scholarly journals Graphical One-Sided Markets

2019 ◽  
Author(s):  
Sagar Massand ◽  
Sunil Simon
Keyword(s):  
Solution Concept ◽  
Weighted Graph ◽  
Rational Agents ◽  
Indivisible Objects ◽  
Stable Allocation ◽  
Np Complete ◽  
Neighbourhood Graph ◽  
Stable Allocations ◽  
Rational Players

We study the problem of allocating indivisible objects to a set of rational agents where each agent's final utility depends on the intrinsic valuation of the allocated item as well as the allocation within the agent's local neighbourhood. We specify agents' local neighbourhood in terms of a weighted graph. This extends the model of one-sided markets to incorporate neighbourhood externalities. We consider the solution concept of stability and show that, unlike in the case of one-sided markets, stable allocations may not always exist. When the underlying local neighbourhood graph is symmetric, a 2-stable allocation is guaranteed to exist and any decentralised mechanism where pairs of rational players agree to exchange objects terminates in such an allocation. We show that computing a 2-stable allocation is PLS-complete and further identify subclasses which are tractable. In the case of asymmetric neighbourhood structures, we show that it is NP-complete to check if a 2-stable allocation exists. We then identify structural restrictions where stable allocations always exist and can be computed efficiently. Finally, we study the notion of envy-freeness in this framework.

scholarly journals A Heuristic Algorithm for Optimal Hamiltonian Cycles in Weighted Graphs

2015 ◽  
Vol 11 (6) ◽  
pp. 5300-5305
Author(s):  
Tadeusz Ostrowski ◽  
Petroula Mavrikiou
Keyword(s):  
Heuristic Algorithm ◽  
Weighted Graph ◽  
Payoff Matrix ◽  
Difficulty Level ◽  
Hamiltonian Cycles ◽  
Weight Losses ◽  
Time Distance ◽  
Np Complete ◽  
The Traveling Salesman Problem ◽  
General Graphs

Abstract. The paper focuses on finding of the optimal Hamiltonian cycle, when it is regarded with respect to cost, time, distance or difficulty level of the route. The problem is strictly related to the traveling salesman problem proved to be NP-complete for general graphs. The paper gives a heuristic algorithm for finding the optimal spanning cycle in a weighted graph. Its idea is based on optimization of weight losses and reduction the complexity of a problem by reduction the dimension of the graph payoff matrix. 

scholarly journals Matching with Contracts: Comment

2008 ◽  
Vol 98 (3) ◽  
pp. 1189-1194 ◽  
Author(s):  
John William Hatfield ◽  
Fuhito Kojima
Keyword(s):  
Unified Model ◽  
The Other ◽  
Auction Models ◽  
Stable Allocation ◽  
Stable Allocations ◽  
Package Auction

Hatfield and Milgrom (2005) present a unified model of matching with contracts phrased in terms of hospitals and doctors, which subsumes the standard two-sided matching and some package auction models. They show that a stable allocation exists if contracts are substitutes for each hospital. They further claim that if a hospital's preferences violate the substitutes condition, there exist singleton preferences for the other hospitals and doctors such that no stable allocation exists. We show this last claim does not hold in general. We further present a weaker condition that is necessary to guarantee the existence of stable allocations. (JEL C78, D86, J41)

scholarly journals Coarse Correlation in Extensive-Form Games

2020 ◽  
Vol 34 (02) ◽  
pp. 1934-1941
Author(s):  
Gabriele Farina ◽  
Tommaso Bianchi ◽  
Tuomas Sandholm
Keyword(s):  
Normal Form ◽  
Saddle Points ◽  
Solution Concept ◽  
Correlated Equilibrium ◽  
Extensive Form ◽  
Extensive Form Games ◽  
Rational Agents ◽  
Correlation Models ◽  
Correlated Equilibria ◽  
Special Case

Coarse correlation models strategic interactions of rational agents complemented by a correlation device which is a mediator that can recommend behavior but not enforce it. Despite being a classical concept in the theory of normal-form games since 1978, not much is known about the merits of coarse correlation in extensive-form settings. In this paper, we consider two instantiations of the idea of coarse correlation in extensive-form games: normal-form coarse-correlated equilibrium (NFCCE), already defined in the literature, and extensive-form coarse-correlated equilibrium (EFCCE), a new solution concept that we introduce. We show that EFCCEs are a subset of NFCCEs and a superset of the related extensive-form correlated equilibria. We also show that, in n-player extensive-form games, social-welfare-maximizing EFCCEs and NFCCEs are bilinear saddle points, and give new efficient algorithms for the special case of two-player games with no chance moves. Experimentally, our proposed algorithm for NFCCE is two to four orders of magnitude faster than the prior state of the art.

