The LP formulation of finite zero-sum games with incomplete information

This paper gives the LP formulation for finite zero sum games with incomplete information using Bayesian mixed strategies. This formulation is then used to derive general properties for the value of such games, the well known concave-convex property but also the “piecewise bilinearity”. These properties may offer considerable help for computational purposes but also provide structural guidelines for the analysis of special classes of games with incomplete information.     

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 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.     

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.     

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.     

Some results on the existence of Nash equilibria for non-zero sum games with incomplete information

We prove the existence of Nash equilibria for two person non-zero sum repeated games with lack of information on one side and two states of nature.     

Some thoughts on saddle-point conditions and information structures in zero-sum differential games

For a very simple two-stage, linear-quadratic, zero-sum difference game with dynamic information structure, we show that (i) there exist nonlinear saddle-point strategies which require the same existence conditions as the well-known linear, closed-loop, no-memory solution and (ii) there exist both linear and nonlinear saddle-point strategies which require more stringent conditions than the unique open-loop solution. We then discuss the implication of this result with respect to the existence of saddle points in zero-sum differential games for different information patterns.     

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.     

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     

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.     

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?     

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.

     

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.     

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 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.     

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 \(\left| {v_n - \mathop {\lim }\limits_{n \to \infty } v_n } \right|\) .     

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.     

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.