In many applications the state of a system depends on the behavior of individual participants that act selfishly in order to minimize their own private cost. Non-cooperative game theory uses the concept of Nash equilibria as a tool for the theoretical analysis of such systems. A Nash equilibrium is a state in which no participant has an incentive to deviate to another strategy. While mixed Nash equilibria, i.e., Nash equilibria in randomized strategies, are guaranteed to exist under mild assumptions on the players’ strategy spaces and the private cost functions they are often hard to interpret. As a consequence, attention is often restricted to pure Nash equilibria, i.e., Nash equilibria in deterministic strategies.
Rosenthal  introduced a class of games, called congestion games that models a variety of strategic interactions and is guaranteed to have pure Nash equilibria. In a congestion game, we are given a finite set of players and a finite set of resources . A strategy of each player is to choose a subset of the resources out of a set of subsets of resources allowable to her. In each strategy profile, each player pays for all used resources where the cost of a resource is a function of the number of players using it. Rosenthal used an elegant potential function argument to show that iterative improvement steps by the players converge to a pure Nash equilibrium and hence its existence is guaranteed.
Note that in congestion games each player using a resource has the same influence on the cost of this resource. To alleviate this limitation,  and  studied a natural generalization called weighted congestion games in which each player has a weight and the joint cost of the resource is , where is the total weight of players using . The joint cost of resource has to be covered by the set of players using it, i.e., , where is the cost share of player on resource . The cost sharing method of the game defines how exactly the joint cost of a resource is divided into individual cost shares . For weighted congestion games, the most widely studied cost sharing method is proportional sharing (PS), where the cost share of a player is proportional to her weight, i.e., . Unfortunately, weighted congestion games with proportional sharing in general do not admit a pure Nash equilibrium (see  for a characterization).
Kollias and Roughgarden  proposed to use the Shapley value (SV) for sharing the cost of a resource in weighted congestion games. In the Shapley cost-sharing method, the cost share of a player on a resource is the average marginal cost increase caused by her over all permutations of the players. Using the Shapley value restores the existence of a potential function and therefore the existence of pure Nash equilibria to such games .
Potential functions immediately give rise to a simple and natural search procedure to find an equilibrium by performing iterative improvement steps starting from an arbitrary state. Unfortunately, this process may take exponentially many steps, even in the simple case of unweighted congestion games111Note that in the unweighted case, proportional sharing and Shapley cost sharing coincide. and linear cost functions . Moreover, computing a pure Nash equilibrium in these games is intractable as the problem is PLS-complete , even for affine linear cost functions . This result directly carries over to our game class with Shapley cost-sharing. Given these intractability results, it is natural to ask for approximation which is formally captured by the concept of an -approximate pure Nash equilibrium. This is a state from which no player can improve her cost by a factor of . Recently, Caragiannis et al.  provided an algorithm to compute -approximate Nash equilibria for unweighted congestion games under proportional sharing. They also generalised their technique to weighted congestion games .
1.1 Our Contributions
We present an algorithm to compute -approximate Nash equilibria in weighted congestion games under Shapley cost sharing. In games with polynomial cost functions of degree at most , our algorithm achieves an approximation factor asymptotically close to . Similar to  our algorithm computes a sequence of improvement steps of polynomial length that yields a -approximate Nash equilibrium. Hence, our algorithm performs only a polynomial number of strategy updates. We show that our algorithm can also be used to compute -approximate pure Nash equilibria for weighted congestion games with proportional sharing which improves the approximation factor of in  to .
We note that our method does not immediately yield an algorithm with polynomial running time since computing the Shapley cost share of a player and hence an improvement step is computationally hard. However, we show that there is a polynomial-time randomized approximation scheme that can be used instead. This results in a randomized polynomial time algorithm that computes a strategy profile that is an approximate pure Nash equilibrium with high probability.
In the course of the analysis we exhibit an interesting relation between the Shapley cost share of a player and her proportional share. In the case of polynomial cost functions with constant degree, each of them can be approximated by the other within a constant factor. This insight leads to an alternative proof to  for the existence of approximate pure Nash equilibria in weighted congestion games with proportional cost sharing.
Finally, we derive bounds on the approximate Price of Anarchy which may be of independent interest as they allow to bound the inefficiency of approximately stable states.
