The Efficiency of Best-Response Dynamics

02/26/2020 ∙ by Michal Feldman, et al. ∙ Tel Aviv University IDC Herzliya 0

Best response (BR) dynamics is a natural method by which players proceed toward a pure Nash equilibrium via a local search method. The quality of the equilibrium reached may depend heavily on the order by which players are chosen to perform their best response moves. A deviator ruleS is a method for selecting the next deviating player. We provide a measure for quantifying the performance of different deviator rules. The inefficiency of a deviator rule S is the maximum ratio, over all initial profiles p, between the social cost of the worst equilibrium reachable by S from p and the social cost of the best equilibrium reachable from p. This inefficiency always lies between 1 and the price of anarchy. We study the inefficiency of various deviator rules in network formation games and job scheduling games (both are congestion games, where BR dynamics always converges to a pure NE). For some classes of games, we compute optimal deviator rules. Furthermore, we define and study a new class of deviator rules, called local deviator rules. Such rules choose the next deviator as a function of a restricted set of parameters, and satisfy a natural condition called independence of irrelevant players. We present upper bounds on the inefficiency of some local deviator rules, and also show that for some classes of games, no local deviator rule can guarantee inefficiency lower than the price of anarchy.

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1 Introduction

Nash equilibrium (NE) is perhaps the most popular solution concept in games. It is a strategy profile from which no individual player can benefit by a unilateral deviation. However, a Nash equilibrium is a declarative notion, not an algorithmic one. To justify equilibrium analysis, we have to come up with a natural behavior model that leads the players of a game to a Nash equilibrium. Otherwise, the prediction that players play an equilibrium is highly questionable. Best response dynamics is a simple and natural method by which players proceed toward a NE via the following local search method: as long as the strategy profile is not a NE, an arbitrary player is chosen to improve her utility by deviating to her best strategy given the profile of others.

Work on BR dynamics advanced in two main avenues: The first studies whether BR dynamics converges to a NE, if one exists, e.g., [33, 26] and references therein. The second explores how fast it takes until BR dynamics converges to a NE, e.g., [17, 19, 36, 28].

It is well known that BR dynamics does not always converge to a NE, even if one exists. However, for the class of finite potential games [35, 32], a pure NE always exists, and BR dynamics is guaranteed to converge to one of the equilibria of the game. A potential game is one that admits a potential function — a function that assigns a real value to every strategy profile, and has the miraculous property that for any unilateral deviation, the change in the utility of the deviating player is mirrored accurately in the potential function. The proof follows in a straight forward way from the definition of a potential game. Due to the mirroring effect, and since the game is finite, the process of BR updates must terminate, and this happens at some local minimum of the potential function, which is a NE by definition. While BR dynamics is guaranteed to converge, convergence may take an exponential number of iterations, even in a potential game. [2] showed that a network design game with costs obtained by a sum of construction cost and latency cost, which is a potential game, convergence via best response dynamics can be exponentially long.

Our focus in this work is different than the directions mentioned above. The description of BR dynamics leaves the choice of the deviating player unspecified. Thus, BR dynamics is essentially a large family of dynamics, differing from one another in the choice of who would be the next player to perform her best response move. In this paper, we study how the choice of the deviating player (henceforth a deviator rule) affects the efficiency of the equilibrium reached via BR dynamics.

Our contribution is three fold.

First, we introduce a new measure for quantifying the performance of different deviator rules. This measure can be used to quantify the performance of different deviator rules in various settings, beyond the ones considered in this paper.

Second, we introduce a natural class of simple deviator rules, we refer to as local. Local deviator rules are simpler to apply, since they are based on limited information. In practice, simple deviator rules should be preferred over more complicated ones. Our results help in quantifying the efficiency loss incurred due to simplicity.

Finally, we quantify the inefficiency of various deviator rules in two paradigmatic congestion games, namely network formation games and job scheduling games. Our results distinguish between games where local deviator rules can lead to good outcomes and games for which any local deviator rule performs poorly.

1.1 Model and Problem Statement

A game has a set of players. Each player has a strategy space , from which she chooses a strategy

. A strategy profile is a vector of strategies for each player,

. The strategy profile of all players except player is denoted by , and it is convenient to denote a strategy profile as . Each player has a cost function , where denotes player ’s cost in the strategy profile . Every player wishes to minimize her cost. There is also a social objective function, mapping each strategy profile to a social cost.

