Our model is driven by externalities over the network: the idea that the formation of a link has effects well beyond the nodes or players it connects—not to be confused with network externalities or network effects describing the phenomenon that the value of a product depends on the number of people owning it.
Externalities are often characterised by the sign of the external effects. As such, we may talk about positive externalities when the formation of a link is generally of positive value to others, or about negative externalities when the new link harms others. We will illustrate both cases with a simple story. In the first, second-degree neighbours are of importance and so friendships our friends make are beneficial to us, too. In the second, connections our contacts make increase the risk of an infection, thereby harming us.
Favour network
We consider an even simpler example of a network, where link formation creates externalities and a wider cooperation can result in more efficient outcomes.
We take the example of a favour network consisting of individuals who maintain friendships at some cost c. For simplicity we assume that a friendship is mutual, but the costs of maintaining the friendship are not necessarily shared equally: in the usual TU fashion we envisage a complex system of transfers of who buys which drink to maintain the network. Having many friends is great, but now we are interested in friends’ friends. When a friend’s friend is hiring and we want to apply for the job, the friend can put in a good word for us. The same would not work if we would make direct contact as praising ourselves is not so credible. Similarly, more distant relations may have too little information about us. In sum, the benefit of the network is the number of secondary friends a player has. Examples of such a network include the referral network studied by Stupnytska and Zaharieva (2017). In the following we formalise this rule, define the payoff function and determine the emerging equilibrium networks.
Let \(N_i=\left\{ j\vert j\in N, \exists {\overline{ij}}\in \ell \right\} \) denote the neighbours of node i and let \(N_i^2=\bigcup _{j\in N_i}N_j{\setminus }\left\{ i\right\} \) denote the secondary connections of i. Let \(d_i=\left| N_i\right| \) denote the degree of node i. Similarly, let \(d^2_i=\left| N^2_i\right| \) denote the secondary degree of node i. Note that the payoff does not depend on the coalition structure. The payoff of coalition C embedded in partition \({\mathcal {P}}\) given the network \(\ell \) is
$$\begin{aligned} V_C(\ell )=\sum _{i\in C}\left( d^2_i-\frac{d_ic}{2}\right) . \end{aligned}$$
(5.1)
Note the absence of outside nodes.
We would like to find the core of this game.
Proposition 4
The core of the favour network game may only contain efficient networks.
Proof
Since the payoff function does not depend on the partition of the players, the game is cohesive, that is, the grand coalition can achieve any configuration. If the network is not efficient, a deviation by the grand coalition can strictly improve it and can strictly increase the payoff of each of the players. \(\square \)
In the following we determine the efficient network. We discuss three main cases:
Tree If the underlying network \(\ell \) is a tree we show that it must be a star. Assume that this is not the case and that \(i\in \arg \max _j d_j\). Moreover let k be a leaf not connected to i, but to j. Since k is a leaf, \(d_k^2=d_j-1\). Now modify \(\ell \) such that the link between j and k is moved to i and k. Since k is still a leaf we get \(d_k^{'2}=d'_i-1=d_i>d_j-1=d_k^2\), where the \(d'\)-s refer to values in the modified network. Since only these and the reciprocal indirect connections are affected, the net gain is positive. Therefore all nodes must be connected to the node with the highest degree resulting in a star.
Graph with triangles Now consider the case when the network is not a tree. Firstly assume that the underlying graph \(\ell \) contains triangles. Consider a triangle \(T=\left\{ f,g,h\right\} \), such that, without loss of generality \(h\in \arg \max _{i\in T}d_i\). We will show that the value of the network increases if we move the links (except from those from f and h) pointing to g to h instead. To be more precise: if there are such i that are not connected to h then the value of the network can be increased. We discuss 5 cases (Fig. 11).
Case 1: No complications Consider a node i such that \({\overline{ig}}\in \ell \). Moving this link from g to h the number of secondary contacts in T remain the same: previously f and h, now f and g. On the other hand if we move the similar links for all \(i\in N_g{\setminus } T\), the former secondary contacts via g remain secondary contacts via h. Those in \(N_h{\setminus } T\) are new secondary contacts, while the number of links has not increased. If we have players outside T, the gain is strictly positive.
Case 2: Already connected If some of these i nodes are already connected to h there is no benefit to moving the links to h: while double links are permitted by our formalism, in this example they bring no benefits. In this case no links are shifted. If all such i nodes are connected to h then the value cannot be increased, at least this way.
