Search

The solution concept of iterated strict dominance for static games with complete information recursively deletes choices that are inferior. Here, we devise such an algorithm for the more general case of incomplete information. The ensuing solution concept of generalized iterated strict dominance is characterized in terms of common belief in rationality as well as in terms of best response sets. Besides, we provide doxastic conditions that are necessary and sufficient for modelling complete information from a one-person perspective.

Battigalli (1997) has shown that in dynamic games with perfect information and without relevant ties, the forward induction concept of extensive-form rationalizability yields the backward induction outcome. In this paper we provide a new proof for this remarkable result, based on four steps. We first show that extensive-form rationalizability can be characterized by the iterated application of a special reduction operator, the strong belief reduction operator. We next prove that this operator satisfies a mild version of monotonicity, which we call monotonicity on reachable histories. This property is used to show that for this operator, every possible order of elimination leads to the same set of outcomes. We finally show that backward induction yields a possible order of elimination for the strong belief reduction operator. These four properties together imply Battigalli's theorem.

After a slight reformulation of the sequential equilibrium concept, we display the set of sequential equilibria of a finite extensive form game as the set of fixed points of a upper hemicontinuous contractible valued best response correspondence. A simple signalling game illustrates the main concepts, after which we lay out the formalism of extensive form games. A behavior strategy for an agent is an assignment of a probability distribution over the available actions at each of the agent’s information sets. A belief is an assignment to each information set of a probability distribution over the information set. When play is governed by an interior behavior strategy profile, each node occurs with positive probability, so there is an induced belief given by taking conditional probabilities. A consistent assessment is a behavior strategy profile-belief pair that is a limit of a sequence of interior behavior strategy profile-induced belief pairs. A sequential equilibrium is a consistent assessment in which behavior at each information set maximizes expected utility, as computed from the belief at the information set and the strategy profile. A consistent conditional system includes additional conditional probabilities beyond those in a consistent assessment. We show that the best response correspondence for consistent conditional systems is contractible valued; in a sense, including additional information “unfolds” the best response correspondence for consistent assessments. Existence proofs for previously known refinements of the sequential equilibrium concept are obtained by showing that certain subcorrespondences are upper hemicontinuous and contractible valued.

In game theory, basic solution concepts often conflict with experimental findings or intuitive reasoning. This fact is possibly due to the requirement that zero probability is assigned to irrational choices in these concepts. Here, we introduce the epistemic notion of common belief in utility proportional beliefs which also attributes positive probability to irrational choices, restricted however by the natural postulate that the probabilities should be proportional to the utilities the respective choices generate. Besides, we propose a procedural characterization of our epistemic concept. With regards to experimental findings common belief in utility proportional beliefs fares well in explaining observed behavior.

For dynamic games we consider the idea that a player, at every stage of the game, will always believe that his opponents will choose rationally in the future. This is the basis for the concept of common belief in future rationality, which we formalize within an epistemic model. We present an iterative procedure, backward dominance, that proceeds by eliminating strategies from the game, based on strict dominance arguments. We show that the backward dominance procedure selects precisely those strategies that can rationally be chosen under common belief in future rationality if we would not impose (common belief in) Bayesian updating.

The Rubinstein alternating offers bargaining game is reconsidered under the assumption that each player is loss averse and the associated reference point is equal to the highest turned down offer of the opponent in the past. This makes the payoffs and therefore potential equilibrium strategies dependent on the history of play. A subgame perfect equilibrium is constructed, in which the strategies depend on the history of play through the current reference points. It is shown that this equilibrium is unique under some assumptions that it shares with the equilibrium in the classical model: immediate acceptance of equilibrium offers, indifference between acceptance and rejection of such offers, and strategies depending only on the current reference points. It is also shown that in this equilibrium loss aversion is a disadvantage. Moreover, a relation with asymmetric Nash bargaining is established, where a player's bargaining power is negatively related to own loss aversion and positively to the opponent's loss aversion. (C) 2011 Elsevier B.V. All rights reserved.

