Bandwidth-sharing game
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. The game involves formula_1 players. Each player formula_2 has utility formula_3 for formula_4 units of bandwidth. Player formula_2 pays formula_6 for formula_4 units of bandwidth and receives net utility of formula_8. The total amount of bandwidth available is formula_9. Regarding formula_3, we assume The game arises from trying to find a price  formula_14 so that every player individually optimizes their own welfare. This implies every player must individually find formula_15. Solving for the maximum yields formula_16. 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. Possible solution. A popular idea to find the price is a method called fair sharing. In this game, every player formula_2 is asked for the amount they are willing to pay for the given resource denoted by formula_6. The resource is then distributed in formula_19 amounts by the  formula formula_20. This method yields an effective price formula_21. This price can proven to be market clearing; thus, the distribution formula_22 is optimal. The proof is as so: Proof. We have formula_23 formula_24. Hence, from which we conclude and thus formula_27 Comparing this result to the equilibrium condition above, we see that when formula_28 is very small, the two conditions equal each other and thus, the fair sharing game is almost optimal.