Given a strategy profile , the best response of player is ; i.e., the set of strategies that minimize player ’s cost, fixing the strategies of all other players. Player is said to be suboptimal in if the player can reduce her cost by a unilateral deviation, i.e., if . If no player is suboptimal in , then is a Nash equilibrium (NE) (in this paper we restrict attention to pure NE; i.e., an equilibrium in pure strategies).

Given an initial strategy profile , a best response sequence from is a sequence in which for every there exists a player such that , where . In this paper we restrict attention to games in which every BR sequence is guaranteed to converge to a NE.

Deviator rules and their inefficiency. A deviator rule is a function that given a profile , chooses a deviator among all suboptimal players in . The chosen player then performs a best response move (breaking ties arbitrarily). Given an initial strategy profile and a deviator rule we denote by the set of NE that can be obtained as the final profile of a BR sequence , where for every , is a profile resulting from a deviation of (recall that players break ties arbitrarily, thus this is a set of possible Nash equilibria).

Given an initial profile , let be the set of Nash equilibria reachable from via a BR sequence, and let be the best NE reachable from via a BR sequence, that is, , where is some social cost function.

The inefficiency of a deviator rule in a game , denoted , is defined as the worst ratio, among all initial profiles , and all NE in , between the social cost of the worst NE reachable by (from ) and the social cost of the best NE reachable from . I.e.,

For a class of games , the inefficiency of a deviator rule with respect to is defined as the worst case inefficiency over all games in : A deviator rule with inefficiency is said to be optimal, i.e., an optimal deviator rule is one that for every initial profile reaches a best equilibrium reachable from that initial profile.

The following observation, shows that the inefficiency of every deviator rule is bounded from above by the price of anarchy (PoA) [31, 34]. Recall that the PoA is the ratio between the cost of the worst NE and the cost of the social optimum, and is used to quantify the loss incurred due to selfish behavior.

Observation 1.1

For every game and for every deviator rule it holds that the inefficiency of is at least and bounded from above by the PoA.

Proof:    For every game , initial profile , and deviator rule , by definition, . Since we know that and therefore . Also, . Since , we conclude and therefore .  

Local deviator rules. We define and study a class of simple deviator rules, called local deviator rules. Local deviator rules are defined with respect to state vectors, that represent the state of the players in a particular strategy profile. Given a profile , every player is associated with a state vector , consisting of several parameters that describe her state in and in the strategy profile obtained by her best response. The specific parameters may vary from one application to another. A vector profile is a vector , consisting of the state vectors of all players. A deviator rule is said to be local if it satisfies the independence of irrelevant players condition, defined below.

Definition 1.1

A deviator rule satisfies independence of irrelevant players (IIP) if for every two state vectors , and every two vector profiles such that , and 222Note that the profiles and may correspond to different sets of players., if , then .

The IIP condition means that if the deviator rule chooses a state vector over a state vector in one profile, then, whenever these two state vectors exist, the deviator rule would not choose vector over . Note that this condition should hold even across different game instances and even when the number of players is different. Many natural deviator rules satisfy the IIP condition. For example, suppose that the state vector of a player contains her cost in the current profile and her cost in the profile obtained by her best response; then, both max-cost, which chooses the player with the maximum current cost, and max-improvement, which chooses the player with the maximum improvement, are local deviator rules.

Congestion games. A congestion game has a set of resources, and the strategy space of every player is a collection of sets of resources; i.e., . Every resource has a cost function , where is the cost of resource if players use resource . The cost of player in a strategy profile is , where is the number of players that use resource in the profile . Every congestion game is a potential game [32], thus admits a pure NE, and moreover, every BR sequence converges to a pure NE. In this paper we study the efficiency of deviator rules in the following congestion games:

Network formation games [2]: There is an underlying graph, and every player is associated with a pair of source and target nodes . The strategy space of every player is the set of paths from to . The resources are the edges of the graph, every edge is associated with some fixed cost , which is evenly distributed by the players using it. That is, the cost of an edge in a profile is . In network formation games the cost of a resource decreases in the number of players using it. We also consider a weighted version of network formation games on parallel edge networks, where players have weights and the cost of an edge is shared proportionally by its users. The social cost function here is the sum of the players’ costs; that is .

The state vector of a player in a network formation game, in a profile , consists of:

  1. Player ’s cost in :

  2. The cost of player ’s path:

  3. Player ’s cost in the profile obtained from a best response of : (where is the profile obtained from a best response of ).

  4. The cost of player ’s path in the profile : .

In weighted instances, the state vector includes player ’s weight as well.