Case 3: g is already a secondary neighbour When g is already a secondary neighbour moving the link to h loses h as a secondary neighbour, but to no gain, as g is already in \(N^2_i\). How is this possible? There exists j with \({\overline{ij}}, {\overline{gj}}\in \ell \). But then following case 1 we move both links: as a result we do lose both g and h as secondary links, but get them back both. At the same time the benefits of \(N_h{\setminus } T\) as new connections still apply.
Case 4: j is already connected to h If j is already connected to h we cannot move both links, but, like in Case 2, the link is already there and so, if we wish, the roles of the links \({\overline{gj}}\) and \({\overline{hj}}\) can be switched.
Case 5: \(j=f\) It is perhaps useful to specially mention the case when \(j=f\). Actually, this case is no different from the rest. Of course, f is connected to both g and h, so we really have a special case of Case 4.
In a similar fashion we can move links to f to h, too. As a result triangles are connected to the rest of the network via one of their vertices only.
Larger cycles Now we show that larger cycles cannot be part of an efficient network. For the moment assume that there are larger cycles, too. Due to the previous result, the cycle may only share vertices and not arcs with triangles. Consider the smallest cycle of length at least 4, C—this has at least four nodes: let h, i, j and k nodes following each other on the cycle and let \(h\in \arg \max _{m\in C} d_m\) be one of the points with the highest degree in the cycle. By the result that \({\overline{hi}}\) is not part of a triangle, \(N_h\) and \(N_i\) are disjoint. Then consider the following modification to the network: move the arc linking j and k to link j and h. After the change j has \(d_h+d_i\) secondary neighbours, while before the change at mostFootnote 1\(d_k+d_i<d_h+d_i\). Therefore if the graph has larger cycles, it can be made more efficient by creating a triangle and thereby breaking the cycle. A repetition of this step eliminates all cycles of length 4 or more.
After the elimination of large cycles, and following the recommended improvements, we get a graph, which looks a bit like a tree, but with some triangles attached to some vertices. Thanks to this similarity, we can improve this graph similarly to the improvement applied for trees:
Select \(i\in \arg \max _j d_j\). With more than 2 players and a connected graph we either do not have triangles or \(d_i>2\) in which case i cannot be one of the non-connecting vertices (the f’s and the g’s) of a triangle. Let f and g such non-connecting vertices of a triangle \(T=\left\{ f,g,h\right\} \). Now modify the graph so that \({\overline{fh}}\) and \({\overline{gh}}\) are moved to \({\overline{fi}}\), \({\overline{gi}}\). As before, by moving to a node with a higher degree, both f and g have more secondary connections. While the direct connections \(N_h{\setminus }\left\{ f,g\right\} \) of h lose them, those in \(N_i\) gain them and by assumption \(d_i\ge d_h\).
Once we are done with the triangles, we have a node with many triangles attached to it, but for the rest, the graph is just like a tree. So let k be a leaf not connected to i, but to j. Since k is a leaf, \(d_k^2=d_j-1\). Now modify \(\ell \) such that the link between j and k is moved to i and k. Since i is still a leaf we get \(d_k^{'2}=d'_i-1=d_i>d_j-1=d_k^2\), where the \(d'\)-s refer to values in the modified network. Since no other indirect connections are affected, the net gain is positive. Therefore all nodes must be connected to the node with the highest degree.
So far we have only looked at improvements that did not affect the number of direct connections, we merely rearranged them to have a more efficient structure. As a result we have a player at the centre and all other \(n-1\) players are linked to it. Some of these outer players f, g are directly connected. Such connections are never needed to have each other as secondary connections as this works via the central player. Outer links are used to have the central player as a secondary connection. The added value of such a link is therefore \(4-c\) if neither f nor g is connected to other non-central players, the value is \(2-c\) if one of them is connected and \(-c\) if both. For high c these links are severed, for low values of c non-central players link up in pairs, and if n is even (so that \(n-1\) is odd) the remaining non-central player links to another only if \(c<2\). The links to the centre only break if \(c>2(n-2)\) (Fig. 12).
Therefore if \(c>2(n-2)\) we get an empty network, for \(2(n-2)> c> 4\) we get a star, for \(4> c\) we get a flower: if n is even, for \(c>2\) it has a stem, otherwise a double petal.
Proposition 5
The core of the favour network game is empty.
