A Model of Minimal Probabilistic Belief Revision

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).microeconomics ;

Epistemic Foundations for Backward Induction: An Overview

In this survey we analyze, and compare, various sufficient epistemic conditions for backward induction that have been proposed in the literature. To this purpose we present a simple epistemic base model for games with perfect information, and translate the different models into the language of this base model. As such, we formulate the various sufficient conditions for backward induction in a uniform language, which enables us to explictly analyze their differences and similarities.mathematical economics;

Rationalizability and Minimal Complexity in Dynamic Games

This paper presents a formal epistemic framework for dynamic games in which players, during the course of the game, may revise their beliefs about the opponents'' utility functions. We impose three key conditions upon the players'' beliefs: (a) throughout the game, every move by the opponent should be interpreted as being part of a rational strategy, (b) the belief about the opponents'' relative ranking of two strategies should not be revised unless one is certain that the opponent has decided not to choose one of these strategies, and (c) the players'' initial beliefs about the opponents'' utility functions should agree on a given profile u of utility functions. Types that, throughout the game, respect common belief about these three events, are called persistently rationalizable for the profile u of utility functions. It is shown that persistent rationalizability implies the backward induction procedure in generic games with perfect information. We next focus on persistently rationalizable types for u that hold a theory about the opponents of ``minimal complexity'''', resulting in the concept of minimal rationalizability. For two-player simultaneous move games, minimal rationalizability is equivalent to the concept of Nash equilibrium strategy. In every outside option game, as defined by van Damme (1989), minimal rationalizability uniquely selects the forward induction outcome.microeconomics ;

Proper Rationalizability and Belief Revision in Dynamic Games

In this paper we develop an epistemic model for dynamic games in which players may revise their beliefs about the opponents'' preferences (including the opponents'' utility functions) as the game proceeds. Within this framework, we propose a rationalizability concept that is based upon the following three principles: (1) at every instance of the game, a player should believe that his opponents are carrying out optimal strategies, (2) a player should only revise his belief about an opponent''s relative ranking of two strategies if he is certain that the opponent has decided not to choose one of these strategies, and (3) the players'' initial beliefs about the opponents'' utility functions should agree on a given profile u of utility functions. Common belief about these events leads to the concept of persistent rationalizability for the profile u of utility functions. It is shown that for a given profile u of utility functions, every properly rationalizable strategy for ``types with non-increasing type supports'''' is a persistently rationalizable strategy for u. This result implies that persistently rationalizable strategies always exist for all game trees and all profiles of utility functions.mathematical economics;

Nash Equilibrium as an Expression of Self-Referential Reasoning

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 should believe 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 equilibrium strategy (that is, a strategy that is optimal in some Nash equilibrium). We thus provide a set of sufficient conditions for Nash equilibrium strategy choice. We also show that none of these seven conditions can be dropped.mathematical economics;

Strategic Disclosure of Random Variables

We consider a game G(n) played by two players. There are n independent random variables Z(1),...,Z(n), each of which is uniformly distributed on [0,1]. Both players know n, the independence and the distribution of these random variables, but only player 1 knows the vector of realizations z := (z(1),...,z(n)) of them. Player 1 begins by choosing an order z(k(1)),...,z(k(n)) of the realizations. Player 2, who does not know the realizations, faces a stopping problem. At period 1, player 2 learns z(k(1)). If player 2 accepts, then player 1 pays z(k(1)) euros to player 2 and play ends. Otherwise, if player 2 rejects, play continues similarly at period 2 with player 1 offering z(k(2)) euros to player 2. Play continues until player 2 accepts an offer. If player 2 has rejected n-1 times, player 2 has to accept the last offer at period n. This model extends Moser''s (1956) problem, which assumes a non-strategic player 1.We examine different types of strategies for the players and determine their guarantee levels. Although we do not find the exact value v(n) of the game G(n) in general, we provide an interval I(n) = [a(n),b(n)] containing v(n) such that the length of I(n) is at most 0.07 and converges to 0 as n tends to infinity. We also point out strategies, with a relatively simple structure, which guarantee that player 1 has to pay at most b(n) and player 2 receives at least a(n). In addition, we completely solve the special case G(2) where there are only two random variables. We mention a number of intriguing open questions and conjectures, which may initiate further research on this subject.mathematical economics;

Bargaining in networks and the myerson value.

We focus on a multiperson bargaining situation where the negotiation possibilities for the players are represented by a graph, that is, two players can negotiate directly with each other if and only if they are linked directly in the graph. The value of cooperation among players is given by a TU game. For the case where the graph is a tree and the TU game is strictly convex we present a noncooperative bargaining procedure, consisting of a sequence of bilateral negotiations, for which the unique subgame perfect equilibrium outcome coincides with the Myerson value of the induced graph-restricted game. In each bilateral negotiation, the corresponding pair of players bargains about the difference in payoffs to be received at the end. At the beginning of such negotiation there is a bidding stage in which both players announce prices. The player with the highest price becomes the proposer and makes a take-it-or-leave-it offer in terms of difference in payoffs to the other player. If the proposal is rejected, the proposer pays his announced price to the other player, after which this particular link is eliminated from the graph and the mechanism starts all over again for the remaining graph.

Commitment in Alternating Offers Bargaining

We extend the Ståhl-Rubinstein alternating-offer bargaining procedure to allow players, prior to each bargaining round, to simultaneously and visibly commit to some share of the pie. If commitment costs are small but increasing in the committed share, then the unique outcome consistent with common belief in future rationality (Perea, 2009), or more restrictively subgame perfect Nash equilibrium, exhibits a second mover advantage. In particular, as the smallest share of the pie approaches zero, the horizon approaches in…nity, and commitment costs approach zero, the unique bargaining outcome corresponds to the reversed Rubinstein outcome (d/(1 + d); 1/(1 + d)).alternating offer bargaining; bargaining power; commitment; epistemic game theory; patience

Repeated Games with Voluntary Information Purchase

We consider discounted repeated games in which players can voluntarily purchase information about the opponents’ actions at past stages. Information about a stage can be bought at a fixed but arbitrary cost. Opponents cannot observe the information purchase by a player. For our main result, we make the usual assumption that the dimension of the set FIR of feasible and individually rational payoff vectors is equal to the number of players. We show that, if there are at least three players and each player has at least four actions, then every payoff vector in the interior of the set FIR can be achieved by a Nash equilibrium of the discounted repeated game if the discount factor is sufficiently close to 1. Therefore, nearly efficient payoffs can be achieved even if the cost of monitoring is high. We show that the same result holds if there are at least four players and at least three actions for each player. Finally, we indicate how the construction can be extended to sequential equilibrium.mathematical economics;

A non-welfarist solution for two-person bargaining situations.

In this paper we present a non-welfarist solution which is applicable to a broad spectrum of twoagent bargaining problems, such as exchange economies, location problems and division problems. In contrast to welfarist bargaining solutions, it depends only on the agents' preferences. not on their specific utility representation, and takes explicitly into account the underlying space of alternatives. We offer a simple sequential move mechanism, without chance moves, that implements our solution in subgame perfect equilibrium. Moreover, an axiomatic characterization of the solution is provided. It is shown that the solution coincides with the Kalai-Rosenthal bargaining solution after choosing a suitable utility representation of the preferences. When applied to exchange economies with equal initial endowments for both agents, the solution generates envy-free, Pare to efficient egalitarian equivalent allocations.Bargaining; Nash program; Welfarism; Non-welfarism; Exchange economies; Location problems; Implementation;