Repeated two-player games with ruin

The class of repeated two-player games (with long-run average payoff criterion) is extended to accommodate initial holdings of wealth and the possibility of ruin. Equilibria of these games are studied under the assumption that each player regards his own ruin as the worst possible outcome of the game and his opponent's ruin as the best possible outcome.     

Solution methods for two-player differential games

A numerical solution method is proposed for the pursuit-and-evasion game in which the terminal set is the sum of a two-dimensional convex compactum and a linear subspace of codimension 2. The “convexification method” is applied to compute the alternated sums. Simple switching lines are constructed for the case when the set of constraints on pursuer controls is a polyhedron. These simple switching lines essentially simplify the construction of a pursuit strategy in the convex programming problem. Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 2, pp. 49–66, 1994.     

Maximal stable sets of two-player games

If a connected component of perfect equilibria of a two-player game contains a stable set as defined by Mertens, then the component is itself stable. Thus the stable sets maximal under inclusion are connected components of perfect equilibria. Received: October 1999/Revised: February 2001     

Pure strategy equilibria in symmetric two-player zero-sum games

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.     

Equilibrium in two-player non-zero-sum Dynkin games in continuous time

We prove that every two-player non-zero-sum Dynkin game in continuous time admits an ?-equilibrium in randomized stopping times. We provide a condition that ensures the existence of an ?-equilibrium in non-randomized stopping times.     

Lower equilibrium payoffs in two-player repeated games with non-observable actions

We characterize, by the one-shot game terms, the set of lower equilibrium payoffs of the undiscounted repeated game with non-observable actions.     

Algorithms for uniform optimal strategies in two-player zero-sum stochastic games with perfect information

We deal with zero-sum two-player stochastic games with perfect information. We propose two algorithms to find the uniform optimal strategies and one method to compute the optimality range of discount factors. We prove the convergence in finite time for one algorithm. The uniform optimal strategies are also optimal for the long run average criterion and, in transient games, for the undiscounted criterion as well.     

Mixed strategy solutions forN-person quadratic games

Mixed strategy -equilibrium points are given forN-person games with cost functions consisting of quadratic, bilinear, and linear terms and strategy spaces consisting of closed balls in Hilbert spaces. The results are applied to linear-quadratic differential games with no information and quadratic integral constraints on the control functions.This work was supported by a Commonwealth of Australia, Postgraduate Research Award.     

On the feasible payoff set of two-player repeated games with unequal discounting

We provide a novel characterization of the feasible payoff set of a general two-player repeated game with unequal discounting. In particular, we show that generically the Pareto frontier shifts outwards and the feasible payoff set expands in the sense of set inclusion, as the time horizon increases. This result reinforces and refines the insight in Lehrer and Pauzner (1999) by showing that a longer horizon enables the players to conduct intertemporal trade in a more flexible fashion.     

Mixed strategy solutions for quadratic games

Mixed strategy solutions are given for two-person, zero-sum games with payoff functions consisting of quadratic, bilinear, and linear terms, and strategy spaces consisting of closed balls in a Hilbert space. The results are applied to linear-quadratic differential games with no information, and with quadratic integral constraints on the control functions.     

Continuously stable strategies,neighborhood superiority and two-player games with continuous strategy space

The continuously stable strategy (CSS) concept, originally developed as an intuitive static condition to predict the dynamic stability of a monomorphic population, is shown to be closely related to classical game-theoretic dominance criteria when applied to continuous strategy spaces. Specifically, for symmetric and non symmetric two-player games, a CSS in the interior of the continuous strategy space is equivalent to neighborhood half-superiority which, for a symmetric game, is connected to the half-dominance and/or risk dominance concepts. For non symmetric games where both players have a one-dimensional continuous strategy space, an interior CSS is shown to be given by a local version of dominance solvability (called neighborhood dominance solvable). Finally, the CSS and half-superiority concepts are applied to points in the bargaining set of Nash bargaining problems. R. Cressman thanks the referee for pointing out further connections between neighborhood superiority and other dominance concepts in the literature. This research was supported by a Natural Sciences and Engineering Research Council of Canada Individual Discovery Grant.     

Linear quadratic stochastic integral games and related topics

This paper studies linear quadratic games problem for stochastic Volterra integral equations(SVIEs in short) where necessary and sufficient conditions for the existence of saddle points are derived in two different ways.As a consequence,the open problems raised by Chen and Yong(2007) are solved.To characterize the saddle points more clearly,coupled forward-backward stochastic Volterra integral equations and stochastic Fredholm-Volterra integral equations are introduced.Compared with deterministic game problems,some new terms arising from the procedure of deriving the later equations reflect well the essential nature of stochastic systems.Moreover,our representations and arguments are even new in the classical SDEs case.     

