$\frac{\rho}{1-\epsilon}$-approximate pure Nash equilibria algorithms for weighted congestion games and their runtimes

Our first algorithm is essentially a $\frac{1}{1-\epsilon}$ best response dynamic on a $\hat{\Psi}$-game of a weighted congestion game with polynomial latency functions for an arbitrary constant $\epsilon\in(0,1)$, see Algorithm 1. Caragiannis et al. (2015) have shown that this $\hat{\Psi}$-game has a potential function $\hat{\Phi}(\cdot)$, and its $\rho$-approximate pure Nash equilibria are essentially $d!\cdot\rho$-approximate pure Nash equilibria of the original weighted congestion game for each constant $\rho\geq 1.$

When players share the same strategy set, we prove with similar arguments to these of Chien and Sinclair (2011) that Algorithm 1 outputs a $\frac{1}{1-\epsilon}$-approximate pure Nash equilibrium of the $\hat{\Psi}$-game within $O(\frac{N\cdot(W+d!\cdot W^{d+1})}{\epsilon}\cdot\log(N\cdot\hat{c}_{\max}))$ iterations, where $N$ is the number of players and $\hat{c}_{\max}=\max_{S\in\prod_{v\in\mathcal{N}}\Sigma_{v}}\max_{u\in\mathcal{N}}\ \hat{c}_{u}(S)$ is the players’ maximum cost value. See Theorem 3 for a detailed result.

When players have different strategy sets, Algorithm 1 produces a $\frac{1}{1-\epsilon}$-approximate pure Nash equilibrium of $\hat{\Psi}$-game within $O(\frac{N\cdot\mu}{\epsilon}\cdot\log(N\cdot\hat{c}_{\max}))$ iterations, where $\mu=E\cdot A\cdot(1+d)!\cdot N^{d}\cdot W^{d+1},$ and $E$ is the number of resources, see Theorem 4.

Theorem 3 and Theorem 4 together imply that Algorithm 1 computes a $\frac{d!}{1-\epsilon}$-approximate pure Nash equilibrium of weighted congestion games with polynomial latency functions within an acceptable runtime. In particular, these runtimes are polynomial parameterized by $d$, $A$ and $W$. This improves the recent work of Feldotto et al. (2017) and Giannakopoulos et al. (2022), which compute $d^{d+o(d)}$-approximate pure Nash equilibria with more sophisticated algorithms.

We further improve the result by incorporating the idea of approximate potential function of Christodoulou et al. (2019) into our framework, and propose a refined $\frac{\rho}{1-\epsilon}$ best response dynamic for $\rho=\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1},$ see Algorithm 2. Here, we note that Christodoulou et al. (2019) have proved the existence of $\rho$-approximate pure Nash equilibria for each $\rho\in[\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1},d+1]$.

When players share the same strategy set and $\rho=\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1}$, Algorithm 2 computes a $\frac{\rho}{1-\epsilon}$-approximate pure Nash equilibria within $O(\frac{N\cdot(1+W)^{d+1}}{\epsilon}\cdot\log(N\cdot c_{\max}))$ iterations, where $\epsilon\in(0,1)$ is an arbitrary constant, and $c_{\max}$ is the players’ maximum cost value in the game $\Gamma$. See Theorem 5 for a detailed result. When players have different strategy sets, Algorithm 2 produces a $\frac{\rho}{1-\epsilon}$-approximate pure Nash equilibrium within $O(\frac{N\cdot\varpi}{\epsilon}\cdot\log(N\cdot c_{\max}))$ iterations, where $\varpi=E\cdot A\cdot(1+d)\cdot N^{d}\cdot W^{d+1},$ see Theorem 6.

While weighted congestion games with polynomial latency functions need not have exact potential functions, our results show that the naive idea of best response dynamics still lead to efficient algorithms for the computation of approximate pure Nash equilibria. In particular, the resulting algorithms compute approximate pure Nash equilibria of much higher accuracy than these proposed recently by Caragiannis et al. (2015), Feldotto et al. (2017) and Giannakopoulos et al. (2022), though our algorithms may have longer polynomial runtimes depending on more parameters.

Rosenthal (1973) proved that every unweighted congestion game has a potential function, and thus admits at least one pure Nash equilibrium. For weighted congestion games, Fotakis et al. (2005) and Panagopoulou and Spirakis (2007) showed the existence of potential functions for affine linear and exponential latency functions, respectively. Hence, such weighted congestion games have pure Nash equilibria. Beyond these cases, pure Nash equilibria may not exist, see Goemans et al. (2005) and Harks and Klimm (2012). In fact, Dunkel and Schulz (2008) even proved that deciding if an arbitrary weighted congestion game has a pure Nash equilibrium is a NP-hard problem.

