Multi-unit bilateral trade refers to the setting, where there is a buyer and a seller, who holds a finite number of units of an indivisible item. An automated mechanism has to decide how many units are transferred from the seller to the buyer and the corresponding payment from the buyer to the seller. The buyer and the seller have both either increasing or increasing submodular valuation functions in the number of units in possession. The (single-unit) bilateral trade problem arises as a particular case. We study the problem of social welfare maximisation by establishing the fraction (approximation ratio) of the optimal social welfare that a fixed-price mechanism can recover. Fixed-price mechanisms, understood as per-unit price in the multi-unit setting, have been characterised as the only truthful, individually rational and strongly budget balanced mechanisms. We narrow the gap on the approximation ratio of optimal fixed-price mechanisms for bilateral trade, which has been shown to lie between 0.72 and 0.7381. We show that it must lie between 0.7292 and 0.73805, which leads to improved bounds on the approximation ratio of optimal fixed-price mechanisms for multi-unit bilateral trade. In particular, we show that multi-unit bilateral trade is at least as hard as single-unit bilateral trade, and obtain several hardness results for different numbers of units.

Strategyproofness has been the holy grail in mechanism design for decades, providing strong incentive compatibility guarantees under the assumption of perfectly rational agents. However, this assumption is questionable when agents exhibit bounded rationality. Moreover, strategyproofness often imposes strong impossibility results that prevent mechanisms from surpassing certain approximation barriers. We study this tension in budget-feasible mechanism design, where a designer wants to procure services of maximum value from agents subject to a budget constraint. Here, strategyproofness imposes approximation barriers of 2.41 and 2 for deterministic and randomized mechanisms, respectively. We investigate how much we can potentially gain under bounded rationality. We adopt the weaker notion of not obviously manipulable (NOM), which only prevents "obvious" strategic deviations. We fully resolve the achievable approximation guarantees under NOM: We derive a deterministic 2-approximate NOM mechanism under the general class of monotone subadditive valuations. We also show that this bound is tight (even for additive valuations). Additionally, we provide a simple randomized NOM mechanism that is approximately optimal. These results demonstrate a clear separation between strategyproof and NOM mechanisms. Our mechanisms use Golden Tickets and Wooden Spoons as natural design primitives, arising from our characterization of NOM mechanisms.

Financial networks raise a significant computational challenge in identifying insolvent firms and evaluating their exposure to systemic risk. This task, known as the clearing problem, is computationally tractable when dealing with simple debt contracts. However under the presence of certain derivatives called credit default swaps (CDSes) the clearing problem is $\textsf{FIXP}$-complete. Existing techniques only show $\textsf{PPAD}$-hardness for finding an $ε$-solution for the clearing problem with CDSes within an unspecified small range for $ε$. We present significant progress in both facets of the clearing problem: (i) intractability of approximate solutions; (ii) algorithms and heuristics for computable solutions. Leveraging $\textsf{Pure-Circuit}$ (FOCS'22), we provide the first explicit inapproximability bound for the clearing problem involving CDSes. Our primal contribution is a reduction from $\textsf{Pure-Circuit}$ which establishes that finding approximate solutions is $\textsf{PPAD}$-hard within a range of roughly 5%. To alleviate the complexity of the clearing problem, we identify two meaningful restrictions of the class of financial networks motivated by regulations: (i) the presence of a central clearing authority; and (ii) the restriction to covered CDSes. We provide the following results: (i.) The $\textsf{PPAD}$-hardness of approximation persists when central clearing authorities are introduced; (ii.) An optimisation-based method for solving the clearing problem with central clearing authorities; (iii.) A polynomial-time algorithm when the two restrictions hold simultaneously.

