Nash equilibrium under knightian uncertainty: breaking down backward induction (extensively revised version)

We define Nash equilibrium for two-person normal form games in the presence of uncertainty, in the sense of Knight(1921). We use the fonna1iution of uncertainty due to Schmeidler and Gilboa. We show tbat there exist Nash equilibria for any degree of uncertainty, as measured by the uncertainty aversion (Dow anel Wer1ang(l992a». We show by example tbat prudent behaviour (maxmin) can be obtained as an outcome even when it is not rationaliuble in the usual sense. Next, we break down backward industion in the twice repeated prisoner's dilemma. We link these results with those on cooperation in the finitely repeated prisoner's dilemma obtained by Kreps-Milgrom-Roberts-Wdson(1982), and withthe 1iterature on epistemological conditions underlying Nash equilibrium. The knowledge notion implicit in this mode1 of equilibrium does not display logical omniscience.

A notion of subgame perfect Nash equilibrium under knightian uncertainty

We define a subgame perfect Nash equilibrium under Knightian uncertainty for two players, by means of a recursive backward induction procedure. We prove an extension of the Zermelo-von Neumann-Kuhn Theorem for games of perfect information, i. e., that the recursive procedure generates a Nash equilibrium under uncertainty (Dow and Werlang(1994)) of the whole game. We apply the notion for two well known games: the chain store and the centipede. On the one hand, we show that subgame perfection under Knightian uncertainty explains the chain store paradox in a one shot version. On the other hand, we show that subgame perfection under uncertainty does not account for the leaving behavior observed in the centipede game. This is in contrast to Dow, Orioli and Werlang(1996) where we explain by means of Nash equilibria under uncertainty (but not subgame perfect) the experiments of McKelvey and Palfrey(1992). Finally, we show that there may be nontrivial subgame perfect equilibria under uncertainty in more complex extensive form games, as in the case of the finitely repeated prisoner's dilemma, which accounts for cooperation in early stages of the game.

A notion of subgame perfect nash equilibrium under knightian uncertainty

We define a subgame perfect Nash equilibrium under Knightian uncertainty for two players, by means of a recursive backward induction procedure. We prove an extension of the Zermelo-von Neumann-Kuhn Theorem for games of perfect information, i. e., that the recursive procedure generates a Nash equilibrium under uncertainty (Dow and Werlang(1994)) of the whole game. We apply the notion for two well known games: the chain store and the centipede. On the one hand, we show that subgame perfection under Knightian uncertainty explains the chain store paradox in a one shot version. On the other hand, we show that subgame perfection under uncertainty does not account for the leaving behavior observed in the centipede game. This is in contrast to Dow, Orioli and Werlang(1996) where we explain by means of Nash equilibria under uncertainty (but not subgame perfect) the experiments of McKelvey and Palfrey(1992). Finally, we show that there may be nontrivial subgame perfect equilibria under uncertainty in more complex extensive form games, as in the case of the finitely repeated prisoner's dilemma, which accounts for cooperation in early stages of the game .

Trade or investment? location decisions under regional integration

This paper studies the production and trade patterns that may arise between two different countries if plant location is introduced as a first step in the producers' decision making. A three-stage game is used: the first deals with location and the next two with capacity and final sales decisions. Demand and cost structures differ by country, and the latter contain specific elements related to the foreign operation. The structure of possible Nash-equilibria is examined and an analysis of the changes in the solution, if the countries engage in an integration process, is made. As in previous models, though global welfare gains may not be very high, single country ones may be considerable, due to changes in the location of the plants. However, even if full integration takes place, global Marshallian welfare may decrease. Conditions which determine a tendency towards multinationalisation are obtained. Assuming a move toward integration, conditions are also provided to characterize when exporting will be preferred to local production. The fact that producers may retain a certain discriminating power, notwithstanding the elimination of barriers to arbitrage, creates a tendency to locate production in the country where prices are higher. This explains why welfare gains may not be obvious. An empirical illustration, with real data from two MERCOSUL countries (Brazil and Argentina) illustrates the possible outcomes.

Three principles of competitive nonlinear pricing

We make three contributions to the theory of contracting under asymmetric information. First, we establish a competitive analog to the revelation principIe which we call the implementation principIe. This principIe provides a complete characterization of all incentive compatible, indirect contracting mechanisms in terms of contract catalogs (or menus), and allows us to conclude that in competi tive contracting situations, firms in choosing their contracting strategies can restrict attention, without loss of generality, to contract catalogs. Second, we establish a competi tive taxation principIe. This principIe, a refinement of the implementation principIe, provides a complete characterization of all implementable nonlinear pricing schedules in terms of product-price catalogs and allows us to reduce any game played over nonlinear pricing schedules to a strategically equivalent game played over product-price catalogs. Third, using the competitive taxation principIe and a recent result due to Reny (1999) on the existence of Nash equilibria in discontinuous games, we demonstrate the existence of a N ash equilibrium for the mixed extension of the nonlinear pricing game.

