On the stability of evolutionary dynamics in games with incomplete information

In an interaction it is possible that one agent has features it is aware of but the opponent is not. These features (e.g. cost, valuation or fighting ability) are referred to as the agent’s type. The paper compares two models of evolution in symmetric situations of this kind. In one model the type of an agent is fixed and evolution works on strategies of types. In the other model every agent adopts with fixed probabilities both types, and type-contingent strategies are exposed to evolution. It is shown that the dynamic stability properties of equilibria may differ even when there are only two types and two strategies. However, in this case the dynamic stability properties are generically the same when the payoff of a player does not depend directly on the type of the opponent. Examples illustrating these results are provided.     

Markov games with incomplete information

We consider zero-sum Markov games with incomplete information. Here, the second player is never informed about the current state of the underlying Markov chain. The existence of a value and of optimal strategies for both players is shown. In particular, we present finite algorithms for computing optimal strategies for the informed and uninformed player. The algorithms are based on linear programming results.     

Existence of optimal strategies in Markov games with incomplete information

The existence of a value and optimal strategies is proved for the class of two-person repeated games where the state follows a Markov chain independently of players’ actions and at the beginning of each stage only Player 1 is informed about the state. The results apply to the case of standard signaling where players’ stage actions are observable, as well as to the model with general signals provided that Player 1 has a nonrevealing repeated game strategy. The proofs reduce the analysis of these repeated games to that of classical repeated games with incomplete information on one side. This research was supported in part by Israeli Science Foundation grants 382/98, 263/03, and 1123/06, and by the Zvi Hermann Shapira Research Fund.     

Statistical linear differential games: A treatment of incomplete information

A zero-sum, two-player linear differential game of fixed duration is considered in the case when the information is incomplete but a statistical structure gives both players the possibility tospy the value of an unknown parameter in the payoff. Considerations of topological vector spaces and functional analysis allow one to demonstrate, via a classical Sion's theorem, sufficient conditions for the existence of a value.The author is indebted to Professor J. Fichefet for his helpful remarks and indications.     

Learning algorithms for repeated bimatrix Nash games with incomplete information

The purpose of this paper is to study a particular recursive scheme for updating the actions of two players involved in a Nash game, who do not know the parameters of the game, so that the resulting costs and strategies converge to (or approach a neighborhood of) those that could be calculated in the known parameter case. We study this problem in the context of a matrix Nash game, where the elements of the matrices are unknown to both players. The essence of the contribution of this paper is twofold. On the one hand, it shows that learning algorithms which are known to work for zero-sum games or team problems can also perform well for Nash games. On the other hand, it shows that, if two players act without even knowing that they are involved in a game, but merely thinking that they try to maximize their output using the learning algorithm proposed, they end up being in Nash equilibrium.This research was supported in part by NSF Grant No. ECS-87-14777.     

Equilibria in repeated games of incomplete information: The general symmetric case

Every two person repeated game of symmetric incomplete information, in which the signals sent at each stage to both players are identical and generated by a state and moves dependent probability distribution on a given finite alphabet, has an equilibrium payoff. Received March 1996/Revised version January 1997/Final version May 1997     

A model of interim information sharing under incomplete information

We propose a two-person game-theoretical model to study information sharing decisions at an interim stage when information is incomplete. The two agents have pieces of private information about the state of nature, and that information is improved by combining the pieces. Agents are both senders and receivers of information. There is an institutional arrangement that fixes a transfer of wealth from an agent who lies about her private information. In our model, we show that (1) there is a positive relation between information revelation and the amount of the transfers, and (2) information revelation has a collective action structure, in particular, the incentives of an agent to reveal decrease with respect to the amount of information disclosed by the other.     

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     

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.     

How to estimate the Value at Risk under incomplete information

A key problem in financial and actuarial research, and particularly in the field of risk management, is the choice of models so as to avoid systematic biases in the measurement of risk. An alternative consists of relaxing the assumption that the probability distribution is completely known, leading to interval estimates instead of point estimates. In the present contribution, we show how this is possible for the Value at Risk, by fixing only a small number of parameters of the underlying probability distribution. We start by deriving bounds on tail probabilities, and we show how a conversion leads to bounds for the Value at Risk. It will turn out that with a maximum of three given parameters, the best estimates are always realized in the case of a unimodal random variable for which two moments and the mode are given. It will also be shown that a lognormal model results in estimates for the Value at Risk that are much closer to the upper bound than to the lower bound.     

Interest rate options valuation under incomplete information

This paper aims at examining the term structure of interest rates and European-type interest rate option prices in a partially observable economy. It extends the existing literature on incomplete information by developing a one-factor model which is consistent with the initial yield curve and by providing closed-form solutions for discount bonds and different kinds of options. The model of this paper encompasses Hull and White’s (1990). Moreover, through a numerical example, these two models are compared and the impact of incomplete information on option prices is analysed. This article was completed when the author was at the University of Cergy-Pontoise. He is now at the University of Paris 1 Panth éon-Sorbonne, PRISM, 17, rue de la Sorbonne, 75231 Paris Cedex 05. e-mail: constantin.mellios@univ-paris1.fr.     

