Optimal Truncation in Matching Markets

We consider two-sided many-to-many matching markets in which each worker may work for multiple firms and each firm may hire multiple workers. We study individual and group manipulations in centralized markets that employ (pairwise) stable mechanisms and that require participants to submit rank order lists of agents on the other side of the market. We are interested in simple preference manipulations that have been reported and studied in empirical and theoretical work: truncation strategies, which are the lists obtained by removing a tail of least preferred partners from a preference list, and the more general dropping strategies, which are the lists obtained by only removing partners from a preference list (i.e., no reshuffling). We study when truncation/dropping strategies are exhaustive for a group of agents on the same side of the market, i.e., when each match resulting from preference manipulations can be replicated or improved upon by some truncation/dropping strategies. We prove that for each stable mechanism, dropping strategies are exhaustive for each group of agents on the same side of the market (Theorem 1), i.e., independently of the quotas. Then, we show that for each stable mechanism, truncation strategies are exhaustive for each agent with quota 1 (Theorem 2). Finally, we show that this result cannot be extended neither to individual manipulations when the agent’s quota is larger than 1 (even when all other agents’ quotas equal 1—Example 1), nor to group manipulations (even when all quotas equal 1—Example 2).

On the Exhaustiveness of Truncation and Dropping Strategies in Many-to-Many Matching Markets

Group decision-making, in which a collective decision needs to be derived from individual agents’ preferences has been an active area of research. In group decisionmaking, autonomous agents need some procedures or mechanisms that can collect preference information from them and make a decision; or some protocols that enable them to be involved in the decision process, to interact with each other and to jointly make a collective decision. Examples include preference aggregation mechanisms, negotiation protocols, voting procedures, and auctions. This thesis studies the problem of agent-based group decision-making in various problem settings, proposing several efficient approaches to support multiple agents in reaching efficient and fair collective decisions overmultiple issues. The research work in this thesis can be divided into two parts, addressing the agent-based group decision-making problem in circumstances in which there is, respectively, complete and incomplete information available. The first part of this thesis focuses on preference aggregation in combinatorial domains, provided all agents’ preferences. Structured preferences Conditional Preference Network (CP-nets) and its variant Trade-off enhanced CP-nets (TCP-nets) are used as formal models for representing the agents’ preferences in group decision-making. Before going into the problem of making a group decision from a collection of CP-nets or TCP-nets, the problem of individual preference reasoning is studied and a computationally efficient heuristic algorithm for dominance testing in CP-nets, which forms the basis for many group decision-making mechanisms, is introduced. Subsequently, we consider the group decision-making problem with multiple agents’ CP-nets and TCP-nets. As the space of possible outcomes in combinatorial domains has a size exponential in the number of variables, computational complexity in group decision-making will be one of the issues that need to be addressed. Two computationally efficient approaches are proposed in order to support agent-based group decisionmaking in combinatorial domains with different decision criteria: (i) an approach that converts the qualitative preferences derived from CP-nets or TCP-nets into numerical penalty scoring functions, and then searches for an optimal collective decision based on the individual penalty scores; and (ii) an approach that focuses on combinatorial vote and computes the winners (i.e., winning alternatives) from a collection of CP-nets according to Majority voting rule. The second part of this thesis is concerned with the disadvantage of disclosing preference information in group decision-making and considers the incomplete information setting. Two negotiation approaches are presented in order to support agent-based group decision-making in, respectively, combinatorial domains and in domains with multiple continuous issues. Firstly, a negotiation protocol in combinatorial domains is proposed, which enables agents to distributively make decisions and leads them to Pareto-optimal agreements under incomplete information setting. Then for the domains with multiple continuous issues, a cooperative mediated negotiation approach is proposed to support the agents searching for joint utility gains and reaching an efficient and fair agreement. For every approach introduced in this thesis, experiments are carried out and the results are presented and analysed in order to provide insights into the performance of these approaches.

Optimal truncation in matching markets

Business negotiation is a key technique for the development of electronic commerce.Negotiation is a means for agents to communicate and compromise to reach mutually beneficial agreements. This paper presents a new model for bilateral multi-objective negotiation under fuzzy preferences and time constraints in an incomplete information setting. This paper employs the theory of fuzzy mathematics to analyze negotiation model,In this model, the order in which objects are bargained over and agreements are reached is determined exogenously, as part of the bargaining equilibrium. In this context, This paper proposes nine negotiation scenarios based on the agents’ fuzzy preferences and time constraints. The analysis on this model shows that the negotiation can reach equilibrium before the deadlines in all proposed negotiation scenarios. This result is checked by using a numerical experiment to simulate the negotiation process.

The author discusses the effect of the mimetic contagion on the dynamics of stock prices. Each investor bids and/or asks prices such that adjusts to his present value, calculated from his incomplete information set, and to the average price of others buyers and sellers (mimetic contagion). An original stochastic treatment, based on the equation of motion of the probability density function, is used to calculate the equations of the mean and variance of the market price. As soon as the agents are mimetic and/or have correlated present values then the moments of the market price are driven away from the fundamental ones. Copyright 1991 by Royal Economic Society.

