Informational aspects of a class of subjective games of incomplete information: Static case

Subjective games of incomplete information are formulated where some of the key assumptions of Bayesian games of incomplete information are relaxed. The issues arising because of the new formulation are studied in the context of a class of nonzero-sum, two-person games, where each player has a different model of the game. The static game is investigated in this note. It is shown that the properties of the static subjective game are different from those of the corresponding Bayesian game. Counterintuitive outcomes of the game can occur because of the different beliefs of the players. These outcomes may lead the players to realize the differences in their models.This work was sponsored by the Office of Naval Research under Contract No. N00014-84-C-0485.     

On discounted stochastic games with incomplete information on payoffs and a security application

This paper presents a robust optimization model for n$n$-person finite state/action stochastic games with incomplete information on payoffs. For polytopic uncertainty sets, we propose an explicit mathematical programming formulation for an equilibrium calculation. It turns out that a global optimal of this mathematical program yields an equilibrium point and epsilon-equilibria can be calculated based on this result. We briefly describe an incomplete information version of a security application that can benefit from robust game theory.     

A note on the L-P formulation of zero-sum sequential games with incomplete information

Zero-sum games with incomplete information are formulated as linear programs in which the players' behavioral strategies appear as primal and dual variables. Known properties for these games may then be derived from duality theory.     

New results in the theory of differential games with information time lag

The results contained herein provide a rigorous formulation of a broad class of differential games with information time lag and present a theoretical analysis for treating such games. This analysis extends the so-called Hamilton-Jacobi theory of optimal control and the main equation analysis developed by Isaacs to treat differential games with information time lag. Necessary and sufficient conditions satisfied by thepotential value function are developed to indicate the strategy-synthesis procedure for differential games with information time lag.     

Analysis of a family of two-person bargaining games with incomplete information

This paper computes the Harsanyi-Selten solution for a family of two-person bargaining games with incomplete information where one player hastwo possible types while the other player has onlyone possible type. The actual computation procedure is also outlined.     

Note Equilibrium payoffs in stochastic games of incomplete information: the general symmetric case

We prove that the existence of equilibrium payoffs for stochastic games of incomplete symmetric information follows from the same result for stochastic games with complete information. Received January 1999/Revised October 2001     

Solution for a class of repeated games without a recursive structure

We give the solution for all infinitely repeated two-person zero-sum games of incomplete information characterized by two 2×2 games, with information matrices including one or two absorbing states.     

An evolutionary game theoretic perspective on e-collaboration: The collaboration effort and media relativeness

Studies of e-collaboration from a game-theoretical perspective are practically nonexistent. This article contributes to filling this gap by focusing on the strategic interaction between players as they decide whether and how much to collaborate with each other. We use evolutionary game theory to make predictions about a two-person e-collaboration game. More specifically, we extend the traditional Prisoners’ Dilemma and Snowdrift game theory notions to discrete-strategy e-collaboration games, by explicitly including social punishments into the players’ payoff functions. We also introduce continuous-strategy e-collaboration games with both complete and incomplete information. Finally, we provide two generic dynamic programming models for e-collaboration games with media selection.     

Approximate common knowledge revisited

Suppose we replace “knowledge” by “belief with probability p” in standard definitions of common knowledge. Very different notions arise depending on the exact definition of common knowledge used in the substitution. This paper demonstrates those differences and identifies which notion is relevant in each of three contexts: equilibrium analysis in incomplete information games, best response dynamics in incomplete information games, and agreeing to disagree/no trade results.     

Stochastic pursuit-evasion differential games in the plane

Four stochastic pursuit-evasion differential games involving two players, P and E, moving in the plane are considered. The difference between the games lies in their information structures. In each of the games, sufficient conditions on optimal feedback strategies, in the cases of complete information, and on weak optimal feedback strategies, in the cases of incomplete information, are derived. Optimal strategies are computed for the cases of complete information and weak suboptimal strategies for the cases of incomplete information. The results indicate that the correct measurement of the direction of the segment PE is more important than the measurement of the distance (P, E).     

Approximation and representation of the value for some differential games with asymmetric information

We consider differential games with incomplete information. For special games with dynamics independent of the state of the system and linear payoffs, we give a representation formula for the value similar to the value of repeated games with lack of information on both sides. For general games, this representation formula does not hold and we introduce an approximation of the value: we build a sequence of functions converging to the value function.     

A fundamental study for partially defined cooperative games

The payoff of each coalition has been assumed to be known precisely in the conventional cooperative games. However, we may come across situations where some coalitional values remain unknown. This paper treats cooperative games whose coalitional values are not known completely. In the cooperative games it is assumed that some of coalitional values are known precisely but others remain unknown. Some complete games associated with such incomplete games are proposed. Solution concepts are studied in a special case where only values of the grand coalition and singleton coalitions are known. Through the investigations of solutions of complete games associated with the given incomplete game, we show a focal point solution suggested commonly from different viewpoints.     

