|
on Game Theory |
| By: | Guilherme Carmona (Universidade Nova de Lisboa) |
| Abstract: | We consider an asymptotic version of Mas-Colell's theorem on the existence of pure strategy Nash equilibria in large games. Our result states that, if players' payoff functions are selected from an equicontinuous family, then all sufficiently large games have an epsilon - pure, epsilon - equilibrium for all epsilon greater than 0. We also show that our result is equivalent to Mas-Colell's existence theorem, implying that it can properly be considered as its asymptotic version. |
| Keywords: | Nash Equilibrium; Asymptotic Results; Pure Strategies; Approximate equilibria; Equicontinuity; Purification |
| JEL: | C72 |
| Date: | 2004–12–21 |
| URL: | http://d.repec.org/n?u=RePEc:wpa:wuwpga:0412008&r=gth |
| By: | Guilherme Carmona (Universidade Nova de Lisboa) |
| Abstract: | We characterize Nash equilibria of games with a continuum of players (Mas-Colell (1984)) in terms of approximate equilibria of large finite games. For the concept of $(\epsilon,\epsilon)$ - equilibrium --- in which the fraction of players not $\epsilon$ - optimizing is less than $\epsilon$ --- we show that a strategy is a Nash equilibrium in a game with a continuum of players if and only if there exists a sequence of finite games such that its restriction is an $(\epsilon_n,\epsilon_n)$ - equilibria, with $\epsilon_n$ converging to zero. The same holds for $\epsilon$ - equilibrium --- in which almost all players are $\epsilon$ - optimizing --- provided that either players' payoff functions are equicontinuous or players' action space is finite. Furthermore, we give conditions under which the above results hold for all approximating sequences of games. In our characterizations, a sequence of finite games approaches the continuum game in the sense that the number of players converges to infinity and the distribution of characteristics and actions in the finite games converges to that of the continuum game. These results render approximate equilibria of large finite economies as an alternative way of obtaining strategic insignificance. |
| Keywords: | Nash equilibrium, Games with a continuum of players, Games with finitely many players, approximate equilibria. |
| JEL: | C72 |
| Date: | 2004–12–21 |
| URL: | http://d.repec.org/n?u=RePEc:wpa:wuwpga:0412009&r=gth |
| By: | Grigorieva,Elena; Herings,Jean-Jacques; Müller,Rudolf; Vermeulen,Dries (METEOR) |
| Abstract: | In this paper we present a new auction, the bisection auction, that can be used for the sale of a single indivisible object. We discuss the issue concerning the information revelation requirement of this auction and the associated amount of data that needs to be transmitted. We show that in the truthtelling equilibrium the bisection auction is economical in its demand for information on the valuations of the players. It requires the players to transmit less information bits to the auctioneer than the Vickrey and English auctions. In particular, we prove that for integer valuations uniformly distributed on the interval [0,L) the bisection auction of n players requires in expectation transmission of at most 2n + log L information bits by the players. Compared with the corresponding number in the Vickrey auction which is n log L, and in the English auction which is on average at least (1/3) nL, the bisection auction turns out to be the best performer. |
| Keywords: | mathematical economics; |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:umamet:2004052&r=gth |
| By: | V. Bhaskar (University of Essex); George J. Mailath (Department of Economics, University of Pennsylvania); Stephen Morris (Yale University) |
| Abstract: | This paper investigates the Harsanyi (1973)-purifiability of mixed strategies in the repeated prisoners’ dilemma with perfect monitoring. We perturb the game so that in each period, a player receives a private payoff shock which is independently and identically distributed across players and periods. We focus on the purifiability of a class of one-period memory mixed strategy equilibria used by Ely and Välimäki (2002) in their study of the repeated prisoners’ dilemma with private Monitoring. We find that the strategy profile is purifiable by perturbed-game finite-memory strategies if and only if it is strongly symmetric, in the sense that after every history, both players play the same mixed action. Thus “most” strategy profiles are not purifiable by finite memory strategies. However, if we allow infinite memory strategies in the perturbed game, then any completely-mixed equilibrium is purifiable. |
| Keywords: | Purification, repeated games, belief-free equilibria, imperfect monitoring. |
| JEL: | C72 C73 |
| Date: | 2004–01–14 |
| URL: | http://d.repec.org/n?u=RePEc:pen:papers:04-004&r=gth |
| By: | Estevez-Fernandez; Borm,P.E.M.; Calleja,P.; Hamers,H.J.M. (Tilburg University, Center for Economic Research) |
