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on Game Theory |
| By: | Alan F. Breardon; Colin Rowat |
| Abstract: | It is known that, in one-good pillage games, stable sets are finite. For m goods, it has been conjectured that the stable sets have measure zero. We introduce a class of sets, termed pseudo-indifference sets, which includes level sets of utility functions, quasi-indifference classes associated with a preference relation not given by a utility function, production possibility frontiers, and Pareto efficient sets. We establish the truth of the conjecture by proving that pseudo-indifference sets in Rp have p-dimensional measure zero. This implies that stable sets in n-agent, m-good pillage games have m(n - 1)-dimensional measure zero. We then prove that each pseudo-indifference set in Rp has Hausdorff dimension at most p - 1, a much stronger result than measure zero. Finally, we establish a stronger version of the conjecture: stable sets in n-agent, m-good pillage games have dimension at most m(n-1)-1. |
| Keywords: | pillage games, cooperative game theory, stable sets, Hausdorff dimension |
| JEL: | C71 D51 P14 |
| Date: | 2010–02 |
| URL: | http://d.repec.org/n?u=RePEc:bir:birmec:10-05&r=gth |
| By: | Jordi Massó; Alejandro Neme |
| Abstract: | We study two cooperative solutions of a market with indivisible goods modeled as a generalized assignment game: Set-wise stability and Core. We first establish that the Set-wise stable set is contained in the Core and it contains the non-empty set of competitive equilibrium payoffs. We then state and prove three limit results for replicated markets. First, the sequence of Cores of replicated markets converges to the set of competitive equilibrium payoffs when the number of replicas tends to infinity. Second, the Set-wise stable set of a two-fold replicated market already coincides with the set of competitive equilibrium payoffs. Third, for any number of replicas there is a market with a Core payoff that is not a competitive equilibrium payoff. |
| Keywords: | Assignment game; Core; Set-wise stability; Competitive equilibrium. |
| JEL: | C78 D78 |
| Date: | 2010–03–05 |
| URL: | http://d.repec.org/n?u=RePEc:aub:autbar:810.10&r=gth |
| By: | Daniel Jaume; Jordi Massó; Alejandro Neme |
| Abstract: | A multiple-partners assignment game with heterogeneous sales and multiunit demands consists of a set of sellers that own a given number of indivisible units of (potentially many different) goods and a set of buyers who value those units and want to buy at most an exogenously fixed number of units. We define a competitive equilibrium for this generalized assignment game and prove its existence by using only linear programming. In particular, we show how to compute equilibrium price vectors from the solutions of the dual linear program associated to the primal linear program defined to find optimal assignments. Using only linear programming tools, we also show (i) that the set of competitive equilibria (pairs of price vectors and assignments) has a Cartesian product structure: each equilibrium price vector is part of a competitive equilibrium with all optimal assignments, and vice versa; (ii) that the set of (restricted) equilibrium price vectors has a natural lattice structure; and (iii) how this structure is translated into the set of agents' utilities that are attainable at equilibrium. |
| Keywords: | Matching; Assignment Game; Indivisible Goods; Competitive Equilibrium; Lattice. |
| JEL: | C78 D78 |
| Date: | 2010–03–05 |
| URL: | http://d.repec.org/n?u=RePEc:aub:autbar:808.10&r=gth |
| By: | Jordi Massó; Inés Moreno de Barreda |
| Abstract: | We characterize the class of strategy-proof social choice functions on the domain of symmetric single-peaked preferences. This class is strictly larger than the set of generalized median voter schemes (the class of strategy-proof and tops-only social choice functions on the domain of single-peaked preferences characterized by Moulin (1980)) since, under the domain of symmetric single-peaked preferences, generalized median voter schemes can be disturbed by discontinuity points and remain strategy-proof on the smaller domain. Our result identifies the specific nature of these discontinuities which allow to design non-onto social choice functions to deal with feasibility constraints. |
| Keywords: | Strategy-proofness, Single-peaked Preferences, Median Voter, Feasibility Constraints. |
| JEL: | D7 |
| Date: | 2010–03–05 |
| URL: | http://d.repec.org/n?u=RePEc:aub:autbar:809.10&r=gth |
| By: | Yan Dolinsky |