Andrés Perea of Maastricht University reviews “The Theory of Extensive Form Games,” by Carlos Alós-Ferrer and Klaus Ritzberger. The Econlit abstract of this book begins: “Extends the theory of extensive-form games to the general case without any finiteness and synthesizes different models that have been proposed for extensive-form games. Discusses game trees; pseudotrees and order theory; extensive decision problems; extensive forms; discrete extensive forms; and equilibrium.”

We consider bargaining problems under the assumption that players are loss averse, i.e., experience disutility from obtaining an outcome lower than some reference point. We follow the approach of Shalev (2002) by imposing the self-supporting condition on an outcome: an outcome z in a bargaining problem is self-supporting under a given bargaining solution, whenever transforming the problem using outcome z as a reference point, yields a transformed problem in which the solution is z.We show that n-player bargaining problems have a unique self-supporting outcome under the Kalai-Smorodinsky solution. For all possible loss aversion coefficients we determine the bargaining solutions that give exactly these outcomes, and characterize them by the standard axioms of Scale Invariance, Individual Monotonicity, and Strong Individual Rationality, and a new axiom called Proportional Concession Invariance (PCI). A bargaining solution satisfies PCI if moving the utopia point in the direction of the solution outcome does not change this outcome.

We investigate the problem in which an agent has to find an object that moves between two locations according to a discrete Markov process (see Pollock, 1970). At every period, the agent has three options: searching left, searching right, and waiting. We assume that waiting is costless whereas searching is costly. Waiting can be useful because it could induce a more favorable probability distribution over the two locations next period. We find an essentially unique (nearly) optimal strategy, and prove that it is characterized by two thresholds (as conjectured by Weber, 1986). We show, moreover, that it can never be optimal to search the location with the lower probability of containing the object. The latter result is far from obvious and is in clear contrast with the example in Ross (1983) for the model without waiting. We also analyze the case of multiple agents. This makes the problem a more strategic one, since now the agents not only compete against time but also against each other in finding the object. We find different kinds of subgame perfect equilibria, possibly containing strategies that are not optimal in the oneagent case. We compare the various equilibria in terms of cost-effectiveness.

Entre las medidas para divulgar la Hidrologia Medica en la sociedad espanola se han realizado camoanas de difusion al ciudadano en medios de comunicadion, ademas de actividades dirigidas a universitarios y a profesionales sanitarios. Ahora, para potenciar la cultura del agua se propone tambien hacer llegar su conocimiento a los estudiantes de la E.S.O. (Ensenanza Secundaria Obligatoria), especialmente a los de Bachillerato. Por este motivo, se presenta la siguiente programacion-tipo de una unidad didactica en la que se exponen los conceptos fundamentales relacionados con la Hidrologia en general y com la Hidrologia Medica en particular. Esta unidad didactica se ha cocebido como parte integrante de la asignatura de Biologia de 2ocurso de Bachillerato. En este trabajo se sigue una metodologia clasica en el desarrollo de la unidad didactica.

An allocation rule is called Bayes–Nash incentive compatible, if there exists a payment rule, such that truthful reports of agents' types form a Bayes–Nash equilibrium in the direct revelation mechanism consisting of the allocation rule and the payment rule. This paper provides a characterization of Bayes–Nash incentive compatible allocation rules in social choice settings where agents have multi-dimensional types, quasi-linear utility functions and interdependent valuations. The characterization is derived by constructing complete directed graphs on agents' type spaces with cost of manipulation as lengths of edges. Weak monotonicity of the allocation rule corresponds to the condition that all 2-cycles in these graphs have non-negative length. For the case that type spaces are convex and the valuation for each outcome is a linear function in the agent's type, we show that weak monotonicity of the allocation rule together with an integrability condition is a necessary and sufficient condition for Bayes–Nash incentive compatibility.

