Tight Bounds for the Price of Anarchy and Stability in Sequential Transportation Games

First, we define a new social cost function $U$ (from utilitarian), which is the sum of the cost of the users, and we present the sequential version of the transportation games. In particular, we give bounds for the $\mathrm{SPoA}$ and $\mathrm{SPoS}$ for all social cost functions considered in this paper ($U$, $E$, and $D$). Most of these bounds are tight or asymptotically tight, others are asymptotically tight if the number of buses is constant.

In short, in Section 3, we first show that the value of the $\mathrm{SPoS}$ is unbounded for non-metric instances for all the three social cost functions and, then, we proceed to show the value of the $\mathrm{SPoS}$ and $\mathrm{SPoA}$ for metric instances. In Table 1, we summarize the lower (LB) and upper (UB) bounds for the inefficiency of equilibria for the metric instances of transportation games, where the columns of this table represent all measures we use to analyze the inefficiency of equilibria: Price of Anarchy ($\mathrm{PoA}$), Price of Stability ($\mathrm{PoS}$), Sequential Price of Anarchy ($\mathrm{SPoA}$), and Sequential Price of Stability ($\mathrm{SPoS}$).

Finally, we extend the results of Fotakis et al. [1] analyzing the inefficiency of equilibria associated with function $U$, by giving bounds on the Price of Stability ($\mathrm{PoS}$) and Price of Anarchy ($\mathrm{PoA}$) for it (see Table 1 and Section 4).

We first focus on simultaneous transportation games, as introduced by Fotakis et al. [1]. An instance $\Gamma$ of a simultaneous transportation game, is a tuple $(N,M,G)$, where ${N=\{1,\ldots,n\}}$ is a set of $n$ players; $M=\{1,\ldots,m\}$ is a set of $m\geq 2$ buses; and $G=(V,E)$ is a complete undirected graph with a source node $s$ and a destination node $t$, where $V=N\cup\{s,t\}$; and $d:E\rightarrow\mathbb{R}_{+}$ is an associated distance function. Each player $i\in N$ is placed in its corresponding vertex in $G$, and they have as a goal to be transported from their location to $t$, while minimizing their cost.

An outcome in $\Gamma$ is an assignment $\sigma:N\rightarrow M$ in which every player $i\in N$ chooses one bus $j\in M$ (the bus that will pick her up), denoted by $\sigma_{i}$. We call $\mathcal{P}$ the set of all outcomes and, considering an outcome $\sigma\in\mathcal{P}$, player’s $i$ cost under $\sigma$, denoted by $c_{i}(\sigma)$, is the distance traveled by $\sigma_{i}$ from location $i$ to destination $t$.

An outcome $\sigma\in\mathcal{P}$ is a (Nash) equilibrium if no player can decrease her cost by changing her action, while the action of every other player remains the same. That is, $\sigma$ is an equilibrium if, for every $i\in N$ and every $\sigma^{\prime}\in\mathcal{P}$ such that $\sigma_{i}\neq\sigma_{i}^{\prime}$ and $\sigma_{j}=\sigma_{j}^{\prime}$ for all $j\in N\setminus\{i\}$, we have that $c_{i}(\sigma)\leq c_{i}(\sigma^{\prime})$.

In order to determine the routes for the buses, we consider that each bus ${j\in M}$ has an algorithm ${\cal A}_{j}$, which, given $V^{\prime}\subseteq V$, calculates its route which starts on node $s$, goes through all vertices of $V^{\prime}$ and finishes its route on node $t$. We consider that, as in Fotakis et al. [1], each algorithm ${\cal A}_{j}$, for $j\in M$, is based on a permutation $\pi_{j}:N\rightarrow N$, given as input, such that a bus $j$ will only visit players that have chosen bus $j$, following the order given by its permutation. That is, bus $j$ will do “shortcuts” in its permutation whenever it is possible. See the example below.

Given an outcome $\sigma$, we denote the $i$-th player picked up by bus $j$ by $p_{i}^{j}$ and $|\{i:\sigma_{i}=j\}|$ by $n_{j}$. Also, we abuse notation and consider $p_{n_{j}+1}^{j}=t$ for all bus $j\in M$.