Job scheduling games [38]: The resources are machines, and players are jobs that need to be processed on one of the machines. Each job has some length, and the strategy space of every player is the set of the machines. The load on a machine in a strategy profile is the total length of the jobs assigned to it. The cost of a job is the load on its chosen machine. We also consider games with conflicting congestion effect ([23, 10]), where jobs have unit length and in addition to the cost associated with the load, every machine has an activation cost , shared by the jobs assigned to it. The social cost function here is the makespan, that is . The state vector of a job (player) in a job scheduling game, in a profile , consists of the job’s length, its current machine, and the loads on the machines.

1.2 Our Results

In Section 2 we present our results for network formation games. Some of our results refer to restricted graph topologies. These topologies are defined in Appendix A. We first show that in general network formation games, finding a BRD-sequence that approximates the optimum better than the PoA is NP-hard. Therefore, we restrict attention to subclasses: We study symmetric games, where all the players share the same source and target nodes. We show that the local Min-Path deviator rule, which chooses a player with the cheapest best response path, is optimal. In contrast, the local Max-Cost deviator rule has the worst possible inefficiency, (which matches the PoA for this game). We then consider asymmetric network formation games. Unfortunately, the optimality of Min-Path does not carry over to asymmetric network formation games, even when played on series of parallel paths (SPP) networks. In particular, the inefficiency of Min-Path in single-source multi-targets instances is , and for multi-sources multi-targets instances, it further grows to . On the positive side, we show a dynamic-programming algorithm for finding an optimal BR sequence for network formation games played on SPP networks in single-source multi-targets instances. We also show a such algorithm for multi-sources multi-targets instances that admit “proper intervals” (see Section 2.3.2 for formal definitions). For network formation games played on extension-parallel networks we show that every local deviator rule has an inefficiency of .

In Section 2.5 we study network formation games with weighted players. It turns out that weighted players lead to quite negative results. We show that even in the simplest case of parallel-edge networks, it is NP-hard to find an optimal BR-sequence, and no local deviator rule can ensure a constant inefficiency. Moreover, the Min-Path deviator rule has inefficiency , even in symmetric games on SPP networks, and even if the ratio between the maximal and minimal weights approaches . This result is quite surprising in light of the fact that Min-Path is optimal (i.e., has inefficiency ) for any symmetric network formation game with unweighted players.

Job scheduling games with identical machines are studied in Section 3. A job’s ( player’s) state vector in job scheduling games includes the job’s length, its machine, and the machines’ loads. Thus, local deviator rules capture many natural rules, such as Longest-Job, Max-Cost, Max-Improvement, and more. We show that no local deviator rule can guarantee inefficiency better than the in this class of games (which is ). In contrast, for job scheduling games with conflicting congestion effects [23] we present an optimal local deviator rule.

Positive results on local deviator rules imply that a centralized authority that can control the order of deviations can lead the population to a good outcome, by considering merely local information captured in the close neighborhood of the current state. In contrast, negative results for local deviator rules imply that even if a centralized authority can control the order of deviations, in order to converge to a good outcome, it cannot rely only on local information; rather, it must be able to perform complex calculations and to consider a large search space.

1.3 Related work

A lot of research has been conducted on the analysis of congestion games using a game-theoretic approach. The questions that are commonly analyzed are Nash equilibrium existence, the convergence of BR dynamics to a NE, and the loss incurred due to selfish behavior – commonly quantified according to the price of anarchy [31, 34] and price of stability (PoS) [2] measures.

Our work addresses mainly congestion games. Specifically, we consider network formation games and job scheduling games and variants thereof. It is well known that every congestion game is a potential game [35, 32] and therefore admits a PNE. In potential games every BR dynamics converges to a PNE. However, the convergence time may, in general, be exponentially long. It has been shown in [14] that in random potential games with players in which every player has at most strategies, the worst case convergence time is . The average case convergence time in random potential games is sublinear but along with the computation complexity of finding the best response has running time of . Results regarding the convergence time in network formation games are shown in [2], [19]. They showed that in general, finding a PNE is PLS-complete. In the restricted case of symmetric players, in a game with both positive and negative congestion effects, the convergence is polynomial. Analysis of job scheduling games is presented in [16], where it is shown that it may take exponential number of steps to converge in general cases.