A probabilistic belief revision function assigns to every initial probabilistic belief and every observable event some revised probabilistic belief that only attaches positive probability to states in this event. We propose three axioms for belief revision functions: (1) linearity, meaning that if the decision maker observes that the true state is in {a,b}, and hence state c is impossible, then the proportions of c''s initial probability that are shifted to a and b, respectively, should be independent of c''s initial probability; (2) transitivity, stating that if the decision maker deems belief β equally similar to states a and b, and deems β equally similar to states b and c, then he should deem β equally similar to states a and c; (3) information-order independence, stating that the way in which information is received should not matter for the eventual revised belief. We show that a belief revision function satisfies the three axioms above if and only if there is some linear one-to-one function ϕ, transforming the belief simplex into a polytope that is closed under orthogonal projections, such that the belief revision function satisfies minimal belief revision with respect to ϕ. By the latter, we mean that the decision maker, when having initial belief β₁ and observing the event E, always chooses the revised belief β₂ that attaches positive probability only to states in E and for which ϕ(β₂) has minimal Euclidean distance to ϕ(β₁).

En este trabajo se ha realizado una revision bibliografica de las epigrafias votivas referentes a las ninfas, pertenecientes a eras encontradas en los puntos de emergencia de agua mineromedicinal localizados en la region geografica de la antigua Hispania, y de la actual Espana. La revision se ha centrado especialmente en las publicaciones de los ultimos 20 anos, mediante la consulta de los fondos bibliograficos de las facultades de Medicina, Geografia e Historia de la Universidad Complutense de Madrid, y a traves de la busqueda en internet con el navegador Google.

In this chapter we focus on the epistemic concept of common belief in future rationality (Perea [37]), which describes a backward induction type of reasoning for general dynamic games. It states that a player always believes that his opponents will choose rationally now and in the future, always believes that his opponents always believe that their opponents choose rationally now and in the future, and so on, ad infinitum. It thus involves infinitely many conditions, which might suggest that this concept is too demanding for real players in a game. In this chapter we show, however, that this is not true. For finite dynamic games we present a finite reasoning procedure that a player can use to reason his way towards common belief in future rationality.

By a player splitting we mean a mechanism that distributes the information sets of a player among so-called agents. A player splitting is called independent if each path in the game tree contains at most one agent of every player. Following Mertens (1989), a solution is said to have the player splitting property if, roughly speaking, the solution of an extensive form game does not change by applying independent player splittings. We show that Nash equilibria, perfect equilibria, Kohlberg-Mertens stable sets and Mertens stable sets have the player splitting property. An example is given to show that the proper equilibrium concept does not satisfy the player splitting property. Next, we give a definition of invariance under (general) player splittings which is an extension of the player splitting property to the situation where we also allow for dependent player splittings. We come to the conclusion that, for any given dependent player splitting, each of the above solutions is not invariant under this player splitting. The results are used to give several characterizations of the class of independent player splittings and the class of single appearance structures by means of invariance of solution concepts under player splittings.

We consider two-player repeated games with nonobservable actions (cf. Lehrer 1989). An information mechanism for a player is a function which assigns a private signal to every action-pair of the one-shot game. In this paper, we extend the model to a situation in which both players can buy an information mechanism before playing the repeated game. Within this model, we provide a characterization of the lower equilibrium payoffs in terms of the one-shot game for the case that both players choose a nontrivial information mechanism with probability one. Moreover, we construct a lower equilibrium in a repeated game in which one of the players strictly randomizes between information mechanisms. It is shown that the corresponding payoffs cannot be induced by a lower equilibrium in which players choose a particular information mechanism with probability one.

In everyday life we must often reach decisions while knowing that the outcome will not only depend on our own choice, but also on the choices of others. These situations are the focus of epistemic game theory. Unlike classical game theory, it explores how people may reason about their opponents before they make their final choice in a game. Packed with examples and practical problems based on stories from everyday life, this is the first textbook to explain the principles of epistemic game theory. Each chapter is dedicated to one particular, natural way of reasoning. The book then shows how each of these ways of reasoning will affect the final choices that can rationally be made and how these choices can be found by iterative procedures. Moreover, it does so in a way that uses elementary mathematics and does not presuppose any previous knowledge of game theory.

