Desirable Fairness Properties Research Articles

The Shapley value for a fair division of group discounts for coordinating cooling loads.

We consider a demand response program in which a block of apartments receive a discount from their electricity supplier if they ensure that their aggregate load from air conditioning does not exceed a predetermined threshold. The goal of the participants is to obtain the discount, while ensuring that their individual temperature preferences are also satisfied. As such, the apartments need to collectively optimise their use of air conditioning so as to satisfy these constraints and minimise their costs. Given an optimal cooling profile that secures the discount, the problem that the apartments face then is to divide the total discounted cost in a fair way. To achieve this, we take a coalitional game approach and propose the use of the Shapley value from cooperative game theory, which is the normative payoff division mechanism that offers a unique set of desirable fairness properties. However, applying the Shapley value in this setting presents a novel computational challenge. This is because its calculation requires, as input, the cost of every subset of apartments, which means solving an exponential number of collective optimisations, each of which is a computationally intensive problem. To address this, we propose solving the optimisation problem of each subset suboptimally, to allow for acceptable solutions that require less computation. We show that, due to the linearity property of the Shapley value, if suboptimal costs are used rather than optimal ones, the division of the discount will be fair in the following sense: each apartment is fairly “rewarded” for its contribution to the optimal cost and, at the same time, is fairly “penalised” for its contribution to the discrepancy between the suboptimal and the optimal costs. Importantly, this is achieved without requiring the optimal solutions.

Walrasian Dynamics in Multi-Unit Markets

In a multi-unit market, a seller brings multiple units of a good and tries to sell them to a set of buyers that have monetary endowments. While a Walrasian equilibrium does not always exist in this model, natural relaxations of the concept that retain its desirable fairness properties do exist. We study the dynamics of (Walrasian) envy-free pricing mechanisms in this environment, showing that for any such pricing mechanism, the best response dynamic starting from truth-telling converges to a pure Nash equilibrium with small loss in revenue and welfare. Moreover, we generalize these bounds to capture all the (reasonable) Nash equilibria for a large class of (monotone) pricing mechanisms. We also identify a natural mechanism, which selects the minimum Walrasian envy-free price, in which for n=2 buyers the best response dynamic converges from any starting profile. We conjecture convergence of the mechanism for any number of buyers and provide simulation results to support our conjecture.

Iterative Local Voting for Collective Decision-making in Continuous Spaces

Many societal decision problems lie in high-dimensional continuous spaces not amenable to the voting techniques common for their discrete or single-dimensional counterparts. These problems are typically discretized before running an election or decided upon through negotiation by representatives. We propose a algorithm called Iterative Local Voting for collective decision-making in this setting. In this algorithm, voters are sequentially sampled and asked to modify a candidate solution within some local neighborhood of its current value, as defined by a ball in some chosen norm, with the size of the ball shrinking at a specified rate. We first prove the convergence of this algorithm under appropriate choices of neighborhoods to Pareto optimal solutions with desirable fairness properties in certain natural settings: when the voters' utilities can be expressed in terms of some form of distance from their ideal solution, and when these utilities are additively decomposable across dimensions. In many of these cases, we obtain convergence to the societal welfare maximizing solution.We then describe an experiment in which we test our algorithm for the decision of the U.S. Federal Budget on Mechanical Turk with over 2,000 workers, employing neighborhoods defined by various L-Norm balls. We make several observations that inform future implementations of such a procedure.

Rank Maximal Equal Contribution: A Probabilistic Social Choice Function

When aggregating preferences of agents via voting, two desirable goals are to incentivize agents to participate in the voting process and then identify outcomes that are Pareto efficient. We consider participation as formalized by Brandl, Brandt, and Hofbauer (2015) based on the stochastic dominance (SD) relation. We formulate a new rule called RMEC (Rank Maximal Equal Contribution) that is polynomial-time computable, ex post efficient and satisfies the strongest notion of participation. It also satisfies many other desirable fairness properties. The rule suggests a general approach to achieving very strong participation, ex post efficiency and fairness.