The observation that convergence via best response dynamics can be exponentially long has led to a large amount of work aiming to identify special classes of congestion games, where BRD converges to a Nash equilibrium in polynomial time or even linear time, as shown by [2] for games with positive congestion effects and by
[16] for games with negative congestion effects. This agenda has been the focus of [25] that considered symmetric network formation games with negative congestion effect played on an extension-parallel graph, and showed that the convergence is bounded by steps. For resource selection games (i.e., where feasible strategies are composed of singletons), it has been shown in [28] that better-response dynamics converges within at most steps for general cost functions (where is the number of resources).

Another direction of research considers an approximate Nash equilibrium, also known as -Nash equilibrium. An approximate-NE is a strategy profile in which no player can significantly improve her utility by a unilateral deviation. [9] presented an algorithm that identifies a polynomially long sequence of BR moves that leads to an approximate NE in congestion games with linear latency functions. [8] showed for the more general case of polynomially decreasing resources’ cost functions, given an algorithm to find -best response, convergence time is polynomial. In weighted network formation games which are known to not always have a PNE, [12] study the relations between the needed to ensure existence of an -PNE and the inefficiency of the solution. [11] showed that it takes steps to converge to a PNE in symmetric network formation games, and that even when using different thresholds the amount of deviations conducted by player will be .

Another technique for controlling the convergence time of BRD is using a specific deviator rule. [16] compared the convergence time of different deviator rules for different variants of job scheduling games. For example, for instances of identical machines they show that the Max-Weight-Job deviator rule ensures convergence in at most steps while convergence by the Min-Weight-Job can be exponential. They also show that the FIFO deviator rule converges within time and the expected convergence time of a random order is also . [23] and [10] considered a variant of job scheduling with both negative and positive congestion effects and analyzed their PoA and PoS. We will refer to that model in this work. [22] showed that using the Max-Cost deviator rule, tightly bounds the convergence rate to (worst-case) in these games. The Max-Cost deviator rule was also considered in [30] for Swap-Games [4]. In Swap-Games with initial strategies corresponding to a tree it has been shown that the convergence time is , but using the Max-Cost deviator rule improves the bound to .

Although every congestion game have a PNE and BRD always converges to one, examples for variants of congestion games that do not have a PNE or that BRD does not necessarily converge to one are shown in the literature. In particular, [12] presents a network-formation game with weighted players that does not have a PNE;
[26] presents a characterization of the resources’ cost functions that ensures existence of PNE, and proves various conditions for the finite-improvement-property (FIP) that implies that every best response dynamic converges to a PNE. [33] shows that in a variant of congestion games in which every player has a specific decreasing cost function for every resource, the FIP condition does not necessarily hold. Though there is always a PNE and best response dynamics can always converges to it, it is shown that there can be infinitely long best response sequences and conditions for them to occur are presented. [5] presents network formation games with a variation that players do not aim to connect a source and a sink, but to select a path corresponding to a word in a regular language, assuming the edges are labeled by an alphabet. They showed cases with no PNE and moreover, that computing a player’s best response is NP-hard, as was shown for another variant of network formation games presented in [18]. A variant of job scheduling game on unrelated parallel machines, studied in [6] is shown to have a PNE only for unit-cost machines. All the above works imply that the existence of a PNE, as well as the convergence of BRD to one are not trivial. Moreover, even when a PNE exists and BRD is guaranteed to converge, the implementation of best response dynamics can be computationally hard.

BRD has been studied also in games that do not converge to a PNE. The notion of dynamic inefficiency was defined in [7] as the average of social costs in a best response infinite sequence (for games that do not possess the finite improvement property). They considered the effect of the chosen player on the obtained efficiency in games where BR dynamics can cycle indefinitely. They consider job scheduling games, hotelling model and facility location games and study the dynamic inefficiency of Random Walk, Round Robin and Best Improvement deviator rules.

2 Network Formation Games

In this section we study network formation games. We consider two natural local deviator rules, namely Max-Cost and Min-Path. The Max-Cost deviator rule chooses a suboptimal player that incurs the highest cost in the current strategy profile , i.e., . The Min-Path deviator rule chooses a suboptimal player whose path in the profile obtained from a best response move is cheapest, i.e., . Both rules are local deviator rules.

Recall that some of our results refer to restricted graph topologies, defined in Appendix A. This section is organized as follows. Section 2.2 includes our analysis of symmetric games. In Sections 2.3 and 2.4 we study games played on series of parallel paths networks and extension parallel networks, respectively. Finally, in Section 2.5 we study weighted network formation games.

2.1 The BRD-Inefficiency in General Network Formation Games

We start with the most general form of network formation games, with no restriction on the network topology or the players’ objectives. We prove that for the general case it is NP-hard to find a deviator rule whose inefficiency is lower than the PoA.

