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A bandwidth-sharing game is a type of resource allocation game designed to model the real-world allocation of bandwidth to many users in a network. The game is popular in game theory because the conclusions can be applied to real-life networks. The game is described as follows:

The game

• $n$ players
• each player $i$ has utility $U_i(x)$ for amount $x$ of bandwidth
• user $i$ pays $w_i$ for amount $x$ of bandwidth and receives net utility of $U_i(x)-w_i$
• the total amount of bandwidth available is $B$

We also use assumptions regarding $U_i(x)$

• $U_i(x)\ge 0$
• $U_i(x)$ is increasing and concave
• $U(x)$ is continuous

The game arises from trying to find a price $p$ so that every player individually optimizes their own welfare. This implies every player must individually find $argma{x}_x{U}_i(x)-px$. Solving for the maximum yields $U_i^{\text{'}}(x)=p$.

The problem

With this maximum condition, the game then becomes a matter of finding a price that satisfies an equilibrium. Such a price is called a market clearing price.

A possible solution

A popular idea to find the price is a method called fair sharing. In this game, every player $i$ is asked for amount they are willing to pay for the given resource denoted by $w_i$. The resource is then distributed in $x_i$ amounts by the formula $x_i=(\frac{w_i}{\sum _j{w}_j})*(B)$. This method yields an effective price $p=\frac{\sum _j{w}_j}{B}$. This price can proven to be market clearing thus the distribution $x_1,...,x_n$ is optimal. The proof is as so:

Proof

$argma{x}_{x_i}{U}_i(x_i)-w_i$

$⟹argma{x}_{w_i}{U}_i(\frac{w_i}{\sum _j{w}_j}*B)-w_i$
$⟹U_i^{\text{'}}(\frac{w_i}{\sum _j{w}_j}*B)(\frac{1}{\sum _j{w}_j}*B-\frac{w_i}{(\sum _j{w}_j{)}^2}*B)-1=0$
$⟹U_i^{\text{'}}(x_i)(\frac{1}{p}-\frac{1}{p}*\frac{x_i}{B})-1=0$
$⟹U_i^{\text{'}}(x_i)(1-\frac{x_i}{B})=p$

Comparing this result to the equilibrium condition above, we see that when $\frac{x_i}{B}$ is very small, the two conditions equal each other and thus, the fair sharing game is almost optimal.