A Relationship Among Gentzen's Proof-Reduction,Kirby-Paris' Hydra Game and Buchholz's Hydra Game

We first note that Gentzen's proof-reduction for his consistency proof of PA can be directly interpreted as moves of Kirby-Paris' Hydra Game, which implies a direct independence proof of the game (Section 1 and Appendix). Buchholz's Hydra Game for labeled hydras is known to be much stronger than PA. However, we show that the one-dimensional version of Buchholz's Game can be exactly identified to Kirby-Paris' Game (which is two-dimensional but without labels), by a simple and natural interpretation (Section 2). Jervell proposed another type of a combinatorial game, by abstracting Gentzen's proof-reductions and showed that his game is independent of PA. We show (Section 3) that this Jervell's game is actually much stronger than PA, by showing that the critical ordinal of Jervell's game is φ_ω (0) (while that of PA or of Kirby-Paris' Game is φ₁ (0) = ?₀) in the Veblen hierarchy of ordinals.     

Cooperation in spatial prisoner’s dilemma game with delayed decisions

Phenomena that time delays of information lead to delayed decisions are extensive in reality. The effect of delayed decisions on the evolution of cooperation in the spatial prisoner’s dilemma game is explored in this work. Players with memory are located on a two dimensional square lattice, and they can keep the payoff information of his neighbors and his own in every historic generation in memory. Every player uses the payoff information in some generation from his memory and the strategy information in current generation to determine which strategy to choose in next generation. The time interval between two generations is set by the parameter m. For the payoff information is used to determine the role model for the focal player when changing strategies, the focal player’s decision to learn from which neighbor is delayed by m generations. Simulations show that cooperation can be enhanced with the increase of m. In addition, just like the original evolutionary game model (m = 0), pretty dynamic fractal patterns featuring symmetry can be obtained when m > 0 if we simulate the invasion of a single defector in world of cooperators on square lattice.     

Optimal Claiming in an Automobile Insurance System with Bonus-Malus Structure

A motorist involved in an accident will have to decide whether to claim from his insurance company or not when he is at fault. An optimal decision rule can only be determined in the light of future developments and future decisions, since the consequences of claiming or not claiming are felt in the subsequent year's premiums. In this paper, optimal no-claim limits are determined for a common Dutch type of insurance policy with bonus-malus structures, using generalized Markovian programming. The computational results are given for various values of the expected number of accidents per year.     

On Simple MIX game

We consider zero-sum game which is called Simple MIX game. Each of two players (I and II) draws a number (x andy respectively) according to a uniform distribution on [0, 1]. After observing his number each player can then choose to offer or not offer to exchange his number for the other player's number. Conditions for an exchange are the following: 1) both players must offer for a trade to occur with certainty; 2) if only one player offers, a trade occurs with probabilityp. A player's payoff is equal to 1, 0 or — 1 if the value of the number which he finally gets is greater, equal or less than the number of his opponent. In the present paper we shall investigate Simple MIX game in which both of the players can obtain additional information about the opponent's number. Besides, we consider two-stage variant of this game.     

Research and Development with Stock-Dependent Spillovers and Price Competition in a Duopoly

This paper investigates the research and development accumulation and pricing strategies of two firms competing for consumer demand in a dynamic framework. A firm’s research and development is production-cost-reducing and can benefit from part of the competitor’s research and development stock without payment. We consider decisions in a game characterized by Nash equilibrium. In this dynamic game, a player’s action depends on whether the competitor’s current research and development stock are observable. If the competitor’s current research and development stock are not observable or observable only after a certain time lag, a player’s action can be solely based on the information on the current period t (open-loop strategy). In the converse case, it can also include the information on the competitor’s reaction to a change in the current value of the state vector (closed-loop strategy), which allows for strategic interaction to take place throughout the game. Given the cumulative nature of research and development activities, a primary goal of this paper is to determine whether, regardless of the observability of the competitor’s current research and development stock, free research and development spillovers generate a lower level of scientific knowledge than research and development appropriability. A second objective of the paper is to determine how the observability of the rival’s current research and development stock affects a firm’s research and development and pricing decisions and payoffs under imperfect research and development appropriability.     

