A generalization of Ehrenfeucht’s game and some applications

The concept of α, β-elementary equivalence of two relational structures is defined. Sufficient and necessary conditions are given for this notion by generalizing Ehrenfeucht’s game in algebraic terms. Some results, in a first order language with generalized quantifier, are obtained. This paper is part of the author’s doctoral dissertation which was prepared under the supervision of Professor Azriel Levy at the Hebrew University of Jerusalem. The author wishes to thank Professor Levy for his kind assistance.     

Feedback Stackelberg equilibrium strategies when the private label competes with the national brand

We consider a noncooperative differential game where a retailer sells her own private label in addition to the manufacturer’s brand. We assume that each brand’s goodwill evolves according to a modified Nerlove-Arrow dynamics, in such a way that the advertising effort of one brand hurts the competitor’s goodwill stock. We characterize Feedback-Stackelberg pricing and advertising strategies and employ simulations to analyze their sensitivity to the main model parameters. We thank an anonymous Reviewer for very helpful comments. Research supported by NSERC, Canada, and FQRSC, Quebec. Research completed when the second author was a visiting professor at GERAD, HEC Montréal. The second author’s research was partially supported by MEC and JCYL under projects SEJ2005-03858/ECON and VA045A06, co-financed by FEDER funds.     

Less is more: an observability paradox in repeated games

We present a repeated prisoners’ dilemma game with imperfect public monitoring, which exhibits the following paradoxical feature: the (limit) equilibrium payoff set achieves full efficiency asymptotically as the public signal becomes insensitive to the hidden actions of the players. The basic logic behind this result also provides an example where the Folk theorem is obtained, while Fudenberg et al.’s (in Econometrica 62:997–1040, 1994) sufficient conditions for Folk theorem are violated.We would like to thank Drew Fudenberg, David Levine, an associate editor and two anonymous referees for their helpful comments and suggestions.     

Analysis of various optimal contracts in the problem of stimulation of agents by a principal in a model with two agents

This work is devoted to one section of a game theory popular in recent years, the theory of contracts and the problem of an agent and a principal. A principal tries to stimulate agents into making a number of contracts with clients using his resources in order to maximize his profits as a result. At the same time, the principal has limited information about the agents. He knows only the results he has obtained; he does not know their type or the efforts made. The strategy of the principal is an incentive scheme that shows the agent’s reward depending on his results. In turn, the agents try to maximize their benefits while having the right to turn down a contract. We do not consider models with random events, and we restrict ourselves to the deterministic case. The model thus does not fall under the heading of a moral hazard but is related to a situation of unfavorable selection.     

Cournot–Walras equilibrium as a subgame perfect equilibrium

In this paper, we investigate the problem of the strategic foundation of the Cournot–Walras equilibrium approach. To this end, we respecify à la Cournot–Walras the mixed version of a model of simultaneous, noncooperative exchange, originally proposed by Lloyd S. Shapley. We show, through an example, that the set of the Cournot–Walras equilibrium allocations of this respecification does not coincide with the set of the Cournot–Nash equilibrium allocations of the mixed version of the original Shapley’s model. As the nonequivalence, in a one-stage setting, can be explained by the intrinsic two-stage nature of the Cournot–Walras equilibrium concept, we are led to consider a further reformulation of the Shapley’s model as a two-stage game, where the atoms move in the first stage and the atomless sector moves in the second stage. Our main result shows that the set of the Cournot–Walras equilibrium allocations coincides with a specific set of subgame perfect equilibrium allocations of this two-stage game, which we call the set of the Pseudo–Markov perfect equilibrium allocations. We would like to thank Pierpaolo Battigalli, Marcellino Gaudenzi, and an anonymous referee for their comments and suggestions.     

The role of utility interdependence in public good experiments

A popular approach to explain over-contribution in public good games is based on the assumption that people care (either positively or negatively) about the utility of other participants. Over-contribution then is an outcome of utility maximization where utility depends on subjects’ own payoffs as well as on the payoffs of other players. In this paper, I study to what extent this assumption of utility interdependence is responsible for over-contribution. I design three treatments where subjects’ decisions cannot affect opponents’ payoffs and thus utility interdependence cannot explain cooperative behavior. The main result is that while average contribution in these treatments is below the benchmark it nonetheless stays well above zero. Even when no one benefits from subjects’ generosity the average contributions are as high as one third of the endowment and are only 25% below those in the benchmark level. This suggests that utility interdependence is not the main factor responsible for over-contribution. I would like to thank my dissertation advisor Shyam Sunder for his valuable suggestions that helped improve this paper. I am also grateful to Dan Levin, Ben Polak, Klaus Schmidt, Andrew Schotter, Anat Bracha, Danielle Catambay, Rodney Chan, two anonymous referees and the anonymous Associate Editor for their comments and suggestions. Participants of the 17th International Conference on Game Theory at Stony Brook University, 2006 ESA International Meeting and 10th Biennial Behavioral Decision Research in Management Conference provided a valuable feedback at earlier stages of the paper. Finally, I would like to thank the Whitebox Fellowship for its generous support of this study.     