1.2 Further Related Work
Congestion games have been introduced by Rosenthal  who proved the existence of pure Nash equilibria by an exact potential function. Games admitting a potential function are called potential games and each potential game is isomorphic to a congestion game . Weighted congestion games were introduced by Milchtaich  and studied by Fotakis et al. . Based on the Shapley value , the class of weighted congestion games using Shapley values (instead of proportional shares) was introduced by  and it was shown that such games are potential games.  extends this result by proving that a weighted generalisation of Shapley values is the only method that guarantee pure Nash equilibria. In contrast, proportional sharing does not guarantee existence of equilibria in general . Further research focuses on the quality of equilibria, measured by the Price of Anarchy (PoA) . For proportional sharing, Aland et al.  show tight bounds on the PoA. Gkatzelis et al.  show that, among all cost-sharing methods that guarantee existence of pure Nash equilibria, Shapley values minimise the worst PoA. Furthermore, tight bounds on PoA for general cost-sharing methods were given . For the extended model with non-anonymous costs by using set functions it was also shown that Shapley cost-sharing is the best method and tight results are given [18, 27].
Computing a pure Nash equilibrium for congestion games was shown to be PLS-complete  even for games with linear cost function  or games with only three players . Chien and Sinclair  study the convergence towards -approximate pure Nash equilibria in symmetric congestion games in polynomial time under a mild assumption on the cost functions. In contrast, Skopalik and Vöcking show that this result cannot be generalized to asymmetric games and that computing a -approximate pure Nash equilibrium is PLS-hard in general . Caragiannis et al.  give an algorithm which computes an -approximate equilibrium for linear cost functions and an -approximate equilibrium for polynomial cost functions with degree of . Weighted congestion games with proportional sharing do not posses pure Nash equilibria in general . However, the existence of -approximate equilibria for polynomial cost functions and -approximate equilibria for concave cost functions was shown  and Caragiannis et al.  present an algorithm for weighted congestion games and proportional sharing that computes -approximate equilibria for linear cost functions and -approximate equilibria for polynomial cost functions.
The computation of approximate equilibria requires the computation of Shapley values. In general, the exact computation is too complex. Mann and Shapley  suggest a sampling algorithm which was later analyzed by Bachrach et al.  for simple coalitional games and by Aziz and de Keijzer  for matching games. Finally, Liben-Nowell et al.  and Maleki  consider cooperative games with supermodular functions which correspond to our class.
2 Our Model
A weighted congestion game is defined as , where is the set of players, the set of resources, is the positive weight of player , the strategy set of player and the cost function of resource (drawn from a set of allowable cost functions). In this work, is the set of polynomial functions with maximum degree and non-negative coefficients. The set of outcomes of this game is given by , for an outcome, we write , where . Let be the outcome that results when player changes her strategy from to and let be the outcome that results when players play their strategies in and players the strategies in . The set of users of resource is defined by and the total weight on by . Furthermore, let and be variants of these definitions with a restricted player set . The Shapley cost of a player on a resource is given as a function of the player’s identity, the resource’s cost function and her users , i.e., . For simplicity, let be an abbreviation if all players are considered in a state . Let . Then, the joint cost on a resource is given by and the costs of players are such that . The total cost of a player equals the sum of her costs in the resources she uses, i.e. . The social cost of the game is given by Further define the social costs of a subset of players with .
The cost-sharing method is important for our analysis, as it defines how the joint cost on a resource is distributed among her users. In this paper, the methods we focus on are the Shapley value and the proportional cost-sharing, which we introduce in detail.
Shapley values. For a set of players , let be the set of permutations . For a , define as the set of players preceding player in and as the sum of their weights.
For the uniform distribution over, the Shapley value of a player on resource is given by
Proportional sharing. The cost of a player on a resource under proportional sharing is given by . For the rest of the paper, we write to indicate when we switch to proportional sharing.
-approximate pure Nash equilibrium. Given a parameter and an outcome , we call as -move a deviation from to where the player improves her cost by more than a factor , formally . We call the state an -approximate pure Nash equilibrium (-PNE) if and only if no player is able to perform a -move, formally it holds for every player and any other strategy that .
-approximate Price of Anarchy. Given a parameter , let be the set of -approximate pure Nash equilibria and the state of optimum, i.e., . Then the -approximate price of anarchy (-PoA) is defined as .
Kollias and Roughgarden  prove that weighted congestion games under Shapley values are potential games using the following potential.
Potential Function. Given an outcome and an arbitrary ordering of the players in , the potential is given by
-limited potential. We now restrict this potential function by allowing only a subset of players to participate and define the -limited potential as
-partial potential. Consider sets and such that . Then the -partial potential of set is defined by
If the set contains only one player, i.e., , then we write . In case of , . Intuitively, is the value that the players in contribute to the -limited potential.