Population monotonicity and consistency in convex games: Some logical relations

On the domain of convex games, many desirable properties of solutions are compatible and there are many single-valued solutions that are intuitively appealing. We establish some interesting logical relations among properties of single-valued solutions. In particular, we introduce a new property, weak contribution-monotonicity, and show that this property is a key property that links other properties such as population-monotonicity, max consistency, converse max consistency, and dummy-player-out. Received: July 2002/Revised: March 2003     

A simple model of corporate international investment under incomplete information and taxes

This paper extends the theory of corporate international investment in Choi (J. Int. Bus. Stud. 20: 145–155, 1989) in an environment where the segmentation of international capital markets for investors or the presence of agency costs provide some independence to corporate decisions. The model shows that the real exchange risk, the competition between firms in different markets and diversification gains affect corporate international investment. By accounting for the role of information as defined in the models of Merton (J. Finance 42: 483–510, 1987), Bellalah (Int. J. Finance Econ. 6: 59–67, 2001a) and Bellalah and Wu (Int. J. Theor. Appl. Finance 5(5): 479–495, 2002), the model embodies different existing explanations based on economic and behavioral variables. We show in a “two-country” firm model that real exchange risk, diversification motives and information costs are important elements in the determination of corporate international investment decisions. The dynamic portfolio model reflects the main results in several theories of foreign direct investment. Our model accounts for the role of information in explaining foreign investments. It provides simple explanations which are useful in explaining the home bias puzzle in international finance. Using the dynamical programming principle method, we provide the general solution for the proportion of firm’s total capital budget. We also use a new method to get explicit solutions in some special cases. This new method can be applied to solve other financial control problems. The simulating results are given to show our conclusion and the influence of some parameters to the optimal solution. The economic results can be seen as a generalization of the model in Solnik (J. Econ. Theory 8: 500–524, 1974).     

Preference Programming with incomplete ordinal information

This paper extends possibilities for analyzing incomplete ordinal information about the parameters of an additive value function. Such information is modeled through preference statements which associate sets of alternatives or attributes with corresponding sets of rankings. These preference statements can be particularly helpful in developing a joint preference representation for a group of decision-makers who may find difficulties in agreeing on numerical parameter values. Because these statements can lead to a non-convex set of feasible parameters, a mixed integer linear formulation is developed to establish a linear model for the computation of decision recommendations. This makes it possible to complete incomplete ordinal information with other forms of incomplete information.     

A differential game of incomplete information

In this paper, we study a differential game of incomplete information. In such a game, the cost function depends on parameter . At the start of the game, only one of the players knows the value of this parameter, while the other player has only a (subjective) probability distribution for the parameter. We obtain explicit expressions for both the value of the game and the two players' optimal strategies.     

The impacts of uncertainties in a real options model under incomplete information

This paper examines the impact with respect to the uncertainty of the underlying state variable, profit uncertainty, on the real options model in a situation of incomplete information. Profit uncertainty has not incorporated into the real options model under incomplete information, in that the underlying state variable is not formulated as the stochastic process (see, e.g., Bernardo, A. E., Chowdhry, E. B., 2002. Resources, real options, and corporate strategy. Journal of Financial Economics, 63, 211–234). We extend the model developed by Bernardo and Chowdhry to formulate the underlying state variable as the stochastic process. We conclude that profit uncertainty has the same type of impact on the real options value and its triggers, both under complete and incomplete information.     

The game for the speed of convergence in repeated games of incomplete information

We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value v _∞ (p) for all 0≤p≤1. The informed player can guarantee that all along the game the average payoff per stage will be greater than or equal to v _∞ (p) (and will converge from above to v _∞ (p) if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payoffs-to the value v _∞ (p). In the context of such repeated games, we define a game for the speed of convergence, denoted SG _∞ (p), and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which v _n (p)− v _∞ (p) is of the order of magnitude of . In that case the value of SG _∞ (p) is of the order of magnitude of . We then show a class of games for which the value does not exist. Given any infinite martingale 𝔛^∞={X _k }^∞ _k=1, one defines for each n : V _n (𝔛^∞) ≔E∑ⁿ _k=1 |X _k+1 − X _k|. For our first result we prove that for a uniformly bounded, infinite martingale 𝔛^∞, V _n (𝔛^∞) can be of the order of magnitude of n ^1/2−ε, for arbitrarily small ε>0. Received January 1999/Final version April 2002     

Subgame-consistent cooperative solutions in randomly furcating stochastic differential games

The paradigm of randomly-furcating stochastic differential games incorporates additional stochastic elements via randomly branching payoffs in stochastic differential games. This paper considers dynamically stable cooperative solutions in randomly furcating stochastic differential games. Analytically tractable payoff distribution procedures contingent upon specific random realizations of the state and payoff structure are derived. This new approach widens the application of cooperative differential game theory to problems where the evolution of the state and future environments are not known with certainty. Important cases abound in regional economic cooperation, corporate joint ventures and environmental control. An illustration in cooperative resource extraction is presented.     

Eigenvector method,consistency test and inconsistency repairing for an incomplete fuzzy preference relation

In this paper, we extend the eigenvector method (EM) to priority for an incomplete fuzzy preference relation. We give a reasonable definition of multiplicative consistency for an incomplete fuzzy preference relation. We also give an approach to judge whether an incomplete fuzzy relation is acceptable or not. We develop the acceptable consistency ratio for an incomplete multiplicative fuzzy preference relation, which is simple and similar to Saaty’s consistency ratio (CR) for the multiplicative preference relation. If the incomplete fuzzy preference relation is not of acceptable consistency, we define a criterion to find the unusual and false element (UFE) in the preference relation, and present an algorithm to repair an inconsistent fuzzy preference relation until its consistency is satisfied with the consistency ratio. As a result, our improvement method cannot only satisfy the consistency requirement, but also preserve the initial preference information as much as possible. Finally, an example is illustrated to show that our method is simple, efficiency, and can be performed on computer easily.     

The simple geometry of perfect information games

Perfect information games have a particularly simple structure of equilibria in the associated normal form. For generic such games each of the finitely many connected components of Nash equilibria is contractible. For every perfect information game there is a unique connected and contractible component of subgame perfect equilibria. Finally, the graph of the subgame perfect equilibrium correspondence, after a very mild deformation, looks like the space of perfect information extensive form games.