We continue the study of the effects of selfish behavior in the network design problem. We provide new bounds for the price of stability for network design with fair cost allocation for undirected graphs. We consider the most general case, for which the best known upper bound is the Harmonic number H n , where n is the number of agents, and the best previously known lower bound is 12/7 ≈ 1.778.We present a nontrivial lower bound of 42/23 ≈ 1.8261. Furthermore, we show that for two players, the price of stability is exactly 4/3, while for three players it is at least 74/48 ≈ 1.542 and at most 1.65. These are the first improvements on the bound of H n for general networks. In particular, this demonstrates a separation between the price of stability on undirected graphs and that on directed graphs, where H n is tight. Previously, such a gap was only known for the cases where all players have a shared source, and for weighted players.Keywords Algorithmic Game Theory ⋅ Price of Stability ⋅ Network Design A preliminary version of this work appeared in [8].

We consider the solution concept of stochastic stability, and propose the price of stochastic anarchy as an alternative to the price of (Nash) anarchy for quantifying the cost of selfishness and lack of coordination in games. As a solution concept, the Nash equilibrium has disadvantages that the set of stochastically stable states of a game avoid: unlike Nash equilibria, stochastically stable states are the result of natural dynamics of computationally bounded and decentralized agents, and are resilient to small perturbations from ideal play. The price of stochastic anarchy can be viewed as a smoothed analysis of the price of anarchy, distinguishing equilibria that are resilient to noise from those that are not. To illustrate the utility of stochastic stability, we study the load balancing game on unrelated machines. This game has an unboundedly large price of Nash anarchy even when restricted to two players and two machines. We show that in the two player case, the price of stochastic anarchy is 2, and that even in the general case, the price of stochastic anarchy is bounded. We conjecture that the price of stochastic anarchy is O(m), matching the price of strong Nash anarchy without requiring player coordination. We expect that stochastic stability will be useful in understanding the relative stability of Nash equilibria in other games where the worst equilibria seem to be inherently brittle.

We consider the classical problem of scheduling preemptible jobs, that arrive over time, on identical parallel machines. The goal is to minimize the total completion time of the jobs. In standard scheduling notation of Graham et al. [5], this problem is denoted P | rj, pmtn | P j cj . A popular algorithm called SRPT, which always schedules the unfinished jobs with shortest remaining processing time, is known to be 2-competitive, see Phillips et al. [13,14]. This is also the best known competitive ratio for any online algorithm. However, it is conjectured that the competitive ratio of SRPT is significantly less than 2. Even breaking the barrier of 2 is considered a significant step towards the final answer of this classical online problem. We improve on this open problem by showing that SRPT is 1.86-competitive. This result is obtained using the following method, which might be of general interest: We define two dependent random variables that sum up to the difference between the cost of an SRPT schedule and the cost of an optimal schedule. Then we bound the sum of the expected values of these random variables with respect to the cost of the optimal schedule, yielding the claimed competitiveness. Furthermore, we show a lower bound of 21/19 for SRPT, improving on the previously best known 12/11 due to Lu et al. [11].

We present a poly-log-competitive deterministic online algorithm for the online transportation problem on hierarchically separated trees when the online algorithm has one extra server per site. Using metric embedding results in the literature, one can then obtain a poly-log-competitive randomized online algorithm for the online transportation on an arbitrary metric space when the online algorithm has one extra server per site.

In this work we simulate algorithmic trading (AT) in asset markets to clarify its impact. Our markets consist of human and algorithmic counterparts of traders that trade based on technical and fundamental analysis, and statistical arbitrage strategies. Our specific contributions are: (1) directly analyze AT behavior to connect AT trading strategies to specific outcomes in the market; (2) measure the impact of AT on market quality; and (3) test the sensitivity of our findings to variations in market conditions and possible future events of interest. Examples of such variations and future events are the level of market uncertainty and the degree of algorithmic versus human trading. Our results show that liquidity increases initially as AT rises to about 10% share of the market; beyond this point, liquidity increases only marginally. Statistical arbitrage appears to lead to significant deviation from fundamentals. Our results can facilitate market oversight and provide hypotheses for future empirical work charting the path for developing countries where AT is still at a nascent stage.