Strategic Formation of Coalitions

Consider a society with a finite number of individuals. A coalition structure is a partition of the set of individuals. Each individual has personal preferences over the set of all coalition structures. We study the strategy proof core and von Neumann and Morgenstern (vN&M) solutions. A roommate problem is a problem in which each coalition in each coalition structure contains at most two members. We show that as long as the core is single-valued, the core is coalitionally strategy proof. Moreover the core mechanism is completely characterized by three properties: strategy proofness, Pareto optimality and individual rationality, in the domain with single-valued core. The single-valued core defines the largest domain one may obtain such a mechanism. We show in an example that the single-valued core is manipulable if coalition structures allow more than two members. Nevertheless we show that the single-valued vN&M solution is coalitionally stratey proof and it is individually rational and Pareto optimal. In fact the vN&M solution is the only mechanism with the three properties in the domain with single-valued vN&M solution

English Auctions and Walrasian Equilibria with Multiple Objects: a dynamic approach

This paper studies the English (progressive) auction for an exchange economy with multiple objects. The English auction is a tatonnement process and lasts multiple rounds. It is modeled as a sequence of round games. Each round game is a normal form game in which an agent's strategies are his bids and his payoff is his trading profits of his winning bundle at that round. Among all these normal form games, all intermediary round games are in fact the ''virtual'' games because the payoffs to agents are not finalized unless the auction closes. We show that any ascending price sequence obtained from the English auction converges to a Walrasian equilibrium (if any) within finite rounds when agents submit their bids that consist of a Nash equilibrium in each round game. We also provide a sufficient condition for the English auction to converge to a Walrasian equilibrium in finite rounds. But this condition is weaker than the Nash equilibrium. This shows that the Nash equilibrium is not necessary (though sufficient) for the English auction to converge to a Walrasian equilibrium

Competitive Equilibrium with Indivisibilities

This paper studies an exchange economy with a finite number of agents in which each agent is initially endowed with a finite number of (personalized) indivisible commodities. We observe that the core equivalence theorem may not hold for this economy when the coalitional form game is generated in the standard manner. We provide an alternative definition of the coalitional form game to resolve this problem so that the balancedness of the new defined game provides a useful necessary and sufficient condition for the existence of competitive equilibrium for the original economy. We also observe that the nice strategy proof property of the minimum competitive price mechanism in the assignment problem and the Vickrey auction model does not carry over to the above economy. We show that examples of exchange economies exist for which no competitive price mechanism is individually (coalitionally) strategy proof

Walrasian Equilibria in a Production Economy with Indivisibilities

This paper studies a production economy with indivisibilities. We provide a characterization for all Walraian equilibria and a necessary and sufficient condition for the existence of Walrasian equilibrium. We find a sufficient condition for every descening (ascending) price process to converge to a Walrasian equilibrium within finite periods of time

Manipulation and Stability in the College Admissions Problem

Roth and Vande Vate (1991) studied the marriage problem and introduced the notion of truncation strategies and showed in an example that the unstable matchings can arise at Nash equilibria in truncations. This paper studies the college admissions problem and shows that all rematching proof or strong equilibria in truncations produce stable matchings, even though the equilibrium profiles are manipulated, and all stable matchings can be achieved in rematching proof or strong equilibria in truncations. It is shown that a preference profile that is rematchinng proof or strong equilibrium for one stable matching mechanism is also rematching proof or strong equilibrium for all stable matching mechanisms. This result shows that there is no difference among all stable matching mechanisms in rematching proof or strong equilibria in truncations, which is in the contrast to the situation in which agents report their preferences in a straightforward manner

Walrasian equilibrium in an exchange economy with indivisibilities

This paper studies an exchange economy with indivisibilities. Our main goal is to see if a price system can function well in an economy (e.g., an economy with complementary preferences) that does not have a Walrasian equilibrium. We study the price adjustment processes governed by the Euler iterative scheme. We show that in an economy that has a Walrasian equilibrium, our price adjustment processes have a common uniform limit that is {\em unique} and converges to a Walrasian equilibrium price vector in finite time. Surprisingly, in an economy that does not have a Walrasian equilibrium, our price adjustment processes also have a common uniform limit that is {\em unique} and converges to a market equilibrium price vector in finite time. Moreover, market equilibrium prices coincide with Walrasian equilibrium ones in an economy that has a Walrasian equilibrium. Further, there are no prices other than the Walrasian or market equilibrium ones that have such a property of global stability

