An Iterated Game of Uncoordinated Sharing of Licensed Spectrum Using Zero-Determinant Strategies

We consider private commons for secondary sharing of licensed spectrum bands with no access coordination provided by the primary license holder. In such environments, heterogeneity in demand patterns of the secondary users can lead to constant changes in the interference levels, and thus can be a source of volatility to the utilities of the users. In this paper, we consider secondary users to be service providers that provide downlink services. We formulate the spectrum sharing problem as a non-cooperative iterated game of power control where service providers change their power levels to fix their long-term average rates at utility-maximizing values. First, we show that in any iterated 2x2 game, the structure of the single-stage game dictates the degree of control that a service provider can exert on the long-term outcome of the game. Then we show that if service providers use binary actions either to access or not to access the channel at any round of the game, then the long-term rate can be fixed regardless of the strategy of the opponent. We identify these rates and show that they can be achieved using mixed Markovian strategies that will be clearly identified in the paper.

💡 Research Summary

The paper addresses the problem of secondary users sharing licensed spectrum without any coordination from the primary license holder, a scenario often referred to as a “private commons”. In such environments, heterogeneous demand patterns cause fluctuating interference, which in turn makes the utilities of the service providers volatile. The authors model the interaction between two downlink service providers as an iterated non‑cooperative game in which each provider chooses a transmission power level in every round. The key contribution is the application of zero‑determinant (ZD) strategies—originally discovered for the iterated Prisoner’s Dilemma—to this spectrum‑sharing setting.

First, the authors formalize any 2×2 repeated game as a Markov chain whose state is the pair of actions taken in the previous round. The transition probabilities are denoted by p_k for player X and q_k for player Y, where k indexes the four possible previous states. The stationary distribution π of this chain determines the long‑run average payoffs u_X and u_Y as linear functions of π and the one‑shot payoff matrices X and Y. By expressing π·f as the determinant of a modified transition matrix, they show that a player can enforce a linear relation a·u_X + b·u_Y = c by fixing one column of the matrix to a linear combination of the payoff vector and a constant. When the relation is chosen as a·u_X + b = 0, player X can fix its own long‑term payoff u_X to any value within a feasible interval, regardless of the opponent’s strategy.

Theorem 1 characterizes precisely when such control is possible. Let X_k,min and X_k,max be the minimum and maximum entries in row k of X. If there exist rows k_max and k_min such that X_kmax,max ≤ X_kmin,min, then player X can achieve any payoff u_X in the interval