Within an epistemic model for two-player extensive games, we formalize the event that each player believes that his opponent chooses rationally at all information sets. Letting this event be common ...certain belief yields the concept of
sequential rationalizability. Adding preference for cautious behavior to this event likewise yields the concept of
quasi-perfect rationalizability. These concepts are shown to (a) imply backward induction in generic perfect information games, and (b) be non-equilibrium analogues to sequential and quasi-perfect equilibrium, leading to epistemic characterizations of the latter concepts. Conditional beliefs are described by the novel concept of a system of
conditional lexicographic probabilities.
Type structures are a simple device to describe higher-order beliefs. However, how can we check whether two types generate the same belief hierarchy? This paper generalizes the concept of a type ...morphism and shows that one type structure is contained in another if and only if the former can be mapped into the other using a generalized type morphism. Hence, every generalized type morphism is a hierarchy morphism and vice versa. Importantly, generalized type morphisms do not make reference to belief hierarchies. We use our results to characterize the conditions under which types generate the same belief hierarchy.
We provide comparable algorithms for the Dekel–Fudenberg procedure, iterated admissibility, proper rationalizability and full permissibility by means of the notions of likelihood orderings and ...preference restrictions. The algorithms model reasoning processes whereby each player’s preferences over his own strategies are completed by eliminating likelihood orderings. We apply the algorithms for comparing iterated admissibility, proper rationalizability and full permissibility, and provide a sufficient condition under which iterated admissibility does not rule out properly rationalizable strategies. We also use the algorithms to examine an economically relevant strategic situation, namely a bilateral commitment bargaining game. Finally, we discuss the relevance of our algorithms for epistemic analysis.
We extend the Ståhl–Rubinstein alternating-offer bargaining procedure to allow players to simultaneously and visibly commit to some share of the pie prior to, and for the duration of, each bargaining ...round. If commitment costs are small but increasing in the committed share, then the unique subgame perfect equilibrium outcome exhibits a second mover advantage. In particular, as the horizon approaches infinity, and commitment costs approach zero, the unique bargaining outcome corresponds to the reversed Rubinstein outcome (δ/(1+δ),1/(1+δ)), where δ is the common discount factor.
•We extend the alternating-offer bargaining model.•At the start of each bargaining round, each party may commit to a share of the pie.•When commitment costs are small but increasing, there is a second mover advantage.•This reverses the sharing of Rubinstein (1982).
Common belief in approximate rationality Mounir, Angie; Perea, Andrés; Tsakas, Elias
Mathematical social sciences,
January 2018, 2018-01-00, 20180101, Letnik:
91
Journal Article
Recenzirano
This paper substitutes the standard rationality assumption with approximate rationality in normal form games. We assume that players believe that their opponents might be ε-rational, i.e. willing to ...settle for a suboptimal choice, and so give up an amount ε of expected utility, in response to the belief they hold. For every player i and every opponents’ degree of rationality ε, we require player i to attach at least probability Fi(ε) to his opponent being ε-rational, where the functions Fi are assumed to be common knowledge amongst the players. We refer to this event as belief in F-rationality. The notion of Common Belief in F-Rationality (CBFR) is then introduced as an approximate rationality counterpart of the established Common Belief in Rationality. Finally, a corresponding recursive procedure is designed that characterizes those beliefs players can hold under CBFR.
•We allow for the possibility that players may err when making a choice.•Players’ error margins are their private information.•Lower bounds of probabilities assigned to margins of errors are common knowledge.•Common belief in F-rationality is a generalization of common belief in rationality.
Proper rationalizability (
Schuhmacher, 1999; Asheim, 2001) is a concept in epistemic game theory based on the following two conditions: (a) a player should be
cautious, that is, should not exclude ...any opponentʼs strategy from consideration; and (b) a player should
respect the opponentsʼ preferences, that is, should deem an opponentʼs strategy
s
i
infinitely more likely than
s
i
′
if he believes the opponent to prefer
s
i
to
s
i
′
. A strategy is properly rationalizable if it can optimally be chosen under common belief in the events (a) and (b). In this paper we present an algorithm that for every finite game computes the set of all properly rationalizable strategies. The algorithm is based on the new idea of a
preference restriction, which is a pair
(
s
i
,
A
i
)
consisting of a strategy
s
i
, and a subset of strategies
A
i
, for player
i. The interpretation is that player
i prefers some strategy in
A
i
to
s
i
. The algorithm proceeds by successively adding preference restrictions to the game.
Within a formal epistemic model for simultaneous-move games, we present the following conditions: (1) belief in the opponents' rationality (BOR), stating that a player believes that every opponent ...chooses an optimal strategy, (2) self-referential beliefs (SRB), stating that a player believes that his opponents hold correct beliefs about his own beliefs, (3) projective beliefs (PB), stating that i believes that j's belief about k's choice is the same as i's belief about k's choice, and (4) conditionally independent beliefs (CIB), stating that a player believes that opponents' types choose their strategies independently. We show that, if a player satisfies BOR, SRB and CIB, and believes that every opponent satisfies BOR, SRB, PB and CIB, then he will choose a Nash strategy (that is, a strategy that is optimal in some Nash equilibrium). We thus provide a sufficient collection of one-person conditions for Nash strategy choice. We also show that none of these seven conditions can be dropped.
In an extensive form game, an assessment is said to satisfy the
one-deviation property if for all possible payoffs at the terminal nodes the following holds: if a player at each of his information ...sets cannot improve upon his expected payoff by deviating unilaterally at this information set only, he cannot do so by deviating at any arbitrary collection of information sets. Hendon et al. (1996. Games Econom. Behav. 12, 274–282) have shown that pre-consistency of assessments implies the one-deviation property. In this note, it is shown that an appropriate weakening of pre-consistency, termed
updating consistency, is both a sufficient and necessary condition for the one-deviation property. The result is extended to the context of rationalizability.
In this paper we want to shed some light on what we mean by backward induction and forward induction reasoning in dynamic games. To that purpose, we take the concepts of common belief in future ...rationality (Perea 1) and extensive form rationalizability (Pearce 2, Battigalli 3, Battigalli and Siniscalchi 4) as possible representatives for backward induction and forward induction reasoning. We compare both concepts on a conceptual, epistemic and an algorithm level, thereby highlighting some of the crucial differences between backward and forward induction reasoning in dynamic games.
Utility proportional beliefs Bach, Christian W.; Perea, Andrés
International journal of game theory,
11/2014, Letnik:
43, Številka:
4
Journal Article
Recenzirano
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.
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