Netflix Games: Local Public Goods with Capacity Constraints
NNetflix Games: Local Public Goods withCapacity Constraints
Stefanie Gerke ∗ Gregory Gutin † Sung-Ha Hwang ‡ Philip Neary § August 27, 2019
Abstract
This paper considers incentives to provide goods that are partially excludablealong social links. Individuals face a capacity constraint in that, conditional uponproviding, they may nominate only a subset of neighbours as co-beneficiaries.Our model has two typically incompatible ingredients: (i) a graphical game (in-dividuals decide how much of the good to provide), and (ii) graph formation(individuals decide which subset of neighbours to nominate as co-beneficiaries).For any capacity constraints and any graph, we show the existence of specialised pure strategy Nash equilibria - those in which some individuals (the ‘Drivers’, D )contribute while the remaining individuals (the ‘Passengers’, P ) free ride. Theproof is constructive and corresponds to showing, for a given capacity, the exis-tence of a new kind of spanning bipartite subgraph , a DP - subgraph , with partitesets D and P . We consider how the set of efficient equilibria (those with mini-mal number of drivers) vary as the capacity constraints are relaxed and show aweak monotonicity result. Finally, we introduce dynamics and show that onlyspecialised equilibria are stable against individuals unilaterally changing theirprovision level. ∗ Mathematics Department, Royal Holloway University of London, Egham TW20 0EX, UK. † Computer Science Department, Royal Holloway University of London, Egham TW20 0EX, UK. ‡ College of Business, Korea Advanced Institute of Science and Technology (KAIST), Seoul, Korea. § Economics Department, Royal Holloway University of London, Egham TW20 0EX, UK. a r X i v : . [ ec on . T H ] A ug Introduction
Since the observations of British economist William Forster Lloyd over 180 years ago(referred to as ‘the tragedy of the commons’ by Hardin (1968)), economists have beenaware of difficulties that arise when shareable resources come up against capacity con-straints. Examples appear everywhere: local schools have only so many classrooms andso many teachers, public parks can become congested on sunny days, six friends cannotall fit into a five-seater car, the online streaming provider Netflix only allows four de-vices stream simultaneously so families of five or more may experience disagreements,and so on. Even for the classic public goods example of a fireworks display, there areoften superior vantage points for which people compete. In this paper we tackle issuesof this type. That is, what happens when individuals who provide a costly good can’tshare with everyone?To address the above, we develop a model wherein individuals live on a connectedgraph G (vertices represent individuals and edges represent connections between pairs)and each must make a two-pronged decision: (i) choose how much of a costly goodto provide, and (ii) choose a subset of neighbours to nominate as co-beneficiaries. Inthe simplest version of the model, that we term the “Netflix Game” as it is inspiredby the online streaming provider Netflix, the quantity choice is binary: each individualsimply decides whether to purchase an account or not. If individual i purchases anaccount then she nominates min { κ ( i ) , d G ( i ) } neighbours as co-beneficiaries, where κ ( i )is an exogenously given number known as i ’s capacity and d G ( i ) is i ’s degree in G . Preferences are homogeneous such that it is always better to have access to Netflix thannot, but due to its cost it is preferable for a neighbour to purchase and nominate youas a co-beneficiary of their account than to purchase an account yourself.Our focus is on pure strategy Nash equilibria, wherein every individual is eithera ‘Driver’, D , who purchases a Netflix account, or a ‘Passenger’, P , who free rides.Our first result, Theorem 1, shows the existence of a pure strategy Nash equilibriumfor any graph and any capacity function κ on the vertices of the graph. The proof isconstructive and amounts to showing, for a given capacity function κ , the existence ofa κ - DP -subgraph of G : a spanning bipartite subgraph H of G with partite sets P and D where for each i ∈ D the degree of i in H is min { κ ( i ) , d G ( i ) } and for every i ∈ P The reader unfamiliar with graph theoretic terminology can skip ahead to the beginning of Section3 for the formal definitions. i in H is positive. While a κ - DP -subgraph is purely graph theoretic, ithas an intuitive economic interpretation: in any pure strategy Nash equilibrium, everyagent must be either a driver or a passenger who is nominated (by a neighbour whois a driver), and no agent can be both. Given its constructive nature, the proof alsosuggests an algorithm that finds a pure strategy equilibrium in polynomial time.While the model is too rich for formal theories of equilibrium selection, we canrelate equilibria to efficiency. We employ the utilitarian social welfare criterion as ourmeasure of efficiency. Since agents share a common preference over the public good, thisis equivalent to saying that a specialised equilibrium is efficient (inefficient) if the set ofDrivers, the ‘ D -set’, is minimal (maximal) amongst all such equilibrium outcomes. Onemight conjecture that the sizes of both maximal and minimal D -sets are monotonicallydecreasing in capacity since at least as much sharing is possible. We show via someexamples that this is not the case. However, for any two ordered capacity functions κ and κ (cid:48) (i.e., κ ( i ) ≤ κ (cid:48) ( i ) for every vertex i ), our second result, Theorem 2, shows thata minimal D -set for κ (cid:48) is never larger than a maximal D -set for κ . With the above ideas fixed we can now introduce the general model. The differenceis that the choice of quantity is not simply 0 or 1, but rather any non-negative integer.Preferences are now defined by a quantity q ∗ at which an individual becomes satiated(in the Netflix Game q ∗ = 1 since no additional benefit is accrued from access to morethan one account). The main difference in this richer set up is that there can be purestrategy Nash equilibria in which everybody contributes a strictly positive quantity.However, our focus is on so-called specialised pure strategy Nash equilibria, whereinDrivers contribute the optimal quantity of the good q ∗ and Passengers free ride (in thebinary action Netflix Game every pure strategy equilibrium is specialised by definition).The reason for this focus becomes clear when we repeat the game and introduce best-response dynamics. However, since an individual’s choice of nomination is not payoff Our model does not admit a potential function (Shapley and Monderer, 1996) which would renderthe existence of a pure strategy equilibrium immediate. An alternative criterion would be to consider Pareto-efficiency. That is, those specialised equilibriain which the D -set does not contain a proper subset that supports another specialised equilibriumoutcome. However, as we will see in some examples in Section 2 and examine more thoroughly inSection 4, Pareto-efficient equilibria can be very inefficient from a utilitarian perspective. D -sets are closely related to the well-studied graph theoretic concepts of “maximal independentsets” and “minimal dominating sets”, see, e.g., Goddard and Henning (2013). However, by definitionno two maximally independent sets nor minimally dominating sets can exhibit set inclusion, whereasit is possible for two D -sets to be so ordered. nicely balanced specialised equilibria - those in which every individual in P nominates at least one of the individuals in D who nominated them. We then fix thenominations and consider only deviations in action choice. We show that only nicelybalanced specialised equilibria are locally stable to unilateral deviations in action choice.In particular non-specialised equilibria are not robust.Without the nominating component, or equivalently when each agent’s capacity isat least equal to their degree, our model is a graphical game (Kearns et al., 2013)in which actions are strategic substitutes, and in fact is equivalent to a discretisedversion of that in Bramoull´e and Kranton (2007). In particular when the action choiceis { , } our model reduces to the best-shot game of Galeotti et al. (2010). Thechange generated by capacity constraints is most sharply viewed in the equilibriumpredictions: not every specialised equilibrium in our model is Pareto-efficient whichshould be contrasted with Bramoull´e and Kranton (2007) since each equilibrium intheir model is associated with a maximal independent set (see footnotes 3 and 4).Without the action choice component, our model falls under the umbrella of “networkformation”. The network formation component to our model is perhaps closest to the“Announcement Game” in Myerson (1991) but with two main differences. First, inMyerson’s model each agent may nominate any subset of individuals whereas in ourmodel each agent i must nominate precisely min { κ ( i ) , d G ( i ) } others. Second, in ourmodel who you nominate is not payoff relevant, rather all that matters is who nominatesyou.There are a host of existing papers that examine issues associated with local publicgood provision in a set up similar to Bramoull´e and Kranton (2007). The closestto ours is that of Galeotti