1 Introduction
Static and dynamic properties of networks not controlled by any centralized authority attracts much attention in last two decades as selforganizing largescale networks play a critical role in a variety of information systems, for example, the Internet, PeertoPeer networks, adhoc networks, wireless sensor networks, social networks, viral networks, and so on. In these networks, participants selfishly and rationally change a part of the network structure to minimize their cost and maximize their gain. Controlling such networks is essentially impossible and many theoretical and empirical studies have been conducted; stochastic network construction models such as the Barabási–Albert model were proposed, and key structural properties such as the small world networks [23] and the scalefree networks [3] have been discovered. Stochastic communication models such as the voting models [10, 21], the random phone call model [14], and the rewiring model [12]
were proposed and many phase transition phenomena have been reported. Many problems related to broadcasting, gossiping, and viral marketing were also proposed
[4, 11, 16].In this paper, we take a gametheoretic approach to analyze dynamics and efficiency of the network structure resulting from local reconstruction by selfish agents. The network creation game (NCG) considers players forming a network [13]. Each player is associated with a vertex of the network, can construct a communication edge connecting itself to another player at the cost of , and can remove an adjacent edge for free. The cost of a player is the sum of the construction cost for edges and the communication cost, which is the sum of distances to all other players in the current network, i.e., the average distance to other players. Each player selfishly changes its strategy to minimize its cost and the social cost of a network is the sum of all players’ costs. The Price of Anarchy (PoA) of NCG is constant for almost all values of [1, 9, 18, 19], yet the PoA is not known for some values of . However, computing the best response in NCG is NPhard [13], and this fact makes the NCG unrealistic in largescale networks. The NCG with another type of communication cost is proposed in [9], where the cost of a player is the maximum distance to other players. We call this game the Max Network Creation Game (MAXNCG) and the original NCG the Sum Network Creation Game (SUMNCG). However, the SUMNCG and the MAXNCG ignores one of the most critical limitations in largescale networks; each player cannot obtain “global” information. This type of locality is a fundamental limitation in distributed computing [22], although players can neither compute its cost nor the best response without global information.
In this paper, we focus on games in such a distributed environment where each player cannot obtain the current strategy of all players nor have enough local memory to store the global information. Rather, players can access only local information. The NCG by players with local information is first proposed in [6]. Each player can observe a subgraph of the current graph induced by the players within distance . We call this information the local information. The players are pessimistic in the sense that they consider the worstcase global graph when they examine a new strategy. Computing the best response for MAXNCG is still NPhard because local information may contain the entire network. For small , more specifically, for , for MAXNCG and for for SUMNCG. These results contrast global information with local information. The SUMNCG and MAXNCG by players with global traceroute based information is proposed, yet for some values of [5]. The NCG for more powerful players with local information is considered in [8], where the players can probe the cost of a new strategy. Computing the best response is NPhard for any while there exists tree equilibrium that achieves and for and for . For nontree networks, depending on the values of and , we have .
The swap game (SG) restricts strategy changes to edge swaps, i.e., simultaneously removing an edge and creating a new edge [2]. Thus, any strategy change does not change the number of edges in the network and the best response can be computed in polynomial time. Additionally, when we restrict initial networks to trees, a star achieves the minimum social cost. Above mentioned cost functions were adopted and these SGs are called the SUMSG and the MAXSG, respectively. The aim of SG is to omit parameter from NCG with keeping the essence of NCG. The authors showed that the diameter of a tree equilibrium is two for the SUMSG and at most three for the MAXSG, while there exists an equilibrium with a large diameter in general networks. Thus, PoA of a tree equilibrium is always constant. Moreover, any SUMSG and MAXSG starting from a tree converges to an equilibrium within edge swaps while they admit best response cycles starting from a general graph [15, 17]. Consequently, local search at players with global information achieves efficient network construction for initial tree networks. The SUMSG and MAXSG with “powerful” players with local information is investigated in [8]. For , the SUMSG and MAXSG starting from general networks admits best response cycles while convergence within moves is guaranteed for tree networks. However, to the best of our knowledge, SG with local information has not been considered.