Information market games

In this paper information markets with perfect patent protection and only one initial owner of the information are studied by means of cooperative game theory. To each information market of this type a cooperative game with sidepayments is constructed. These cooperative games are called information (market) games. The set of all information games with fixed player set is a cone in the set of all cooperative games with the same player set. Necessary and sufficient conditions are given in order that a cooperative game is an information game. The core of this kind of games is not empty and is also the minimal subsolution of the game. The core is the image of an (n-1)-dimensional hypercube under an affine transformation, (= hyperparallellopiped), the nucleolus and -value coincide with the center of the core. The Shapley value is computed and may lie inside or outside the core. The Shapley value coincides with the nucleolus and the -value if and only if the information game is convex. In this case the core is also a stable set.     

Singular linear quadratic differential games with bounds on control

A class of two-persons zero-sum differential games with the possibility of singular arcs is considered. Under some assumptions, a sufficient condition for the existence of a saddle-point solution is derived. In this approach, the properties of convexity and concavity of the performance index are used to find the sufficient condition. It is observed that, unlike total singularity, the occurrence of partially singular arcs does not cause any problems for the existence of a unique saddle-point solution.     

Optimal information acquisition for a linear quadratic control problem

This paper develops a generalization of the linear quadratic control problem with partial information. As in the standard partial information setting, it is assumed that the state variable is only observed with noise. The idea in this paper is that the information level may be chosen optimally. In real life information is costly to acquire. It is therefore a trade off between the costs of getting detailed information and the increased value this information gives. We believe that the technique we present should have potential for application within both economics and engineering.     

Compensatory transfers in two-player decision problems

This paper presents an axiomatic characterization of a family of solutions to two-player quasi-linear social choice problems. In these problems the players select a single action from a set available to them. They may also transfer money between themselves.The solutions form a one-parameter family, where the parameter is a non-negative number, t.The solutions can be interpreted as follows: Any efficient action can be selected. Based on this action, compute for each player a “best claim for compensation”. A claim for compensation is the difference between the value of an alternative action and the selected efficient action, minus a penalty proportional to the extent to which the alternative action is inefficient. The coefficient of proportionality of this penalty is t. The best claim for compensation for a player is the maximum of this computed claim over all possible alternative actions. The solution, at the parameter value t, is to implement the chosen efficient action and make a monetary transfer equal to the average of these two best claims. The characterization relies on three main axioms. The paper presents and justifies these axioms and compares them to related conditions used in other bargaining contexts. In Nash Bargaining Theory, the axioms analogous to these three are in conflict with each other. In contrast, in the quasi-linear social choice setting of this paper, all three conditions can be satisfied simultaneously.This work was supported by the Division of Research at the Harvard Business School. Thanks are due to the Cowles Foundation for Research in Economics at Yale University for its kind hospitality during the Spring of 2002. I have received helpful advice and comments from Youngsub Chun, Ehud Kalai, Herve Moulin, Al Roth, Ilya Segal, Adam Szeidl, Richard Zeckhauser, and other members of the Theory Seminars at Harvard, MIT, Princeton, Rice and Northwestern.     

Information acquisition in new product introduction

In the presence of huge losses from unsuccessful new product introductions, companies often seek forecast information from various sources. As the information can be costly, companies need to determine how much effort to put into acquiring the information. Such a decision is strategically important because an insufficient investment may cause lack of knowledge of product profitability, which in turn may lead to introducing a loss-making product or scrapping a potentially profitable one. In this paper, we use decision analytical models to study information acquisition for new product introduction. Specifically, we consider a decision maker (DM) who, prior to introducing a new product, can purchase forecasts and use the information to update his knowledge of the market demand. We analyze and compare two approaches: The first approach is to determine the total amount of forecasts to purchase all at once. The second one is to purchase forecasts sequentially and, based on the purchased forecasts, determine whether those forecasts are informative enough for making an introduction decision or an additional forecast is needed. We present dynamic programming formulations for both approaches and derive the optimal policies. Via a numerical study, we find the second approach, i.e., purchasing forecasts sequentially, can generate a significant profit advantage over the first one when (1) the cost of acquiring forecasts is neither too high nor too low, (2) the precision of the forecasts is of a moderate level, and (3) the profit margin of the new product is small.     

Information acquisition and transparency in committees

We study an intertemporal model of committee decision-making where members differ in their levels of efficiency. They may acquire costly information that enhances their ability to make a correct decision. We focus on the impact of transparency. We show that the principal’s initial utility is higher under transparency, because members exert more effort, which makes correct decisions more likely. The principal also benefits from transparency later, unless transparency leads to an alignment of the signal qualities of highly efficient and less efficient committee members. In general, committee members are harmed by transparency. Together with the insights from the literature, our results may help to decide when transparency in committees is desirable.