Approachability of Convex Sets in Games with Partial Monitoring

We provide a necessary and sufficient condition under which a convex set is approachable in a game with partial monitoring, i.e. where players do not observe their opponents’ moves but receive random signals. This condition is an extension of Blackwell’s Criterion in the full monitoring framework, where players observe at least their payoffs. When our condition is fulfilled, we construct explicitly an approachability strategy, derived from a strategy satisfying some internal consistency property in an auxiliary game.     

基于有信息委托人视角的创意转化契约设计研究

为了形式化一个有信息的委托人在创意的转化过程中,与代理人缔约时所面临的信号显示问题,以创意的特性为基础,通过建立Stackelberg信号博弈模型,证明了创意转化契约分离均衡的存在性,并分析了知识产权保护强度和产业成熟度对均衡契约的影响。研究还发现,当同时存在委托人的逆向选择和代理人的道德风险时,分离均衡契约使得代理人的努力水平会产生向下的扭曲。     

Semidefinite complementarity reformulation for robust Nash equilibrium problems with Euclidean uncertainty sets

Consider the N-person non-cooperative game in which each player’s cost function and the opponents’ strategies are uncertain. For such an incomplete information game, the new solution concept called a robust Nash equilibrium has attracted much attention over the past several years. The robust Nash equilibrium results from each player’s decision-making based on the robust optimization policy. In this paper, we focus on the robust Nash equilibrium problem in which each player’s cost function is quadratic, and the uncertainty sets for the opponents’ strategies and the cost matrices are represented by means of Euclidean and Frobenius norms, respectively. Then, we show that the robust Nash equilibrium problem can be reformulated as a semidefinite complementarity problem (SDCP), by utilizing the semidefinite programming (SDP) reformulation technique in robust optimization. We also give some numerical example to illustrate the behavior of robust Nash equilibria.     

Is There a Conceptual Connection Between Intentionality and Consciousness?

It is argued that Searle’s argument for the thesis that there is a conceptual connection between intentionality and consciousness suffers from serious ambiguities. Searle’s argument is reconstructed as consisting of three premises that contain the notions of aspectual shape and irreducible subjectivity. Having identified two different readings of ‘aspectual shape’ and ‘irreducibly subjective,’ I conclude that each version of Searle’s argument incorporates at least one false premise. This paper is a descendant of a paper presented at the Wittgenstein Symposium at Kirchberg am Wechsel in August 2004. I would like to thank the reviewer of this journal for his/her helpful comments on a previous draft.     

Optimal Supply Functions in Electricity Markets with Option Contracts and Non-smooth Costs

In this paper we investigate the optimal supply function for a generator who sells electricity into a wholesale electricity spot market and whose profit function is not smooth. In previous work in this area, the generator’s profit function has usually been assumed to be continuously differentiable. However in some interesting instances, this assumption is not satisfied. These include the case when a generator signs a one-way hedge contract before bidding into the spot market, as well as a situation in which a generator owns several generation units with different marginal costs. To deal with the non-smooth problem, we use the model of Anderson and Philpott, in which the generator’s objective function is formulated as a Stieltjes integral of the generator’s profit function along his supply curve. We establish the form of the optimal supply function when there are one-way contracts and also when the marginal cost is piecewise smooth.We would like to thank two anonymous referees for careful reading of the paper and helpful comments which lead to a significant improvement of this paper.     

Asymmetric Information in a Capital Accumulation Differential Game with Spillover and Learning Effects

Journal of Optimization Theory and Applications - This paper considers a capital accumulation game where the installation costs of investments are lowered by the firm’s own capital stock...     

Prediction with Expert Advice: A PDE Perspective

This work addresses a classic problem of online prediction with expert advice. We assume an adversarial opponent, and we consider both the finite horizon and random stopping versions of this zero-sum, two-person game. Focusing on an appropriate continuum limit and using methods from optimal control, we characterize the value of the game as the viscosity solution of a certain nonlinear partial differential equation. The analysis also reveals the predictor’s and the opponent’s minimax optimal strategies. Our work provides, in particular, a continuum perspective on recent work of Gravin et al. (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, SODA ’16, (Philadelphia, PA, USA), Society for Industrial and Applied Mathematics, 2016). Our techniques are similar to those of Kohn and Serfaty (Commun Pure Appl Math 63(10):1298–1350, 2010), where scaling limits of some two-person games led to elliptic or parabolic PDEs.     