Resultados uniformemente seguros e equilíbrio de Nash em jogos compactos

Nós introduzimos uma condição, resultados uniformemente seguros, para jogos compactos e resultados (“payoffs”) limitados e mensur´aveis nas estrat´egias. Demonstramos que se um jogo compacto tem resultados uniformemente seguros, ent˜ao sua extens˜ao mista tem resultados seguros.

Equilibria in security markets with a continuum of agents

We prove the existence of a competitive equilibrium for exchange economies with a measure space of agents and for which the commodity space is ` p, 1 < p < +∞. A vector x = (xn) in ` p may be interpreted as a security which promises to deliver xn units of numeraire at state (or date) n. Under assumptions imposing uniform bounds on marginal rates of substitution, positive results on core-Walras equivalence were established in Rustichini–Yannelis [21] and Podczeck [20]. In this paper we prove that under similar assumptions on marginal rates of substitution, the set of competitive equilibria (and thus the core) is non-empty.

First-price auction symmetric equilibria with a general distribution

In this paper I obtain the mixed strategy symmetric equilibria of the first-price auction for any distribution. The equilibrium is unique. The solution turns out to be a combination of absolutely continuous distributions case and the discrete distributions case.

The Bayesian Foundations of solution concepts of games

We transform a non co-operati ve game into a -Bayesian decision problem for each player where the uncertainty faced by a player is the strategy choices of the other players, the pr iors of other players on the choice of other players, the priors over priors and so on.We provide a complete characterization between the extent of knowledge about the rationality of players and their ability to successfulIy eliminate strategies which are not best responses. This paper therefore provides the informational foundations of iteratively unàominated strategies and rationalizable strategic behavior (Bernheim (1984) and Pearce (1984». Moreover, sufficient condi tions are also found for Nash equilibrium behavior. We also provide Aumann's (1985) results on correlated equilibria .

Nash equilibrium under knightian uncertainty: breaking-down backward induction

We present two alternative definitions of Nash equilibrium for two person games in the presence af uncertainty, in the sense of Knight. We use the formalization of uncertainty due to Schmeidler and Gilboa. We show that, with one of the definitions, prudent behaviour (maxmin) can be obtained as an outcome even when it is not rationalizable in the usual sense. Most striking is that with the Same definition we break down backward induction in the twice repeated prisoner's dilemma. We also link these results with the Kreps-Milgrom-Roberts-Wilson explanation of cooperation in the finitely repeated prisoner's dilemma.

A comment on "Rational learning lead to nash equilibrium" by professors Ehud Kalai and Ehud Lehrer

Kalai and Lebrer (93a, b) have recently show that for the case of infinitely repeated games, a coordination assumption on beliefs and optimal strategies ensures convergence to Nash equilibrium. In this paper, we show that for the case of repeated games with long (but finite) horizon, their condition does not imply approximate Nash equilibrium play. Recently Kalai and Lehrer (93a, b) proved that a coordination assumption on beliefs and optimal strategies, ensures that pIayers of an infinitely repeated game eventually pIay 'E-close" to an E-Nash equilibrium. Their coordination assumption requires that if players believes that certain set of outcomes have positive probability then it must be the case that this set of outcomes have, in fact, positive probability. This coordination assumption is called absolute continuity. For the case of finitely repeated games, the absolute continuity assumption is a quite innocuous assumption that just ensures that pIayers' can revise their priors by Bayes' Law. However, for the case of infinitely repeated games, the absolute continuity assumption is a stronger requirement because it also refers to events that can never be observed in finite time.

Pure strategy equilibria of multidimensional and non-monotonic auctions

We give necessary and sufficient conditions for the existence of symmetric equilibrium without ties in interdependent values auctions, with multidimensional independent types and no monotonic assumptions. In this case, non-monotonic equilibria might happen. When the necessary and sufficient conditions are not satisfied, there are ties with positive probability. In such case, we are still able to prove the existence of pure strategy equilibrium with an all-pay auction tie-breaking rule. As a direct implication of these results, we obtain a generalization of the Revenue Equivalence Theorem. From the robustness of equilibrium existence for all-pay auctions in multidimensional setting, an interpretation of our results can give a new justification to the use of tournaments in practice.

The set of equilibria of first-price auctions

In this note I specify the class of functions that are equilibria of symmetric first-price auctions.

Competitive equilibria in infinite-horizon collateralized economies with default penalties

Araújo, Páscoa and Torres-Martinez (2002) have shown that, without imposing either debt constraints or transversality conditions, Ponzi schemes are ruled out in infinite horizon economies with default when collateral is the only mechanism that partially secures loans. Páscoa and Seghir (2008) subsequently show that Ponzi schemes may reappear if, additionally to the seizure of the collateral, there are sufficiently harsh default penalties assessed (directly in terms of utility) against the defaulters. They also claim that if default penalties are moderate then Ponzi schemes are ruled out and existence of a competitive equilibrium is ensured. The objective of this paper is two fold. First, contrary to what is claimed by Páscoa and Seghir (2008), we show that moderate default penalties do not always prevent agents to run a Ponzi scheme. Second, we provide an alternative condition on default penalties that is sufficient to rule out Ponzi schemes and ensure the existence of a competitive equilibrium.