Financial networks model a set of financial institutions (firms) interconnected by obligations. Recent work has introduced to this model a class of obligations called credit default swaps , a well-known type of financial derivative. The main computational challenge for such systems is known as the clearing problem . This problem involves the task of determining insolvent firms and quantifying their exposure to systemic risk. The technical term used to describe this exposure is the clearing recovery rate . In essence, the clearing problem involves computing the clearing recovery rates of all financial institutions in a given network. We address the clearing problem in financial networks containing simple debt contracts and credit default swaps. Our work builds on the model proposed by Schuldenzucker et al. 2016, 2017 and 2020 who analysed the complexity of the \(\epsilon\) -weak (almost)-approximation version of the problem. In this paper, we study the complexity of the problem from the point of view of exact computation, approximation strength and numerically irrational solutions. Our main result establishes FIXP -completeness for the exact computation version of the problem. Consequently, we infer FIXP \({}_{a}\) -completeness for finding a strongly (or ‘near’) approximate solution as a direct consequence of our main result, while we legitimise the significance of the strong approximation variant through an observation that weakly approximate solutions may ‘severely’ misrepresent the actual financial state of an institution. Finally, we study the structural properties required for irrationality, and we identify necessary conditions for numerically irrational solutions to emerge: The presence of certain types of cycles in a financial network forces the recovery rates to take the form of roots of second- or higher-degree polynomials. In the absence of a large subclass of such cycles, we study the complexity of finding an exact solution, which we show to be a problem close to, albeit outside of, PPAD .

We study the exploration-exploitation trade-off for large multiplayer coordination games where players strategise via Q-Learning, a common learning framework in multi-agent reinforcement learning. Q-Learning is known to have two shortcomings, namely non-convergence and potential equilibrium selection problems, when there are multiple fixed points, called Quantal Response Equilibria (QRE). Furthermore, whilst QRE have full support for finite games, it is not clear how Q-Learning behaves as the game becomes large. In this paper, we characterise the critical exploration rate that guarantees convergence to a unique fixed point, addressing the two shortcomings above. Using a generating-functional method, we show that this rate increases with the number of players and the alignment of their payoffs. For many-player coordination games with perfectly aligned payoffs, this exploration rate is roughly twice that of p-player zero-sum games. As for large games, we provide a structural result for QRE, which suggests that as the game size increases, Q-Learning converges to a QRE near the boundary of the simplex of the action space, a phenomenon we term asymptotic extinction, where a constant fraction of the actions are played with zero probability at a rate o(1/N) for an N -action game.

We study strategic candidate positioning in multidimensional spatial-voting elections. Voters and candidates are represented as points in Rd and each voter supports the candidate that is closest under a distance induced by an ℓp-norm. We prove that computing an optimal location for a new candidate is NP-hard already against a single opponent, whereas for a constant number of issues the problem is tractable: an O(nd + 1) hyperplane-enumeration algorithm and an O(n log n) radial-sweep routine for d = 2 solve the task exactly. We further derive the first approximation guarantees for the general multi-candidate case and show how our geometric approach extends seamlessly to positional scoring rules such as k-approval and Borda. These results clarify the algorithmic landscape of multi-dimensional spatial elections and provide practically implementable tools for campaign strategy.

Abstract We study the following natural variant of the budgeted maximum coverage problem: We are given a budget B and a hypergraph $$G = (V, E)$$ , where each vertex has a non-negative cost and a non-negative profit. The goal is to select a set of hyperedges $$T \subseteq E$$ such that the total cost of the vertices covered by T is at most B and the total profit of all covered vertices is maximized. This is a natural generalization of the maximum coverage problem. Our interest in this problem stems from its application to bid optimization in sponsored search auctions. It is easily seen that this problem is at least as hard as budgeted maximum coverage (where the costs are associated with the selected hyperedges instead of the covered vertices). This implies $$(1-1/e+\epsilon )$$ -inapproximability for any $$\epsilon> 0$$ . Furthermore, standard greedy approaches do not yield constant factor approximations for our variant of the problem. In fact, through a reduction from Densest k -Subgraph, it can be established that our problem is inapproximable up to a constant factor, conditional on the exponential time hypothesis. Our main results are as follows: (i.) We obtain a $$(1 - 1/\sqrt{e})/2$$ -approximation algorithm for graphs. (ii.) We derive a fully polynomial-time approximation scheme (FPTAS) if the incidence graph of the hypergraph is a forest (i.e., the hypergraph is Berge-acyclic ). We extend this result to incidence graphs with a fixed-size feedback hyperedge node set. (iii.) We give a $$(1-\varepsilon )/(2d^2)$$ -approximation algorithm for all $$\varepsilon> 0$$ , where d is the maximum vertex degree.

