/pdf/the-theory-of-industrial-organization-19bejph0ej.pdf

The Theory of Industrial Organization

We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times---the total latency---is minimized.In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance.In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total latency of the routes chosen by unregulated selfish network users may be arbitrarily larger than the minimum possible total latency; however, we prove that it is no more than the total latency incurred by optimally routing twice as much traffic.

/pdf/how-bad-is-selfish-routing-g644vu88rg.pdf

How bad is selfish routing

Information, trade and common knowledge

A major challenge in the operation of wireless communications systems is the efficient use of radio resources. One important component of radio resource management is power control, which has been studied extensively in the context of voice communications. With the increasing demand for wireless data services, it is necessary to establish power control algorithms for information sources other than voice. We present a power control solution for wireless data in the analytical setting of a game theoretic framework. In this context, the quality of service (QoS) a wireless terminal receives is referred to as the utility and distributed power control is a noncooperative power control game where users maximize their utility. The outcome of the game results in a Nash (1951) equilibrium that is inefficient. We introduce pricing of transmit powers in order to obtain Pareto improvement of the noncooperative power control game, i.e., to obtain improvements in user utilities relative to the case with no pricing. Specifically, we consider a pricing function that is a linear function of the transmit power. The simplicity of the pricing function allows a distributed implementation where the price can be broadcast by the base station to all the terminals. We see that pricing is especially helpful in a heavily loaded system.

/pdf/efficient-power-control-via-pricing-in-wireless-data-34gk0jwxel.pdf

Efficient power control via pricing in wireless data networks

The History of Economic Analysis

: Our aim is to explore efficiency, as well as strongness, of the Nash Equilibria (N.E.) of finite-person noncooperative games in strategic form. We show that--with smooth strategy sets and payoff functions--it is almost always the case that the N.E. are finite in number; efficient N.E. are extremal points; and strong N.E. are inactive points. Extremal (inactive) points are points in the Cartesian product of the players' strategy sets at which at least one (at most one) of the players is (is not) at a 'vertex' of her/his strategy set. Both sets of points are therefore thin (they are nonexistent if the strategy-sets are vertex-free). This result is not very surprising because efficiency or strongness is generally an outcome of cooperation. And, indeed, it has been part of the folklore of Game Theory (witness: the Prisoners' Dilemma).

/pdf/finiteness-and-inefficiency-of-nash-equilibria-3u0zsqb41z.pdf

Finiteness and Inefficiency of Nash Equilibria

Consider agents who undertake costly effort to produce stochastic outputs observable by a principal. The principal can award a prize deterministically to the agent with the highest output, or to all of them with probabilities that are proportional to their outputs. We show that, if there is sufficient diversity in agents' skills relative to the noise on output, then the proportional prize will, in a precise sense, elicit more output on average, than the deterministic prize. Indeed, assuming agents know each others' skills (the complete information case), this result holds when any Nash equilibrium selection, under the proportional prize, is compared with any individually rational selection under the deterministic prize. When there is incomplete information, the result is still true but now we must restrict to Nash selections for both prizes.We also compute the optimal scheme, from among a natural class of probabilistic schemes, for awarding the prize; namely that which elicits maximal effort from the agents for the least prize. In general the optimal scheme is a monotonic step function which lies "between" the proportional and deterministic schemes. When the competition is over small fractional increments, as happens in the presence of strong contestants whose base levels of production are high, the optimal scheme awards the prize according to the "log of the odds," with odds based upon the proportional prize.

/pdf/the-allocation-of-a-prize-r-1daisqauob.pdf

The Allocation of a Prize (R)

A note on an optimal garnishing rule

Consider agents who undertake costly effort to produce stochastic outputs observable by a principal. The principal can award a prize deterministically to the agent with the highest output, or to all of them with probabilities that are proportional to their outputs. We show that, if there is sufficient diversity in agents' skills relative to the noise on output, then the proportional prize will, in a precise sense, elicit more output on average, than the deterministic prize. Indeed, assuming agents know each others?skills (the complete information case), this result holds when any Nash equilibrium selection, under the proportional prize, is compared with any individually rational selection under the deterministic prize. When there is incomplete information, the result is still true but now we must restrict to Nash selections for both prizes. We also compute the optimal scheme, from among a natural class of probabilistic schemes, for awarding the prize; namely that which elicits maximal effort from the agents for the least prize. In general the optimal scheme is a monotonic step function which lies ?between?the proportional and deterministic schemes. When the competition is over small fractional increments, as happens in the presence of strong contestants whose base levels of production are high, the optimal scheme awards the prize according to the "log of the odds", with odds based upon the proportional prize.

/pdf/the-allocation-of-a-prize-expanded-28sgw68o3x.pdf

The Allocation of a Prize (Expanded)

There are many situations in which a customer’s proclivity to buy the product of any firm depends not only on the classical attributes of the product such as its price and quality, but also on who else is buying the same product. Under quite general circumstances, it turns out that customers’ influence on each other dynamically converges to a steady state. Thus we can model these situations as games in which firms compete for customers located in a "social network." A canonical example is provided by competition for advertisement on the web. Nash Equilibrium (NE) in pure strategies exist in general. In the quasi-linear version of the model, NE turn out to be unique and can be precisely characterized. If there are no a priori biases between customers and firms, then there is a cut-off level above which high cost firms are blockaded at an NE, while the rest compete uniformly throughout the network. Otherwise there is a tendency towards regionalization, with firms dominating disjoint territories. We also explore the relation between the connectivity of a customer and the money firms spend on him. This relation becomes particularly transparent when externalities are dominant: NE can be characterized in terms of the invariant measures on the recurrent classes of the Markov chain underlying the social network. Finally we consider convex (instead of linear) cost functions for the firms. Here NE need not be unique as we show via an example. But uniqueness is restored if there is enough competition between firms or if their valations of clients are anonymous.

Pradeep Dubey

Papers

Finiteness and Inefficiency of Nash Equilibria

The Allocation of a Prize (R)

A note on an optimal garnishing rule

The Allocation of a Prize (Expanded)

Competing for Customers in a Social Network