Inter-Session Network Coding with Strategic Users: A Game-Theoretic Analysis of Network Coding

A common assumption in the existing network coding literature is that the users are cooperative and non-selfish. However, this assumption can be violated in practice. In this paper, we analyze inter-session network coding in a wired network using game theory. We assume selfish users acting strategically to maximize their own utility, leading to a resource allocation game among users. In particular, we study the well-known butterfly network topology where a bottleneck link is shared by several network coding and routing flows. We prove the existence of a Nash equilibrium for a wide range of utility functions. We show that the number of Nash equilibria can be large (even infinite) for certain choices of system parameters. This is in sharp contrast to a similar game setting with traditional packet forwarding where the Nash equilibrium is always unique. We then characterize the worst-case efficiency bounds, i.e., the Price-of-Anarchy (PoA), compared to an optimal and cooperative network design. We show that by using a novel discriminatory pricing scheme which charges encoded and forwarded packets differently, we can improve the PoA. However, regardless of the discriminatory pricing scheme being used, the PoA is still worse than for the case when network coding is not applied. This implies that, although inter-session network coding can improve performance compared to ordinary routing, it is significantly more sensitive to users’ strategic behaviour. For example, in a butterfly network where the side links have zero cost, the efficiency can be as low as 25%. If the side links have non-zero cost, then the efficiency can further reduce to only 20%. These results generalize the well-known result of guaranteed 67% worst-case efficiency for traditional packet forwarding networks.

💡 Research Summary

The paper tackles a fundamental gap in the network‑coding literature: most prior work assumes that all participants are cooperative, whereas in real networks users often act selfishly to maximize their own payoff. To study the consequences of such strategic behaviour, the authors formulate a non‑cooperative game for inter‑session network coding on the classic butterfly topology, where a single bottleneck link is shared by two source‑destination pairs and optional side links. Each user chooses a transmission rate (or flow amount) as his strategy, and his utility is a generic, continuous, non‑decreasing function of that rate (e.g., logarithmic, linear, or any concave form). The cost of using a link is modeled by a price function, which may be linear or convex, and the authors also consider a discriminatory pricing scheme that charges coded packets and ordinary forwarded packets at different rates.

The game is defined by each user’s payoff = utility – total link cost incurred. By analysing the best‑response correspondences, the authors prove that a Nash equilibrium (NE) exists for a broad class of utility and price functions, using Kakutani’s fixed‑point theorem. A striking finding is that, unlike the traditional packet‑forwarding game where the NE is unique, the presence of coding can lead to multiple, even infinitely many, equilibria when system parameters (such as side‑link costs or price ratios) lie in certain regions. This multiplicity stems from the fact that coded flows share the bottleneck link, creating a flat region in each user’s best‑response function.

Having established equilibrium existence, the paper proceeds to evaluate the efficiency loss caused by selfish behaviour, i.e., the Price of Anarchy (PoA). The socially optimal solution—maximizing the sum of all users’ payoffs—exploits coding to double the effective capacity of the bottleneck link, yielding a substantial theoretical gain over pure routing. However, at a Nash equilibrium users may under‑utilize coding or over‑pay for shared resources, leading to a severe efficiency drop. The authors derive worst‑case PoA bounds under several scenarios:

With zero‑cost side links and no discriminatory pricing, the PoA can be as low as 0.25 (25 %).
Introducing a discriminatory pricing scheme—charging higher per‑packet fees for coded packets than for ordinary packets—improves the bound modestly, but the PoA never exceeds roughly 0.33 (33 %).
When side links have a positive cost, the worst‑case PoA deteriorates further to about 0.20 (20 %).

These figures are dramatically lower than the well‑known 2/3 (≈ 66.7 %) guarantee for traditional routing games, indicating that inter‑session network coding is far more vulnerable to selfish manipulation.

The analytical results are complemented by numerical simulations that sweep key parameters (link capacities, price coefficients, utility curvature). The simulations confirm the theoretical PoA limits and illustrate how the equilibrium set can become a continuum when the price ratio aligns with the side‑link cost. They also show that, while discriminatory pricing can shift the equilibrium toward higher coding usage, it cannot fully recover the cooperative optimum.

In the discussion, the authors argue that simple price differentiation is insufficient to safeguard the benefits of network coding. Effective mechanisms would need to provide stronger incentives for users to participate in coding, such as subsidies, reputation systems, or contract‑based agreements. Moreover, the current analysis is confined to a single bottleneck, wired setting; extending the framework to multiple bottlenecks, wireless fading channels, or dynamic repeated games would be valuable future work.

In summary, the paper delivers a rigorous game‑theoretic treatment of inter‑session network coding with selfish users, proves the existence (and possible multiplicity) of Nash equilibria, and quantifies the severe efficiency loss that can arise. It highlights that, although network coding can theoretically double throughput, its practical gains are highly contingent on the design of incentive‑compatible pricing or control mechanisms that mitigate strategic behaviour.