The exposure of banks to systemic risk in financial networks usually requires large bailouts of taxpayer money with long-lasting and damaging societal consequences. We examine whether the banking network can reduce systemic risk from within by selfishly cancelling the debts of banks in distress. This operation can in principle reduce losses and prevent default cascades. We define an abstract model to simulate the ensuing strategic game on randomly generated financial networks, where each systemically important bank independently decides how likely it is to cancel some debts of insolvent banks. We compute the equilibrium of the induced empirical game with the empirical game-theoretic analysis and analyse its efficiency by measuring the price of anarchy. Our results show that selfish debt cancellation can reduce systemic risk when adopting the equilibrium strategy profile. However, our results also indicate that the efficiency of the equilibrium can be low and relatively few banks cancel debts at equilibrium, and we explain the reason for this through analysis of the banks’ incentives and game dynamics.

We examine the extent to which rescue strategies within a banking system can reduce systemic risk. We focus on donations from solvent banks to banks in distress, which can in principle reduce losses and prevent default cascades. We build an agent-based model to simulate the ensuing strategic game on a randomly generated financial network, where nodes represent banks and edges represent inter-bank liabilities. Each bank independently decides whether to rescue (and whom) to maximise their payoffs. We analyse the rescue strategies adopted by the banks at equilibrium, using empirical game-theoretic analysis. Our results show that donations can indeed reduce systemic risk when the equilibrium strategy profile is adopted. Individual donations can benefit multiple banks in the network. Our results also indicate that lower default costs and small-variance liabilities tend to decrease the incentives to donate. We furthermore examine the impact of the banks’ rationality on the effects of rescue, finding that banks behaving rationally use their funds for rescues more efficiently than banks that behave irrationally.

Recent work in algorithmic mechanism design focuses on designing mechanisms for agents with bounded rationality, modifying the constraints that must be satisfied in order to achieve incentive compatibility. Starting with Li's strengthening of strategyproofness, obvious strategyproofness (OSP) requires truthtelling to be "obvious" over dishonesty, roughly meaning that the worst outcome from truthful actions must be no worse than the best outcome for dishonest ones. A celebrated result for dominant-strategy incentive-compatible mechanisms that allows us to restrict attention to direct mechanisms, known as the revelation principle, does not hold for OSP: the implementation details matter for the obvious incentive properties of the mechanism. Studying agent strategies in real-life mechanisms, Troyan and Morrill introduce a relaxation of strategyproofness known as non-obvious manipulability, which only requires comparing certain extrema of the agents' utility functions in order for a mechanism to be incentive-compatible. Specifically a mechanism is not obviously manipulable (NOM) if the best and worst outcomes when acting truthfully are no worse than the best and worst outcomes when acting dishonestly. In this work we first extend the cycle monotonicity framework for direct-revelation NOM mechanism design to indirect mechanisms. We then apply this to two settings, single-parameter agents and mechanisms for two agents in which one has a two-value domain, and show that under these models the revelation principle holds: direct mechanisms are just as powerful as indirect ones.

Financial networks model debt obligations between economic firms. Computational and game-theoretic analyses of these networks have been recent focus of the literature. The main computational challenge in this context is the clearing problem, a fixed point search problem that essentially determines insolvent firms and their exposure to systemic risk, technically known as recovery rates. When Credit Default Swaps, a derivative connected to the 2008 financial crisis, are factored into the obligations, the clearing problem becomes more complex. Specifically, whenever insolvent firms pay their debts proportionally to their recovery rates, computing a weakly approximate solution was shown by Schuldenzucker et al. (2017) to be PPAD-complete. Additionally, Ioannidis et al. (2022) showed that computing a strongly approximate solution in the same framework is FIXP-complete. This paper addresses the computational complexity of the clearing problem in financial networks with derivatives, whenever payment priorities among creditors are applied. This practically relevant model has only been studied from a game-theoretic standpoint. We explicitly study the clearing problem whenever the firms pay according to a singleton liability priority list and prove that it is FIXP-complete. Finally, we provide a number of NP-hardness results for the computation of priority lists that optimise specific objectives of importance in the domain.