Assuming the opponent's rationality In the previous chapter we discussed the idea of respecting the opponent's preferences , which states that you deem an opponent's choice a infinitely more likely than some other choice b whenever you believe that the opponent prefers a to b . That is, you deem a better choice infinitely more likely than a worse choice, even if you believe that both choices are not optimal for the opponent. Various examples illustrated that this idea may put severe restrictions on your lexicographic belief – not only on your primary belief but also on your secondary and higher-level beliefs. And these restrictions – we saw – may be crucial for the eventual choice you make in a game. The concept of respecting the opponent's preferences is just one reasonable condition we may impose on a lexicographic belief, but it is certainly not the only one. To illustrate this, let us return to the example “Spy game” we introduced in the previous chapter. Example 7.1 Spy game Recall the story from Example 6.5. For convenience, we have reproduced the utilities for you and Barbara in Table 7.1. As we have seen, common full belief in “caution and respect of preferences” leads you to choose pub c . In a nutshell, the reasoning was as follows: It is clear that Barbara prefers pub a to pub b . Hence, you must deem the event that Barbara will visit pub a infinitely more likely than the event that Barbara will visit pub b if you respect her preferences. This, in turn, implies that you will prefer pub b to pub a .

We consider situations where the cooperation and negotiation possibilities between pairs of agents are given by an undirected graph. Every connected component of agents has a value, which is the total surplus the agents can generate by working together. We present a simple, sequential, bilateral bargaining procedure, in which at every stage the two agents in a link (i, j) bargain about their share from cooperation in the connected component they are part of. We show that, if the marginal value of a link is increasing in the number of links in the connected component it belongs to, then this procedure yields exactly the Myerson value payoff (Myerson, 1977) for every player.

We adapt the core concept to deal with economies in which trade in assets takes place at period 1, uncertainty about asset payoffs is released at period 2, and agents trade in commodities afterwards. We define the weak sequential core as the set of allocations that are stable against coalitional deviations ex ante, and moreover cannot be improved upon by any coalition once the uncertainty is being released. We restrict ourselves to credible deviations, i.e. coalitional deviations at period 1 that cannot be counterblocked by some subcoalition at period 2. We study the relationship of the resulting core concept with other sequential core concepts, give sufficient conditions under which the weak sequential core is non-empty, but show that it is possible to give reasonable examples where it is empty.

In Chapter 3 we have discussed some restrictions on assessments which, roughly speaking, reflect the requirement that players should believe that opponents act optimally. In the concepts of Nash equilibrium and normal form perfect equilibrium, this requirement is only imposed at the beginning of the game, whereas in other concepts such as backward induction, subgame perfect equilibrium and quasi-perfect equilibrium, it is imposed at every stage of the game. The concepts discussed so far did not explicitly involve the belief system, however, since explicit restrictions were put on the behavioral conjecture profile only. In this chapter we discuss some rationality criteria which impose explicit conditions on the belief system as well. We consider two types of restrictions. The first type, which we refer to as consistency conditions, should guarantee that the behavioral conjectures and the beliefs in a given assessment should not contradict one another. For instance, the beliefs held by a player at an information set should be in accordance with his beliefs held at earlier information set, which, in turn, should be compatible with the behavioral conjecture held by this player about the opponents’ behavior. Conditions of the second type are similar to those presented in Chapter 3, as they require players to attach positive probability only to those opponents’ strategies that constitute sequential best responses. By imposing this requirement at every information set we obtain the notion of sequential rationality. If we impose it only at those information sets that are not avoided by the initial conjectures, the weaker notion of weak sequential rationality results.

In this chapter we investigate how rationality criteria for extensive form games react to certain transformations of games, that is, functions which map an extensive form game to some new game. In particular, it is analyzed whether the set of conjecture profiles selected by the criterion in the new game “coincides” with the set of conjecture profiles selected in the original game. If this is the case, we say that the criterion is invariant to the transformation under consideration. Invariance is a desirable property for a rationality criterion if one believes that the transformation does not affect the “strategic features” of the game.

In this paper we look for a solution to a land division problem that could be applied to different types of disputes when the arbitrator has a very limited information about the agents’ preferences. The solution must be fair and efficient under the constraint of the limited information available to the arbitrator. To this scope, we propose to use the concept of equal-opportunity equivalence defined by Thomson (Soc Choice Welfare 11:137–156 1994). We prove the existence of an efficient and equal opportunity equivalent allocation for a land division problem and we present a simple procedure to implement a rule that selects such allocation at each preference profile.