and Goyal (2010). Like our model they consider allowingindividuals to nominate, at a cost, others in society. The difference is that in their The original best-shot game is due to Hirshleifer (1983), but it did not have a network component.The best-shot game is often used to study costly information acquisition. See for example Foster andRosenzweig (1995) and Conley and Udry (2010) who study whether a new crop-harvesting technologyis shared between farmers. Network formation is typically modelled either as the realisation of a random process (originatingwith the Erd˝os-Renyi model (Erd˝os and R´enyi, 1959; Gilbert, 1959)) or via a non-cooperative game-theoretic formulation as in Jackson and Wolinsky (1996) and the Myerson (1991). See Chapters 4-6of the textbook Jackson (2008). See for example, Allouch (2015, 2017), Baetz (2015), Cho (2010), Elliott and Golub (2019), andKinateder and Merlino (2017), piggy back on the (potentiallylarge) contributions of other agents helps the system rule out inefficient equilibria. The remainder of the paper is organised as follows. Section 2 motivates our anal-ysis with three examples. The first shows how the set of equilibrium outcomes to thebest-shot game (our model with no capacity constraints) can change dramatically oncecapacity constraints are imposed. The second shows that finding a monotonic sequenceof D -sets for increasing capacities is not possible and also that the number of equilibrianeed not increase with relaxing capacity constraints. The third shows a graph whereina nicely balanced specialised profile is robust, but another pure strategy equilibrium isnot. Section 3 introduces the model and proves existence of a specialised Nash equi-librium for every capacity function and every graph. Section 4 examines comparativestatics and efficiency. Section 5 introduces dynamics and shows that specialised equi-libria are necessary for stability. Section 6 concludes with a summary of our results andsome suggestions for further research on this topic. This section discusses three examples that illustrate features of the model and highlightssome of our main results. The first example shows that specialised equilibrium outcomesto local public good provision can change dramatically with the introduction of capacityconstraints. The second example shows how, within the class of specialised equilibria,the non-monotonicity of D -sets (those who contribute) can evolve as the constraintson capacity are relaxed. In particular, the most efficient equilibrium outcomes (thatwe define as those with the smallest D -sets) may not occur when capacity is maximal.The third example illustrates how only specialised equilibria are candidates for beingdynamically stable in the long run. Example 1.
There is a social network of 5 individuals arranged in a star as depictedin Figure 1. We label the peripheral players h, i, j , and k and the central player (cid:96) .Each individual wishes to utilise the online media services provider Netflix. The Here efficiency is the same as our usage (utilitarian). Formally, Galeotti and Goyal (2010) showthat the fraction of agents contributing positive quantities of the good goes to zero as n gets large.Galeotti and Goyal (2010) have no mention of Pareto-efficiency. k ji(cid:96) Figure 1: A 5-person star networkcompany’s rules permit any individual who purchases an account to stream simultane-ously on a maximum of five devices. We assume that edges in the network representclose friendships so that any person who purchases will and can share with each of hisfriends. Formally, this is modelled as the simultaneous-move ‘best-shot game’ of Gale-otti et al. (2010) where each agent has strategy set { , } , with 1 meaning purchase aNetflix account and 0 meaning don’t.It is a best-response for each agent to purchase a Netflix account if and only ifno neighbour does. With this in mind, there are two pure strategy equilibria to theabove. In the first, only the individual in the centre, (cid:96) , purchases. In the second, all theperipheral individuals, h, i, j, k purchase and the central individual (cid:96) does not. Thesetwo equilibria are depicted in Figure 2 below, with adopters in blue and non adoptersin red. The direction of sharing is further indicated by arrows, with the tail of anyarrow originating at a purchaser and the head of an arrow pointing to those with whomshe shares. In each equilibrium the set of adopters forms a maximal independent set soboth equilibria are Pareto-efficient. hk ji(cid:96) hk ji(cid:96) Figure 2: Equilibria for best-shot game on 5-person star networkNow let consider what would happen if Netflix altered the number of devices that oneaccount can simultaneously access. Let the number of people that may simultaneously6se the service other than the account holder be denoted by κ . An individual whopurchases an account can nominate only κ of her neighbours (and will nominate all ofher neighbours if she has less than κ ). For κ = 1 , ,
3, the only equilibrium outcomeis for the peripheral individuals to purchase an account (and each to nominate thecentral individual, (cid:96) , as the friend who may use the account free of charge). Theseequilibrium outcomes have a D -set of size 4 and are each depicted in the right handpanel of Figure 2. The outcome depicted in the left hand panel is no longer supportedby an equilibrium, the reason being that if (cid:96) purchases an account then she can onlynominate 3 of her 4 neighbours which will leave one without access. It is then optimalfor this un-nominated neighbour to purchase an account, but he in turn must nominate (cid:96) who subsequently would no longer want to purchase. So in this example, restrictingattention to equilibrium outcomes, the number of adopters in equilibrium can onlydecrease as Netflix permit more “shareability”. Example 2.
The previous example suggests a natural conjecture. This is, loosely put,that the size of D -sets will not increase as capacity increases. The following exampleshows that this conjecture is not correct and also highlights some other interestingfeatures of the set up.We imagine a couple Isabella and John, I and J , who are planning on introducingtheir friends to one another at a gathering. We refer to Isabella’s friends as i , i , and i , and John’s friends as j , j , and j . Isabella knows her friends and John knowshis friends; no other pair of individuals have previously met. This social network isdepicted in Figure 3. i i i I J j j j Figure 3: Two star graphs with central vertices connectedThe gathering will take place at a restaurant outside of town and so people musttravel by car. While everybody owns a car and will drive if needs be, it is preferable to7et a ride than to drive oneself (one can then drink alcohol, save on fuel costs, save onparking costs, etc.).In this example, a person’s capacity is the number of passenger seats in their car.For the sake of simplicity, let us assume that everybody’s car is the same size andconsider how the equilibria vary as capacities are increased. (Note that to economiseon space we are somewhat loose and describe equilibria by naming only the drivers,i.e., the D -set.) When everyone’s car can give a ride to only one passenger, κ is equalto 1 for everybody, there is only one equilibrium with D -set D = { i , i , i , j , j , j } .For κ equal to 2, D is again the only D -set. When κ is increased to 3, three newequilibria emerge. This collection of equilibria have D -sets given by D , D = { I, J } with Isabella offering a ride only to her three friends and John offering a ride to histhree friends, D = { i , i , i , J } and D = { I, j , j , j } . Thus for κ equal to 3, there isan equilibrium with only two people driving. This equilibrium is depicted in Figure 4below as a subgraph of the social network with drivers again in blue and passengers inred, and arrows between the two originating at drivers offering a ride. When κ increasesto 4, the model reduces to the best-shot game for which adopters in equilibrium form amaximal independent set (the collection of which is D , D , and D ) for which at least4 people drive. i i i I J j j j Figure 4: Equilibrium with minimal number of adopters for κ = 3We note some other interesting features illustrated by this example. First, when κ is equal to 1 or 2, there is no equilibrium in which I or J drive which highlightsa difference between D -sets and maximal independent sets, since for any graph everyvertex is part of at least one maximal independent set. Second, and referencing theoriginal conjecture, the equilibrium with the fewest number of drivers occurs with κ equal to 3 for each individual, at which point individuals I and J both have degree8reater than their capacity. Third, the number of equilibrium outcomes decreased as κ increased from 3 to 4. Furthermore, while increasing capacity may lead to an increasein the number of equilibria, it may remove the most efficient equilibrium (that with thefewest drivers). Lastly, consider an amendment to the graph in Figure 3 such that i and i are friends. For κ = 1 there is an equilibrium with D -set D where 6 peopledrive, but when capacity is maximal the largest D -set is the maximal independent setof maximum size which has only 5 elements. These two equilibria can be Pareto rankedsince in the second case either i or i is no longer driving. Example 3.
