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.     

Some results on zero-sum games with incomplete information: The dependent case

In games with incomplete information, the players' states of information may be determined either through independent chance moves or through a unique one. Generally, a unique chance move generates some dependance in the players' state of information thus giving rise to significant complications in the analysis. However, it turns out that many results obtained in the simpler independent case have their counterpart in the dependent one. This is proved in this paper for several previous results of the authors.     

The value of two-person zero-sum repeated games with incomplete information and uncertain duration

Fix a zero-sum repeated game Γ with incomplete information on both sides. It is known that the value of the infinitely repeated game Γ_∞ need not exist (Aumann and Maschler 1995). It is proved that any number between the minmax and the maxmin of Γ_∞ is the value of a long finitely repeated game Γ _n where players’ information about the uncertain number of repetitions n is asymmetric.     

A note on the value of zero-sum sequential repeated games with incomplete information

We consider repeated two-person zero-sum games with lack of information on both sides. If the one shot game is played sequentially, it is proved that the sequencev _n is monotonic,v _n being the value of then shot game. Moreover the speed of convergence is bounded byK/n, and this is the best bound.     

Generic uniqueness of the bias vector of finite zero-sum stochastic games with perfect information

Mean-payoff zero-sum stochastic games can be studied by means of a nonlinear spectral problem. When the state space is finite, the latter consists in finding an eigenpair $(u,\lambda )$ solution of $T(u)=\lambda e+u$, where $T:{\mathbb{R}}^n\to {\mathbb{R}}^n$ is the Shapley (or dynamic programming) operator, λ is a scalar, e is the unit vector, and $u\in {\mathbb{R}}^n$. The scalar λ yields the mean payoff per time unit, and the vector u, called bias, allows one to determine optimal stationary strategies in the mean-payoff game. The existence of the eigenpair $(u,\lambda )$ is generally related to ergodicity conditions. A basic issue is to understand for which classes of games the bias vector is unique (up to an additive constant). In this paper, we consider perfect-information zero-sum stochastic games with finite state and action spaces, thinking of the transition payments as variable parameters, transition probabilities being fixed. We show that the bias vector, thought of as a function of the transition payments, is generically unique (up to an additive constant). The proof uses techniques of nonlinear Perron–Frobenius theory. As an application of our results, we obtain an explicit perturbation scheme allowing one to solve degenerate instances of stochastic games by policy iteration.     

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.     

Finite approximations to a zero-sum game with incomplete information

In this paper, we investigate a scheme for approximating a two-person zero-sum gameG of incomplete information by means of a natural systemG _mn of its finite subgames. The main question is: For largem andn, is an optimal strategy forG _mn necessarily anε-optimal strategy forG?     

Zero-sum sequential games with incomplete information

Repeated zero-sum two-person games of incomplete information on one side are considered. If the one-shot game is played sequentially, the informed player moving first, it is proved that the value of then-shot game is constant inn and is equal to the concavification of the game in which the informed player disregards his extra information. This is a strengthening ofAumann andMaschler's results for simultaneous games. Optimal strategies for both players are constructed explicitly.     

Statistically varyingN-person games with incomplete information

The paper provides a functional relationship between the amount of information available to the players and an equilibrium payoff in a game with statistically varying states of nature. Moreover, it is shown that for a supergame consisting of independent repetitions of a common matrix game equilibrium strategies are available which provide only few sequences of actions with positive probability.

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Zusammenfassung Es werdenN-Personenspiele betrachtet, in denen die Auszahlungen sowohl von den Aktionen der Spieler, als auch von einem im Zeitablauf variierenden, a priori unbekannten, Zustandsparameter beeinflußt werden. Im vorliegenden Ansatz wird die erhaltbare Information in expliziter Weise als strategischer Parameter der Spieler aufgefaßt. Die Spieler sind bei der Beschaffung von Information informationstheoretisch modellierten Beschränkungen and die Menge verfügbarer Information unterworfen. Es gelingt in dem Modell, asymptotisch den funktionalen Zusammenhang zwischen der Menge verfügbarer Information und den erreichbaren Gleichgewichtsauszahlungen inT-fach replizierten Spielen zu bestimmen.

     

The mixed expansion of differential games with incomplete information

Kuhn's theorem [1953] about behavioral strategies is proved for the mixed expansion of a differential game.     

Perfect information two-person zero-sum markov games with imprecise transition probabilities

Based on an extension of the controlled Markov set-chain model by Kurano et al. (in J Appl Prob 35:293–302, 1998) into competitive two-player game setting, we provide a model of perfect information two-person zero-sum Markov games with imprecise transition probabilities. We define an equilibrium value for the games formulated with the model in terms of a partial order and then establish the existence of an equilibrium policy pair that achieves the equilibrium value. We further analyze finite-approximation error bounds obtained from a value iteration-type algorithm and discuss some applications of the model.     

The information revealed in infinitely-repeated games of incomplete information

Zero-sum two-person games with incomplete information on one side are considered. It is shown that the information revealed by the informed player in an infinitely-repeated game with simultaneous moves is essentially the same as the information revealed by him in a one-stage game in which he must move first.     

The value of two-person zero-sum repeated games with lack of information on both sides

We consider repeated two-person zero-sum games in which each player has only partial information about a chance move that takes place at the beginning of the game. Under some conditions on the information pattern it is proved that \(\mathop {\lim }\limits_{n \to \infty } v_n\) exists,v _n being the value of the game withn repetitions. Two functional equations are given for which \(\mathop {\lim }\limits_{n \to \infty } v_n\) is the only simultaneous solutions. We also find the least upper bound for the error term \(\left| {v_n - \mathop {\lim }\limits_{n \to \infty } v_n } \right|\) .     

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.     

Infinite sequential games with perfect but incomplete information

Infinite sequential games, in which Nature chooses a Borel winning set and reveals it to one of the players, do not necessarily have a value if Nature has 3 or more choices. The value does exist if Nature has 2 choices. The value also does not necessarily exist if Nature chooses from 2 Borel payoff functions. Similarly, if Player 1 chooses the Borel winning set and does not reveal his selection to Player 2, then the game does not necessarily have a value if there are 3 or more choices; it does have a value if there are only 2 choices. If Player 1 chooses from 2 Borel payoff functions and does not reveal his choice, the game need not have a value either.     

Optimal strategies in repeated games with incomplete information

The paper is concerned with zero-sum two-person repeated games with lack of information on one side. The main result in the construction of an optimal strategy for the uninformed player in the infinitely repeated game.     

Repeated games with incomplete information and transportation problems

We consider two person zero-sum repeated games with lack of information on one side and with payoffs of special "separable" form. The solutions for these games are reduced to the solutions for families of special transportation type problems with recursive structure. We illustrate our approach applying it to the game introduced by Mertens/Zamir [1976] and later studied by several authors. The "symmetric" subclass of games under consideration was studied in Domansky, Kreps [1995].     

Formulation of Bayesian analysis for games with incomplete information

A formal model is given of Harsanyi's infinite hierarchies of beliefs. It is shown that the model closes with some Bayesian game with incomplete information, and that any such game can be approximated by one with a finite number of states of world.