While each unweighted congestion game has a pure Nash equilibrium, Fabrikant et al. (2004) have proved that computing such an equilibrium state is of PLS-complete (Johnson et al. (1988)), which is a complexity class locating in between the two well-known classes P and NP. Moreover, Ackermann et al. (2008) even showed that finding a pure Nash equilibrium in a unweighted congestion game with affine linear latency functions is PLS-complete. This becomes more even tough for weighted congestion games, for which Skopalik and Vöcking (2008) have proved that finding a $\rho$-approximate pure Nash equilibrium with a constant $\rho>1$ has already been PLS-complete.

Due to this extreme complexity, we may have to consider particular cases when we compute approximate pure Nash equilibria in weighted congestion games. For an arbitrary constant $\epsilon\in(0,1),$ Chien and Sinclair (2011) have shown a $\frac{1}{1-\epsilon}$ best response dynamic, which computes a $\frac{1}{1-\epsilon}$-approximate pure Nash equilibrium within $O(\frac{N\cdot\alpha}{\epsilon}\log(\frac{\Phi(S^{(0)})}{\Phi_{\min}}))$ iterations when the weighted congestion game is symmetric and the latency functions fulfill a so-called $\alpha$ bounded jump condition, where $\Phi(\cdot)$ is the potential function of the game and $\Phi_{\min}$ is the minimum values of the potential function.

For an arbitrary weighted congestion game $\Gamma$ with polynomial latency functions of degree at most an integer $d\geq 1,$ Caragiannis et al. (2015) defined a $\hat{\Psi}$-game with so-called $\hat{\Psi}$-functions. This $\hat{\Psi}$-game is essentially a potential game, and shares all other features with $\Gamma$, but has latency functions dominating the polynomial latency functions of $\Gamma$. With the help of this $\hat{\Psi}$-game, Caragiannis et al. (2015) then showed an algorithm for computing a $d^{O(d^{2})}$-approximate pure Nash equilibrium of $\Gamma$ within $O(g\cdot(1+\frac{m}{\gamma}))$ iterations with an arbitrary small $\gamma>0$, where $m=\text{log}\frac{\hat{c}_{\text{max}}}{\hat{c}_{\text{min}}}$, $\hat{c}_{\text{max}}$ and $\hat{c}_{\min}$ are the respective maximum and minimum cost of players of the $\hat{\Psi}$-game, $g=(1+m\cdot(1+\gamma^{-1}))^{d}\cdot d^{d}\cdot N^{2}\cdot\gamma^{-4}$, and $N$ is the number of players. Coupling with certain random mechanisms, Feldotto et al. (2017) improved this result of Caragiannis et al. (2015), and showed a randomized algorithm that efficiently outputs a $d^{d+o(d)}$-approximate pure Nash equilibrium of $\Gamma$ with a high probability.

Recently, Christodoulou et al. (2019) initiated a different research with the notion of an approximate potential function, which was first proposed by Chen and Roughgarden (2009). They proved that every weighted congestion game with polynomial latency functions of degree at most $d\geq 1$ has a $\rho$-approximate pure Nash equilibrium for each $\rho\in[\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1},d+1],$ where $W$ is the common upper bound of players’ weights. Inspired by Christodoulou et al. (2019), Giannakopoulos et al. (2022) showed a deterministic polynomial-time algorithm for computing $d^{d+o(d)}$-approximate pure Nash equilibrium of weighted congestion games within $O\left(g\cdot(1+(d+1)\cdot p)\cdot(m\cdot p+d+1)\right)$ iterations when the latency functions are polynomial with degree at most $d\geq 1,$ where $m=\text{log}\frac{c_{\text{max}}}{c_{\text{min}}}$, $c_{\text{max}}$ and $c_{\min}$ are the respective maximum and minimum cost of players, $g=N^{2}\cdot p^{3}\cdot(1+m\cdot(1+p))^{d}\cdot d^{d}+1$, $p=(2\cdot d+3)\cdot(d+1)\cdot(4\cdot d)^{d+1}=d^{d+o(d)}$, and $N$ is the number of players. To our knowledge, this is the best known result in the literature for the computation of approximate pure Nash equilibria in weighted congestion games.

We generalize the runtime analysis of $\frac{1}{1-\epsilon}$ best response dynamics in Chien and Sinclair (2011) from symmetric weighted congested games with $\alpha$ bounded jump latency functions to arbitrary weighted congestion games with polynomial latency functions. In particular, our results outperform these in the literature from the perspective of obtaining a preciser approximation of pure Nash equilibria.