The previous examples were special cases of the general model - thosewhere the action set is simply { , } (i.e., do not purchase / purchase, and do not drive/ drive). In the general model, the action set is the set of non-negative integers andthere is a most-desired quantity that we denote by q ∗ . By “most desired” we mean thatonce an individual’s total quantity received (his own action choice added to the actionchoices of those who nominated him) reaches q ∗ , he is satiated.In this richer model, there can be pure strategy equilibria where some individualschoose a strictly positive action choice that is less than q ∗ . We call equilibria in whichindividuals choose either action 0 or action q ∗ specialised . (In the first two examples q ∗ was equal to 1, so all pure strategy equilibria were specialised.) We will show via anexample that pure strategy equilibria that are not specialised may not be stable.We consider a social network with 6 agents denoted i, j, k, (cid:96), m , and n . Individual j is linked to everybody, individuals i and k are linked to j and one other, while allother individuals are linked to j and exactly two others. This is depicted in Panel A ofFigure 5 below.In addition to the larger action space, this example is more complex in that weallow capacities to differ across agents. Specifically, we suppose that the capacity for j is equal to 3 and the capacity for all other individuals is equal to 1. Lastly, we supposethat the optimal quantity of the good is q ∗ = 4.Panel B of Figure 5 presents a pure strategy that is not specialised, while PanelC presents a specialised equilibrium. Our focus is on the equilibrium in Panel B. Allagents choose positive quantities where the quantity is given by the number beside their Note that the size of the smallest D -set can not only change with capacity but can do so to anarbitrary extent. To see this suppose in the example above that I and J each have k friends. Thenfor κ = k −
1, the smallest D -set is the set of the peripheral agents with size 2 k −
2, for κ = k , thesmallest D -set is { I, J } , but for κ = k + 1 the smallest D -set has size k + 1. jklmn i jklmn i jklmn Figure 5: A 6 person networkname. Arrows depict nominations with the number of arrows originating at each vertexequal to that individual’s capacity. It is a pure strategy since, for every agent, thesum of their quantity choice and the in-flow of quantities from other individuals whonominate them is equal to 4 (= q ∗ ).Now we consider dynamics. We imagine that agent i unilaterally decreases his actionchoice from 3 to 2. We label the time, t , at which this happens by 0. We label theaction profile at this time by x (0) , where ordering the players as before we have that x (0) = ( x (0) i , x (0) j , x (0) k , x (0) (cid:96) , x (0) m , x (0) n ) = (2 , , , , , . Our interest is in the evolution ofthe sequence of action profiles (cid:8) x ( t ) (cid:9) t ≥ where for all t ≥
1, elements in the profile x ( t ) is the best-action response for the relevant agent to x ( t − . Importantly, we imaginethat the nominations of each agent are held fixed.It is also important to emphasise that when it comes to studying dynamics, themodeller must be aware of who each player nominates. The reason can be seen fromconsidering a specialised equilibrium. In the static case, it did not matter who thosethat made an action choice of 0 nominated. But in the dynamic environment, if onethose individuals deviates in their action choice, then this will have repercussions thatcan only be analysed if the nominations are known.In turns of updating their action choice, the behavioural rule is one of myopicbest-response (in action, not nomination as this is held fixed). Specifically, each agentexamines the total currently being supplied to him. If this total is less than 4, thenhe makes up the difference by increasing his own supply; if this total is more than 4then he decreases his own supply. Thus in period 1, the only individuals who will alter10heir action are i and j as each receive a total of 3 ( i provides 2 himself and receives1 from n who nominated him, while j provides 1 himself and receives 2 from agent i who nominated him). Since both are 1 unit short of q ∗ = 4, they will each increasetheir period 0 action by 1. We thus get that x (1) = ( x (1) i , x (1) j , x (1) k , x (1) (cid:96) , x (1) m , x (1) n ) =(3 , , , , , t x ( t ) i x ( t ) j x ( t ) k x ( t ) (cid:96) x ( t ) m x ( t ) n q ∗ is perturbed slightly, then this will always occur.11 The Model
We begin with the graph theoretic terminology required to describe the model.An undirected graph G = ( V, E ) consists of a nonempty finite set V = V ( G ) ofelements called vertices and a finite set E = E ( G ) of unordered pairs of distinct verticescalled edges . We call V ( G ) the vertex set of G and E ( G ) the edge set of G . In otherwords, an edge { i, j } is a 2-element subset of V ( G ). We will often denote an edge { i, j } by ij . For edge ij ∈ E ( G ) we say that i and j are the end-vertices , and say that end-vertices are adjacent . We say that vertex i is incident to edge e if it is an end-vertexof e . A graph G on n vertices is called complete if every two distinct vertices in G areadjacent; G will be denoted by K n . A path in a G is a finite sequence of edges which connect a sequence of distinctvertices. A graph is connected if there is at least one path containing each pair ofvertices. We define the neighbourhood of a vertex i , N G ( i ), in a graph G to be the setof vertices that vertex i is adjacent to, N G ( i ) = { j ∈ V : ij ∈ E } , and we say thatvertex j ∈ N G ( i ) is a neighbour of vertex i ; we write d G ( i ) for the cardinality of N G ( i ).For a connected graph with at least two vertices, the neighbourhood of every vertex isnonempty.With the above we can now introduce the game theoretic model. In the model,vertices are interpreted as players and edges represent connections between pairs ofplayers. We assume the population (vertex set) is of size n and that the graph G isconnected.Let X = { , , . . . , ¯ x } denote the finite set of actions common to each agent. Actionshave both a private and (local) public benefit, but only a private cost. Writing N forthe set of non-negative integers, there is a capacity function κ : V → N that specifies,for each player, how many of his neighbours may benefit from his action choice. Thissubset of neighbours are said to be nominated . If a player’s capacity is zero then henominates no neighbours; if a player’s capacity is at least as great as his degree then allof his neighbours are nominated. Formally, for any nonempty set A and nonnegativeinteger k , we denote by [ A ] k the collection of k -subsets of A . That is, [ A ] k = { A } when k ≥ | A | , [ A ] k = { S ⊆ A : | S | = k } when 0 < k < | A | , and [ A ] k = ∅ when k = 0.With this, player i ’s set of pure strategies is given by X × M ( κ ) i , with M ( κ ) i = [ N G ( i )] κ ( i ) representing the collection of subsets of N G ( i ) of size κ ( i ).12e write x i for individual i ’s action choice from X , and m i for his nominating choicefrom M i . (Note we omit the superscript ( κ ) of M ( κ ) i whenever no confusion arises.) Apure strategy profile is represented by a vector ( x , m ) = (cid:0) ( x , . . . , x n ) , ( m , . . . , m n ) (cid:1) specifying an action and a set of nominees for each agent. (We call x the action profile and m the nomination profile .) Given the above, the utility to player i , U i , from strategy profile ( x , m ) is given by U i (cid:0) x , m (cid:1) = f (cid:16) x i + (cid:88) { j ∈ N G ( i ) : i ∈ m j } x j (cid:17) − cx i (1)where we assume that (i) c >
0, and (ii) there exists a q ∗ ∈ X such that q ∗ ∈ argmax x ∈ X ( f ( x ) − cx ) and f ( x ) − cx is non-increasing for x ≥ q ∗ . Note that this definition implies that it makes no sense for a player i to increase x i if x i + (cid:80) { j ∈ N G ( i ) : i ∈ m j } = y ≥ q ∗ as f ( y + t ) − c ( y + t ) ≤ f ( y ) − cy and thus f ( y + t ) − ct ≤ f ( y ). Thus, each player i chooses an action x i ∈ X i and decides whichneighbours to share with via the choice m i ∈ M i . Note that player i ’s utility solelydepends upon his own action choice and the action choices of his neighbours in G whonominate him. Player i ’s utility does not depend upon who he himself nominates.The utility function as defined in (1) is very general. It does not require concavityof f as in that of Bramoull´e and Kranton (2007). If we take X = { , } , f ( x ) = 1for all x ≥
1, and 0 < c <
1, then the game becomes the “Netflix Game with κ -usersharing rule”.A pure strategy Nash equilibrium is defined in the usual way. Definition 1.