Theorem 2.1

In network formation games, it is NP-hard to find a deviator rule with inefficiency lower than .

Proof:    Given a game, an initial strategy profile, and a value , the associated decision problem is whether there exists an order according to which the players are activated and results in a NE whose social cost is at most . We show a reduction from the Partition problem: Given a set of numbers such that , the goal is to find a subset such that . This problem is NP-hard even if it is known that such a subset exists. Given an instance of Partition, such that for some , it holds that , consider the network depicted in Figure 1, with the following initial strategy profile of players:

  • Players : Each Player , has a source and a target and she initially uses the upper edge of cost by herself. The alternative edge for Player costs .

  • Players : players whose objective is an -path. Initially, they all use the lower -edge of cost , thus, each of them pays .

  • Player whose objective is an -path. She initially uses the upper path and share its second edge with Player , so her initial cost is .

  • Player whose objective is an -path. She initially uses the upper path and share its second edge with Player , so her initial cost is .

Figure 1: The network constructed for a given Partition instance. Every edge is labeled by its cost and (in brackets) the number of players using it in the initial strategy profile.

The following simple observations limit the possible BRD sequences of the above game instance:

  1. Since , for all , each of the first players uses the -edge in any NE.

  2. The -players would benefit from a deviation only after the edge of cost is utilized by some other player.

  3. Players and will never use edges of cost . If they select a path that passes through , it will consist of all the -edges. The initial cost of this alternative path for Player is . She will deviate to this path only after her total share in the -edges is at most .

  4. As long as Player is using the upper path, deviating to a path through is not beneficial for Player , since such a path will incur her a cost of at least . Also, deviating to the lower path will incur her a cost of , which is more than her current cost. Therefore, Player will not deviate before Player .

Using the above observations we prove the statement of the reduction.

Claim 2.2

Let be a subset of such that . A deviator rule that knows can determine a BR-sequence that ends up with a NE whose social cost is . If is not detected then the final NE will have social cost .

Proof:    Assume that a subset such that is known. Consider a BR-sequence in which the players in deviate first (in an arbitrary order), then Player deviates, then Player , and then all the suboptimal players (in an arbitrary order). Since , the players in utilize -edges of total cost . When Player gets to deviate, she selects a path that passes through . Her share in the sub-path consisting of the -edges is for the utilized edges, and for the non-utilized edges. Since her current cost is , this deviation is indeed beneficial.

When Player gets to deviate, her share in the sub-path consisting of the -edges would be . Her best response is therefore the lower path since .

After Player deviates all the players using the expensive -edge will join Player , and the rest of the first players will deviate to their corresponding -edge. The social cost of the resulting NE would be .

For the other direction of the reduction assume that the social cost of some NE reachable from the given initial profile is a constant. We show that the deviator rule must activate a subset of the players , corresponding to partition elements of total size before Player deviates. In order to end up with a constant social cost, the expensive lower -edge was abandoned. By the above observations, the players using this edge deviated to the -edge, and in order for that to happen, Player must deviate to this -edge first. As stated, Player will not deviate before Player . Also, players must deviate sometime along the dynamics. Let () be the total cost of -edges utilized by Players before Player () deviated. Since Player must deviate before Player it holds that .

In order for the deviation to be profitable for Player , it must be that , since her newly incurred cost consists of her share in the cost of edges she share with players that already deviated and the cost of edges she uses alone. We conclude that .

In order for Player to deviate to the -edge it must be more beneficial than deviating to a path through . The latter would consists of the edge of cost , edges she share with Player and players in that already deviated, and edges that she share only with Player . Formally, . That is, .

Combining the above observations, we get that . This implies that for , we have . The set of players that utilize the edges of total cost correspond to a set of sum , which induces a partition.

Finally, note that any other BR-sequence leaves the lower expensive edge activated, and therefore, has total cost .    

2.2 Symmetric Network Formation Games

A network-formation game is symmetric if all the players have the same source and target nodes. Recall that the inefficiency of any deviator rule is upper bounded by the price of anarchy of the game. It is well known that the PoA of network formation games is (i.e., the number of players).

We first show that the Max-Cost rule may perform as poorly as the PoA, even in symmetric games on parallel-edge networks.

Observation 2.3

The inefficiency of Max-Cost in symmetric network formation games on parallel-edge networks is .

Proof:    Given , we present an instance over players for which . Consider the network depicted in Figure 2.