Proof
The next question is stability. Consider a deviation by a single player forming a singleton coalition. If this player forms or keeps no links, it has a zero payoff. Let us see if it can have a higher payoff. Suppose it keeps a link with its highest-degree neighbour. Since the residual game will be similar to the original one, the players form a star or a flower. If so, it is always better to form it “around” the player with the external link. Thereby the deviating player becomes a peripheral player in a star with a payoff \(n-2-\frac{c}{2}\). The total value of a star is \((n-1)(n-2)-(n-1)c\). Since the star is formed by n players, there is a player with a payoff of at most \(\frac{(n-1)(n-2)-(n-1)c}{n}\), therefore the deviation is profitable if
$$\begin{aligned} n-2-\frac{c}{2}>\frac{(n-1)(n-2)-(n-1)c}{n} \end{aligned}$$
This is satisfied when \(c>2(n-2)\), but we have not tested the stability of the residual core. If it empty, the deviating player must expect the worst of all possible reactions, including the one where links to it are broken and therefore his payoff is 0. To check this, consider a more general case with k players deviating. It is easy to see that these players will all be peripheral players who do not want to change the underlying network, only the distribution of the payoffs, so that all these players will keep their links to the central player and then efficient and therefore only possible reaction in the residual game is a star around that player. The question is: will this player keep the links to the deviated players. What causes the problems? While the total value of the network does not change, the central player, by maintaining the external links, subsidizes the deviated players more and more. As the number of departed players increases the residual players’ benefit per link to the deviating players decreases, while the associated costs remain the same. The links remain profitable only if
$$\begin{aligned} n-k-1>\frac{c}{2} \end{aligned}$$
where \(0<k<n\). For some k this will be violated and then the deviations are not profitable any more. Consider a deviation by \(k-1\) peripheral players: the residual core is nonempty and the deviation will be profitable. Therefore the recursive core of this game is empty. \(\square \)
Note that this finding is driven by the fact that the central player must sacrifice himself to the benefit of others: Normally others compensate him for this, but selfish players may deviate and stop such transfers. In reality such a central player has a very strong position and gets rewarded for the favours he can provide. In the following example we make these rewards explicit by assuming that, upon forming a link between players i and j, player i must pay a transfer to j that is proportional to \(d_j-1\). As a result, a central player gets a high transfer, while a leaf gets nothing. Then the payoff of coalition C embedded in partition \({\mathcal {P}}\) given the network \(\ell \) is
$$\begin{aligned} V_C(\ell )=\sum _{i\in C}\left( d^2_i-\frac{d_ic}{2}+\left( d_i(d_i-1)-\sum _{j\in N_i}(d_j-1)\right) t\right) , \end{aligned}$$
(5.2)
where \(t<1\) is the compulsory transfer for using an intermediary.
Proposition 6
The core of the modified favour game is not empty if t is sufficiently high \(t>\frac{c+2}{2n}\).
Proof
Firstly observe that the modification merely introduces transfers among players, so that the value of the grand coalition does not change. In particular, the efficient structures remain the same. We may therefore focus on the issue of stability. We limit our attention to star structures; the case when \(c<4\) is similar.
Consider a star, and consider a deviation by k peripheral players. What happens in the residual game? The former central player has already k connections to the deviating players. Due to our assumptions that no new links may form between coalitions, no other player can have external links. By linking to this player the remaining \(n-k-1\) players do not only get a very high payoff, but they also increase the value of this central player’s services to the deviating players. Formerly this was positive externality they could not benefit from, but now the deviators must pay a fee for it. So if the residual core is not empty, it keeps the pre-deviation structure. Is this core non-empty? To see this, first compare the payoffs of players in different positions (without the possible transfers within the coalition). We will show that the central player earns more. To see this, observe the following: What a player earns only depends on the network. The network has not changed due to the deviations. At last: the network, and the payoffs (recall we ignore transfers) are symmetric among the peripheral players. Therefore if we show that the average payoff is higher than the peripheral players’ payoff this shows the result.
A player on the periphery has a value \(n-2-(n-2)t-\frac{c}{2}\), while an average player has \(\frac{(n-1)(n-2)-(n-1)c}{n}\). We want to show
$$\begin{aligned} n-2-(n-2)t-\frac{c}{2}< & {} \frac{(n-1)(n-2)-(n-1)c}{n} \end{aligned}$$
(5.3)
$$\begin{aligned} \frac{c+2}{2n}< & {} t \end{aligned}$$
(5.4)
That is, if t is sufficiently large, the central player earns more. In such a case the central player has no incentives to deviate and become a peripheral player, while a player can only become central by cooperation with all other players. This holds both in the original game and in the residual game, since the underlying networks are the same.