A strategy profile ( x ∗ , m ∗ ) is a pure strategy Nash equilibrium if forevery i = 1 , . . . , n , and every x i ∈ X i and every m i ∈ M i we have U i (( x ∗ , m ∗ )) ≥ U i (cid:0) ( x ∗ , . . . , x ∗ i − , x i , x ∗ i +1 , . . . , x ∗ n )( m ∗ , . . . , m ∗ i − , m i , m ∗ i +1 , . . . , m ∗ n ) (cid:1) . We have the following: While we have assumed that the graph G is exogenously given, the model permits an alternativeinterpretation. Specifically, suppose initially that there is no network so that nobody is connected.Under this interpretation each agent i is simply required to nominate any m i others to whom he wantsto link. Thus the nominations induce the formation of a social network. roposition 1. A pure strategy Nash equilibrium exists for any graph G and anycapacity function κ : V → N . While the above is a strong result, we will focus on what, following Bramoull´e andKranton (2007), we term specialised strategy profiles - those in which each agent eitherchoose action x i = 0 or x i = q ∗ . We have the following definition. Definition 2.
A specialised strategy profile is a pure strategy profile in which for all i ∈ V we have either x i = 0 or x i = q ∗ .We emphasise that specialised strategy profiles do not say anything about equilib-rium since for that we must also know who nominated who. For example, all individualschoose action 0 is specialised but clearly not an equilibrium. We begin building up tosuch a definition now.For a given specialised strategy profile ( x , m ), let D ( x , m ) = { i ∈ V : x i = q ∗ } bethose agents who supply q ∗ , and P ( x , m ) = { i ∈ V : x i = 0 } , where we interpret D and P as Drivers and Passengers respectively (Drivers provide while Passengers freeride). Clearly, at any specialised strategy profile, we have that both D ∩ P = ∅ and D ∪ P = V . However, we aim to go further. In words, we wish to find a strategyprofile such that nobody in D is nominated by somebody else in D , and everyone in P is nominated by at least on person in D .Some additional graph theoretic terminology is required. Given a graph G =( V ( G ) , E ( G )), we say that H = ( V ( H ) , E ( H )) is a subgraph of G if both V ( H ) ⊆ V ( G )and E ( H ) ⊆ E ( G ). If V ( H ) = V ( G ) we say that H is a spanning subgraph of G . Asubgraph H is said to be induced by a subset S of vertices of G = ( V, E ), if the vertexset of H is S and the edge set consists of all edges in E that have both end-verticesin S . If G = ( V, E ) is a graph and S ⊆ V ( G ), then G − S is the subgraph inducedby V ( G ) \ S . For a subgraph H of G , we define G − H as G − V ( H ). Similarly, foredges, if B ⊆ E ( G ), then G − B is the spanning subgraph of G with edge set E ( G ) − B .A bipartite graph is a graph whose vertices can be partitioned into two disjoint sets(called partite sets ) A and B such that every edge has one end-vertex in A and theother in B . A bipartite graph G with partite sets A and B is called complete bipartite if ab ∈ E ( G ) for every a ∈ A and b ∈ B . Then G is denoted by K | A | , | B | , where | A | and | B | are the cardinalities of sets A and B. A complete bipartite graph K ,p ( p ≥ star , the vertex adjacent to all other vertices the center , all other vertices leaves . For example the graph in Figure 1 is a star with center (cid:96) .Abstracting from the actions chosen by the individuals, we wish find a spanningbipartite subgraph, H , of G , with partite sets D and P such that there does not existan i ∈ D such that i ∈ m j for any j ∈ D , and for all i ∈ P there exists at least one j ∈ D such that i ∈ m j . This leads us to the following definition. Definition 3.
A spanning bipartite subgraph H of G with partite sets P and D iscalled a κ - DP -subgraph (or, simply a DP -subgraph if the function κ is fixed) if for each i ∈ D the degree of i in H is min { κ ( i ) , d G ( i ) } and for every i ∈ P the degree of i in H is positive.With this, we can define what it means for a strategy profile to be a balancedspecialised profile as follows. Definition 4.
A specialised profile ( x , m ) is a balanced specialised profile supportedby κ - DP -subgraph H of G if x i = q ∗ , m i = N H ( i ) if i ∈ Dx i = 0 , m i = S for some S ∈ M ( κ ) i if i ∈ P When it comes to adding dynamics, we will not only examine balanced specialisedprofiles, but we will also require that every vertex in P nominates at least one of thevertices from D who nominated him. This gives us the following stronger definition. Definition 5.
A specialised profile ( x , m ) is a nicely balanced specialised profile sup-ported by κ - DP -subgraph H of G if x i = q ∗ , m i = N H ( i ) if i ∈ Dx i = 0 , m i = S for some S ∈ M ( κ ) i such that S ∩ N H ( i ) (cid:54) = ∅ if i ∈ P We now have the following theorem, which is an improvement of Proposition 1.
Theorem 1.