We use three different social cost functions in order to measure the inefficiency of equilibria, with two of them ($D$ and $E$) already defined and analyzed by Fotakis et al. [1]. Since all three functions neglect the distance between $s$ and the first player picked up by a bus, we will ignore $s$ from now on.

The first social cost function is described as Vehicle Kilometers Traveled, which reflects the environmental impact of the game’s outcome. We define

Indeed, $D(\sigma)$ represents the total distance traveled by the buses, except the distance from $s$ to the first player of each bus.

The Egalitarian social cost function is a classical social cost function in game theory literature which, in our context, it is the worst traveling distance of a player. It is defined as

Notice that this value is also the maximum distance traveled by a single bus (again, ignoring $s$).

In this paper, we introduce the analysis of another social cost function, called the Utilitarian social cost function, which is also very common in algorithmic game-theory [13, 14, 15]. This function represents, in the context of transportation games, the sum of the distances travelled by all players, and it is defined as

Thus, $U(\sigma)/n$ can be seen as the average distance a player will travel to reach her final destination.

The Price of Anarchy and the Price of Stability of a class $\mathcal{G}$ of transportation games are defined by ${\mathrm{PoA}{}(f,\mathcal{G})=\sup_{\Gamma\in\mathcal{G}}\mathrm{PoA}{}(f,\Gamma)}$ and ${\mathrm{PoS}{}(f,\mathcal{G})=\sup_{\Gamma\in\mathcal{G}}\mathrm{PoS}{}(f,\Gamma)}$, respectively. We will use $\mathrm{PoA}{}(f)$ and $\mathrm{PoS}{}(f)$ when the class of transportation games $\mathcal{G}$ is clear from the context.

Fotakis et al. [1] showed that a NE always exists when all the buses have the same permutation or in metric instances with two buses, but they also proved that not all metric instances of this game possess a $\mathrm{NE}$. Because of this, they analyzed the inefficiency of equilibria of only the instances of the game that do have a $\mathrm{NE}$. We also make this assumption when analyzing the $\mathrm{PoA}{}(U)$ and the $\mathrm{PoS}{}(U)$ in Section 4.

In the sequential version of the transportation game, players still choose a bus $j\in M$ but, instead of announcing their actions simultaneously, they choose their actions following an arbitrary predefined order. For simplicity, we consider this order to be $(1,2,\ldots,n)$ as one can always change buses’ permutation accordingly. Therefore, player $i$ has to choose an action $\sigma_{i}$ knowing only the actions taken by the players $1,2,\ldots,i-1$. However, because we are considering a game with full information, when choosing an action, players will indeed take into consideration their successors’ behavior, so they will be able to fully anticipate their actions considering that they will behave selfishly.

Because of the game’s sequentiality, for each set of possible actions ${\sigma_{<i}=(\sigma_{1},\sigma_{2},\ldots,\sigma_{i-1})}$ chosen by the predecessors of player $i$, player $i$ needs to specify an action $\lambda_{i}(\sigma_{<i})$ to deal with all of those actions, that is, specify which bus should be chosen if player $1$ to $i-1$ chooses actions $\sigma_{<i}$. We call $\lambda_{i}$ a strategy, since it defines an action for every possible predecessors’ choices. Also, we use $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})$ to refer to the strategies’ choice of all players, which we call a strategy profile and we denote the set of all strategy profiles by $\mathcal{S}$.

Finally, the outcome of $\lambda$ is an outcome $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{n})$ such that $\sigma_{i}=\lambda_{i}(\sigma_{<i})$ for all $i\in N$, where, again, ${\sigma_{<i}=(\sigma_{1},\sigma_{2},\ldots,\sigma_{i-1})}$. That is, the outcome of $\lambda$ is such that $\sigma_{i}$ is the action chosen by player $i$ (according to $\lambda$) when player $j$ chooses $\sigma_{j}$ for $1\leq j<i$.

