Lecture 16 Selfish Network Routing, Congestion Games, and the Price of Anarchy

(Selfish) Network Routing as a Game

Congestion Games

A Congestion Game,

has:

• A finite set $N = \lbrace 1, \cdots, n\rbrace $ of players.
• A finite set of $R = \lbrace 1, \cdots, m\rbrace $ of resources, (could be edges).
• For each player, $i$, a set $Z_i \subseteq 2^{R}$, of admissible strategies for player $i$. So a pure strategy $s_i\in Z_i$ is a set of resources (edges).
• Each resource $r \in R$ has a cost function: $d_r: \mathbb{N} \to \mathbb{Z}$. Intuitively, $d_r(j)$ is the cost of using resource $r$ if there are $j$ agents simultaneously using $r$.
• For a pure strategy profile $s = (s_1, \cdots, s_n) \in Z_1 \times \cdots \times Z_n$, the congestion (how many agents) on resource $r$ is: $n_r(s) = \lvert\lbrace i\mid r\in s_i\rbrace \rvert$. I.e., if agent $i$ has resource $r$, then $n_r(s_i) = \lvert \lbrace i\rbrace \rvert = 1$.

• Under strategy profiles $s = (s_1, \cdots, s_n)$, the total cost to player $i$ is

  \[C_i(s) \doteq \sum_{r\in s_i}d_r(n_r(s))\]

  Every player $i$ wants to minimize its own (expected) cost.

Best response dynamics and pure Nash Equilibria

In a congestion game $G$, for any pure strategy profile \(s = (s_1, \cdots, s_n)\), suppose that some player $i$ has a better alternative strategy, \(s^\prime_i\in Z_i\), such that \(C_i(s_{-i}; s_i^\prime) < C_i(s)\).

Player $i$ can switch (unilaterally) from $s_i$ to $s_i^\prime$. This takes us from profile perform a sequence of such (strict) improvement steps.

Theorem: ([Rosenthal’73]) In any congestion game, every sequence of strict improvement steps is necessarily finite, and terminates in a pure NE.

Thus, in particular, every congestion game has a pure strategy NE.

Proof: Consider the following potential function (from potential energy):

This function computes the total cost or benefit of the players^\prime choices in the game. Sometimes called the social cost function. the Let $s^\prime := (s_{-i}; s_i^\prime)$.

Claim: $\varphi(s) - \varphi(s^\prime) = C_i(s) - C_i(s^\prime)$.

For $i^\prime\in \lbrace 1, \cdots, n \rbrace$, define

By exchanging the order of summation in equation $(1)$ for

\(\varphi(s) = \sum_{i=1}^n\sum_{r\in s_i} d_r(n^{(i)}_r(s))\).

Now note that $n^{(n)}_r(s) = n_r(s)$. Thus

So, if player $n$ switches from strategy $s_n$ to $s_n^\prime$, leading us from profile $s$ to $s^\prime$, then $\varphi(s) - \varphi(s^\prime) = C_i(s) - C_i(s^\prime)$.

$n_r(s)$ is the total congestion on resource $r$. The formula $\sum_{j=1}^{n_r(s)}$ doesn’t need have a intuitively explanation. This constructed formula represents a sense of potential which is quite like the gravitational potential energy.

The claim shows that $\varphi$ is a so-called exact potential, i.e., if a single player decreases its latency by a value of $\Delta > 0$, then $\varphi$ decreases by exactly the same amount.

Continue to proof: Observe that every strict improvement step must decreases the value of the potential function $\varphi(s)$ by at least $1$ (the costs $d_r(s)$ are all integers). Furthermore, there are only finitely many pure strategies $s$, so there are finite integers:

$a = \min_s\varphi(s)$ and $b = \max_s\varphi(s)$. Thus, every improvement sequence is finite.

Finally, note that the last profile $s$ in any improvement sequence which can not be further improved is, by definition, a pure Nash equilibrium.

A flow network game and Braess’s paradox

    graph LR
        s -- 1 --> t
        s -- x --> t
    

    graph LR
        s -- 1 --> b -- x --> t
        s -- x --> a -- 1 --> t
    

Social welfare and the price of anarchy

Price of anarchy in auctions: The Price of Anarchy (PoA) measures how the social welfare of a system degrades due to selfish behavior of its agents.

Recall that in a strategic game $\Gamma$, “utilitarian social welfare”, ${\rm welfare}(x)$, under a particular profile of mixed strategies $x\in X$, is defined as ${\rm welfare(x)}:= \sum_{i=1}^n U_i(x)$. For a game $\Gamma$, let ${\rm NE}(\Gamma)$ be the set of NE’s of $\Gamma$.

For our next definition suppose ${\rm welfare}(x) > 0$ for all $x\in X$.

One approach to maximizing the social welfare is designing a truthful mechanism. In such a mechanism, each agent is incentivized to report his true valuations to the items. Then, the auctioneer can calculate and implement an allocation that maximizes the sum of values. An example to such a mechanism is the VCG auctions.

A version of “the price of anarchy” can be defined as the ratio between the optimal social welfare and the social welfare in the worst equilibrium (where does the social welfare degrade to): ([Koutsoupias-Papadimitriou98])

Note: this ratio is $\geq 1$ and larger means “worse”.

Pure price of anarchy

In some settings, such as congestion games, where we know that a pure equilibrium exists, it is sometimes more sensible to compare the best overall outcome to the worst pure-NE outcome.

Let pure-${\rm NE}(\Gamma)$ denote the set of pure NEs in the game $\Gamma$. For settings (such as congestion games) where we know pure-${\rm NE}(\Gamma)$ is non-empty, we define “the pure price of anarchy” as:

Price of anarchy in the flow network game

• For flow $f$ let ${\rm welfare}(f):= 1/(\text{average s-t-delay})$
• In Braess’s paradox, the price of anarchy is $\frac{4}{3}$: by playing the NE the average delay is 2, but playing half-and-half on the upper and lower route, the average delay is $\frac{3}{2}$ (and that’s optimal).
• [Roughgarden-Tardos’00] showed that in a more general flow network setting (where there can be multiple source-destination pairs $(s_j, t_j)$), as long as congestions labeling edges are linear functions of $x$, the worst-case price of anarchy is $\frac{4}{3}$.

Back to atomic network congestion games

By an “atomic” network congestion game, we simply mean a standard network congestion game with a finite number of players, where each aims to minimize its own cost. (Whereas in non-atomic network flow games that average cost in equilibrium is uniquely determined, this is not the case with atomic network congestion games.)

Theorem: [Christodoulou-Koutsoupias’2005]. The pure price of anarchy for a pure NE in atomic network congestion games with linear utilities is $\frac{5}{2}$.