For any graph G and common utility function as given by (1) , thereexists a nicely balanced specialised profile and all nicely balanced strategy profiles arepure strategy Nash equilibria. roof. The proof has two parts. The first is to show that every graph G possesses atleast one κ - DP -subgraph . The second is then to show that a nicely balanced strategyprofile induced by κ - DP -subgraph involves each agent choosing the optimal action.Here we prove the following statement: If G is a graph and κ : V ( G ) → N is afunction, then G has a DP -subgraph.The proof proceeds by induction on the number n of vertices of G. If n = 1, then G is a DP subgraph with D = V ( G ) and P = ∅ .Now assume the claim is true for all graphs with fewer than n ≥ G be a graph on n vertices. Case 1: There is a vertex i of degree at most κ ( i ) . Let B be the star with cen-ter i and leaves N G ( i ) , where N G ( i ) is the neighbourhood of i in G . Let G (cid:48) = G − B .Set D = { i } and P = N G ( i ) . If G (cid:48) has no vertices then B is clearly a DP -subgraphof G. Otherwise, by induction hypothesis, G (cid:48) has a DP -subgraph H (cid:48) with partite sets P (cid:48) and D (cid:48) . Construct a subgraph H of G from the disjoint union of H (cid:48) and B byadding to it for every j ∈ D (cid:48) with d G (cid:48) ( j ) < κ ( v ) , exactly min { d G ( j ) , κ ( j ) }− d G (cid:48) ( j )edges of G between j and N ( i ) . Set D = D (cid:48) ∪ { i } and P = P (cid:48) ∪ N ( i ) . To see that H is a DP -subgraph of G, observe that (a) H is a spanning bipartitesubgraph of G as H (cid:48) and B are bipartite and the added edges are between D and P only, (b) every vertex j ∈ D has degree in H equal to min { κ ( j ) , d G ( j ) } , (c)every vertex k ∈ P is of positive degree (since it is so in both B and H (cid:48) ). Case 2: For every vertex j ∈ V ( G ) , d G ( j ) > κ ( j ) . Let i be an arbitrary vertex.Delete d G ( i ) − κ ( i ) edges incident to i and denote the resulting graph by L .Observe that every DP -subgraph of L is a DP -subgraph of G since no vertex in L has degree less than κ ( i ) . This reduces Case 2 to Case 1.To complete the proof it remains to show that a nicely balanced strategy profileinduced by a κ - DP -subgraph is a Nash equilibrium. Let ( x ∗ , m ∗ ) be a nicely balancedspecialised profile. As mentioned before, we note that the utility function defined in(1) does not depend on m i . Thus player i cannot increase her payoff by deviatingfrom m ∗ i . If i ∈ D then by the definition of a nicely specialised profile induced by a κ -DP subgraph, we have x ∗ i = q ∗ and (cid:80) { j ∈ N G ( i ): i ∈ m ∗ j } x ∗ j = 0. Thus is follows from16he definition of the utility of i that for all x that are obtained from x ∗ by replacing x ∗ i = q ∗ by any t >
0, we have U i ( x ∗ , m ∗ ) = f ( q ∗ ) − cq ∗ ≥ f ( t ) − ct = U i ( x , m ∗ ) . If i ∈ P then x ∗ i = 0 and (cid:80) { j ∈ N G ( i ): i ∈ m ∗ j } x ∗ j = sq ∗ for some s >
1. Thus by theobservation just after the definition of the utility function, for all x that are obtainedfrom x ∗ by replacing x ∗ i = 0 by any t > U i ( x ∗ , m ∗ ) = f ( sq ∗ ) ≥ f ( sq ∗ + t ) − ct = U i ( x , m ∗ ) . We now make some observations about Theorem 1 and in particular κ - DP -subgraphsand their associated D -sets and P -sets.Fix a graph G = ( V, E ). An independent set is a subset of vertices no pair of whichare adjacent, and a maximal independent set is an independent set that is not a propersubset of any other. A dominating set is a set of vertices such that every vertex in V is either in the set or has a neighbour in the set, and a minimal dominating set is adominating set that does not contain a proper subset that is dominating. (Note thatthe notion of dominating is defined only for sets whereas our concept of nominatingis defined for individual vertices and by considering multiple nominating vertices canbe extended to sets.) We now relate D -sets to maximal independent sets and minimaldominating sets. For a further discussion of some of the properties of D -sets, refer toAppendix A.Clearly a D -set is a dominating set though the reverse need not hold. Indeed, ina complete graph K n every vertex forms a dominating set, but if κ ( i ) < d ( i ) for every i ∈ V ( K n ) , then no singleton can be a D -set. We will add further observations. First,as with dominating sets but not independent sets, it is possible that two vertices in D are adjacent in G . An instance of this was seen in the second example of Section 2 forthe equilibrium in which the only adopters two central vertices were the only adopters.Second, when κ ( i ) ≥ d G ( i ) for all i , we have that D is a maximal independent set(which is by definition dominating). Third, and related to the previous observation,is that unlike maximal independent sets, for a given graph G and capacity function κ ,17ne D -set may be a strict subset of another and if so it will be possible to Pareto-rankthe two corresponding specialised equilibria. As an example consider a complete graphwith 5 vertices i, j, k, l , and m with κ = 2 for each vertex. One such D -set is { i, j, k } with each nominating l and m , while another D -set is { i, j } with i nominating k and l and j nominating l and m . Finally we note that the procedure described in the proofof Theorem 1 allows us to find a D -set in polynomial time. Note, however, that such aprocedure may not find all D -sets of a given graph G. For example the D -set given inExample 2 consisting of I and J would never be found. In this section we focus on the efficiency of balanced specialised profiles. For a givengraph and a given capacity function κ, there are often multiple balanced strategy pro-files. We adopt the utilitarian social welfare criterion as our measure of efficiency, so ourattention is on those equilibria with the smallest and largest D -sets. We then considerhow incremental amendments to the model will affect the set of (in)efficient specialisedequilibria. The most natural way to do this is to incrementally increase the capacitiesof the agents.Recall that for a specialised strategy profile ( x ∗ , m ∗ ), D ( x , m ) denotes the set ofindividuals who adopt. We say that a balanced specialised profile ( x , m ) is utilitarianwelfare maximising / efficient if its associated D -set is of minimal size, and inefficientif it is of maximal size. , Clearly not every nicely balanced strategy profile is efficient.For a graph G and capacity function κ : V → N , let δ κ min ( G ) and δ κ max ( G ) denote theminimum and maximum sizes of D -sets of G .First, we will show that computing δ κ min ( G ) and δ κ max ( G ) are unfortunately NP-hardproblems. Indeed, if κ ( i ) ≥ d ( i ) for every vertex i in a graph G , then δ κ min ( G ) ( δ κ max ( G ),respectively) equals the minimum (maximum, respectively ) size of an independent setof vertices in G , whose computation is NP-hard (see, e.g., Garey and Johnson (1979))for both minimum and maximum. Note that this is a very strong notion of efficiency. An alternative criterion to employ would be
Pareto- efficiency where we say that ( x , m ) is Pareto-efficient if there does not exist another balancedspecialised profile ( x ∗∗ , m ∗∗ ) such that D ( x ∗∗ , m ∗∗ ) ⊂ D ( x ∗ , m ∗ ). Note further that Pareto efficiencyis not useful in refining the equilibrium set in either the best-shot game or the model of Bramoull´eand Kranton (2007) since by definition any two maximal independent sets of a graph do not have theset inclusion property.
18e wish to see how the sizes of such sets co-evolve as the player’s capacities areincreased.To this end, let κ (cid:48) : V ( G ) → N be a function such that κ ( i ) ≤ κ (cid:48) ( i ) forevery i ∈ V ( G ). We compare δ κ min ( G ) and δ κ max ( G ) with δ κ (cid:48) min ( G ) and δ κ (cid:48) max ( G ) . Theorem2 shows a particular inequality holds for every graph G . Unfortunately, none of theother three possible inequalities can hold as we have seen with some examples. Theorem 2.