Financial networks model a set of financial institutions (firms) interconnected by obligations. Recent work has introduced to this model a class of obligations called credit default swaps, a certain kind of financial derivatives. The main computational challenge for such systems is known as the clearing problem, which is to determine which firms are in default and to compute their exposure to systemic risk, technically known as their recovery rates. It is known that the recovery rates form the set of fixed points of a simple function, and that these fixed points can be irrational. Furthermore, Schuldenzucker et al. (2016) have shown that finding a weakly (or "almost") approximate (rational) fixed point is PPAD-complete. We further study the clearing problem from the point of view of irrationality and approximation strength. Firstly, we observe that weakly approximate solutions may misrepresent the actual financial state of an institution. On this basis, we study the complexity of finding a strongly (or "near") approximate solution, and show FIXP-completeness. We then study the structural properties required for irrationality, and we give necessary conditions for irrational solutions to emerge: The presence of certain types of cycles in a financial network forces the recovery rates to take the form of roots of non-linear polynomials. In the absence of a large subclass of such cycles, we study the complexity of finding an exact fixed point, which we show to be a problem close to, albeit outside of, PPAD.

Abstract We consider a multi-stage facility reallocation problems on the real line, where a facility is being moved between time stages based on the locations reported by n agents. The aim of the reallocation algorithm is to minimise the social cost, i.e., the sum over the total distance between the facility and all agents at all stages, plus the cost incurred for moving the facility. We study this problem both in the offline setting and online setting. In the offline case the algorithm has full knowledge of the agent locations in all future stages, and in the online setting the algorithm does not know these future locations and must decide the location of the facility on a stage-per-stage basis. We derive the optimal algorithm in both cases. For the online setting we show that its competitive ratio is $$(n+2)/(n+1)$$ ( n + 2 ) / ( n + 1 ) . As neither of these algorithms turns out to yield a strategy-proof mechanism, we propose another strategy-proof mechanism which has a competitive ratio of $$(n+3)/(n+1)$$ ( n + 3 ) / ( n + 1 ) for odd n and $$(n+4)/n$$ ( n + 4 ) / n for even n, which we conjecture to be the best possible. We also consider a generalisation with multiple facilities and weighted agents, for which we show that the optimum can be computed in polynomial time for a fixed number of facilities.

We consider the setting of combinatorial auctions when the agents are single-minded and have no contingent reasoning skills. We are interested in mechanisms that provide the right incentives to these imperfectly rational agents, and therefore focus our attention to obviously strategyproof (OSP) mechanisms. These mechanisms require that at each point during the execution where an agent is queried to communicate information, it should be "obvious" for the agent what strategy to adopt in order to maximise her utility. In this paper we study the potential of OSP mechanisms with respect to the approximability of the optimal social welfare. We consider two cases depending on whether the desired bundles of the agents are known or unknown to the mechanism. For the case of known-bundle single-minded agents we show that OSP can actually be as powerful as (plain) strategyproofness (SP). In particular, we show that we can implement the very same algorithm used for SP to achieve a √m-approximation of the optimal social welfare with an OSP mechanism, m being the total number of items. Restricting our attention to declaration domains with two values, we provide a 2-approximate OSP mechanism, and prove that this approximation bound is tight. We also present a randomised mechanism that is universally OSP and achieves a finite approximation of the optimal social welfare for the case of arbitrary size finite domains. This mechanism also provides a bounded approximation ratio when the valuations lie in a bounded interval (even if the declaration domain is infinitely large). For the case of unknown-bundle single-minded agents, we show how we can achieve an approximation ratio equal to the size of the largest desired set, in an OSP way. We remark this is the first known application of OSP to multi-dimensional settings, i.e., settings where agents have to declare more than one parameter. Our results paint a rather positive picture regarding the power of OSP mechanisms in this context, particularly for known-bundle single-minded agents. All our results are constructive, and even though some known strategyproof algorithms are used, implementing them in an OSP way is a non-trivial task.

We characterise the set of dominant strategy incentive compatible (DSIC), strongly budget balanced (SBB), and ex-post individually rational (IR) mechanisms for the multi-unit bilateral trade setting. In such a setting there is a single buyer and a single seller who holds a finite number k of identical items. The mechanism has to decide how many units of the item are transferred from the seller to the buyer and how much money is transferred from the buyer to the seller. We consider two classes of valuation functions for the buyer and seller: Valuations that are increasing in the number of units in possession, and the more specific class of valuations that are increasing and submodular.Furthermore, we present some approximation results about the performance of certain such mechanisms, in terms of social welfare: For increasing submodular valuation functions, we show the existence of a deterministic 2-approximation mechanism and a randomised e/(1 − e) approximation mechanism, matching the best known bounds for the single-item setting.

