Congestion Games with Variable Demands

We initiate the study of congestion games with variable demands where the (variable) demand has to be assigned to exactly one subset of resources. The players’ incentives to use higher demands are stimulated by non-decreasing and concave utility functions. The payoff for a player is defined as the difference between the utility of the demand and the associated cost on the used resources. Although this class of non-cooperative games captures many elements of real-world applications, it has not been studied in this generality, to our knowledge, in the past. We study the fundamental problem of the existence of pure Nash equilibria (PNE for short) in congestion games with variable demands. We call a set of cost functions C consistent if every congestion game with variable demands and cost functions in C possesses a PNE. We say that C is FIP consistent if every such game possesses the alpha-Finite Improvement Property for every alpha>0. Our main results are structural characterizations of consistency and FIP consistency for twice continuously differentiable cost functions. Specifically, we show 1. C is consistent if and only if C contains either only affine functions or only homogeneously exponential functions (c(x) = a exp(p x)). 2. C is FIP consistent if and only if C contains only affine functions. Our results provide a complete characterization of consistency of cost functions revealing structural differences to congestion games with fixed demands (weighted congestion games), where in the latter even inhomogeneously exponential functions are FIP consistent. Finally, we study consistency and FIP consistency of cost functions in a slightly different class of games, where every player experiences the same cost on a resource (uniform cost model). We give a characterization of consistency and FIP consistency showing that only homogeneously exponential functions are consistent.

💡 Research Summary

This paper initiates a systematic study of congestion games in which each player must allocate a single, possibly variable, amount of demand to exactly one subset of resources. Unlike classical congestion games where players have fixed demands (or can split demand across multiple resources), the model captures realistic settings such as single‑path routing in communication networks or traffic flow where splitting is infeasible. Each player i chooses a strategy (x_i, d_i) where x_i ⊆ R is a set of resources (or a path) and d_i ≥ 0 is the demand placed on every resource in x_i. The player receives a non‑decreasing, concave utility U_i(d_i) that depends only on the amount of demand, and incurs a cost equal to the sum over used resources of d_i·c_r(ℓ_r), where c_r is the resource‑specific cost function and ℓ_r = Σ_{j: r∈x_j} d_j is the total load on r. The payoff is π_i = U_i(d_i) – Σ_{r∈x_i} d_i·c_r(ℓ_r). All cost functions are assumed to be twice continuously differentiable, strictly increasing, and non‑negative.

The central research questions are: (1) For which families of cost functions C does every game built from C admit a pure Nash equilibrium (PNE)? This property is called “consistency”. (2) For which C does every such game satisfy the α‑Finite Improvement Property (α‑FIP), i.e., every sequence of unilateral moves that improve a player’s payoff by at least α > 0 must be finite? This is called “FIP‑consistency”. The paper provides complete structural characterizations of both notions.

Main Results (Proportional Cost Model).
Let C be a set of non‑negative, strictly increasing, twice differentiable cost functions.

1. Consistency: C is consistent if and only if it falls into exactly one of the following two categories:
   a) Affine functions: Every c ∈ C has the form c(ℓ) = a·ℓ + b with a > 0, b ≥ 0.
   b) Homogeneously exponential functions: Every c ∈ C has the form c(ℓ) = a·e^{φℓ} where a > 0 may vary across resources but the exponent φ > 0 is common to all functions in C.
2. FIP‑consistency: C is FIP‑consistent if and only if C consists solely of affine functions. Homogeneously exponential functions do not guarantee α‑FIP, even though they guarantee existence of a PNE.

These results settle an open question raised by Orda et al. (2003) concerning the existence of PNE in single‑path routing games with variable demand. The proofs combine two novel ideas. For the “only‑if” direction, the authors start from known weighted congestion games (fixed demands) that lack a PNE, and construct corresponding variable‑demand games that inherit the same improvement cycles by carefully designing concave utility functions. This reduction shows that any cost function family that fails consistency in the weighted setting also fails it here, and it specifically excludes inhomogeneously exponential functions. For the “if” direction, the paper introduces essential generalized ordinal potentials—real‑valued functions that increase on a subset of improving moves. For affine costs the game is an exact potential game; for homogeneous exponential costs the authors construct a local essential potential, establishing the existence of a PNE.

Uniform Cost Model.
In a variant called the uniform cost model, the cost incurred on a resource does not depend on the player’s demand; each player simply pays c_r(ℓ_r) when using r. This model is motivated by large‑scale telecommunication pricing and certain scheduling contexts. The authors prove a contrasting characterization:

• Consistency holds iff C consists exclusively of homogeneously exponential functions (the same form as above). Notably, affine costs no longer guarantee a PNE in this model.
• FIP‑consistency is trivial: it holds only when C is empty. Nevertheless, for homogeneous exponential costs the improvement graph is weakly acyclic, and a convergent improvement dynamics can be defined.

A comprehensive table summarizes the existence of PNE and α‑FIP across four classes of cost functions (affine, homogeneous exponential, inhomogeneous exponential, and non‑affine/non‑exponential) for both variable‑demand and fixed‑demand (weighted) games, highlighting the structural differences introduced by variable demand.

Implications and Future Work.
The paper demonstrates that allowing variable demand dramatically restricts the set of cost functions that ensure equilibrium existence, compared with the weighted setting. This has practical relevance: system designers must choose cost functions (e.g., queuing delay functions, pricing schemes) that belong to the identified families to guarantee stable outcomes. Open directions include extending the analysis to non‑differentiable or non‑convex cost functions, exploring hybrid models where demand can be split across a limited number of paths, and incorporating dynamic demand adjustment mechanisms. Overall, the work establishes a foundational theory for congestion games with variable demand and provides clear guidelines for designing stable networked systems.