| Abstract: | In this paper we consider Dynkin's games with payoffs which are functions of an underlying process. Assuming extended weak convergence of underlying processes $\{S^{(n)}\}_{n=0}^\infty$ to a limit process $S$ we prove convegence Dynkin's games values corresponding to $\{S^{(n)}\}_{n=0}^\infty$ to the Dynkin's game value corresponding to $S$. We use these results to approximate game options prices with path dependent payoffs in continuous time models by a sequence of game options prices in discrete time models which can be calculated by dynamical programming algorithms. In comparison to previous papers we work under more general convergence of underlying processes, as well as weaker conditions on the payoffs. |
| Date: | 2009–08 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:0908.3661&r=gth |
| By: | A. Arrighetti; S. Curatolo |
| Abstract: | In the real world, many social and economic interactions are highly affected by coordination problems. These, in turn, emerge from the trial to dynamically organize strategies of collective action in complex contexts where agents and groups are heterogeneous and information is only imperfectly transmitted. In such an environment, rational strategies of coordination games cannot be set ex-ante because, even if benefits from collective coordination are common knowledge, yet there exist many unknown ex-post costs to be sustained. Agent-based simulations done in this paper show how these costs impact the net payoff in different stages of the game with different weights depending on structure of the environment and nature of co-players. With perfect information, coordination is the outcome of the game, as game theory predicts. On the contrary, if coordination costs are positive, coordination failures frequently emerge, even in absence of opportunism (as postulated in this paper). Moreover, our simulations show that information costs are more important, in determining the success of coordination, than both organization and supervision costs. Finally, a new kind of coordination failure can emerge from the dynamic interaction among agents even in contexts where ex-ante gross payoffs are sufficiently high. |
| Keywords: | Coordination Games, Agent-based Models, Coordination Costs |
| JEL: | B4 C15 C71 |
| Date: | 2010 |
| URL: | http://d.repec.org/n?u=RePEc:par:dipeco:2010-ep01&r=gth |
| By: | Britz Volker (METEOR) |
| Abstract: | Two impatient players bargain about the division of a pie under a standard bargaining protocol in discrete time with time-invariant recognition probabilities. Instantaneous utility is linear, but players discount the future by a constant factor. Before bargaining starts, a player can commit to a utility level. This commitment is perfectly binding initially. However, once so much time has passed that even receiving the entire pie would yield less than thecommitted level of utility, then the commitment becomes void. Intuitively, this simply means that no player can remain committed to something which has become impossible. If only one player can commit, his subgame--perfect equilibrium payoff varies between one half and the entire pie, depending on the distribution of proposal power. If both players commit sequentially before the bargaining starts, we find a unique perfect equilibrium division. If both players commit simultaneously, there is a range of perfect equilibrium divisions. However, no player obtains less than one third of the pie, even with arbitrarily small proposal power. The equal split is the only division supported by a perfect equilibrium for any choice of the discount factor and the recognition probabilities. |
| Keywords: | microeconomics ; |
| Date: | 2010 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:umamet:2010013&r=gth |
| By: | Gustavo Bergantiños; Jordi Massó; Alejandro Neme |
| Abstract: | The division problem consists of allocating a given amount of an homogeneous and perfectly divisible good among a group of agents with single-peaked preferences on the set of their potential shares. A rule proposes a vector of shares for each division problem. The literature has implicitly assumed that agents will find acceptable any share they are assigned to. In this paper we consider the division problem when agents' participation is voluntary. Each agent has an idiosyncratic interval of acceptable shares where his preferences are single-peaked. A rule has to propose to each agent either to not participate or an acceptable share because otherwise he would opt out and this would require to reassign some of the remaining agents' shares. We study a subclass of efficient and consistent rules and characterize extensions of the uniform rule that deal explicitly with agents' voluntary participation. |
| Keywords: | Division Problem, Single-peaked Preferences, Uniform Rule, Voluntary Participation |
| JEL: | D71 |
| Date: | 2010–03–05 |
| URL: | http://d.repec.org/n?u=RePEc:aub:autbar:807.10&r=gth |