1.1 Our results
In this paper, we investigate the convergence property and PoA of SGs by players with local information. First, we consider pessimistic players and demonstrate that starting from an initial tree, any SUMSG and MAXSG converge to an equilibrium within edge swaps in the same manner as [17], i.e., we present a generalized ordinal potential function for the two games. We also show that convergence from a general network is not always guaranteed. Then, we present a clear phase transition phenomenon caused by the locality.

When , pessimistic players never perform any edge swap in the SUMSG and in MAXSG. Any network is an equilibrium of the two games, thus .

When , in the SUMSG and MAXSG, there exists an equilibrium of diameter , thus .

When , in the SUMSG and MAXSG the diameter of every equilibrium is constant, thus PoA is constant.
We then introduce weaklypessimistic players and optimistic players to obtain a better PoA for . A weakly pessimistic player performs an edge swap even when its cost does not decrease. This relaxation results in a constant PoA of the MAXSG when at the cost of best response cycles. An optimistic player assumes the bestcase global graph for an edge swap and this optimism results in a constant PoA of the SUMSG and MAXSG for any value of . Consequently, the combination of locality for and pessimism enables distributed construction of efficient trees by selfish players.
1.2 Related works
We briefly survey existing results of the NCG and SG for players with global information. Regarding the SUMNCG, when for the PoA is [9]. Thus, when is sufficiently large, the PoA is bound by a constant. When , the PoA is at most [7]. In addition, any constant upper bound of PoA for is not known and the best upper bound is [9]. If every equilibrium is a tree, then and an interesting conjecture is that every equilibrium is a tree for sufficiently large [13]. Regarding the MAXNCG, the PoA is and it is constant when or [19].
Regarding the SUMSG, there exists an equilibrium with diameter while the diameter of any equilibrium is at most two (thus, a star) if an initial graph is a tree [2]. Regarding the MAXSG, there exists an equilibrium with diameter while the diameter of any equilibrium is at most three if an initial graph is a tree [2].
1.3 Organization
Preliminary section introduces the SGs and pessimistic players with local information. In the next section, we analyze the dynamics and PoA of SGs by pessimistic players. In the following section, we introduce less pessimistic players and present best response cycles and equilibria with small diameter. Finally, we conclude this paper with open problems.
2 Preliminaries
A swap game (SG) by players with local information is denoted by , where is an initial network and integer is the size of each player’s “visibility”. is a simple undirected connected graph, where and . We say is adjacent to if edge is an element of . Each player is associated to a vertex in and the strategy of a player is the set of its incident edges.
Each player can change its strategy by an edge swap, i.e., removing one incident edge and creating a new edge. Starting from , a sequence of edge swaps generates a network evolution .
Let be the set of adjacent vertices of in and be the distance between in . When is not connected and is not reachable from , . The cost of a player depends on the current graph . We consider two different types of cost functions, and defined as follows:
When is not connected, and . We call a swap game where each player uses the sum swap game (SUMSG) and a swap game where each player uses the max swap game (MAXSG). When it is clear from the context, we omit the name of the game and use .
Each player can access local information determined by . Let denote the set of vertices within distance from in (thus, the neighborhood of ). Player can observe the subgraph of induced by and we call this subgraph the view of . We say the information at is local and we call its view the local information of . We assume that each player does not know any global information such as the values of and .
In a transition from to , a single player performs an edge swap. Consider the case where a player performs an edge swap in . We call the moving player in . The resulting graph is . Note that the number of edges does not change in a SG.