For lower c values we must also check the incentives to keep peripheral links but these will be kept for the very same reason they were created in the efficient network. \(\square \)
Contagion network with social preferences
Our next application is motivated by the recent Covid-19 epidemic and is a clear example of negative externalities. While various mathematical models have been introduced to study the optimal response to an epidemic (Parvin et al. 2012; Sharomi and Malik 2017), our model takes the citizens’ perspective. The network models social contacts of individuals, where the tradeoff is between the benefit of having friends and the risk of getting infected. It is assumed that an individual may get infected by an involuntary contact and may spread the infection to others in his or her social network.
This problem is rather different from other instances of bad networks. It is not really related to the literature on dark or covert networks (Milward and Raab 2006; Husslage et al. 2013), where the nodes of the bad network are aware of the fact that their contacts are bad and are contributing to the maintenance of this network. In our model the network is a positive message but carries the risk of spreading the disease. Even if only a small fraction of the population is infected, network results, known as the small world phenomenon suggest that the social distance from infected people may be smaller than what one would think (Vieira et al. 2010). Put it differently: nodes do not know if they or their neighbours are infected and the threat is near.
We use the notation of the previous subsection by letting \(d_i\) denote the degree of node i. In addition we use \(d^{\infty }_i\) to denote the number of nodes connected to i: in other words the size of i’s component.
For a coalition C the payoff is
$$\begin{aligned} V_C(\ell )=\sum _{i\in C}\left( s_i(d_i)-d^{\infty }_i\right) , \end{aligned}$$
(5.5)
where \(s_i\) is a concave, weakly increasing function that we call player i’s sociability function. The sociability function expresses the benefit from being social, keeping in touch with people. We assume that the function is increasing (the more, the merrier). At this point we are also very pragmatic and assume that it is only the number of contacts that matters.
Expression \(V_C(\ell )\) models the payoff from a long-term strategy of keeping in touch with a chosen set of other players (and avoiding contact with everyone else). At the same time, people who contact the epidemic via involuntary encounters will eventually spread the infection to everyone in the same network component.
It is clear from looking at an individual’s payoff (coalition \(\left\{ i\right\} \)) that a typical player is either extremely social, wanting to keep in touch with everyone or has a bound on the optimal number of connections where the additional risk exceeds the benefit of seeing one more person. We will ignore extremely social people or just assume that their bound is the number of players.
In the following we present some simple results regarding this problem. The first result is almost trivial.
Lemma 7
An core outcome has a network with fully connected components.
Proof
It is clear that each player benefits from having more contacts while keeping the risk of infection at the same level. The network with fully connected components can actually be obtained by a deviation by N partitioned into coalitions corresponding to the components. The coalitions will continue to have no external connections but will build all internal links. No player is harmed by the change so the deviation is profitable. \(\square \)
Homogeneous players
First we look at the case where \(s_i=s_j\) for all \(i,j\in N\). In this case all players have the same bliss point. Let k denote this number of ideal neighbours, where the marginal benefit of an additional neighbour is less than the cost of additional risk.
Consider a deviation by some coalition S forming partition \(\mathcal {S}\). Then each coalition \(C\in \mathcal {S}\) with \(\left| C\right| >k\) will severe all external links. To see this, we construct a network that gives a higher payoff for the coalition. If there is a player i with more than k neighbours including x outside C, then severing \({\overline{ix}}\) increases the coalitional payoff. Now consider player j with less than k neighbours, including y in \(N{\setminus } C\). Since j has less than k neighbours, there is also an \(h\in C\), such that \({\overline{jh}}\not \in \ell \). Then the network, where \({\overline{jy}}\) is severed and \({\overline{jh}}\) is created gives a higher payoff: the number of j’s neighbours does not change, \(h's\) increase, while \(n^{\infty }\) stays or decreases for all in C. On the other hand it may be that \(y\in C'\subset S\) is harmed. Notice, however that as soon as C breaks all external links, it does not experience externalities from other coalitions. In other words, if \(\mathcal {S}\) can deviate profitably, then a profitable deviation for \(\left\{ C\right\} \) exists, where it separates itself from all other players.
When k is the ideal number of neighbours, a player has the highest payoff in a component with \(k+1\) players, where he is connected to all other members. In the symmetric case the same applies to all other members of this component. Therefore we have the following observation.
Remark 1
The per-member payoff is the highest in a coalition corresponding to a component with exactly \(k+1\) players.
Corollary 8
A deviation by a coalition of size \(k+1\) forming a fully connected component is profitable if at least one of the members belongs to a coalition of a different size.