For every graph G , δ κ (cid:48) min ( G ) ≤ δ κ max ( G ) .Proof. Let κ + : V ( G ) → N be function such that κ + ( i ) = κ ( i ) for i ∈ V ( G ) − j and κ + ( j ) = κ ( j ) + 1 for some j ∈ V ( G ) . To prove the theorem is sufficient to show that δ κ + min ( G ) ≤ δ κ max ( G ).We proceed by induction on n + m, where n is the number of vertices of G and m is the number of edges in G . If n + m = 1, then G consists of a single vertex andsetting D = V ( G ) and P = ∅ gives the only DP -subgraph for both κ and κ + . We mayassume that G is connected as otherwise we can consider its components and applythe induction hypothesis on the component containing j and the vertices in the othercomponents have the same values for κ and κ + . Let G = K ,n − , where n ≥ j isthe center of the star. If κ + ( j ) ≥ n −
1, then δ κ + min ( G ) = 1 and we are done. Otherwise, V ( G ) − j is a D -set for both κ and κ + .Now we may assume that n ≥ G is connected and there is an edge in G which isnot incident to j . Consider two cases. Case 1: There is a vertex i ∈ V ( G ) − j of degree at most κ ( i ) . Let B be the thestar with center i and leaves N G ( i ) , where N G ( i ) is the neighbourhood of i in G .Let G (cid:48) = G − B . Set D = { i } and P = N ( i ) . If G (cid:48) has no vertices then B isclearly a κ - DP -subgraph and κ + - DP -subgraph of G. Otherwise, by induction hypothesis, δ k + min ( G (cid:48) ) ≤ δ k max ( G (cid:48) ), where k is κ restricted to G (cid:48) . The two corresponding DP -subgraphs of G (cid:48) can be extended to those of G byadding i to their D -sets and N ( i ) to their P -sets and adding to every (cid:96) ∈ D with The graph in Example 2 showed that the smallest D -set can both decrease and increase as capacityincreases. We also saw that the largest D -set can decrease in the same example by adding an edgebetween any pair of Isabella’s friends. To see that the largest D -sets can increase in size consider thecomplete graph K on 7 vertices. Pick two 2 vertices i, j and let κ ( i ) = κ ( j ) = 2 and κ ( x ) = 6 for allother vertices x . Then all equilibria have D -sets of size 1 (and all vertices except i and j can form a D -set) and thus δ κ max ( G ) = 1. But if we increase κ ( i ) by one to obtain a new capacity function κ + then i, j form a D -set and δ κ + max ( G ) = 2. G (cid:48) ( (cid:96) ) < κ ( (cid:96) ) , and d G (cid:48) ( (cid:96) ) < κ + ( (cid:96) ) , respectively, exactly min { d G ( (cid:96) ) , κ ( (cid:96) ) } − d G (cid:48) ( (cid:96) )or exactly min { d G ( (cid:96) ) , κ + ( (cid:96) ) } − d G (cid:48) ( (cid:96) ) edges of G between (cid:96) and N ( i ) . Thus, δ κ + min ( G ) ≤ δ k + min ( G (cid:48) ) + 1 ≤ δ k max ( G ) + 1 ≤ δ κ max ( G ) . Case 2: The degree of every vertex i ∈ V ( G ) − j is larger than κ ( i ) . Choose anyedge i(cid:96) such that j (cid:54)∈ { i, (cid:96) } and delete it from G . By induction hypothesis, forthe resulting graph G (cid:48) we have δ k + min ( G (cid:48) ) ≤ δ k max ( G (cid:48) ) . It remains to observe thatthe two DP -subgraphs of G (cid:48) are also DP -subgraphs of G (the functions κ and κ + were not changed and the deleted edges are not needed). In this section we introduce dynamics with the goal of examining which strategy profilesare robust to unilateral deviations. Recalling that an agent’s utility is unaffected by hischoice of nomination, the number of best-responses for each agent may be enormous. Assuch, throughout we will assume the nominating profile is fixed and consider only thewhat action choices are optimal given the nominating profile and the action choices ofothers. We call this restricted best-response a best-action response . Given a nominationprofile m and the action profile x , it is not hard to see that the best-action responseof agent i , B i, m ( x ), is given by B i, m ( x ) = max q ∗ − (cid:88) { j ∈ N i ( G ) : i ∈ m j } x j , (2)We extend this to the best-action reply dynamic B m : X → X as B m ( x ) = (cid:16) B , m ( x ) , B , m ( x ) , . . . , B n, m ( x ) (cid:17) Definition 6.
Given an action profile x we define the best action evolution of x recur-sively by x (0) = x and for t ≥ x ( t ) = B m ( x t − ).We are interested in comparing population level contributions. As such we will wishto order action profiles wherever possible. For any two action profiles x , x (cid:48) ∈ X , we20ay x ≥ x (cid:48) if x i ≥ x (cid:48) i for all i ∈ { , . . . , n } , and x > x (cid:48) if x i ≥ y i for all i ∈ { , . . . , n } and x j > y j for at least one j ∈ { , . . . , n } .We then have the following results. Lemma 1.
Suppose that ( x ∗ , m ∗ ) is a pure strategy Nash equilibrium and let x ≤ x ∗ .Then the best action evolution of x satisfies for all t ≥ (i) x ( t +1) ≥ x ∗ ≥ x ( t ) if t is even. (ii) x ( t +1) ≤ x ∗ ≤ x ( t ) if t is odd.Proof. Let t = 0. Then by assumption x (0) ≤ x ∗ . If we choose i to be an individualwith x ∗ i = 0, then clearly we have x (1) i ≥ x ∗ i = 0. Now, let i be an individual with x ∗ i (cid:54) = 0. Then x ∗ i = q ∗ − (cid:88) { j ∈ N G ( i ): i ∈ m j } x ∗ j ≤ q ∗ − (cid:88) { j ∈ N G ( i ): i ∈ m j } x (0) j = x (1) i . Thus x (1) i ≥ x ∗ i and x (1) ≥ x ∗ . By considering individuals with x (2) (cid:54) = 0 separately, wecan use the same argument to shows that x (2) ≤ x ∗ . Thus we have x (2) ≤ x ∗ ≤ x (1) .The result then follows easily by induction.Lemma 1 implies that if we take a pure strategy Nash equilibrium action profile x ∗ ,for any x ≤ x ∗ the best action evolution of x either oscillate around x ∗ forever, or thereexists an t such that for all t ≥ t , x ( t ) = x ∗ . Note that this is the case as we consideronly finite networks and so there are only a finite number of different possibilities forthe x t and therefore they have to repeat. This motivates the following definition. Definition 7.
We say that the best action evolution of x settles in x ∗ if there existsan t such that for all t ≥ t , we have x ( t ) = x ∗ .We emphasise that Lemma 1 holds for all Nash equilibrium strategy profiles andnot simply specialised ones. While Lemma 1 is a statement about the populationaction profiles, the next two results, Lemmas 2 and 3 are statements about best-actionresponses at the level of the individual. We emphasise again that both results applyto all pure strategy Nash equilibria, not simply specialised ones. The first lemma is animmediate corollary of Lemma 1. 21 emma 2. Suppose that ( x ∗ , m ∗ ) is a pure strategy Nash equilibrium. Let x < x ∗ .Then the best action evolution of x satisfies the following: (i) If t is odd, then x ( t ) i (cid:54) = 0 , for all i such that x ∗ i > , and (ii) If t is even, then x ( t ) i (cid:54) = q ∗ , for all i such that x i < q ∗ . We will need the following lemma which essentially says that a state (cid:96) reacts to anychanges of states that affect it unless (cid:96) does not need to provide anything, that is x (cid:96) can remain zero and is still satisfied. Lemma 3.
Suppose that ( x ∗ , m ∗ ) is a pure strategy Nash equilibrium. Let x ≤ x ∗ bean action profile and consider the best action evolution of x . Let (cid:96) be nominated by i ,that is (cid:96) ∈ m i . If x ( t ) (cid:96) > and x ( t ) i > x ( t − i , then x ( t +1) (cid:96) < x ( t ) (cid:96) (3) If x ( t +1) (cid:96) > and x ( t ) i < x ( t − i , then x ( t +1) (cid:96) > x ( t ) (cid:96) (4) Proof.
We first show (3). So assume x ( t ) i > x ( t − i . Then, by Lemma 1, t must be odd.Now, again using Lemma 1, we have for (cid:96) ∈ m i (cid:88) { j ∈ N G ( (cid:96) ): (cid:96) ∈ m j } x ( t ) j − (cid:88) { j ∈ N G ( (cid:96) ): (cid:96) ∈ m j } x ( t − j = x ( t ) i − x ( t − i + (cid:88) { j ∈ N G ( (cid:96) ) : (cid:96) ∈ m j , j (cid:54) = i } x ( t ) j − (cid:88) { j ∈ N G ( (cid:96) ) : (cid:96) ∈ m j , j (cid:54) = i } x ( t − j ≥ x ( t ) i − x ( t − i > q ∗ to both sides of this inequality gives that q ∗ − (cid:88) { j ∈ N G ( (cid:96) ): (cid:96) ∈ m j } x ( t ) j < q ∗ − (cid:88) { j ∈ N G ( (cid:96) ): (cid:96) ∈ m j } x ( t − j . But note that by the best-action response (2) and because we assume that x ( t ) (cid:96) > x ( t ) (cid:96) . Also x ( t +1) (cid:96) equals either the left hand side of the aboveinequality or is equal to zero and in both cases is smaller than x ( t ) (cid:96) . The inequality in(4) is shown in a similar manner. 22e now introduce our notion of stability when the nominating profile is held fixedand actions are updated according to the best-action reply dynamic. In words, we saythat action profile x is stable relative to the nomination profile m , if, when the actionof any individual is changed by some strictly positive amount, repeated application ofthe best-action reply dynamic will lead population behaviour back to action profile x . Definition 8.