| Abstract: | Two classes of one machine sequencing situations are considered in which each job corresponds to exactly one player but a player may have more than one job to be processed, so called RP(repeated player) sequencing situations. In max-RP sequencing situations it is assumed that each player's cost function is linear with respect to the maximum completion time of his jobs, whereas in min-RP sequencing situations the cost functions are linear with respect to the minimum completion times. For both classes, following explicit procedures to go from the initial processing order to an optimal order for the coalition of all players, equal gain splitting rules are defined. It is shown that these rules lead to core elements of the associated RP sequencing games. Moreover, it is seen that min-RP sequencing games are convex. |
| JEL: | C71 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:kubcen:2004128&r=gth |
| By: | Slikker,M.; Norde,H.W. (Tilburg University, Center for Economic Research) |
| Abstract: | We study the model of link formation that was introduced by Aumann and Myerson (1988) and focus on symmetric convex games with transferable utilities. We answer an open question in the literature by showing that in a specific symmetric convex game with six players a structure that results in the same payoffs as the full cooperation structure can be formed according to a subgame perfect Nash equilibrium. |
| JEL: | C71 C72 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:kubcen:2004114&r=gth |
| By: | Dreef,M.; Tijs,S.H. (Tilburg University, Center for Economic Research) |
| Abstract: | This paper studies two classes of two-person zero-sum games in which the strategies of both players are of a special type. Each strategy can be split into two parts, a taking and a guessing part. In these games two types of asymmetry between the players can occur. In the first place, the number of objects available for taking does not need to be the same for both players. In the second place, the players can be guessing sequentially instead of simultaneously; the result is asymmetric information. The paper studies the value and equilibria of these games, for all possible numbers of objects available to the players, for the case with simultaneous guessing as well as for the variant with sequential guessing. |
| JEL: | C72 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:kubcen:2004129&r=gth |
| By: | Nikolai S. Kukushkin (Russian Academy of Sciences, Dorodnicyn Computing Center) |
| Abstract: | Strategic games are considered where the players derive their utilities from participation in certain 'processes.' Two subclasses consisting exclusively of potential games are singled out. In the first, players choose where to participate, but there is a unique way of participation, the same for all players. In the second, the participation structure is fixed, but each player may have an arbitrary set of strategies. In both cases, the players sum up the intermediate utilities; thus the first class essentially coincides with that of congestion games. The necessity of additivity in either case is proven. |
| Keywords: | Nash equilibrium; Potential games; Congestion games; Additive aggregation |
| JEL: | C72 |
| Date: | 2004–12–22 |
| URL: | http://d.repec.org/n?u=RePEc:wpa:wuwpga:0412010&r=gth |
| By: | Frank Heinemann; Rosemarie Nagel; Peter Ockenfels |
| Abstract: | This paper explores three aspects of strategic uncertainty: its relation to risk, predictability of behavior and subjective beliefs of players. In a laboratory experiment we measure subjects’ certainty equivalents for three coordination games and one lottery. Behavior in coordination games is related to risk aversion, experience seeking, and age. From the distribution of certainty equivalents we estimate probabilities for successful coordination in a wide range of games. For many games, success of coordination is predictable with a reasonable error rate. The best response to observed behavior is close to the global-game solution. Comparing choices in coordination games with revealed risk aversion, we estimate subjective probabilities for successful coordination. In games with a low coordination requirement, most subjects underestimate the probability of success. In games with a high coordination requirement, most subjects overestimate this probability. Estimating probabilistic decision models, we show that the quality of predictions can be improved when individual characteristics are taken into account. Subjects’ behavior is consistent with probabilistic beliefs about the aggregate outcome, but inconsistent with probabilistic beliefs about individual behavior. |
| JEL: | C72 C91 D81 D84 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:ces:ceswps:_1364&r=gth |
| By: | Oliver Hart; John Moore |
| Abstract: | We view a contract as a list of outcomes. Ex ante, the parties commit not to consider outcomes not on the list, i.e., these are "ruled out". Ex post, they freely bargain over outcomes on the list, i.e., the contract specifies no mechanism to structure their choice; in this sense outcomes on the list are not "ruled out". A "loose" contract (long list) maximizes flexibility but may interfere with ex ante investment incentives. When these incentives are important enough, the parties may write a "tight" contract (short list), even though this leads to ex post inefficiency. |