Due to local information, each player cannot compute its current cost nor the improvement by a strategy change. We first consider pessimistic players that consider the worstcase improvement for each possible edge swap and select one that achieves positive improvement. A player is unhappy if it has an edge swap that decreases its cost in the worstcase global graph. In other words, there exists at least one edge swap at that satisfies
where is the set of simple undirected connected graphs consisting of finite number of vertices and compatible with ’s local view, and is a graph obtained by the edge swap at in . We assume that a moving player always performs an edge swap with . When every player is not unhappy with respect to in graph , we call a sumswap equilibrium. When every player is not unhappy with respect to in graph , we call a maxswap equilibrium. When a graph is a sumswap equilibrium and a maxswap equilibrium we simply call the graph swap equilibrium.
We define the social cost of a graph as the sum of all players’ costs, i.e., . Let be the set of simple undirected connected graphs of players and edges and be the set of sumswap equilibrium graphs of players with local information and edges. The Price of Anarchy (PoA) of the SUMSG is defined as follows:
In the same way, the PoA of the MAXSG is defined for the set of maxswap equilibrium graphs of players with local information and edges. The PoA of the SUMSG (and MAXSG) starting from a tree is denoted by (, respectively).
A strategic game has the finite improvement property (FIP) if every sequence of improving strategy changes is finite [20]. Thus, from any initial state, any sequence of finite improving strategy changes reaches an equilibrium. Monderer and Shapley showed that a strategic game has the FIP if and only if it has a generalized ordinal potential function. Regarding a swap game, a function is a generalized ordinal potential function if we have the following property for every graph , every unhappy player , and every edge swap that makes unhappy,
where is a graph obtained by the edge swap at . That is, any transition in the SUMSG and MAXSG satisfies the above property for the moving player.
The best response of a player in is an edge swap that maximizes . We call an evolution a best response cycle when each moving player in performs a best response for .
We further introduce some notations for graph . For a set of vertices the graph obtained by removing vertices in and their incident edges is denoted by . Additionally, for a set of edges the graph obtained by removing edges in is denoted by . The vertex set and the edge set of a graph is denoted by and , respectively.
3 Convergence properties for pessimistic players
In this section, we investigate the dynamics of the SUMSG and MAXSG by pessimistic players with local information. We first consider general settings where the initial graph is not a tree and multiple players perform edge swaps simultaneously. We show that the two games admit best response cycles. We then demonstrate that when the initial graph is a tree, the SUMSG and MAXSG have the FIP and converges to an equilibrium within polynomial number of edge swaps.
3.1 Impossibility in general settings
We first present several necessary conditions for an evolution of the SUMSG and MAXSG by players with local information to reach an equilibrium. We first present the necessary visibility for each player to change their strategies.
Theorem 1
In the SUMSG and MAXSG, when , no player is unhappy in an arbitrary graph. Thus, any graph is a swap equilibrium.
Proof.
When , no player can perform an edge swap because at any .
When , we first consider the SUMSG. Assume player is unhappy because of edge swap in graph . Let be the graph obtained by this edge swap. Thus, and . In a worstcase global graph, has no adjacent vertex other than those in and the cost of decreases by at most one by this edge swap. In , must be reachable from . There exists at least one player that is in and adjacent to , otherwise is not reachable from in a worstcase global graph. Hence, . Additionally, for any . Consequently, and is not unhappy in .
Next, we consider the MAXSG. Assume player is unhappy because of edge swap in graph . Thus, and is the only player at distance from in otherwise is not unhappy because of the edge swap in . Let be the graph obtained by this edge swap. In a worstcase global graph, has no adjacent vertex other than those in and the cost of is expected to be reduced to . By the same discussion above, is reachable from in , however . Hence, the maximum distance from to players in is still two, thus . Hence, is not unhappy in . ∎
The following theorem justifies our assumption of a single edge swap in each transition.
Theorem 2
When , if multiple players change their strategies simultaneously, the SUMSG and MAXSG admit best response cycles.
Proof.
We first consider the SUMSG. Consider a path of four players , , , and aligned in this order. When , the two endpoint players and are unhappy because of the edge swap and , respectively. If the two players perform the edge swaps simultaneously, the resulting graph is again a path graph, where and are unhappy.