We say that strategy profile ( x , m ) is action stable if there exists δ ≥ i = 1 , . . . , n and any action profile x (cid:48) with x j = x (cid:48) j for j (cid:54) = i and | x (cid:48) i − x i | ≤ δ the best action evolution of x (cid:48) settles in x .In other words a strategy profile is action stable if one can change any one coordinateby at most δ and the best action evolution will settle in x . The following result showsthat any pure strategy Nash equilibrium profile that is not balanced specialised is notaction stable and thus, a strategy profile being balanced and specialised, supported bya κ - DP -subgraph, is necessary for action stability. Proposition 2.
Suppose that ( x ∗ , m ∗ ) is a pure strategy Nash equilibrium such that < x ∗ (cid:96) < q ∗ for some (cid:96) . Then ( x ∗ , m ∗ ) is not action stable.Proof. Define L := (cid:8) j : 0 < x ∗ j < q ∗ (cid:9) . By assumption (cid:96) ∈ L . First we will show that (cid:96) cannot be the only element in the set L . This is immediate since0 < x ∗ (cid:96) = q ∗ − (cid:88) { j ∈ N G ( (cid:96) ) : (cid:96) ∈ m ∗ j } x ∗ j < q ∗ = ⇒ < (cid:88) { j ∈ N G ( (cid:96) ) : (cid:96) ∈ m ∗ j } x ∗ j < q ∗ . Thus, there exists some j ∈ L such that j ∈ N G ( (cid:96) ), (cid:96) ∈ m ∗ j . It follows that there existindices i , i , . . . , i k such that i ∈ m ∗ i , i ∈ m ∗ i , . . . , i k − ∈ m ∗ i k and i k ∈ m ∗ i . (For thereaders familiar with graph theory we can build a directed graph on L with an arc from j to i if i ∈ m ∗ j . The above result implies that every vertex has at least one in-neighbourand hence the graph contains a cycle.) By renaming if necessary we may assume that i j = j for all j = 1 , . . . , k , so for i = 1 , . . . , k we have i ∈ L , i ∈ m ∗ i +1 and k ∈ m ∗ . Wedefine x by subtracting 1 from the first coordinate of x ∗ , that is, x = ( x ∗ − , x ∗ . . . , x ∗ n ) . We claim that x does not settle in x ∗ . We prove the claim by contradiction. Inparticular we assume that there exists a (minimal) t such that for all t ≥ t and for all23 = 1 , . . . , k , we have x ( t ) i = x ( t +1) i . Note that t ≥ x (0)1 = x ∗ − (cid:54) = x ∗ = x (1)1 . Alsofor all i = 1 , . . . , k , we have x ( t ) i = x ( t +1) i = x ∗ i >
0. The contradiction now followsfrom Lemma 3 by choosing i ∈ { , . . . , k } such that x t − i (cid:54) = x t i and (cid:96) = i + 1. Such an i exists as we have chosen t minimal.While Proposition 2 says that balanced specialised profiles are necessary for sta-bility, the following result, Proposition 3 shows that they are not sufficient. However,Proposition 4 shows that balanced specialised profiles supported by a κ - DP -subgraphswith a mild density condition are sufficient for stability. Proposition 3.
Suppose ( x ∗ , m ∗ ) is a nicely balanced specialised profile induced by κ - DP -subgraph H of G , and suppose further that d H ( i ) = 1 for some i ∈ P . Then ( x ∗ , m ∗ ) is not stable.Proof. Without loss of generality we may assume that 1 ∈ P , d H (1) = 1 and that 2 isthe neighbour of 1 in H . Note that x ∗ = q ∗ as the strategy profile is nicely balancedand specialised. Consider x = ( x ∗ , x ∗ − , . . . , x ∗ n ) = ( x ∗ , q ∗ − , . . . , x ∗ n ) . We claim that x does not settle in x ∗ . To see this we prove by induction that forall odd t we have x ( t )1 > t we have x t < q ∗ , Clearly this is true for t = 0 and also for t = 1 as the best action response for 1 is x (1)1 = 1 >
0. Now assumethat the statement is true for all t ≤ t with t ≥ t is even then by induction hypothesis x t < q ∗ and by Lemma 1 for all j ∈ P we have x ( t ) j ≤ x ∗ j = 0 and thus x ( t ) j = 0. Since 1 is only nominated by 2 and perhapselements in P , the best action response for 1 is x ( t +1)1 = q ∗ − x ( t )2 > t is odd then by induction hypothesis x t > j ∈ D wehave x ( t ) j ≥ x ∗ j = q ∗ and thus x ( t )2 = q ∗ . Since 2 is nominated by 1 the best actionresponse for 2 is x ( t +1)2 ≤ q ∗ − x ( t )1 < q ∗ .Thus, if there exists a nicely balanced specialised profile with an individual in P who is nominated by only one agent in D , then the equilibrium is not stable. However,the following proposition shows that if all individuals in P are nominated by at least 2individuals in D , then the equilibrium is action-stable.24 roposition 4. Let q ∗ ≥ . Suppose ( x ∗ , m ∗ ) is a nicely balanced specialised profileinduced by κ - DP -subgraph H of G , and suppose further that d H ( i ) = 2 for all i ∈ P .Then ( x ∗ , m ∗ ) is stable.Proof. Choose ε = 1. We must consider deviations by those in D and those in P . Webegin with those in D . Case 1: i ∈ D. By definition x ∗ i = q ∗ . There are two subcases: Either we increase x ∗ j by one or we decrease x ∗ j by one. In the first case, i.e. x satisfies x (0) i = x ∗ i + 1 = q ∗ + 1 and for all j (cid:54) = i , x (0) j = x ∗ j , it is immediate that B m ∗ ( x ) = x ∗ .In the second case let x satisfy x (0) i = x ∗ i − q ∗ − j (cid:54) = i , x (0) j = x ∗ j .By our assumption for all j ∈ P there is at least one neighbour k (cid:54) = i , k ∈ D with j ∈ m ∗ k and thus max (cid:110) q ∗ − (cid:80) { k ∈ N j ( G ) : j ∈ m k } x k , (cid:111) = 0. It follows that x (1) j = 0 for all j ∈ P . For all j ∈ D we have x (1) j = q ∗ by Lemma 1 (or one canjust observe only vertices in k ∈ P are nominating j and all these vertices satisfy x (0) k = 0). Case 2: i ∈ P. By definition x ∗ i = 0, so the only deviation is to choose x (0) i = 1 andfor all j (cid:54) = i , x (0) j = x ∗ j . Then B (cid:96), m ∗ ( x (0) ) = , (cid:96) ∈ P,q ∗ − > , (cid:96) ∈ N H ( i ) ∩ D,q ∗ , (cid:96) ∈ D \ N H ( i )Consider an individual h ∈ P with h (cid:54) = i , and consider how all of H ’s neighboursin the nominating network will behave in period 1. We have (cid:88) { j ∈ N G ( h ) : h ∈ m j } x (1) j = (cid:88) { j ∈ N G ( h ) : h ∈ m j } B j, m ∗ ( x (0) ) ≥ d H ( h )( q ∗ − ≥ q ∗ − ≥ q ∗ where the last inequality follows since q ∗ ≥
2. And thus, B h, m ∗ ( x (0) ) = 0.Furthermore, Since H is a κ - DP -subgraph, we have that N H ( i ) ⊆ P for all i ∈ D ,25nd thus the set { j ∈ N G ( i ) : i ∈ m j } ⊆ P for all i in D . Thus, for all (cid:96) ∈ D , wemust have that B (cid:96), m ∗ ( x (0) ) = q ∗ since B i, m ∗ ( x (0) ) = 0 for all i ∈ P .Therefore, ( x ∗ , m ∗ ) is stable. This paper introduces a network model of public goods in which individuals face acapacity constraint as to how many neighbours they may share with. Each agentmuch decide how much of the good to provide and which subset of his neighbours toshare with. We focus on a subclass of pure strategy equilibria, that we term specialisedequilibria, wherein each agent provides either the most desirable quantity (is a ‘Driver’)or provides nothing (is a ‘Passenger’). Interestingly, as capacities are increased thenumber of Drivers across specialised equilibria need not decrease.The model has a second interpretation wherein the nomination component is viewedas pure network formation, with public good provision then occurring via the realisednetwork. Our model is thus one of the first to have a network formation componentcoupled with a rich underlying game. While we introduce dynamics later in the paper,we do so with the nomination component held fixed. Relaxing this would yield thefirst model (to our knowledge) that captures the coevolution of behaviour and networkformation. This is left to future research. 26
PPENDIXA Properties of D -sets First we show some upper and lower bounds on the sizes of D -sets for all graphs. Subsequently we lookat specific classes of graphs that are commonly studied. The following definitions are required. For a G = ( V, E ) be a graph and capacity function κ : V → N , we define κ = min i κ ( i ), ¯ κ = max i κ ( i ), and d G := min i d G ( i ). Furthermore, letting R denote the real line, we define the ceiling and floor functionsas follows: for any x ∈ R , let (cid:100) x (cid:101) := min { n ∈ N | x ≤ n } and let (cid:98) x (cid:99) := max { n ∈ N | x ≥ n } . A.1 Bounds on D -sets Lemma 4.