An Analysis of the Seasonal Cycle and the Business Cycle

Robert Barsky and Jeffrey Miron (1989) revealed the seasonal cycle of the U.S. economy from 1948 to 1985 was characterized by a “bubble-like” expansion in the second and fourth quarters, a “crash-like” contraction in the first quarter, and a mild contraction in the third quarter. We replicate, in part, their seasonal cycle analysis from 1946 to 2001. Our results are largely in line with theirs. Nonetheless, we find the seasonal cycle is not stable and can evolve across time. In particular, the Great Moderation affected both the business cycle and the seasonal cycle. Robert Barsky and Jeffrey Miron also found real aggregates, like the output, move together in the seasonal cycle across broadly defined sectors, similar to a phenomenon observed under the conventional business cycle. They posed a challenge question concerning why “the seasonal and the conventional business cycles are so similar.” To answer their question, we focus on a number of aggregate variables with a recursive application of the HP filter and find that aggregates, such as the GDP, consumption, the S&P 500 Index, and so forth, have a “bubble-like” expansion and a “crash-like” contraction in their cyclical trends in business cycle frequencies. Although preference shifts and production synergy are the two major forces that drive the seasonal cycle, we find the time-varying stochastic discount factor is the main cause of the business cycle and plays a more important role in macroeconomic fluctuations in business cycle frequencies than other factors

Capturing Topology in Graph Pattern Matching

Graph pattern matching is often defined in terms of subgraph isomorphism, an NP-complete problem. To lower its complexity, various extensions of graph simulation have been considered instead. These extensions allow pattern matching to be conducted in cubic-time. However, they fall short of capturing the topology of data graphs, i.e., graphs may have a structure drastically different from pattern graphs they match, and the matches found are often too large to understand and analyze. To rectify these problems, this paper proposes a notion of strong simulation, a revision of graph simulation, for graph pattern matching. (1) We identify a set of criteria for preserving the topology of graphs matched. We show that strong simulation preserves the topology of data graphs and finds a bounded number of matches. (2) We show that strong simulation retains the same complexity as earlier extensions of simulation, by providing a cubic-time algorithm for computing strong simulation. (3) We present the locality property of strong simulation, which allows us to effectively conduct pattern matching on distributed graphs. (4) We experimentally verify the effectiveness and efficiency of these algorithms, using real-life data and synthetic data.Comment: VLDB201

Stable Matchings and the Small Core in Nash Equilibrium in the College Admissions Problem

Both rematching proof and strong equilibrium outcomes are stable with respect to the true preferences in the marriage problem. We show that not all rematching proof or strong equilibrium outcomes are stable in the college admissions problem. But we show that both rematching proof and strong equilibrium outcomes with truncations at the match point are all stable in the college admissions problem. Further, all true stable matchings can be achieved in both rematching proof and strong equilibrium with truncations at the match point. We show that any Nash equilibrium in truncations admits one and only one matching, stable or not. Therefore, the core at a Nash equilibrium in truncations must be small. But examples exist such that the set of stable matchings with respect to a Nash equilibrium may contain more than one matching. Nevertheless, each Nash equilibrium can only admit at most one true stable matching. If, indeed, there is a true stable matching at a Nash equilibrium, then the only possible equilibrium outcome will be the true stable matching, no matter how players manipulate their equilibrium strategies and how many other unstable matchings are there at the Nash equilibrium. Thus, we show that a necessary and sufficient condition for the stable matching rule to be implemented in a subset of Nash equilibria by a direct revelation game induced by a stable matching mechanism is that every Nash equilibrium profile in that subset admits one and only one true stable matching.