The above example is also a best response cycle in the MAXSG. ∎
Finally, we consider dynamics of SGs starting from an arbitrary initial graph. Lenzner presented a best response cycle for the SUMSG by players with global information [17]. During the evolution, the distance to any player from a moving player is always less than four and we can apply the result to the SUMSG by pessimistic players with local information for . In addition, the edge swaps are also best responses in the MAXSG. Hence, we can also apply the result to the MAXSG by pessimistic players with local information for . We have the following theorem.
Theorem 3
When , there exists an initial graph from which the SUMSG and MAXSG admit a best response cycle.
In the following, we concentrate on the SUMSG and MAXSG by pessimistic players with local information for starting from a tree. As defined in the preliminary, a single player changes its strategy in each transition.
3.2 Convergence from an initial tree
In this section, we show that the SUMSG and MAXSG have the FIP. For players with global information, generalized ordinal potential functions for the SUMSG [17] and MAXSG [15] have been proposed. We can use these generalized ordinal potential functions for pessimistic players with local information.
Theorem 4
If is a tree, any SUMSG has the FIP and reaches a sumswap equilibrium within edge swaps.
Proof.
We show that is a generalized ordinal potential function for the SUMSG irrespective of the value of . Consider a tree where an arbitrary unhappy player performs an edge swap that yields a new graph . We have .
Lenzner showed that for players with global information holds if [17]. Since considers the worst case graph, holds. Consequently, is a generalized ordinal potential function for the SUMSG.
Lenzner also showed that when the graph is a path of vertices, achieves the maximum value of , and if the graph is a star of vertices, achieves the minimum value of . Hence, the number of edge swaps is . ∎
We next show the FIP of the MAXSG. Kawald and Lenzner presented a generalized ordinal function for the MAXSG by players with global information [15]. Their generalized ordinal function is an tuple of players’ costs, where the players are sorted in the descending order of their costs. We apply their function to the MAXSG by pessimistic players with local information, however, we found that their proof needs small correction. In the following, we provide a new proof for their function.
Consider the case where an unhappy player performs an edge swap in and a new graph is formed. Graph consists of two trees and let be the tree containing vertex and be the tree containing vertex . We have the following two lemmas.
Lemma 5
[15] Any player satisfies .
Lemma 6
Any player satisfies at least one of the following two conditions; (i) there exists a player that satisfies and (ii) .
Proof.
For an arbitrary player , let
We have the following three equations;
If , we have
The third equation holds because during the transition from to any distance between vertices in (and , respectively) does not change. The last line is bounded by because
otherwise was not unhappy in . Thus, the first condition is satisfied.
If , we have
and the second condition is satisfied. ∎
We define for a graph as an tuple where for . We assume that ties are broken arbitrarily. We then consider lexicographic ordering of tuples. For two tuples and where for , when and the leftmost nonzero entry of is positive, we say is lexicographically larger than , denoted by .
Theorem 7
If is a tree, a MAXSG has the FIP and reaches a maxswap equilibrium within edge swaps.
Proof.
We demonstrate that any transition from to satisfies . Let be the moving player in and be the player with the maximum cost in . Then, in , there may exist a player with larger cost than . We sort these players in the descending order of their costs and let be the obtained sequence of players and be the remaining players in .
We first show that any () satisfies . If the second condition of Lemma 6 holds for all , we have the property. Otherwise, there exists that does not satisfy the second condition but the first condition. However, we have and by the proof of Lemma 6, holds. This is a contradiction and all satisfies .
Then we consider a player . By Lemma 5 and Lemma 6 such player satisfies . Consequently, we have , and .
We can bound the number of edge swaps in the same manner as [15]. ∎
4 PoA for pessimistic players
In this section, we analyze PoA of the SUMSG and MAXSG by pessimistic players with local information. Alon et al. showed that for players with global information, the diameter of a tree swap equilibrium is constant for the two cost functions, thus PoA is also constant [2]. On the other hand, our results show the clear contrast by the value of . When , there exists a sumswap equilibrium of diameter and a maxswap equilibrium of diameter . Thus, PoA is for both games. When , the diameter of any sumswap equilibrium is at most two and that of any maxswap equilibrium is at most three. Thus, the PoA is bounded by a constant for both games.