Theorem 9
The core consists of a single outcome with the players partitioned into fully connected components of size \(k+1\) and each player getting the same payoff.
Corollary 10
The core is nonempty if and only if \(\left| N\right| \) is divisible by \(k+1\).
Heterogeneous players
Just as in the homogeneous case, each player i has an optimal number of contacts \(k_i\). The following result is fairly trivial.
Lemma 11
If there exists a partition \({\mathcal {P}}\) of N such that each coalition \(S\in {\mathcal {P}}\) contains only players with optimal number \(\left| S\right| -1\) then the outcome where the network consists of fully connected components according to \({\mathcal {P}}\) and each player getting his own payoff is the only element of the core.
These conditions are rather special, it is generally not possible to group players into such components. Consider the following, very simple example.
Example 4
Consider a network with only 2 players, 1 and 2. Without loss of generality \(s_1(0)=s_2(0)=2.8\), but \(s_1(1)=3\) and \(s_2(1)=4\). Clearly, Player 1 prefers to be solitary as the additional risk is higher than the marginal benefit of keeping in touch with a friend. Therefore \(k_1=0\). On the other hand, 2 is happy to be in touch with another player. Two networks are feasible corresponding to two singletons (\(\ell _1\)) or a pair (\(\ell _2\)). Then \(v(\left\{ 1\right\} ,\ell _1)=2.8\), \(v(\left\{ 2\right\} ,\ell _2)=2.8\), while \(v(N,\ell _2)=3+4-2=5\), therefore the core network is segregated.
On the other hand, if the payoffs are somewhat different, \(s_1(0)=s_2(0)=2.8\), but \(s_1(1)=4\) and \(s_2(1)=5\), the payoffs of the singletons remain the same, but the pair gets \(v(N,\ell _2)=4+5-2=7\), therefore the more social player can compensate the other and form a component of her ideal size.
It is good to see that a nonempty core does not require the extremely rare structure required in Lemma 11: so an equilibrium network may contain components containing heterogenous players. Do those components at least contain similar players? In some sense, probably yes, but their preferred component size is a poor indicator for that. A player with a very small optimal size may have nearly the same benefit for every additional contact, while another characterised by a larger k may get practically no benefit from additional contacts. The second has a higher k value, but is very difficult to compensate for the larger component, while the first prefers larger components when these come with a little transfer.
Let \(\Delta s_i(m)=s_i(m)-s_i(m-1)\).
Theorem 12
Consider a player set N such that \(\Delta s_1(m)\le \Delta s_2(m)\le \dots \le \Delta s_{\left| N\right| }(m)\) for all m. Then if the core is not empty then there exists a core outcome with a network that sorts players into components according to their sociability function s.
Proof
There is a natural ordering of coalitions according to size. Players with a low index prefer to be in small coalitions, while those with a large index, in large coalitions. Now assume that the theorem is false. This means that some social players ended up in small, and more introvert players in large coalitions. In particular, there are two players, i and j in two neighbouring coalitions \({\mathcal {P}}(i)\) and \({\mathcal {P}}(j)\) of different sizes (\(d_i+1> d_j+1\)), such that they are “mixed up”: for example \(s(i)> s(j)\), while \(d_i+1=\left| {\mathcal {P}}(i)\right| \le \left| {\mathcal {P}}(j)\right| =d_j+1\). Take this pair and consider a deviation with the components as coalitions but two players exchanged. In this deviation, the degrees of other players do not change, so we can focus on i and j, where \(d'_i=d_j\) and \(d'_j=d_i\). Then \(V_{{\mathcal {P}}(i)\cup {\mathcal {P}}(j)}(\ell ')-V_{{\mathcal {P}}(i)}(\ell )-V_{{\mathcal {P}}(j)}(\ell ) =s_i(d'_i)-d_i^{'\infty }+s_j(d'_j)-d_i^{'\infty }-(s_i(d_i)-d_i^{\infty }+s_j(d_j)-d_i^{\infty }) =s_i(d_j)+s_j(d_i)-s_i(d_i)-s_j(d_j) =(s_i(d_j)-s_i(d_i))-(s_j(d_j)-s_j(d_i)) =\sum _{m=d_i+1}^{d_j}\left( \Delta s_i(m)-\Delta s_j(m)\right) >0\), where the last inequality follows from our assumption and that \(s(i)> s(j)\). \(\square \)
When there are multiple components of the same size, it does not matter how the players are distributed resulting in additional—payoff-equivalent—core outcomes.