-stretch. Similar to -PoA, we define a ratio with respect to the potential function. Let be the outcome that minimises the potential, i.e., . Then the -stretch is defined as
-limited -stretch. Additionally, we define a -stretch restricted to players in a subset . Let be the set of -approximate pure Nash equilibria where only players in participate. The rest of the players have a fixed strategy . Then we define the -limited -stretch as
3 Algorithmic Approach and Outline
Our algorithm is based on ideas by Caragiannis et al. . Intuitively, we partition the players’ costs into intervals in decreasing order. The cost values in one interval are within a polynomial factor. Note that this ensures that every sequence of -moves for of players with costs in one or two intervals converges in polynomial time.
After an initialization, the algorithm proceeds in phases from to . In each phase , players with costs in the interval do -approximate moves where is close to the desired approximation factor. Players with costs in the interval make -moves for some small . After a polynomial number of steps no such moves are possible and we freeze all players with costs in . These players will never be allowed to move again. We then proceed with the next phase. Note that at the time players are frozen, they are in an -approximate equilibrium. The purpose of the -moves of players of the neighboring interval is to ensure that the costs of frozen players do not change significantly in later phases. To that end we utilize a potential function argument. We argue about the potential of sub games among a subset of players. We can bound the potential value of an arbitrary -approximate equilibrium with the minimal potential value (using the stretch). Compared to the approach in , we directly work with the exact potential function of the game which significantly improves the results, but also requires a more involved analysis. We show that the potential of the sub game in one phase is significantly smaller than . Therefore, the costs experienced by players moving in phase are considerably lower than the costs of any player in the interval . The analysis heavily depends on the stretch of the potential function which we analyze in Section 6. The proof there is based on the technique of Section 5 in which we approximate the Shapley with proportional cost sharing. For the technical details in both sections we need some structural properties of costs-shares and the restricted potentials which we show in the next section.
4 Shapley and Potential Properties
The following properties of the Shapley values are extensively used in our proofs.
Fix a resource . Then for any set of players and , we have for :
, with and ,
, with ,
, with .
We proceed to the properties of the restricted types of potential defined before.
Let and be sets of players such that , and outcomes of the game such that the players in use the same strategies in both and , and an arbitrary player. Then
Next, we show that the potential property also holds for the partial potential.
Consider a subset and a player . Given two states, and , that differ only in the strategy of player , then .
The next lemma gives a relation between partial potential and Shapley values.
Given an outcome of the game, a resource and a subset , it holds that .
Summing up over all resources yields the next corollary.
Given an outcome of the game and a subset , it holds that .
5 Approximating Shapley with Proportional Cost-Shares
In this section we approximate the Shapley value of a player with her proportional share. This approximation plays an important role in our proofs of the stretch and for the computation.
For a player , a resource and any state , the following inequality holds between her Shapley and proportional cost:
Summing up over all implies the following corollary.
For a player and any state , the following inequality holds between her Shapley and proportional cost:
Any -approximate pure Nash equilibrium for a SV weighted congestion game of degree is a -approximate pure Nash equilibrium for the weighted congestion game with proportional sharing.
6 The Approximate Price of Anarchy and Stretch
Firstly, we upper bound the approximate Price of Anarchy for our game class.
Let and the maximum degree of the polynomial cost functions. Then
Similar to the -PoA, we also derive an upper bound on the -stretch which expresses the ratio between local and global optimum of the potential function.
Let and the maximum degree of the polynomial cost functions. Then an upper bound for the -stretch of polynomial SV weighted congestion games is
Let , the maximum degree of the polynomial cost functions and . Then
Let , the maximum degree of the polynomial cost functions and an arbitrary subset of players. Then
For , the maximum degree of the polynomial cost functions and an arbitrary subset of players,
7 Computation of Approximate Pure Nash Equilibria
To compute -approximate pure Nash equilibria in SV congestion games, we construct an algorithm based on the idea by Caragiannis et al. . The main idea is to separate the players in different blocks depending on their costs. The players who are processed first are the ones with the largest costs followed by the smaller ones. The size of the blocks and the distance between them is polynomially bounded by the number of players and the maximum degree of the polynomial cost functions . Formally, we define as the maximum cost among all players before running the algorithm. Let be a state of the game in which only player participates and plays her best move. Then, define as the minimum possible cost in the game. Let be an arbitrary constant such that , is the number of different blocks and the block size for any , where .
The algorithm is now executed in phases. Let be the current state of the game and, for each phase , let be the state before phase . All players with perform an -move with (almost -approximate moves), while all players with perform a -move with (almost pure moves). Let be the best response of player in state . The phase ends when the first and the second group of players are in an - and -approximate equilibrium, respectively. At the end of the phase, players with have irrevocably decided their strategy and have been added in the list of finished players. In addition, before the described phases are executed, there is an initial phase in which all players with can perform a -move to prepare the first real phase.
