise, if z ℓ = − − λ , one of the players in set R belongs to N r ( z , s ) . The expected cost of player r is E [cost r ( z , s )] = (8 + λ ) + (8 + 5 λ ) + (16 − λ ) + (16 − λ ) = 12 − λ/ , and, hence, players ℓ and r contribute − λ to the expected social cost of z . It remains to show that player r cannotdecrease her expected cost by deviating to another opinion y . The expected cost of player r whendeviating to y is E z − r [cost r (( y, z − r ) , s )] = 12 max {|
16 + 2 λ − y | , | y |} + 12 max {| y + 8 − λ | , | λ − y |}≥
12 (16 + 2 λ − y ) + 12 ( y + 8 − λ )= 12 − λ/ , where the inequality holds since max {| a | , | b |} ≥ a for any a and b . Hence, we conclude that z is amixed Nash equilibrium with expected social cost E [ SC ( z , s )] = 8 k + 16 − λ .As in the proof of Theorem 22, there exists an opinion vector ˜ z with social cost SC (˜ z , s ) =8 + 2 λ for k ≥ and SC (˜ z , s ) = (4 + λ ) for k = 2 . Since SC (˜ z , s ) is an upper bound on theoptimal social cost, we have that the price of anarchy over mixed equilibria is at least k +16 − λ λ for k ≥ and − λ )5(4+ λ ) for k = 2 , and the theorem follows, again by setting λ arbitrarily close to .
9. Open problems and extensions
We have introduced the class of compromising opinion formation games by enriching coevolution-ary opinion games with a cost function that urges players to “meet halfway”. Our findings indicatethat the quality of their equilibria grows linearly with the neighborhood size k . Still, there exists agap between our lower and upper bounds for k ≥ and closing this gap is a challenging technicaltask. Furthermore, we know that mixed equilibria are strictly worse for -COF games but we havebeen unable to prove upper bounds on their price of anarchy. Is their price of anarchy still linear?Another natural question is about the complexity of pure Nash equilibria in k -COF games for k ≥ . We conjecture that there exists a polynomial time algorithm for computing them, but findingsuch an algorithm remains elusive. Similarly, what is the complexity of computing an optimalopinion vector (even for k = 1 )? 33inally, our modeling assumption that the number of neighbors is the same for all players israther restrictive. Extending our results to the general case of different neighborhood size perplayer deserves investigation. One possible such generalization is to combine our approach to theHegselmann-Krause model, i.e., each player’s neighborhood consists solely of those players withopinions sufficiently close to her belief. References
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