Figure 2: A network on which Max-Cost has inefficiency .

The initial profile consists of a single player using the top edge and players using the bottom edge. For the first step, the cost of the player on the top edge is and the costs of the players on the bottom one is . Therefore, Max-Cost chooses the player on the upper edge. That player migrates to the bottom edge, resulting in social cost . However, if the players on the bottom edge deviate first (to the middle edge), then the player on the top edge will also deviate to the middle edge, resulting in a NE profile with a social cost of . The ratio converges to as .  

On the other hand, we show that Min-Path is an optimal deviator rule, i.e., it always reaches the best NE reachable from any initial profile. Our analysis of Min-Path is based on the following Lemma:

Lemma 2.4

In symmetric network formation games, the path chosen by the first deviator is the unique path that will be chosen by all subsequent players, regardless of the order in which they deviate.

Proof:    Assume that player is the first to migrate and it chooses as a best response. Since is ’s best-response, after her deviation she cannot be suboptimal, and so are all the other players who use by symmetry. Assume, by a way of contradiction, that is a best response of a suboptimal player not using . Since can deviate to and incur a cost lower than the cost pays, but prefers to deviate to - a deviation of to will also be beneficial, a contradiction.  

Lemma 2.4 directly implies the optimality of Min-Path:

Theorem 2.5

Min-Path is an optimal deviator rule for symmetric network formation games.

Proof:    By lemma 2.4 after the first deviation, the first deviation dictates the NE to be reached. Thus, the set of BR paths in are the set of reachable NE, and choosing the cheapest one among them is optimal.  

2.3 Series of Parallel Paths (SPP) Networks

In this section we study network formation games played on SPP networks. An SPP network consists of segments, where each segment is a parallel-edge network. Let denote the vertex set, and for every , let denote the set of edges in segment (i.e., the parallel edges connecting and ). For a player , let , denote the set of edges player may choose.

Note that in an SPP network, a player’s choice of an edge in is independent of any other segment in her path. This implies that a network formation game on an SPP network consists of a sequence of symmetric games, where the set of players participating in each game varies. Combining this observation with Lemma 2.4 implies the following.

Lemma 2.6

In every network formation game played on an SPP network with segments, for every , and every BR sequence, let be the first player in the sequence such that , and let be the edge in chosen by player . Then is the unique edge in players deviate to.

2.3.1 Optimal BR sequence for single-source, multi-targets games

We first consider single-source, multi-targets instances. For this case, we devise a dynamic programming algorithm that computes an optimal BR sequence.

Theorem 2.7

For any single-source multi-targets network formation game played on an SPP network, an optimal BR sequence can be computed in time , where is the number of segments and is the number of players.

Proof:    Lemma 2.6 applied to a game with a single source implies that a migration of a player with target determines the path from the source () to that will be used in the NE reached.

Let denote the subnetwork consisting of the last segments in the input SPP. Formally, . Our dynamic programming solution is based on calculating an optimal solution for every suffix of the network. In particular, a solution for is a solution for the input SPP.

Let be the minimum cost of a path that can be reached in the game induced by , and let denote the first player that has to migrate in order to reach it. The base case of the dynamic programming is . Clearly, and can be easily found since the game induced by is symmetric:

For we calculate and as follows: For every player such that , denote by the minimal cost of a path that can be reached in the game induced by , if player is the first to perform her BR. Assume . By Lemma 2.6, the NE corresponding to consists of a prefix in , which is player ’s best response path. That prefix is followed by a suffix of cost . Denote the prefix’s cost by

We can compute using the following formulas:

The algorithm consists of an preprocessing in which the values are calculated for all and . Given , it is possible to calculate each of in time , giving a total of for the whole algorithm. Note that the player is a one determining , that is, . Given for all , the BR sequence begins with as the first player, if its target is then the next player to perform BR will be , etc.  

2.3.2 Optimal BR sequence for multi-sources, multi-targets games with proper intervals

In this section we consider multi-sources, multi-targets instances with proper intervals. A network formation game played on an SPP network is said to have proper intervals if for every two players it holds that if then . We denote by the set of segments from ’s source to ’s target, and call it the interval of player . I.e., .

Theorem 2.8

For any multi-sources multi-targets network formation game played on an SPP network with proper intervals, an optimal BR sequence can be computed in time , where is the number of segments and is the number of players.

