Beyond myopic best response (in Cournot competition)

A Nash Equilibrium is a joint strategy profile at which each agent myopically plays a best response to the other agents' strategies, ignoring the possibility that deviating from the equilibrium could lead to an avalanche of successive changes by other agents. However, such changes could potentially be beneficial to the agent, creating incentive to act non-myopically, so as to take advantage of others' responses. To study this phenomenon, we consider a non-myopic Cournot competition, where each firm selects whether it wants to maximize profit (as in the classical Cournot competition) or to maximize revenue (by masquerading as a firm with zero production costs). The key observation is that profit may actually be higher when acting to maximize revenue, (1) which will depress market prices, (2) which will reduce the production of other firms, (3) which will gain market share for the revenue maximizing firm, (4) which will, overall, increase profits for the revenue maximizing firm. Implicit in this line of thought is that one might take other firms' responses into account when choosing a market strategy. The Nash Equilibria of the non-myopic Cournot competition capture this action/response issue appropriately, and this work is a step towards understanding the impact of such strategic manipulative play in markets. We study the properties of Nash Equilibria of non-myopic Cournot competition with linear demand functions and show existence of pure Nash Equilibria, that simple best response dynamics will produce such an equilibrium, and that for some natural dynamics this convergence is within linear time. This is in contrast to the well known fact that best response dynamics need not converge in the standard myopic Cournot competition. Furthermore, we compare the outcome of the non-myopic Cournot competition with that of the standard myopic Cournot competition. Not surprisingly, perhaps, prices in the non-myopic game are lower and the firms, in total, produce more and have a lower aggregate utility

Truly Online Paging with Locality of Reference

The competitive analysis fails to model locality of reference in the online paging problem. To deal with it, Borodin et. al. introduced the access graph model, which attempts to capture the locality of reference. However, the access graph model has a number of troubling aspects. The access graph has to be known in advance to the paging algorithm and the memory required to represent the access graph itself may be very large. In this paper we present truly online strongly competitive paging algorithms in the access graph model that do not have any prior information on the access sequence. We present both deterministic and randomized algorithms. The algorithms need only O(k log n) bits of memory, where k is the number of page slots available and n is the size of the virtual address space. I.e., asymptotically no more memory than needed to store the virtual address translation table. We also observe that our algorithms adapt themselves to temporal changes in the locality of reference. We model temporal changes in the locality of reference by extending the access graph model to the so called extended access graph model, in which many vertices of the graph can correspond to the same virtual page. We define a measure for the rate of change in the locality of reference in G denoted by Delta(G). We then show our algorithms remain strongly competitive as long as Delta(G) >= (1+ epsilon)k, and no truly online algorithm can be strongly competitive on a class of extended access graphs that includes all graphs G with Delta(G) >= k- o(k).Comment: 37 pages. Preliminary version appeared in FOCS '9

Makespan Minimization via Posted Prices

We consider job scheduling settings, with multiple machines, where jobs arrive online and choose a machine selfishly so as to minimize their cost. Our objective is the classic makespan minimization objective, which corresponds to the completion time of the last job to complete. The incentives of the selfish jobs may lead to poor performance. To reconcile the differing objectives, we introduce posted machine prices. The selfish job seeks to minimize the sum of its completion time on the machine and the posted price for the machine. Prices may be static (i.e., set once and for all before any arrival) or dynamic (i.e., change over time), but they are determined only by the past, assuming nothing about upcoming events. Obviously, such schemes are inherently truthful. We consider the competitive ratio: the ratio between the makespan achievable by the pricing scheme and that of the optimal algorithm. We give tight bounds on the competitive ratio for both dynamic and static pricing schemes for identical, restricted, related, and unrelated machine settings. Our main result is a dynamic pricing scheme for related machines that gives a constant competitive ratio, essentially matching the competitive ratio of online algorithms for this setting. In contrast, dynamic pricing gives poor performance for unrelated machines. This lower bound also exhibits a gap between what can be achieved by pricing versus what can be achieved by online algorithms