Let G = ( V, E ) be a graph and κ : V → N be a capacity function. Then for any κ - DP -subgraph H of G we have, (i) | D ∪ P | ≤ (1 + ¯ κ ) | D | (ii) | D ∪ P | ≥ | D | + min { κ, d G } Proof. (i) We have | V | = | D ∪ P | = | D | + | P |≤ | D | + (cid:88) i ∈ D min { κ ( i ) , d G ( i ) }≤ | D | + (cid:88) i ∈ D κ ( i ) ≤ | D | + ¯ κ | D | = (1 + ¯ κ ) | D | (ii) For any i ∈ D we have that N H ( i ) ⊆ P . As such | V | = | D ∪ P |≥ | D | + N H ( i )= | D | + min { κ ( i ) , N G ( i ) }≥ | D | + min { κ, d G } Recall that for a given graph G and given capacity function κ , δ κ min ( G ) and δ κ max ( G ) represent thesize of minimum and maximum D -sets taken over all κ - DP -subgraphs of G . We have the followinglemma. Lemma 5.
Let G = ( V, E ) be a graph and κ : V → N be a capacity function. Then we have(i) δ κ min ( G ) ≥ | V | κ .(ii) δ κ max ( G ) ≤ | V | − min { κ, d G } .Proof. (i) Suppose to a contradiction that δ κ min ( G ) < | G | κ . Then there exists a κ - DP subgraph of G such that | V | < | G | κ = | D ∪ P | κ which is contradiction to part (i) of Lemma 4. ii) Suppose, again to a contradiction, that δ k max ( G ) > | V | − min { κ, d G } . Then there exists a κ - DP -subgraph H of G with D -set D such that | D | > | V | − min { κ, d G } = | D ∪ P | − min { κ, d G } whichis a contradiction to part (ii) of Lemma 4. Remark . For Lemma 5, we need | G | − min { κ, d G } ≥ | G | κ ⇐⇒ (1 + ¯ κ )( | G | − min { κ, d G } ) ≥ | G | To show this, first observe that¯ κ + ¯ κ ( | V | − ≥ min { κ, d V } + ¯ κ min { κ, d G } (5)because ¯ κ ≥ κ and | V | − ≥ d G ( i ) for all i . By rearranging (5), we find that This follows because¯ κ | V | − (1 + ¯ κ ) min { κ, d G } ≥ ⇐⇒ (1 + ¯ κ )( | V | − min { κ, d G } ) ≥ | V | (cid:4) Now, putting Lemmas 4 and 5 together, we obtain the following proposition.
Proposition 5.
Suppose that κ ( i ) ≤ d G for some i . Then we have (cid:108) | V | κ (cid:109) ≤ δ κ min ( G ) ≤ δ κ max ( G ) ≤ | V | − κ Proof.
The middle inequality is true by definition, while the left and right inequalities come fromLemma 5 parts (i) and (ii) respectively.Proposition 5 holds for all graphs. However, we can improve upon those bounds for various classesof graph. This is the purpose of the following subsection.
A.2 D -Sets for complete graphs In this subsection we suppose that G = ( V, E ) is a complete graph and that all individuals have thesame capacity so that 1 ≤ κ ( i ) = k ≤ | V | − i ∈ V .Given this we have d G = | V | − i { κ ( i ) , d G } = min { κ, d G } = k for all i ∈ V . Thusfrom Proposition 5 we obtain the following bounds, (cid:108) | V | k (cid:109) ≤ δ κ min ( G ) ≤ δ κ max ( G ) ≤ | V | − k (6)In particular, if k = | V | −
1, then δ κ min ( G ) = δ κ max ( G ) = 1We can show that there exist k - DP -subgraphs that achieve the lower and upper bounds, (cid:108) | V | k (cid:109) and | V | − k . o achieve the lower bound, choose (cid:108) | V | k (cid:109) individuals from G and designate these individuals as D . Now define let R = V \ D . Then, using that | V | − k ≥ (cid:108) | V | k (cid:109) ⇐⇒ | V | − (cid:108) | V | k (cid:109) ≥ k, and (cid:108) | V | k (cid:109) ≥ k ≤ | V | −
1) and | V | k ≤ (cid:108) | V | k (cid:109) ⇐⇒ | V | − (cid:108) | G | k (cid:109) ≤ (cid:108) | G | k (cid:109) k we obtain k ≤ | R | ≤ (cid:108) | V | k (cid:109) k. That is, there are least k agents in R and at most (cid:100) | V | k (cid:101) k . Thus, since G is a complete network, wemake each agent in D nominate k agents in R such that D ∪ R = G and obtain κ - DP -subgraph of G .For the upper bound, we choose | V | − k individuals as elements in D . Then there exists exactly k remaining individuals that we refer to as P . Since G is a complete graph, we make it such that allindividuals in D nominate each of the k remaining agents in P and obtain κ - DP -subgraph of G . Thuswe obtain the following proposition. Proposition 6.
Suppose that G = ( V, E ) is a complete graph and ≤ κ ( i ) = k ≤ | G | − for all i .Then we have (cid:108) | V | k (cid:109) ≤ δ κ min ( G ) ≤ δ κmax ( G ) ≤ | G | − k. (7) Moreover there exist bipartite k - DP -subgraphs H and ¯ H with partite sets ( D, P ) and ( ¯ D, ¯ P ) such that | D | = (cid:108) | V | k (cid:109) and | ¯ D | = | V | − k
10 20 30 40 50 Figure 6: The lower and upper bounds for the complete graph with | V | = 50. Horizontalaxis: k = 1 , · · · ,
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