| Keywords: | Agreements to agree, ruling out but not ruling in, bargaining, ex post inefficiency. |
| Date: | 2004–03 |
| URL: | http://d.repec.org/n?u=RePEc:cep:stitep:/2004/472&r=gth |
| By: | Aloisio Pessoa de Araújo (EPGE/FGV); Luciano I. de Castro Filho; Humberto Luiz Ataide Moreira (EPGE/FGV) |
| Date: | 2004–11 |
| URL: | http://d.repec.org/n?u=RePEc:fgv:epgewp:571&r=gth |
| By: | Paulo Klinger Monteiro (EPGE/FGV) |
| Date: | 2004–09 |
| URL: | http://d.repec.org/n?u=RePEc:fgv:epgewp:568&r=gth |
| By: | Quant,M.; Meertens,M.; Reijnierse,J.H. (Tilburg University, Center for Economic Research) |
| Abstract: | A generalization of processing problems with restricted capacities is introduced. In a processing problem there is a finite set of jobs, each requiring a specific amount of effort to be completed, whose costs depend linearly on their completion times. The new aspect is that players have interest in all jobs. The corresponding cooperative game of this generalization is proved to be totally balanced. |
| JEL: | C71 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:kubcen:2004126&r=gth |
| By: | Michael Berlemann; Marcus Dittrich; Gunther Markwardt |
| Abstract: | In this paper we present three simple theoretical models to explain the influence of the possibility to make non-binding announcements on investment behaviour in public goods settings. Our models build on the idea that voluntary contributions to the supply of a public good might be motivated by some form of joy of giving. We show that the possibility to make non-binding announcements has a positive effect on cooperative behaviour, especially if individual announcements and factual investments are communicated to the players after each round. We also show that this result holds true even though the players have an incentive to overstate their true degrees of cooperativeness. Altogether, our theoretical considerations point in the direction that revealing as much information on individual intentions and factual behaviour as possible enhances cooperative behaviour. These conclusions are broadly confirmed by the results of a series of classroom experiments we present. |
| Keywords: | public goods, announcements, joy of giving, experimental economics |
| JEL: | C92 D74 H41 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:ces:ceswps:_1352&r=gth |
| By: | Gonzalez-Diaz,J.; Borm,P.E.M.; Norde,H.W. (Tilburg University, Center for Economic Research) |
| Abstract: | The division of a cake by n players is modeled as a game of timing. We show that such games admit a unique Nash equilibrium. |
| JEL: | C72 D61 D63 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:kubcen:2004119&r=gth |
| By: | Springael J.; De Smet Y.; Kunsch P. |
| Abstract: | Many occurrences in real-life result from decisions taken by a very large number of decision makers. Furthermore, these decisions often depend on the optimisation of several conflicting criteria. The decision or behaviour of every microscopic entity results finally in a global macroscopic behaviour of the whole group of decision makers. In such context, the use of classical decision support tools, such as multicriteria based group decision support systems, to model these macroscopic phenomena is not appropriate. In order to tackle this type of problems, we introduce a tool based on Markov chains to model and manage these large group decisions. Finally, the method results in a statistical distribution of decisions, allowing us to study the impact of policy measures on the global group behaviour. This is illustrated by means of a simple example stemming from the telecommunication sector. |
| Date: | 2004–06 |
| URL: | http://d.repec.org/n?u=RePEc:ant:wpaper:2004011&r=gth |
| By: | George J. Mailath (Department of Economics, University of Pennsylvania); Stephen Morris (Department of Economics, Yale University) |
| Abstract: | Some private-monitoring games, that is, games with no public histories, can have histories that are almost public. These games are the natural result of perturbing public-monitoring games towards private monitoring. We explore the extent to which it is possible to coordinate continuation play in such games. It is always possible to coordinate continuation play by requiring behavior to have bounded recall (i.e., there is a bound L such that in any period, the last L signals are sufficient to determine behavior). We show that, in games with general almost-public private monitoring, this is essentially the only behavior that can coordinate continuation play. |
| Keywords: | repeated games, private monitoring, almost-public monitoring, coordination, bounded recall |
| JEL: | C72 C73 D82 |
| Date: | 2004–08–20 |
| URL: | http://d.repec.org/n?u=RePEc:pen:papers:04-033&r=gth |
| By: | Quant,M.; Borm,P.E.M.; Fiestras-Janeiro,G.; Megen,F. van (Tilburg University, Center for Economic Research) |