We consider a multi-stage facility reallocation problems on the real line, where a facility is being moved between stages based on the locations reported by n agents. The aim of the reallocation mechanism is to minimize the social cost, i.e., the sum over the total distance between the facility and all agents at all stages, plus the cost incurred for moving the facility. We also study this problem both in the offline setting and online setting. In the offline case the mechanism has full knowledge of the agent locations in all future stages, and in the online setting the mechanism does not know these future locations and must decide the location of the facility on a stage-per-stage basis. For both cases, we derive the optimal mechanism, where for the online setting we show that its competitive ratio is (n+2)/(n+1). As neither of these mechanisms turns out to be strategyproof, we propose another strategyproof mechanism which has a competitive ratio of (n+3)/(n+1) for odd n and (n+4)/n for even n, which we conjecture to be the best possible. We also consider a generalization with multiple facilities and weighted agents, for which we show that the optimum can be computed in polynomial time for a fixed number of facilities.

We develop and extend a line of recent work on the design of mechanisms for two-sided markets. The markets we consider consist of buyers and sellers of a number of items, and the aim of a mechanism is to improve the social welfare by arranging purchases and sales of the items. A mechanism is given prior distributions on the agents' valuations of the items, but not the actual valuations; thus the aim is to maximise the expected social welfare over these distributions. As in previous work, we are interested in the worst-case ratio between the social welfare achieved by a truthful mechanism, and the best social welfare possible.

We study the following natural variant of the budgeted maximum coverage problem: We are given a budget B and a hypergraph G = (V, E), where each vertex has a non-negative cost and a non-negative profit. The goal is to select a set of hyperedges T subseteq E such that the total cost of the vertices covered by T is at most B and the total profit of all covered vertices is maximized. Besides being a natural generalization of the well-studied maximum coverage problem, our motivation for investigating this problem originates from its application in the context of bid optimization in sponsored search auctions, such as Google AdWords. It is easily seen that this problem is strictly harder than budgeted max coverage, which means that the problem is (1-1/e)-inapproximable. The difference of our problem to the budgeted maximum coverage problem is that the costs are associated with the covered vertices instead of the selected hyperedges. As it turns out, this difference refutes the applicability of standard greedy approaches which are used to obtain constant factor approximation algorithms for several other variants of the maximum coverage problem. Our main results are as follows: - We obtain a (1 - 1/sqrt(e))/2-approximation algorithm for graphs. - We derive a fully polynomial-time approximation scheme (FPTAS) if the incidence graph of the hypergraph is a forest (i.e., the hypergraph is Berge-acyclic). We also extend this result to incidence graphs with a fixed-size feedback hyperedge node set. - We give a (1-epsilon)/(2d^2)-approximation algorithm for every epsilon > 0, where d is the maximum degree of a vertex in the hypergraph.

Mechanism design for one-sided markets is an area of extensive research in economics and, since more than a decade, in computer science as well. Two-sided markets, on the other hand, have not received the same attention despite the numerous applications to web advertisement, stock exchange, and frequency spectrum allocation. This work studies double auctions, in which unit-demand buyers and unit-supply sellers act strategically. An ideal goal in double auction design is to maximize the social welfare of buyers and sellers with individually rational (IR), incentive compatible (IC) and strongly budget-balanced (SBB) mechanisms. The first two properties are standard. SBB requires that the payments charged to the buyers are entirely handed to the sellers. This property is crucial in all the contexts that do not allow the auctioneer retaining a share of buyers' payments or subsidizing the market. Unfortunately, this goal is known to be unachievable even for the special case of bilateral trade, where there is only one buyer and one seller. Therefore, in subsequent papers, meaningful trade-offs between these requirements have been investigated. Our main contribution is the first IR, IC and SBB mechanism that provides an O(1)-approximation to the optimal social welfare. This result holds for any number of buyers and sellers with arbitrary, independent distributions. Moreover, our result continues to hold when there is an additional matroid constraint on the sets of buyers who may get allocated an item. To prove our main result, we devise an extension of sequential posted price mechanisms to two-sided markets. In addition to this, we improve the best-known approximation bounds for the bilateral trade problem.