As in the single-source case in the previous subsection, every deviation dictates the specific path that will be used in the NE reached. Therefore, the best NE path reached by a BR sequence that starts by a deviation of a specific player, consists of the player’s BR path and the best solution in the subnetworks that the player doesn’t use. Our dynamic programming solution uses this observation to find the player that the NE reached by a BR sequence that starts with her deviation is the optimal. To find that NE, we consider the player’s BR path and the optimal paths on the residual, strictly shorter subnetworks. Denote by the subnetwork consisting of the segments connecting and . That is, . The dynamic programming solution is based on calculating, for every , an optimal solution for the subnetwork by iterating over the subnetworks in ascending lengths. In particular, a solution for is a solution for the input SPP. Let be the minimum social cost in a NE that can be reached in the game induced by . Let denote the first player that has to migrate in order to reach that optimal NE. The base cases of the dynamic programming are all the subnetworks induced by a single segment, i.e., for all . Clearly, and can be easily found since the game induced by is symmetric. So we initialize

In addition, for all , we initialize .

For , we compute and as follows: For every player for which , denote by the minimal cost of a path that can be reached in the game induced by , if player is the first to perform her BR. Denote by the cost of ’s BR on the subnetwork .

When calculating We distinguish between four cases:

  • If then , i.e., if the subnetwork is contained in the ’s interval then its deviation will set the cost of the whole subnetwork.

  • If then , i.e., if the subnetwork has a suffix not contained in ’s interval then its deviation will set the cost of their intersection and we need to add the optimal cost of the suffix to the total cost.

  • If then , residue prefix equivalent to the suffix in the previous condition

  • Otherwise, , combines residue prefix and suffix of the subnetwork over the player’s interval. Note that since we assume a proper interval instance, no player that plays in can play in and vice versa, so the process of computing each of them is completely independent.

In each of the cases, the calculation of requires as sub-problems only values of for which is strictly shorter than . Also, our base cases include and for every , thus calculating can be done by iterating through in increasing length. We get that

The algorithm consists of an -time preprocessing in which the values are calculated for all and . Given , it is possible to calculate each of in time , giving a total of for the whole algorithm. The set of players that are calculated in the process form the optimal BR-sequence.

2.3.3 The performance of Min-Path in SPP networks

After presenting non-local optimal deviator rules, we turn to analyze the inefficiency of Min-Path that was shown to be optimal for symmetric network formation games. Given an SPP network and a BR-sequence, we say that a segment is unresolved if there are at least two players whose interval include the segment, and each of them will select a different edge in the segment if chosen to perform a BR next. The other segments are denoted resolved. By Lemma 2.6, after a player performs her best response, all the segments in her interval are resolved. Thus, no player migrates more than once. By definition, in a resolved segment, the edge chosen in every reachable Nash equilibria is determined already, therefore, migrations of players who use only resolved segments do not influence the reachable Nash equilibria and in the following analyses we ignore them. In other words, all the deviations we consider resolve at least one segment. We denote by the resolved segments after such deviations. Let denote the minimal cost of a NE reachable from by some BR sequence. Formally, .

Lemma 2.9

For any BR sequence of an SPP network instance, as long as there are unresolved segments, there exists a suboptimal player whose interval includes unresolved segments, and if this player is chosen next, then the cost of the unresolved segments she would set (”resolve”) is at most OPT.

Proof:    Let be an intermediate strategy profile in the BR sequence. Consider the players according to the order they deviate in some optimal BR sequence. Let be the first player in this order who is suboptimal in . Since no player prior to in the optimal sequence is suboptimal, the segments that would resolve by a deviation from are a subset of the segments she resolves in the optimal sequence. In the optimal sequence she obviously resolves these segments such that the selected edges are of total cost at most , and therefore this is an upper bound on the total cost of unresolved segments she would set by deviating from .  

Using the above lemma, we provide tight analysis on the performance of Min-Path for SPPs with multi-targets and single or multiple sources. Note that in a single-source instance, every player resolves the prefix of the network corresponding to her interval.

Theorem 2.10

The inefficiency of Min-Path in SPP network formation games with single-source and multi-targets is .

Proof:    We show that the total cost determined for the segments resolved in every iteration is at most . Since at least one segment is resolved in each iteration, the whole network’s cost is bounded by . Let be the -th player chosen to deviate by Min-Path and assume has unresolved segments. Let be the player guaranteed by Lemma 2.9. It may be that . Both players have the same BR path in the resolved segments and therefore differ only in their unresolved segments. Since Min-Path chose , the cost of her unresolved segments is at most the cost of ’s unresolved segments, which is at most OPT by Lemma 2.9.