Stable Governments and the Allocation of Policy Portfolios

1990 ◽  
Vol 84 (3) ◽  
pp. 891-906 ◽  
Author(s):  
David Austen-Smith ◽  
Jeffrey Banks
Keyword(s):  
Government Policy ◽  
Policy Decision ◽  
Portfolio Allocation ◽  
Policy Outcomes ◽  
Allocation Model ◽  
Coalition Governments ◽  
The Core ◽  
Stable Allocation ◽  
Usual Concept ◽  
Stable Allocations

Different members of coalition governments typically have responsibility for different aspects, or dimensions, of policy. Such responsibilities are allocated as portfolios to government members. Given a distribution of such portfolios, final government policy is derived as the accumulation of individual members' decisions in regard to their respective responsibilities. We develop a portfolio allocation model of government formation and policy decision in multiparty legislatures. In particular, we focus on stable portfolio allocations, where a stable allocation is one that yields a policy that no legislative coalition is willing or able to overturn. Several notions of stability are considered and related to the usual concept of the core. Among the results are that although stable allocations are not guaranteed, such allocations can exist with minority governments; and that final policy outcomes associated with stable governments need not be “centrist.“

Modeling Rational Players: Part I

1987 ◽  
Vol 3 (2) ◽  
pp. 179-214 ◽  
Author(s):  
Ken Binmore
Keyword(s):  
Game Theory ◽  
Solution Concept ◽  
Dynamic Structure ◽  
Economic Models ◽  
The Other ◽  
The Right ◽  
Simple Economic ◽  
Rational Players

Game theory has proved a useful tool in the study of simple economic models. However, numerous foundational issues remain unresolved. The situation is particularly confusing in respect of the non-cooperative analysis of games with some dynamic structure in which the choice of one move or another during the play of the game may convey valuable information to the other players. Without pausing for breath, it is easy to name at least 10 rival equilibrium notions for which a serious case can be made that here is the “right” solution concept for such games.

scholarly journals On Non-Cooperativeness in Social Distance Games

2019 ◽  
Vol 66 ◽  
pp. 625-653
Author(s):  
Alkida Balliu ◽  
Michele Flammini ◽  
Giovanna Melideo ◽  
Dennis Olivetti
Keyword(s):  
Nash Equilibrium ◽  
Polynomial Time ◽  
Social Distance ◽  
Price Of Anarchy ◽  
Cluster Formation ◽  
Solution Concept ◽  
Price Of Stability ◽  
The Social ◽  
Np Complete ◽  
Inverse Distance

We consider Social Distance Games (SDGs), that is cluster formation games in which the utility of each agent only depends on the composition of the cluster she belongs to, proportionally to her harmonic centrality, i.e., to the average inverse distance from the other agents in the cluster. Under a non-cooperative perspective, we adopt Nash stable outcomes, in which no agent can improve her utility by unilaterally changing her coalition, as the target solution concept. Although a Nash equilibrium for a SDG can always be computed in polynomial time, we obtain a negative result concerning the game convergence and we prove that computing a Nash equilibrium that maximizes the social welfare is NP-hard by a polynomial time reduction from the NP-complete Restricted Exact Cover by 3-Sets problem. We then focus on the performance of Nash equilibria and provide matching upper bound and lower bounds on the price of anarchy of Θ(n), where n is the number of nodes of the underlying graph. Moreover, we show that there exists a class of SDGs having a lower bound on the price of stability of 6/5 − ε, for any ε > 0. Finally, we characterize the price of stability 5 of SDGs for graphs with girth 4 and girth at least 5, the girth being the length of the shortest cycle in the graph.

scholarly journals Trust-Based Mechanisms for Robust and Efficient Task Allocation in the Presence of Execution Uncertainty

2009 ◽  
Vol 35 ◽  
pp. 119-159 ◽  
Author(s):  
S. D. Ramchurn ◽  
C. Mezzetti ◽  
A. Giovannucci ◽  
J. A. Rodriguez-Aguilar ◽  
R. K. Dash ◽  
...  
Keyword(s):  
Task Allocation ◽  
Social Utility ◽  
Rational Agents ◽  
New Class ◽  
Incentive Compatible ◽  
Subjective Perceptions ◽  
Opt Out ◽  
Real World Applications ◽  
Np Complete ◽  
Programming Solution

Vickrey-Clarke-Groves (VCG) mechanisms are often used to allocate tasks to selfish and rational agents. VCG mechanisms are incentive compatible, direct mechanisms that are efficient (i.e., maximise social utility) and individually rational (i.e., agents prefer to join rather than opt out). However, an important assumption of these mechanisms is that the agents will "always" successfully complete their allocated tasks. Clearly, this assumption is unrealistic in many real-world applications, where agents can, and often do, fail in their endeavours. Moreover, whether an agent is deemed to have failed may be perceived differently by different agents. Such subjective perceptions about an agent's probability of succeeding at a given task are often captured and reasoned about using the notion of "trust". Given this background, in this paper we investigate the design of novel mechanisms that take into account the trust between agents when allocating tasks. Specifically, we develop a new class of mechanisms, called "trust-based mechanisms", that can take into account multiple subjective measures of the probability of an agent succeeding at a given task and produce allocations that maximise social utility, whilst ensuring that no agent obtains a negative utility. We then show that such mechanisms pose a challenging new combinatorial optimisation problem (that is NP-complete), devise a novel representation for solving the problem, and develop an effective integer programming solution (that can solve instances with about 2x10^5 possible allocations in 40 seconds).

Matching with Contracts: Comment

2013 ◽  
Vol 103 (5) ◽  
pp. 2050-2051 ◽  
Author(s):  
Orhan Aygün ◽  
Tayfun Sönmez
Keyword(s):  
Matching Theory ◽  
Stable Allocation ◽  
Stable Allocations

The matching with contracts model (Hatfield and Milgrom 2005) is widely considered to be one of the most important advances of the last two decades in matching theory. One of their main messages is that the set of stable allocations is non-empty under a substitutes condition. We show that an additional irrelevance of rejected contracts (IRC) condition is implicitly assumed throughout their analysis, and in the absence of IRC several of their results, including the guaranteed existence of a stable allocation, fail to hold. (JEL C78, D86)

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