We study the revenue performance of sequential posted price mechanisms and some natural extensions, for a general setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted price mechanisms are conceptually simple mechanisms that work by proposing a “take-it-or-leave-it” offer to each buyer. We apply sequential posted price mechanisms to single-parameter multi-unit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the case of a continuous valuation distribution when various standard assumptions hold simultaneously. In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted price mechanism is proportional to the ratio of the highest and lowest possible valuation. We prove that for two simple generalizations of these mechanisms, a better revenue performance can be achieved: if the sequential posted price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a $$\varOmega (1/\max \{1,d\})$$ fraction of the optimal revenue can be extracted, where d denotes the “degree of dependence” of the valuations, ranging from complete independence ( $$d=0$$ ) to arbitrary dependence ( $$d = n-1$$ ). When we generalize the sequential posted price mechanisms further, such that the mechanism has the ability to make a take-it-or-leave-it offer to the i-th buyer that depends on the valuations of all buyers except i, we prove that a constant fraction $$(2 - \sqrt{e})/4 \approx 0.088$$ of the optimal revenue can be always extracted.

In the “The curse of simultaneity”, Paes Leme et al. show that there are interesting classes of games for which sequential decision making and corresponding subgame perfect equilibria avoid worst case Nash equilibria, resulting in substantial improvements for the price of anarchy. This is called the sequential price of anarchy. A handful of papers have lately analysed it for various problems, yet one of the most interesting open problems was to pin down its value for linear atomic routing (also: network congestion ) games, where the price of anarchy equals 5/2. The main contribution of this paper is the surprising result that the sequential price of anarchy is unbounded even for linear symmetric routing games, thereby showing that sequentiality can be arbitrarily worse than simultaneity for this class of games. Complementing this result we solve an open problem in the area by establishing that the (regular) price of anarchy for linear symmetric routing games equals 5/2. Additionally, we prove that in these games, even with two players, computing the outcome of a subgame perfect equilibrium is \(\mathsf {NP}\) -hard. These 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.

This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.

In many circumstances where multiple agents need to make a joint decision, voting is used to aggregate the agents' preferences. Each agent's vote carries a weight, and if the sum of the weights of the agents in favor of some outcome is larger than or equal to a given quota, then this outcome is decided upon. The distribution of weights leads to a certain distribution of power. Several `power indices' have been proposed to measure such power. In the so-called inverse problem, we are given a target distribution of power, and are asked to come up with a game in the form of a quota, plus an assignment of weights to the players whose power distribution is as close as possible to the target distribution (according to some specied distance measure). Here we study solution approaches for the larger class of voting game design (VGD) problems, one of which is the inverse problem. In the general VGD problem, the goal is to find a voting game (with a given number of players) that optimizes some function over these games. In the inverse problem, for example, we look for a weighted voting game that minimizes the distance between the distribution of power among the players and a given target distribution of power (according to a given distance measure). Our goal is to find algorithms that solve voting game design problems exactly, and we approach this goal by enumerating all games in the class of games of interest. We first present a doubly exponential algorithm for enumerating the set of simple games. We then improve on this algorithm for the class of weighted voting games and obtain a quadratic exponential (i.e., 2^O(n^2)) algorithm for enumerating them. We show that this improved algorithm runs in output-polynomial time, making it the fastest possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime-algorithm that runs in exponential time for the power index weighted voting game design problem (the `inverse problem'). We implement this algorithm to find a weighted voting game with a normalized Banzhaf power distribution closest to a target power index, and perform experiments to obtain some insights about the set of weighted voting games. We remark that our algorithm is applicable to optimizing any exponential-time computable function, the distance of the normalized Banzhaf index to a target power index is merely taken as an example.

The (Shapley-Scarf) housing market is a well-studied and fundamental model of an exchange economy. Each agent owns a single house and the goal is to reallocate the houses to the agents in a mutually beneficial and stable manner. Recently, Alcalde-Unzu and Molis (2011) and Jaramillo and Manjunath (2011) independently examined housing markets in which agents can express indifferences among houses. They proposed two important families of mechanisms, known as TTAS and TCR respectively. We formulate a family of mechanisms which not only includes TTAS and TCR but also satisfies many desirable properties of both families. As a corollary, we show that TCR is strict core selecting (if the strict core is non-empty). Finally, we settle an open question regarding the computational complexity of the TTAS mechanism. Our study also raises a number of interesting research questions.