We attempt to address the challenge of suggesting a useful bidding strategy to an agent in the an ad auction setting. We explore the possibility of using competitive safety strategies in that context; a C-competitive strategy guarantees a payoff which is no less than 1/C of the payoff obtained in a best Nash equilibrium. We adopt the model of ad auctions suggested by Varian and provide analysis of competitive safety strategies in that context. We first show that no useful safety competitive strategies exist in a setting with complete information about the agents’ valuations. Namely, in a setting with N bidders and exponential click-rate functions the ratio can be arbitrarily close to N. We also show that N is a general upper bound for any click-rates and valuations, while \(\sum_{t=1}^N \frac{1}{t}\) is a tight bound for linear click-rates. However, in our main results we show that, surprisingly, useful C-competitive strategies do exist in the incomplete information setting. More specifically, we show that under the assumption that agents’ valuations are uniformly distributed, an e-competitive strategy exists for the case of exponential click-rate functions, and a 2-competitive safety strategy exists for linear click-rate functions.

When ignorance helps: Graphical multicast cost sharing games

We study the price of anarchy (PoA) of simultaneous 2nd price auctions (S2PA) under a new natural condition of no underbidding, meaning that agents never bid on items less than their marginal values. We establish improved (mostly tight) bounds on the PoA of S2PA under no underbidding for different valuation classes (including unit demand, submodular, XOS, subadditive, and general monotone valuations), in both full information and incomplete information settings. To derive our results, we introduce a new parameterized property of auctions, termed (gamma,delta) revenue guaranteed, which implies a PoA of at least gamma/(1+delta). Via extension theorems, this guarantee extends to coarse correlated equilibria (CCE) in full information settings, and to Bayesian PoA (BPoA) in settings with incomplete information and arbitrary (correlated) distributions. We then show that S2PA are (1,1) revenue guaranteed with respect to bids satisfying no underbidding. This implies a PoA of at least 1/2 for general monotone valuation, which extends to BPOA with arbitrary correlated distributions. Moreover, we show that (lambda,mu) smoothness combined with (gamma,delta) revenue guaranteed guarantees a PoA of at least (gamma+lambda)/(1+delta+mu). This implies a host of results, such as a tight PoA of 2/3 for S2PA with submodular (or XOS) valuations, under no overbidding and no underbidding. Beyond establishing improved bounds for S2PA, the no underbidding assumption sheds new light on the performance of S2PA relative to simultaneous 1st price auctions.

Macropudential Policy in a Knightian Uncertainty Model with Credit-, Risk-, and Leverage Cycles

Quantitative results on quadratic semi-algebraic sets

In an instance of the house allocation problem two sets A and B are given. The set A is referred to as applicants and the set B is referred to as houses. We denote by m and n the size of A and B respectively. In the house allocation problem, we assume that every applicant a ∈ A has a preference list over every house b ∈ B. We call an injective mapping τ from A to B a matching. A blocking coalition of τ is a subset A′ of A such that there exists a matching τ′ that differs from τ only on elements of A′, and every element of A′ improves in τ′, compared to τ according to its preference list. If there exists no blocking coalition, we call the matching τ an Pareto optimal matching (POM).A house b ∈ B is reachable if there exists a Pareto optimal matching using b. The set of all reachable houses is denoted by E *. We show $$|E^*| \leq \sum_{i = 1,\ldots, m}{\left\lfloor\frac{m}{i}\right\rfloor} = \Theta(m\log m).$$ This is asymptotically tight. A set E ⊆ B is reachable (respectively exactly reachable) if there exists a Pareto optimal matching τ whose image contains E as a subset (respectively equals E). We give bounds for the number of exactly reachable sets.We find that our results hold in the more general setting of multi-matchings, when each applicant a of A is matched with ℓ a elements of B instead of just one. Further, we give complexity results and algorithms for corresponding algorithmic questions. Finally, we characterize unavoidable houses, i.e., houses that are used by all POM’s. This yields efficient algorithms to determine all unavoidable elements.KeywordsPareto OptimalityInjective MappingStable MatchPreference ListColumn MatriceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Buying the Option to say 'No'

The stable marriage problem (SMP) can be seen as a typical game, where each player wants to obtain the best possible partner by manipulating his/her preference list. Thus the set Q of preference lists submitted to the matching agency may differ from P, the set of true preference lists. In this paper, we study the stability of the stated lists in Q. If Q is not Nash equilibrium, i.e., if a player can obtain a strictly better partner (with respect to the preference order in P) by only changing his/her list, then in the view of standard game theory, Q is vulnerable. In the case of SMP, however, we need to consider another factor, namely that all valid matchings should not include any with respect to P. Thus, if the above manipulation of a player introduces blocking pairs, it would prevent this manipulation. Consequently, we say Q is totally stable if either Q is a Nash equilibrium or if any attempt at manipulation by a single player causes blocking pairs with respect to P. We study the complexity of testing the total stability of a stated strategy. It is known that this question is answered in polynomial time if the instance (P,Q) always satisfies P=Q. We extend this polynomially solvable class to the general one, where P and Q may be arbitrarily different.

In the Hewitt–Savage 0-1 law, symmetric measurable sets are considered in a countable product of a σ-algebra Σ with itself. Therefore, it may be of interest to find simple generators for these sets in terms of the generators of Σ. We put the problem into a more general framework of action groups and as an application find simple generators for the finite product case. We also have a partial result in the countable product case.