The remaining of the paper is organized as follows. Section 2 defines the weighted congestion game model, and presents preliminary properties of $\hat{\Psi}-$games and of approximate potential functions. We propose the two algorithms and analyze their runtimes in Section 3. A short summary is then presented in Section 4.

We represent an arbitrary weighted congestion game $\Gamma$ by tuple $(\mathcal{N},\mathbb{E},(\Sigma_{u})_{u\in\mathcal{N}},\\ (c_{e})_{e\in\mathbb{E}},w)$ with components defined below.

G1.

$\mathcal{N}$ is a collection of $N$ different players.

G2.

$\mathbb{E}$ is a set of $E\geq 1$ different resources.

G3.

$\Sigma_{u}$ is a set of feasible strategies of player $u\in\mathcal{N}$. Here, each strategy $s_{u}\in\Sigma_{u}$ is a nonempty subset of $\mathbb{E}$, that is, $s_{u}\in\Sigma_{u}\subseteq 2^{\mathbb{E}}\setminus\{\emptyset\}=\{B\neq\emptyset:\ B\subseteq\mathbb{E}\}$.

G4.

$w=(w_{u})_{u\in\mathcal{N}}$ is a nonempty and finite vector, where each component $w_{u}>0$ is a positive weight of player $u\in\mathcal{N}$ that denotes an unsplittable demand and will be delivered by a single strategy $s_{u}\in\Sigma_{u}$. When $w_{u}=w_{u^{\prime}}$ for all $u,u^{\prime}\in\mathcal{N},$ then $\Gamma$ is called unweighted. Otherwise, $\Gamma$ is called weighted.

G5.

$c=(c_{e})_{e\in\mathbb{E}}$ is a vector of latency (or cost) functions of resources $e\in\mathbb{E}$. Here, each latency function $c_{e}:[0,\infty)\to[0,\infty)$ is continuous, nonnegative and nondecreasing.

Players are noncooperative, and each of them chooses a strategy independently. This then results in a (pure strategy) profile $S=(s_{u})_{u\in\mathcal{N}}$, where $s_{u}$ denotes the strategy used by player $u\in\mathcal{N}.$ For each resource $e\in\mathbb{E},$ we denote by $U_{e}(S):=\{w_{u}:\ e\in s_{u}\ \forall u\in\mathcal{N}\}$ a multi-set consisting of weights of players $u\in\mathcal{N}$ using resource $e$ in profile $S.$ Moreover, we denote by $L(M)$ the sum $\sum_{m\in M}m$ of all elements in a multi-set $M.$ Then $L(U_{e}(S))$ is the total weight of players using resource $e$ in profile $S.$ Furthermore, resource $e\in\mathbb{E}$ has a latency of $c_{e}(L(U_{e}(S)))$, and player $u\in\mathcal{N}$ has a cost of $c_{u}(S)=w_{u}\cdot\sum_{e\in s_{u}}c_{e}(L(U_{e}(S)))$, when player $u$ uses a strategy $s_{u}\in\Sigma_{u}$ in profile $S.$ Finally, the profile $S$ has a (total) cost of $C(S):=\sum_{u\in\mathcal{N}}c_{u}(S)=\sum_{e\in\mathbb{E}}L(U_{e}(S))\cdot c_{e}(L(U_{e}(S))).$

To facilitate our discussion, for each player $u\in\mathcal{N},$ we denote by $S_{-u}$ a subprofile $(s_{u^{\prime}})_{u^{\prime}\in\mathcal{N}\setminus\{u\}}$ of strategies used by “opponents” $u^{\prime}\in\mathcal{N}\setminus\{u\}$ of player $u,$ and write $S=(s_{u},S_{-u})$ when we need to show explicitly the strategy $s_{u}$ used by player $u.$

We call a profile $S$ a pure Nash equilibrium if

for each player $u\in\mathcal{N}$ and each strategy $s_{u}^{\prime}\in\Sigma_{u}$. Here, we recall that $c_{u}(S)$ is the cost of player $u$ in profile $S,$ and that $c_{u}(s^{\prime}_{u},S_{-u})$ is the resulting cost of player $u$ when player $u$ unilaterally moves from strategy $s_{u}$ to another strategy $s^{\prime}_{u}.$ Inequality (2.1) then simply states that $s_{u}$ is a best-response to $S_{-u}$ for each player $u\in\mathcal{N}$ when $S=(s_{u},S_{-u})$ is a pure Nash equilibrium. Hence, there is no incentive for a player to unilaterally change strategy when the profile is already a pure Nash equilibrium.