小订单优先分配规则对供应链运作的影响

通过理论分析与实验研究相结合,在完全信息和不完全信息两种模式下,探讨供应短缺情景中小订单优先分配规则对零售商订货决策的影响。理论结果表明,小订单优先诱导零售商制定不超过真实市场需求量的订货决策,并且订货量不随运作成本而改变。实验数据表明,完全信息模式下,被试者的订货量高于理论预测值,不同运作成本的订货决策存在显著差异,并且被试者在不断重复决策中存在学习效应;不完全信息模式下,被试者的订货量与理论的预测没有显著偏差。这些研究结论为小订单优先分配规则的实践应用提供了理论和实证依据。     

Solutions for some bargaining games under the Harsanyi-Selten solution theory,part II: Analysis of specific bargaining games

Part II of the paper (for Part I see Harsanyi (1982)) describes the actual solutions the Harsanyi-Selten solution theory provides for some important classes of bargaining games, such as unanimity games; trade between one seller and several potential buyers; and two-person bargaining games with incomplete information on one side or on both sides. It also discusses some concepts and theorems useful in computing the solution; and explains how our concept of risk dominance enables us to analyze game situations in terms of some intuitively very compelling probabilistic (subjective-probability) considerations disallowed by classical game theory.     

Cooperative games with imcomplete information

A bargaining solution concept which generalizes the Nash bargaining solution and the Shapley NTU value is defined for cooperative games with incomplete information. These bargaining solutions are efficient and equitable when interpersonal comparisons are made in terms of certainvirtual utility scales. A player's virtual utility differs from his real utility by exaggerating the difference from the preferences of false types that jeopardize his true type. In any incentive-efficient mechanism, the players always maximize their total virtual utility ex post. Conditionally-transferable virtual utility is the strongest possible transferability assumption for games with incomplete information.     

Infinitely repeated games of incomplete information: Symmetric case with random signals

We consider infinitely repeated two-person zero-sum games of incomplete information in which the signals are the same for both players and consist of probability distributions on a given alphabet. We show that such games have a value.     

On repeated games without a recursive structure: Existence of limV n

We consider repeated zero-sum games with symmetric incomplete information where at each stage the common signal is either non-revealing or completely revealing. We prove that the sequence of values ofn-stage games converges by approximating the repeated game by a sequence of games in continuous time.     

The rise of complex phenomena in Cournot duopoly games due to demand functions without inflection points

In this paper, we propose Cournot duopoly games where quantity-setting firms use non-linear demand functions that have no inflection points. Two different kinds of repeated games are introduced based on rationality process of firms and Puu’s incomplete approach. First, a model of two rational firms that are in competition and produce homogenous commodities is introduced. The equilibrium points of this model are obtained and their dynamical characteristics such as stability, bifurcation and chaos are investigated. By using rationality process firms do not need to solve any optimization problem but they adjust their production based on estimation of the marginal profit. Using Puu’s incomplete information approach a new model is introduced. As in the first model, the equilibrium points are obtained and their dynamical characteristics are investigated. By using Puu’s approach firms only need to know their profits and the quantities produced in the past two times. We compare the properties of the two models under the two approaches. The paper extends and generalizes the results of other authors that consider similar processes.     

Experimental results on ultimatum games with incomplete information

This paper is about experiments on two versions of ultimatum games with incomplete information, called the offer game and the demand game. We apply the strategy method, that is, each subject had to design a complete strategy in advance instead of reacting spontaneously to a situation which occurs in the game. Game theory predicts very similar outcomes for the offer and the demand games. Our experiments, however, show significant differences in behavior between both games. Using the strategy method, allows us to explore the motivations leading to those differences. Since each subject played the same version of the game eight rounds against changing anonymous opponents we can also study subjects' learning behavior. We propose a theory of boundedly rational behavior, called the “anticipation philosophy”, which is well supported by the experimental data.     

Monotonicity and dummy free property for multi-choice cooperative games

Given a coalition of ann-person cooperative game in characteristic function form, we can associate a zero-one vector whose non-zero coordinates identify the players in the given coalition. The cooperative game with this identification is just a map on such vectors. By allowing each coordinate to take finitely many values we can define multi-choice cooperative games. In such multi-choice games we can also define Shapley value axiomatically. We show that this multi-choice Shapley value is dummy free of actions, dummy free of players, non-decreasing for non-decreasing multi-choice games, and strictly increasing for strictly increasing cooperative games. Some of these properties are closely related to some properties of independent exponentially distributed random variables. An advantage of multi-choice formulation is that it allows to model strategic behavior of players within the context of cooperation.Partially funded by the NSF grant DMS-9024408