In the following, we consider a path in a graph. A path of length is denoted by a sequence of vertices on it, i.e., . The set of vertices that appear on is denoted by and the set of edges of is denoted by . Given a tree and a path in , consider the forest and let denote the connected component (thus, a tree) containing . We consider as the root of when we address the depth of . The following lemma provides a basic technique to check the existence of an unhappy player.
Lemma 8
In the SUMSG, when , a player in a tree is unhappy if and only if there exists a path that satisfies the following two conditions; (i) the depth of is at most one, and (ii) .
Proof.
We first show that is unhappy if the two conditions hold. Assume that there is a path satisfying the two conditions. See Figure 1. Let be the graph obtained by the edge swap at in . For every , and for every . By condition (i), knows that the edge swap increases the distance to . By condition (ii), knows that the edge swap decreases its cost by at least . In the worstcase global graph, has no adjacent players other than . Hence,
and is unhappy because of this edge swap .
Next, we show that is unhappy in only if the two conditions hold. Consider the case where for any path , (i’) the depth of is larger than one, or (ii’) holds. We show that any player is not unhappy. We check an arbitrary edge swap at . Thus, and . must have a path between and , otherwise the edge swap disconnects the players. If cannot see this path, in the worst case global graph, is not reachable from . Hence, contains a path or .
If contains a path , the edge swap satisfies , , and . The worstcase global graph for the edge swap is a graph where is not adjacent to any other vertex in . Thus, . Hence, is not unhappy with respect to the edge swap .
If contains a path and condition (i’) holds, there exist vertices that form a path . In the worst case global graph for the edge swap , has many children whose distance from increases by one in . Hence, and is not unhappy with respect to the edge swap .
If contains a path and condition (ii’) holds, in the worstcase global graph for the edge swap , the number of players whose distance from decreases by one with the edge swap is and the number of players whose distance from increases by one is lower bounded by . Thus, holds and player is not unhappy with respect to the edge swap .
Consequently, is not unhappy with respect to any edge swap. ∎
Theorem 9
When and , .
Proof.
We present a sumswap equilibrium of diameter . We define a tree with a spine path of length as follows: For , is a tree, where has four children , , , and . For , is a tree rooted at with three children , c, and . is a tree defined by
Figure 2 shows as an example.
We show that for each (), any path does not satisfy the two conditions of Lemma 8. First, consider the case where is a leaf of . Then, is for some . If is a leaf, the depth of is larger than one, and the first condition of Lemma 8 is not satisfied. If is an internal vertex, and , and the second condition of Lemma 8 is not satisfied.
Second, consider the case where is an internal vertex of . Then, is also an internal vertex otherwise we cannot find . When the depth of is larger than one, the first condition of Lemma 8 is not satisfied. When the depth of is one, is for some and is a leaf. Thus, and , and the second condition of Lemma 8 is not satisfied.
Thus, every player is not unhappy and is a sum swap equilibrium.
We then calculate the social cost in , that consists of vertices.
When for any integer , we have the same bound by attaching extra vertices to some .
Since star is a sumswap equilibrium with the minimum social cost, the PoA of is . ∎
By , we have the following corollary.
Corollary 10
There exists a sumswap equilibrium of diameter for any .
We now demonstrate that when , sumswap equilibrium for pessimistic players with local information achieves the same PoA as that with global information.
We first show the following lemma.
Lemma 11
In an arbitrary tree whose diameter is larger than two, there exists a path that satisfies the following two conditions; (i) , and (ii) the depth of is at most two.
Proof.
We prove the lemma by induction. There exists at least one path of length at least three in . We choose a path arbitrarily. Let and be the depth of and , respectively.
(Base case.) Consider the case where . Let . The two paths and satisfies the second condition and either or satisfies the first condition. Thus, the statement holds when .