Monderer and Shapley (1996) have shown that pure Nash equilibria exist in an arbitrary finite game with a potential function, i.e., a potential game. For our model, a real-valued function $\Phi:\prod_{u\in\mathcal{N}}\Sigma_{u}\to[0,\infty)$ is called an (exact) potential function if

Hence, a potential function quantifies the cost reduction of a unilateral change of strategy. In particular, its global minimizers are pure Nash equilibria.

Rosenthal (1973) has shown that an arbitrary unweighted congestion game has a potential function, and so admits pure Nash equilibria. For weighted congestion games, Panagopoulou and Spirakis (2007) have proved the existence of potential functions for exponential latency functions, and Fotakis et al. (2005) have proved the existence of potential functions for affine linear latency functions. Beyond these cases, potential functions need not exist, and neither need pure Nash equilibria, see, e.g., Harks and Klimm (2012). Then we have to consider a relaxed notion of $\rho$-approximate pure Nash equilibrium instead.

Formally, for an arbitrary constant $\rho\geq 1$, a profile $S$ is called $\rho$-approximate pure Nash equilibrium if

for each player $u\in\mathcal{N}$ and each strategy $s_{u}^{\prime}\in\Sigma_{u}$. Inequality (2.3) means that a unilateral change of strategy reduces the cost at a rate of at most $\frac{c_{u}(S)-c_{u}(s^{\prime}_{u},S_{-u})}{c_{u}(S)}\leq 1-\frac{1}{\rho}$ in a $\rho$-approximate pure Nash equilibrium. Hence, the smaller the constant $\rho$ is, the closer the profile $S$ would approximate a precise pure Nash equilibrium. To facilitate our discussion, we call $\rho$ the approximation ratio of $S$ when $S$ is a $\rho$-approximate pure Nash equilibrium.

As Skopalik and Vöcking (2008) have shown that computing an approximate pure Nash equilibrium in an arbitrary congestion game is PLS-complete, we thus consider such a computation in a parametric setting. We focus only on weighted congestion games with polynomial latency functions fulfilling Conditions 1–2 below.

Condition 1 simply states that each player has a weight in $[1,W]$ for a constant upper bound $W\geq 1.$ Condition 2 requires that each of the polynomial latency functions has a degree at most $d\geq 1,$ and the ratios between their positive coefficients are bounded from above by a constant $A>0.$ Such weighted congestion games need not have precise pure Nash equilibria, see, e.g., Harks and Klimm (2012).

In Section 3, we will propose two algorithms computing $\frac{\rho}{1-\epsilon}$-approximate pure Nash equilibria for $\rho=d!$ and $\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1},$ respectively, and prove that their runtimes are polynomial parameterized by $d,$ $A$ and $W.$ The first algorithm is essentially a $\frac{1}{1-\epsilon}$ best response dynamic on $\hat{\Psi}$-game, while the second algorithm is a refined $\frac{\rho}{1-\epsilon}$ best response dynamic for $\rho=\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1}.$ Their runtime analyses will involve properties of $\hat{\Psi}$-games and of approximate potential functions. Before we formally define these two algorithms, let us first introduce the respective notions of $\hat{\Psi}$-games and of approximate potential functions in Section 2.2 and Section 2.3, so as to facilitate our further discussion.

While weighted congestion games fulfilling Conditions 1–2 need not have potential functions, Caragiannis et al. (2015) have shown that a suitable revision of the latency functions with the so-called $\hat{\Psi}$-functions would result in a $\hat{\Psi}$-game, which has an exact potential function.

Clearly, $\hat{\Psi}_{k}(M)$ coincides with a summation of all monomials of (total) degree $k$ over elements in $M$. In particular, $\hat{\Psi}_{1}(M)=L(M)$. Moreover. $\hat{\Psi}_{k}(M)$ and $L(M)^{k}$ share the same monomial terms, though different coefficients. Lemma 1 below collects some trivial properties of these $\hat{\Psi}$-functions. Readers may refer to Caragiannis (2009) for their proofs.

With these $\hat{\Psi}-$functions, we are now ready to introduce the notion of $\hat{\Psi}$-games proposed by Caragiannis et al. (2015).