Improving performance of backpressured packet networks by integrating with an end-to-end congestion control algorithm

Most of the existing congestion control schemes are classified according to where the control decision is made and what the feedback is. In backpressure mechanism, control decision is made hop-by-hop inside the network. These schemes dynamically react to congestion in a short term time scale. Hop-by-hop backpressure avoids wasting the network resources and also would reduce congestion by taking advantage of additional resources available at upstream switches and stations. Hence, it has potential for improved throughput. On the other hand, in end-to-end congestion control algorithms, congestion information is fed back, either implicitly or explicitly, and the sources decide on how to react. In this paper, we propose an integrated hop-by-hop backpressure mechanism with an end-to-end congestion control algorithm. In this combination, the end-to-end algorithm controls the network’s quasi-static behavior, while the backpressure mechanism prevents occasional congestion and packet losses that may result from traffic bursts. To integrate, we use an appropriate class of MCFC (minimum cost flow control) algorithm with desirable fairness properties. Finally, performance and fairness properties of the integrated algorithm are studied. Simulation results show that in comparison with the rate-based and window-based end-to-end congestion control algorithm, the proposed integrated algorithm is more dynamic in reacting to congestion and also more efficient in terms of avoiding loss and deadlock. Provided that stability is guaranteed, the algorithm is also quite efficient in tracking maximum throughput. The simulation results show significant performance improvement for a hybrid wired-wireless network, in comparison with TCP.

Queue proportional scheduling via geometric programming in fading broadcast channels

For fading broadcast channels (BC), a throughput optimal scheduling policy called queue proportional scheduling (QPS) is presented via geometric programming (GP). QPS finds a data rate vector such that the expected rate vector over all fading states is proportional to the current queue state vector and is on the boundary of the ergodic capacity region of a fading BC. Utilizing the degradedness of BC for each fading state, QPS is formulated as a geometric program that can be solved with efficient algorithms. The GP formulation of QPS is also extended to orthogonal frequency-division multiplexing (OFDM) systems in a fading BC. The throughput optimality of QPS is proved, and it is shown that QPS can arbitrarily scale the ratio of each user's average queueing delay. Throughput, delay, and fairness properties of QPS are numerically evaluated in a fading BC and compared with other scheduling policies such as the well-known maximum weight matching scheduling (MWMS). Simulation results for Poisson packet arrivals and exponentially distributed packet lengths demonstrate that compared with MWMS, QPS provides a significant decrease in average queueing delay and has more desirable fairness properties.

The impact of multicast layering on network fairness

Many definitions of fairness for multicast networks assume that sessions are single rate, requiring that each multicast session transmits data to all of its receivers at the same rate. These definitions do not account for multirate approaches, such as layering, that permit receiving rates within a session to be chosen independently. We identify four desirable fairness properties for multicast networks, derived from properties that hold within the max-min fair allocations of unicast networks. We extend the definition of multicast max-min fairness to networks that contain multirate sessions, and show that all four fairness properties hold in a multirate max-min fair allocation, but need not hold in a single-rate max-min fair allocation. We then show that multirate max-min fair rate allocations can be achieved via intra-session coordinated joins and leaves of multicast groups. However, in the absence of coordination, the resulting max-min fair rate allocation uses link bandwidth inefficiently, and does not exhibit some of the desirable fairness properties. We evaluate this inefficiency for several layered multirate congestion control schemes, and find that, in a protocol where the sender coordinates joins, this inefficiency has minimal impact on desirable fairness properties. Our results indicate that sender-coordinated layered protocols show promise for achieving desirable fairness properties for allocations in large-scale multicast networks.

The impact of multicast layering on network fairness

Many definitions of fairness for multicast networks assume that sessions are single-rate, requiring that each multicast session transmits data to all of its receivers at the same rate. These definitions do not account for multi-rate approaches, such as layering, that permit receiving rates within a session to be chosen independently. We identify four desirable fairness properties for multicast networks, derived from properties that hold within the max-min fair allocations of unicast networks. We extend the definition of multicast max-min fairness to networks that contain multi-rate sessions, and show that all four fairness properties hold in a multi-rate max-min fair allocation, but need not hold in a single-rate max-min fair allocation. We then show that multi-rate max-min fair rate allocations can be achieved via intra-session coordinated joins and leaves of multicast groups. However, in the absence of coordination, the resulting max-min fair rate allocation uses link bandwidth inefficiently, and does not exhibit some of the desirable fairness properties. We evaluate this inefficiency for several layered multi-rate congestion control schemes, and find that, in a protocol where the sender coordinates joins, this inefficiency has minimal impact on desirable fairness properties. Our results indicate that sender-coordinated layered protocols show promise for achieving desirable fairness properties for allocations in large-scale multicast networks.