Figure 3: A network on which Min-Path has inefficiency .

We show that the analysis is tight: Consider the network depicted in Figure 3. There are players, where is the target of player . In the initial strategy profile for every , . Note that in every segment, , connecting and , the upper edge costs and is used by the players and the lower edge costs and is used only by player . Thus, the initial cost of player is .

Clearly, in every segment, the players using the upper edge will benefit from deviating to the lower one and the player using the lower edge will benefit from deviating to the upper one. By Lemma 2.6, the first deviation will determine the edge that will be used in the NE reached.

Note that player ’s best response, is whose cost is . Player will be chosen by the Min-Path deviator rule, and will deviate to . After this deviation is included in the BR of all the players. We can therefore consider the game induced by the nodes , treating as the source. The resulting game is similar to the initial one, and the same analysis shows that the second deviation is of player to . Repeating the process, we get that NE reached by Min-Path consists of the edges and therefore .

On the other hand, another valid BR sequence is a one in which player migrates first. Her BR is deviating to all the lower edges. By Lemma 2.6, other players will follow and the BR sequence will converge to having . The ratio between the two social costs is . Since , we conclude that the inefficiency of Min-Path in SPP networks with single-source and multi-targets is .  

We turn to consider arbitrary instances and show that Min-Path can perform poorly.

Theorem 2.11

The inefficiency of Min-Path in SPP network formation games with multi-sources and multi-targets is .

Proof:    Let denote the total cost of resolved segments after deviations of players whose deviation resolved at least one segment. The segments that are resolved already in will be included in the NE reached, therefore, . We prove that for every ; i.e., . This implies that the total network’s cost is .

Let be the player chosen to deviate by Min-Path that has some unresolved segments. The total cost of the unresolved segments that resolves is . Let be a player guaranteed by Lemma 3. Since was chosen by Min-Path, the cost of ’s BR path is lower than the cost of ’s BR path. But the cost of ’s BR path is at least the cost of the unresolved segments in ’s BR path. On the other hand, the cost of ’s BR path equals the sum of the cost of her resolved segments, which is bounded by and the cost she would set to her unresolved segments, bounded by (by Lemma 3). Putting it all together, we get , as required.

We show that the bound is tight:

Figure 4: Min-Path lower bound for multi-sources multi-targets SPP.

Every edge is labelled by its cost and the number of players using it in the initial strategy profile. E.g., the edge costs and is used by players in the initial strategy profile.

Consider the network depicted in Figure 4. There are players as follows: For : The source of player is and her target is . ’s initial strategy is . Therefore, there is one player in each segment of the network who uses the lower edge. There are other players, the ”upper-players”, with objective who use the upper path . For every segment , the BR path of player is , since and the BR path of the upper-players is the lower path since .

An optimal BR sequence start by a deviation of an upper-player to the lower path , after that deviation all the players do not have a beneficial deviation and all the other upper-players will follow. The NE will be , whose cost .

Consider now the BR sequence induced by the Min-Path deviator rule. Notice that for two players , the BR-paths costs satisfy . The cost of the BR path of the upper-players is . Thus, Player whose BR path cost will be chosen first. After her migration to , the BR path of the upper players becomes . The cost of this path is , and therefore the second player to migrate will be Player .

Continuing in the same manner, the first migrations are of players and the BR path of the upper-players is , having cost . Player is the minimal of the first players that did not deviate and her BR path’s cost is , so the Min-Path deviator rule will choose player .

We conclude by induction that Min-Path deviator rule will select the players in order , and the NE that will be reached is , having . The inefficiency of Min-Path in this game is .

 

2.4 Local Rules for Extension Parallel Graphs

We now show that the inefficiency of any local deviator rule is as high as the PoA, namely, , even in the restricted class of EP networks. Recall that the state vector of a player consists of the player’s cost in her current profile and in the profile obtained by a deviation of the player, and the total cost of the path used by the player in the two profiles.

Theorem 2.12

For the class of single-source network formation games played on extension-parallel networks, the inefficiency of every local deviator rule is .

Proof:    Consider the network depicted in Figure 5(a). There are players, all sharing the source , and the targets are as depicted in the figure. Consider the following profile:

  • Player uses the path .

  • Player uses the path .

  • Players use the path .

  • Players use the path .

Figure 5: A local deviator rule fails if is preferred, and if is preferred.

Every edge is labelled by the edge cost and the number of players using it in the initial strategy profile. E.g., the edge