(Induction step.) Assume the statement holds when for . Consider the case where . Without loss of generality, we assume . Hence, there exists at least one path in . See Figure 3. Let and be the depth of and , respectively. Clearly, and hold and the statement holds by the induction hypothesis.
∎
By Lemma 11, in any graph whose diameter is larger than two there exists an unhappy player.
Theorem 12
When is a tree and , any sumswap equilibrium is a star and .
Proof.
Assume that there exists a sumswap equilibrium whose diameter is larger than two. By Lemma 11, there exists a path that satisfies the two conditions. Hence, is unhappy because
This is a contradiction and is not a sumswap equilibrium. Hence, the diameter of a sumswap equilibrium is smaller than or equal to two and we have the statement. ∎
Consequently, the “visibility” of pessimistic players has a significant effect on the PoA of the SUMSG. We then demonstrate that this is also the case for the MAXSG.
Theorem 13
When and , .
Proof.
We show that a tree shown in Figure 4 is a maxswap equilibrium.
First, consider the two endpoint vertices. Each endpoint player has one incident edge and any edge swap involving this edge does not decrease the maximum distance to the vertices in its view. In the worstcase global graph the player at distance two has no other vertices. Thus, the two endpoint players are not unhappy.
Second, consider other leaves. Each leaf player has one incident edge and any edge swap at a leaf increases the maximum distance to some vertex in its view. In the worstcase global graph, there is a long path starting from such a vertex. Hence, the leaf players are not unhappy.
Finally, consider inner vertices. Each inner player has three edges but it cannot remove the edge connecting it to a leaf because such an edge swap disconnects the graph. If the player remove an edge incident to another inner vertex and create a new edge, by the same discussion as above, this edge swap increases the maximum distance to some vertex in its view and the player is not unhappy.
Consequently, the graph shown in Figure 4 is a maxswap equilibrium. By adding inner vertices (with its child), we have the similar equilibrium for any even
. For odd
, we attach an extra player to an inner vertex and obtain a max swap equilibrium.Since the star graph is an maxswap equilibrium with the minimum social cost, we have . ∎
By the proof of Theorem 13, we have following corollary.
Corollary 14
There exists a maxswap equilibrium of diameter for any .
We now demonstrate that when , any MAXSG by pessimistic players with local information achieves the same PoA as that of players with global information. The following lemma shows that in any tree of diameter larger than three, there is at least one unhappy player.
Lemma 15
In any tree whose diameter is larger than three, there exists a path that satisfies the following two conditions; (i) starts from a leaf , and (ii) the depth of is at most one.
Proof.
There exists at least one path of length at least four in . We arbitrarily choose a path that starts from some leaf . If the depth of is smaller than two, the statement holds. If the depth of is larger than one, choose a leaf in and its parent vertex, say . There exists at least one path and satisfies the second condition. ∎
Theorem 16
When is a tree and , the diameter of any maxswap equilibrium is at most three and .
Proof.
Assume that there exists a maxswap equilibrium whose diameter is larger than three. By Lemma 15, there exists a path such that is a leaf and the depth of is at most one. Player is unhappy because . This is a contradiction and is not a maxswap equilibrium. Thus, the diameter of any maxswap diameter is at most three.
Because a equilibrium with the minimum cost is a star, the PoA is bounded by . ∎
5 Swap games with nonpessimistic players
We demonstrated that when , the PoA for pessimistic players is in the SUMSG and MAXSG. In this section, we introduce less pessimistic players to obtain smaller PoA for these cases. We consider two types of nonpessimistic players: A player is weakly pessimistic if is unhappy when there exists an edge swap at such that .
A player is optimistic if its is defined as
where is a graph obtained by an edge swap at in .
Weakly pessimistic players and optimistic players do not perform any edge swap in the SUMSG and MAXSG when . Different from Theorem 1, weakly pessimistic players change their strategies when . However, when , weakly pessimistic players cause a cycle of edge swaps from an initial path graph.
Example 1
Let