For the analysis, let be the set of deviating players in phase and denote the state after player has done her last move within phase .
An -approximate pure Nash equilibrium with can be computed with a polynomial number of improvement steps.
The main argument follows from bounding the -partial potential of the moving players in each phase (see Lemma 16). To that end, we first prove that the partial potential is bounded by the sum of the costs of players when they did their last move (Lemma 15).
For every phase , it holds that .
We now use the Lemma 15 and the stretch of the previous section to bound the potential of the moving players by the according block size.
For every phase , it holds that .
It remains to show that the running time is bounded and that the approximation factor holds. For the first, since the partial potential is bounded and each deviation decreases the potential, we can limit the number of possible improvement steps (see Lemma 17).
The algorithm uses a polynomial number of improvement steps.
We show next that every player who has already finished his movements will not get much worst costs at the end of the algorithm (see Lemma 18) and that there is no alternative strategy which is more attractive at the end (see Lemma 19).
Let be a player who makes her last move in phase of the algorithm. Then,
Let be a player who makes her last move in phase and let be an arbitrary strategy of . Then,
Next, we bound the approximation factor of the whole algorithm (see Lemma 20).
After the last phase of the algorithm, every player is in an -approximate pure Nash equilibrium with .
The approximation factor is in the order of .
This completes the proof of Theorem 14. ∎
We note that a significant improvement below of the approximation factor would require new algorithmic ideas as the lower bound of the PoA in  immediately yields a corresponding lower bound on the stretch.
This algorithm can be used to compute also approximate pure Nash equilibria in weighted congestion games (with proportional sharing). Such a game can now be approximated by a Shapley game losing only a factor of (by Lemma 8), which is included in .
For any weighted congestion game with proportional sharing, an -approximate pure Nash equilibrium with can be computed with a polynomial number of improvement steps.
7.1 Sampling Shapley Values
The previous section gives an algorithm with polynomial running time with respect to the number of improvement steps. However, each improvement step requires the multiple computations of Shapley values, which are hard to compute. For this reason, one can instead compute an approximated Shapley value with sampling methods. Since we are only interested in approximate equilibria, an execution of the algorithm with approximate steps has a negligible impact on the final result. The technical properties of Shapley values stated in Section 4 also hold for sampled instead of exact Shapley values with high probability.
For any constant , an -approximate pure Nash equilibrium with can be computed in polynomial time with high probability.
Given an arbitrary state and an arbitrary but fixed constant , Algorithm 2 computes a -approximation of for any player in polynomial running time with probability at least
For using the sampling in the computation of an improvement step, a Shapley value has to be approximated for each alternative strategy of a player and for each resource in the strategy. In the worst case, each player has to be checked for an available improvement step.
Given an arbitrary state and running the sampling algorithm at most times computes an improvement step for an arbitrary player with probability at least .
Lemma 17 gives a bound on the number of improvement steps. Using the sampling algorithm for , we can bound the total number of samplings:
During the whole execution of Algorithm 1 the sampling algorithm for is applied at most times and the computation of the approximate pure Nash equilibrium is correct with probability at least for an arbitrary constant c.
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Appendix A Proofs for the Properties in Section 4
Proof of Proposition 1.
Let . By the definition of Shapley values
For (b) and (c), consider . Observe, that only for permutations where either or the corresponding contribution to changes if we change the weight of but keep their sum the same. Fix a permutation with and pair it with the corresponding permutation where only and are swapped. Then the contribution of and to is
Since is convex in , we get that
Part (c) and (b) follow, respectively. Part (d) of the proposition is shown in . ∎
Proof of Proposition 2.
We prove the different parts separately:
For each , let . By definition of the -partial potential (3), we have
By the definition of limited potential (2), for an arbitrary , define , , as
Hart and Mas-Collel  proved that the potential is independent of the ordering that players are considered. As mentioned before, is a restriction of where only players in participate. Thus, independence from also applies to the limited potential.
Firstly, we focus on the first term of (a) and choose an ordering where the players in set are first. Then we observe that by substituting with , the cost share remains the same. This is due to the fact that any player coming after the players in set in the ordering has no impact in the cost computation. These are the players who belong in set (since we assume players in are first). Therefore, the first term of (a) equals to
Following the same technique for the second term of (a), we choose an ordering in which the players in are first. Then we can substitute with without affecting the term’s value. Therefore, (a) is equivalent to
For each , define to be equal to
Note that , . Intuitively, the first term computes the cost with respect to all players using resource , . Regarding the second term, if we take away some of these players, i.e., players in , then due to convexity the costs of the remaining players either remain the same or are reduced. This depends on the position players in had in the ordering. To simplify, for the rest of this proof, let
Since , we get that for each ,