To simplify notation, we write $\hat{c}_{e}(L(U_{e}(S)))$ simply as $\hat{c}_{e}(S)$ when we discuss the latency of a resource $e$ w.r.t. a profile $S$ in $\hat{\Psi}$-game $\hat{\Gamma}.$ Moreover, we employ $\hat{c}_{u}(S)=w_{u}\cdot\sum_{e\in s_{u}}\hat{c}_{e}(S)=w_{u}\cdot\sum_{e\in s_{u}}\sum_{k=0}^{d}a_{e,k}\cdot\hat{\Psi}_{k}(U_{e}(S))$ and $\hat{C}(S)=\sum_{u\in\mathcal{N}}\hat{c}_{u}(S)$ to denote the respective cost of a player $u\in\mathcal{N}$ and of a profile $S$ in $\hat{\Psi}$-game $\hat{\Gamma}.$ Clearly, $\hat{\Psi}$-game $\hat{\Gamma}$ would coincide with $\Gamma,$ and so $\hat{c}_{u}(S)=c_{u}(S)$ and $\hat{C}(S)=C(S),$ when $d=1.$ In general, they are different. Nonetheless, Lemma 1a yields immediately that $\hat{c}_{u}(S)\in[c_{u}(S),\ d!\cdot c_{u}(S)]$ for each $u\in\mathcal{N}$ and each $S\in\prod_{u^{\prime}\in\mathcal{N}}\Sigma_{u^{\prime}}.$ We summarize this in Lemma 2 below.

While Lemma 2 is trivial, it implies that a profile $S$ is a $\frac{d!}{1-\epsilon}$-approximate pure Nash equilibrium of $\Gamma$ if $S$ is a $\frac{1}{1-\epsilon}$-approximate pure Nash equilibrium of $\hat{\Psi}$-game $\hat{\Gamma}$ for each constant $\epsilon\in(0,1).$ This follows immediately from inequality (2.3). We summarize it in Lemma 3 below.

Lemma 3 indicates that we can obtain a $\frac{d!}{1-\epsilon}$-approximate pure Nash equilibrium of a weighted congestion game $\Gamma$ by computing a $\frac{1}{1-\epsilon}$-approximate pure Nash equilibrium of the $\hat{\Psi}$-game $\hat{\Gamma}$ when $\Gamma$ fulfills Conditions 1–2 and when $\hat{\Gamma}$ has a $\frac{1}{1-\epsilon}$-approximate pure Nash equilibrium. Theorem 1 below confirms this idea by showing that $\hat{\Psi}$-game $\hat{\Gamma}$ has a potential function, and so admits a precise pure Nash equilibrium when the weighted congestion game $\Gamma$ fulfills Conditions 1–2. Readers may refer to Caragiannis et al. (2015) for a proof.

With Theorem 1, we will employ a $\frac{1}{1-\epsilon}$ best response dynamic on the $\hat{\Psi}$-game $\hat{\Gamma}$ to compute a $\frac{d!}{1-\epsilon}$-approximate pure Nash equilibrium of $\Gamma.$ This then results in Algorithm 1 in Section 3.1.1.

Lemma 4 below bounds the potential function $\hat{\Phi}(S)$ of an arbitrary profile $S$ with the total cost $\hat{C}(S).$ While this is trivial, it will be very helpful when we derive the runtime of Algorithm 1 in Section 3.1.2. We omit its proof due to its triviality.

Lemma 4 yields immediately that the potential function value $\hat{\Phi}(S)$ is bounded from above by

which implies that the runtime of Algorithm 1 is polynomial parameterized by the three constants $d,$ $A$ and $W$. We will come to this later in Section 3.1.2.

In addition to above idea of $\frac{1}{1-\epsilon}$ best response dynamic on $\hat{\Psi}(\cdot),$ we will also consider a refined best response dynamic with the idea of approximate potential function proposed by Chen and Roughgarden (2009) in Section 3.2.

With this notion, Christodoulou et al. (2019) have shown the existence of $\rho$-approximate pure Nash equilibria in weighted congestion games fulfilling Conditions 1–2 for each constant $\rho\in[\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1},d+1].$ In particular, they showed that

defines a $\rho$-approximate potential function for $\rho=\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1}$ when

With this approximate potential function, we design a refined best response dynamic for computing $\frac{2\cdot W\cdot(d+1)}{(2\cdot W+d+1)\cdot(1-\epsilon)}$-approximate pure Nash equilibria, see Algorithm 2 in Section 3.2.1. Again, to facilitate the resulting runtime analysis, we introduce a trivial upper bound of this approximate potential function in Lemma 5 below.

Similar to the potential function $\hat{\Phi}(\cdot)$ of the $\hat{\Psi}$-game, Lemma 5 implies that the approximate potential function $\Phi(\cdot)$ has an upper bound of

which implies that Algorithm 2 has a polynomial runtime parameterized by the three constants $d,$ $A$ and $W$, see Section 3.2.2 below.