| Abstract: | This paper extends the concepts of proper equilibria, protective behaviour and prudent behaviour to multicriteria games. Three types of proper equilibria based on different types of domination are introduced. It is shown that protective behaviour coincides with prudent behaviour. Possible relations and existence are analyzed. |
| JEL: | C72 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:kubcen:2004127&r=gth |
| By: | Julio Dávila (Department of Economics, University of Pennsylvania); Jan Eeckhout (Department of Economics, University of Pennsylvania) |
| Abstract: | We propose a simple bargaining procedure, the equilibrium of which converges to the Walrasian allocation as the agents become increasingly patient. We thus establish that the competitive outcome obtains even if agents have market power and are not price-takers. Moreover, where in other bargaining protocols the final outcome depends on bargaining power or relative impatience, the outcome here is determinate and depends only on preferences and endowments. This procedure has therefore important implications for policy applications compared to standard bargaining rules. |
| Keywords: | Bargaining, Walrasian Equilibrium, Price-setting, Quantity Constraints, Search, Matching |
| JEL: | C60 C71 D41 D51 |
| Date: | 2004–05–01 |
| URL: | http://d.repec.org/n?u=RePEc:pen:papers:04-024&r=gth |
| By: | Berliant,M.; Raa,M.H. ten (Tilburg University, Center for Economic Research) |
| Abstract: | Conventional wisdom tells us that with no market failure and local non-satiation of preferences, the core is at least as large as the collection of competitive equilibrium allocations. We con.rm this for a standard model featuring land. Next we consider the public land ownership version of the model. If the role of land ownership and rent distribution is assumed by a government that ploughs back rent (at least in excess of its agricultural value) to its citizens, the equilibrium remains efficient, but no longer need be in the core. |
| JEL: | H42 R13 R52 D51 D61 |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:kubcen:2004118&r=gth |
| By: | Hillinger, Claude |
| Abstract: | In his seminal Social Choice and Individual Values, Kenneth Arrow stated that his theory applies to voting. Many voting theorists have been convinced that, on account of Arrow?s theorem, all voting methods must be seriously flawed. Arrow?s theory is strictly ordinal, the cardinal aggregation of preferences being explicitly rejected. In this paper I point out that all voting methods are cardinal and therefore outside the reach of Arrow?s result. Parallel to Arrow?s ordinal approach, there evolved a consistent cardinal theory of collective choice. This theory, most prominently associated with the work of Harsanyi, continued the older utilitarian tradition in a more formal style. The purpose of this paper is to show that various derivations of utilitarian SWFs can also be used to derive utilitarian voting (UV). By this I mean a voting rule that allows the voter to score each alternative in accordance with a given scale. UV-k indicates a scale with k distinct values. The general theory leaves k to be determined on pragmatic grounds. A (1,0) scale gives approval voting. I prefer the scale (1,0,-1) and refer to the resulting voting rule as evaluative voting. A conclusion of the paper is that the defects of conventional voting methods result not from Arrow?s theorem, but rather from restrictions imposed on voters? expression of their preferences. The analysis is extended to strategic voting, utilizing a novel set of assumptions regarding voter behavior. |
| JEL: | D72 D71 |
| Date: | 2004–12 |
| URL: | http://d.repec.org/n?u=RePEc:lmu:muenec:473&r=gth |
| By: | Jansen,Thijs; Lier,Arie,van; Witteloostuijn,Arjen,van (METEOR) |
| Abstract: | In this paper, we consider a two-stage (sequential) game as introduced by Vickers (1985),Fershtman (1985), Fershtman and Judd (1987) and Sklivas (1987). This game models the situation where the owners of competing firms manipulate their managers'' incentive contracts for strategic reasons. Instead of the sales volume as part of these contracts, we introduce market share, besides profit, as a natural part of managers'' incentives. Then we compare the results with those obtained for combinations of profits and sales volume, as well as for the classical Cournot model. Concerning an {\eightit n}-firm oligopoly, and compared to the sales-delegation case, it appears that owners put more emphasis on managerial profit-maximizing behavior, indicated by smaller weights attributed to market share in managerial incentive contracts. Social welfare corresponding to the market share-delegation case almost equals welfare associated with the sales-delegation case. However, its components differ.The case of market share-delegation leads to a higher profitability of incumbent rivals and to a lower consumer surplus, in comparison to the sales-delegation case. One may state that the owner''s strategic use of market share as a managerial incentive leads to a (partial) shift of benefits from consumers to producers. |
| Keywords: | mathematical economics; |
| Date: | 2004 |
| URL: | http://d.repec.org/n?u=RePEc:dgr:umamet:2004051&r=gth |