Optimal Compensation with Hidden Action and Lump-Sum Payment in a Continuous-Time Model

We consider a problem of finding optimal contracts in continuous time, when the agent’s actions are unobservable by the principal, who pays the agent with a one-time payoff at the end of the contract. We fully solve the case of quadratic cost and separable utility, for general utility functions. The optimal contract is, in general, a nonlinear function of the final outcome only, while in the previously solved cases, for exponential and linear utility functions, the optimal contract is linear in the final output value. In a specific example we compute, the first-best principal’s utility is infinite, while it becomes finite with hidden action, which is increasing in value of the output. In the second part of the paper we formulate a general mathematical theory for the problem. We apply the stochastic maximum principle to give necessary conditions for optimal contracts. Sufficient conditions are hard to establish, but we suggest a way to check sufficiency using non-convex optimization.     

Some projection-like methods for the generalized Nash equilibria

A generalized Nash game is an m-person noncooperative game in which each player’s strategy depends on the rivals’ strategies. Based on a quasi-variational inequality formulation for the generalized Nash game, we present two projection-like methods for solving the generalized Nash equilibria in this paper. It is shown that under certain assumptions, these methods are globally convergent. Preliminary computational experience is also reported.     

The Compromise Value for Cooperative Games with Random Payoffs

This paper introduces and studies the compromise value for cooperative games with random payoffs, that is, for cooperative games where the payoff to a coalition of players is a random variable. This value is a compromise between utopia payoffs and minimal rights and its definition is based on the compromise value for NTU games and the τ-value for TU games. It is shown that the nonempty core of a cooperative game with random payoffs is bounded by the utopia payoffs and the minimal rights. Consequently, for such games the compromise value exists. Further, we show that the compromise value of a cooperative game with random payoffs coincides with the τ-value of a related TU game if the players have a certain type of preferences. Finally, the compromise value and the marginal value, which is defined as the average of the marginal vectors, coincide on the class of two-person games. This results in a characterization of the compromise value for two-person games.I thank Peter Borm, Ruud Hendrickx and two anonymous referees for their valuable comments.     

The dynamics of performance space of Major League Baseball pitchers 1871–2006

A central decision-maker, the principal, employs performance evaluation criteria consistent with an organization’s overall goal(s) to measure the effectiveness of the agents who execute decisions and implement strategies within a specified period. The dataset of performance criteria spanning the performance space will change over time to reflect the principal’s strategic modifications. This paper applies a Data Envelopment Analysis (DEA) based approach to reveals the dynamics of the performance space of Major League Baseball (MLB) pitchers with minimum subjective judgment imposed on the data. The proposed approach is applied to data on MLB pitchers from 1871 to 2006. We conclude that many of the findings are consistent with the observations of baseball’s experts. The approach also suggests new directions for investigating a large dataset to identify revealed preferences or strategies by using historical and modern observations.     

Risk minimization and optimal derivative design in a principal agent game

We consider the problem of Adverse Selection and optimal derivative design within a Principal–Agent framework. The principal’s income is exposed to non-hedgeable risk factors arising, for instance, from weather or climate phenomena. She evaluates her risk using a coherent and law invariant risk measure and tries minimize her exposure by selling derivative securities on her income to individual agents. The agents have mean–variance preferences with heterogeneous risk aversion coefficients. An agent’s degree of risk aversion is private information and hidden from the principal who only knows the overall distribution. We show that the principal’s risk minimization problem has a solution and illustrate the effects of risk transfer on her income by means of two specific examples. Our model extends earlier work of Barrieu and El Karoui (in Financ Stochast 9, 269–298, 2005) and Carlier et al. (in Math Financ Econ 1, 57–80, 2007). We thank Guillaume Carlier, Pierre-Andre Chiappori, Ivar Ekeland, Andreas Putz and seminar participants at various institutions for valuable comments and suggestions. Financial support through an NSERC individual discovery grant is gratefully acknowledged.     

Conformal capacity and quasiconformal mappings in\bar R^n

Conformal capacity of some ring domains inn-spaces is computed and applied in estimations of the modulus of Groetzsch and Teichmuller’s domains. Upper bounds for modR are given here by means of the specific diameters of the ring domainR. Convergence theorems of Lehto and Virtanen are generalized here ton-spacesn>2; and are applied to discrete groups ofn-Moebius transformations. This paper is mainly based on the author’s Ph.D. Thesis done at the Technion, Haifa, under the guidance of Professor E. Netanyahu. The author wishes to thank Professor E. Netanyahu for his help in preparing this paper.     

On the complexity of computing the handicap of a sufficient matrix

The class of sufficient matrices is important in the study of the linear complementarity problem (LCP)—some interior point methods (IPM’s) for LCP’s with sufficient data matrices have complexity polynomial in the bit size of the matrix and its handicap.In this paper we show that the handicap of a sufficient matrix may be exponential in its bit size, implying that the known complexity bounds of interior point methods are not polynomial in the input size of the LCP problem. We also introduce a semidefinite programming based heuristic, that provides a finite upper bond on the handicap, for the sub-class of P{\mathcal{P}} -matrices (where all principal minors are positive).