Following the definition given by Shoham and Leyton-Brown [6], we say that a strategy profile $\lambda$ is a Subgame Perfect Equilibrium ($\mathrm{SPE}$) if it induces a Nash Equilibrium in any subgame (the restriction of the game to some $\sigma_{<i}$). That is, $\lambda$ is a $\mathrm{SPE}$ if for all players $i$ and all $\sigma_{<i}$, $i$ cannot decrease her cost by changing to another strategy different from $\lambda(\sigma_{<i})$ in the subgame where $\sigma_{<i}$ is fixed for her predecessors (players $1,2,\ldots,i-1$) and $\lambda_{i+1},\lambda_{i+2},\ldots,\lambda_{n}$ are the strategies chosen by her successor (players $i+1,i+2,\ldots,n$).

Furthermore, we can express the sequential transportation game in its extensive form by considering a rooted $m$-ary tree, wherein each level $i$ ($1\leq i\leq n)$, player $i$ is responsible for taking decisions of all nodes in her level. In other words, she has to choose one of the $m$ buses knowing only the decisions of her $i-1$ predecessors. On the leaves (terminal nodes) is the information about the cost of each player according to that outcome (which can be seen as a path of choices from the root to that leaf). Since we are dealing with a game with full information, it always has at least one $\mathrm{SPE}$, which can be computed by backward induction, using the well-known Zermelo’s algorithm [16].

Regarding the inefficiency of equilibria, Paes Leme et al. [5] introduced the notion of the Sequential Price of Anarchy ($\mathrm{SPoA}$) as a tool to measure the quality of $\mathrm{SPE}$s.

Consider $f$ as one of the previously defined social cost function ($U$, $D$ or $E$). Notice that $f:\mathcal{P}\to\mathbb{R}$ and, thus, it measures the social cost of an outcome of the sequential game. For some strategy profile $\lambda\in\mathcal{S}$, we will abuse notation and write $f(\lambda)$ to denote the social cost of the outcome of $\lambda$ according to $f$.

The Sequential Price of Anarchy and the Sequential Price of Stability of a class of sequential transportation games $\mathcal{G}$ are defined by ${\mathrm{SPoA}{}(f,\mathcal{G})=\sup_{\Gamma\in\mathcal{G}}\mathrm{SPoA}{}(f,\Gamma)}$ and ${\mathrm{SPoS}{}(f,\mathcal{G})=\sup_{\Gamma\in\mathcal{G}}\mathrm{SPoS}{}(f,\Gamma)}$, respectively. Again, we will use $\mathrm{SPoA}{}(f)$ and $\mathrm{SPoS}{}(f)$ when the class of sequential transportation games $\mathcal{G}$ is clear from the context. Next, we show an example of an instance of a sequential transportation game.

As one could imagine, following the results of Fotakis et al. [1] stating that the $\mathrm{PoS}$ is unbounded for non-metric instances of the transportation game in its simultaneous version, we show that the $\mathrm{SPoS}$ is also unbounded for social cost functions $U$, $E$, and $D$ when dealing with non-metric instances, even if all buses’ permutations are equal.

Next, we show the value of the $\mathrm{SPoS}$ and $\mathrm{SPoA}$ for metric instances in relation with social function $D$. It turns out that this value is $n$, which is equal to the value of the $\mathrm{PoA}{}(D)$ and the $\mathrm{PoS}{}(D)$ for its simultaneous version [1]. First, we show that $\mathrm{SPoS}{}(D)\geq n$ and, then, we proceed by showing that $\mathrm{SPoA}{}(D)=n$.

We begin by showing a lower bound on the $\mathrm{SPoS}{}(E)$, and for this, we show in Example 3.1 a family of instances proposed by Fotakis et al. [1] for proving that ${\mathrm{PoS}{}(D)\geq n}$.

Next, in contrast with the results on the value of $\mathrm{PoA}$ related to function $E$ presented by Fotakis et al. [1] ($\mathrm{PoA}{}(E)=2\lceil\frac{n}{m}\rceil-1$ for $n>m$ and $\mathrm{PoA}{}(E)=1$ if $n\leq m$), the next theorem shows that the value of the $\mathrm{SPoA}$ is worse than in its simultaneous version for function $E$ even when ${n=m}$.

We begin by showing a lower bound for $\mathrm{SPoS}{}(U)$.

We now show a lower bound on the value of an optimal outcome for $U$.

Next, we show a general bound for $U$ in any outcome of a transportation game. Then, using this result, we show a tight bound on $\mathrm{SPoA}{}(U)$.