Reciprocity-driven Sparse Network Formation

A resource exchange network is considered, where exchanges among nodes are based on reciprocity. Peers receive from the network an amount of resources commensurate with their contribution. We assume the network is fully connected, and impose sparsity constraints on peer interactions. Finding the sparsest exchanges that achieve a desired level of reciprocity is in general NP-hard. To capture near-optimal allocations, we introduce variants of the Eisenberg-Gale convex program with sparsity penalties. We derive decentralized algorithms, whereby peers approximately compute the sparsest allocations, by reweighted l1 minimization. The algorithms implement new proportional-response dynamics, with nonlinear pricing. The trade-off between sparsity and reciprocity and the properties of graphs induced by sparse exchanges are examined.Comment: 19 page

Market Equilibrium with Transaction Costs

Identical products being sold at different prices in different locations is a common phenomenon. Price differences might occur due to various reasons such as shipping costs, trade restrictions and price discrimination. To model such scenarios, we supplement the classical Fisher model of a market by introducing {\em transaction costs}. For every buyer $i$ and every good $j$, there is a transaction cost of \cij; if the price of good $j$ is $p_j$, then the cost to the buyer $i$ {\em per unit} of $j$ is p_j + \cij. This allows the same good to be sold at different (effective) prices to different buyers. We provide a combinatorial algorithm that computes $\epsilon$-approximate equilibrium prices and allocations in $O\left(\frac{1}{\epsilon}(n+\log{m})mn\log(B/\epsilon)\right)$ operations - where $m$ is the number goods, $n$ is the number of buyers and $B$ is the sum of the budgets of all the buyers

Tit-for-Tat Dynamics and Market Volatility

We study the tit-for-tat dynamic in production markets, where each player can make a good given as input various amounts of goods in the system. In the tit-for-tat dynamic, each player allocates its good to its neighbors in fractions proportional to how much they contributed in its production in the last round. Tit-for-tat does not use money and was studied before in pure exchange settings. We study the phase transitions of this dynamic when the valuations are symmetric (i.e. each good has the same worth to everyone) by characterizing which players grow or vanish over time. We also study how the fractions of their investments evolve in the long term, showing that in the limit the players invest only on players with optimal production capacity

Exchange of Services in Networks: Competition, Cooperation, and Fairness

Exchange of services and resources in, or over, networks is attracting nowadays renewed interest. However, despite the broad applicability and the extensive study of such models, e.g., in the context of P2P networks, many fundamental questions regarding their properties and efficiency remain unanswered. We consider such a service exchange model and analyze the users' interactions under three different approaches. First, we study a centrally designed service allocation policy that yields the fair total service each user should receive based on the service it others to the others. Accordingly, we consider a competitive market where each user determines selfishly its allocation policy so as to maximize the service it receives in return, and a coalitional game model where users are allowed to coordinate their policies. We prove that there is a unique equilibrium exchange allocation for both game theoretic formulations, which also coincides with the central fair service allocation. Furthermore, we characterize its properties in terms of the coalitions that emerge and the equilibrium allocations, and analyze its dependency on the underlying network graph. That servicing policy is the natural reference point to the various mechanisms that are currently proposed to incentivize user participation and improve the efficiency of such networked service (or, resource) exchange markets.Comment: to appear in ACM Sigmetrics 201

On Nash Dynamics of Matching Market Equilibria

In this paper, we study the Nash dynamics of strategic interplays of n buyers in a matching market setup by a seller, the market maker. Taking the standard market equilibrium approach, upon receiving submitted bid vectors from the buyers, the market maker will decide on a price vector to clear the market in such a way that each buyer is allocated an item for which he desires the most (a.k.a., a market equilibrium solution). While such equilibrium outcomes are not unique, the market maker chooses one (maxeq) that optimizes its own objective --- revenue maximization. The buyers in turn change bids to their best interests in order to obtain higher utilities in the next round's market equilibrium solution. This is an (n+1)-person game where buyers place strategic bids to gain the most from the market maker's equilibrium mechanism. The incentives of buyers in deciding their bids and the market maker's choice of using the maxeq mechanism create a wave of Nash dynamics involved in the market. We characterize Nash equilibria in the dynamics in terms of the relationship between maxeq and mineq (i.e., minimum revenue equilibrium), and develop convergence results for Nash dynamics from the maxeq policy to a mineq solution, resulting an outcome equivalent to the truthful VCG mechanism. Our results imply revenue equivalence between maxeq and mineq, and address the question that why short-term revenue maximization is a poor long run strategy, in a deterministic and dynamic setting