Service capacity procurement of logistics service supply chain with demand updating and loss-averse preference

This paper studies the impacts of loss-averse preference on the service capacity procurement decisions with demand updating in a logistics service supply chain, which consists of one logistics service integrator and one functional logistics service provider. It starts from a basic two-stage Stackelberg game model, then, extends to three scenarios where either the integrator or the provider or neither of them has loss-averse preference. The impact of loss-averse preference on the decisions of supply chain members is discussed by comparing the four models. Our results reveal, first, the loss-averse preferences do not always affect the decisions of supply chain members. If certain conditions are satisfied, the logistics service integrator can benefit from its loss-averse preference. Second, the increased service level can affect the logistics service integrator's procurement strategy and the functional logistics service provider's pricing strategy. This effect is only related to the loss-averse preference of the functional logistics service provider. Last, under certain conditions, the total service capacity decreases with the increased service level, regardless of whether or not the supply chain members have loss-averse preferences.     

Conflict dissolution by reframing game payoffs using linear perturbations

Human beings have a prevailing drive to achieve their self-interest goals or equilibrium states, which may subsume their social interests. An ideal working environment or cooperative game situation would be one in which each participant or player maximizes his/her own interest while maximizing his/her contribution to the collective group interest. This paper addresses the feasibility, methods, and bounds for reframing a generaln-person game into an ideal game in which full cooperation or a targeted solution can be induced and maintained by the players' self-interest maximization. Criteria for good reframing are introduced. Monotonic games, self-interest cooperative and noncooperative games, and a decomposition theory of general games are also introduced to facilitate the study. It is shown that everyn-person game can be written as the sum of a self-interest cooperative game and a self-interest noncooperative game. Everyn-person game can be reframed so that full cooperation can be achieved by the players' self-interest maximization. Everyn-person game can be reframed so that a targeted solution can be obtained and maintained through the players' self-interest maximization.     

THE INCOMPLETE‐INFORMATION SPLIT‐STREAM FISH WAR: EXAMINING THE IMPLICATIONS OF COMPETING RISKS

ABSTRACT. . It is now widely recognized that climactic regime shifts, which aperiodically alter a harvested fish stock's biomass and spatial distribution, may lead to distorted fisheries management decisions which negatively impact the fishery, both biologically and economically. This is particularly true for trans‐boundary migratory stocks, where optimal management relies on coordination among independent nation‐states. Unanticipated changes in stock distribution and abundance can upset expectations of national authorities, leading them to sanction inappropriate harvesting levels by their separately managed fleets targeting the same breeding fish stock. Our theoretical studies are based on a spatially‐distributed stochastic model, which we have called the “split‐stream model,‘ where two separately managed fleets harvest simultaneously at two separate sites. Our key assumption is that competing fleet managers, when harvesting noncooperatively, hold incomplete and asymmetric private information of current stock recruitment and spatial distribution. When subsequently negotiating to coordinate their harvests, they agree that they will share their information and then bargain over partition of the gains from their cooperation. This bargaining process takes into account the fleet's relative competitive strengths, particularly due to private information asymmetries. In this present article we introduce a more complex information structure than had been assumed in our earlier work (McKelvey and Golubtsov [2002], McKelvey, Miller and Golubtsov [2003], Mckelvey et al. [2004]). Specifically, both stock‐growth and stock‐split parameters vary stochastically and asynchronously. Thus, when harvesting noncooperatively, each fleet may possess private knowledge which is unavailable to the other. We examine the interplay of the harvesting game's information structure with other fishery characteristics, such as the fleets' economics and operating characteristics and their attitudes toward risk, to determine the implications of such structure for the outcome of the harvesting game. All of these changes are made to capture new conceptual phenomena and expand the range of applicability of the model.     

DYNAMIC BEHAVIOURS OF AN AUTONOMOUS STAGE-STRUCTURED COMPETITIVE SYSTEMS WITH TOXIC EFFECT

It is known that somebody''s behavior (decision) in a stochastic social network may be influenced by that of his (or her) friends. In this paper, we consider two stochastic social network game models (a) and (b) which can be defined respectively by two different utility functions. Some sufficient conditions for the existence of Nash equilibrium (NE) of the two network game models are obtained by analyzing the different effort relation between a player and his (or her) neighbors.     

Production capacity buildup and double marginalization mitigation in a dynamic supply chain

The paper investigates the extent to which capacity investment considerations interact with the double marginalization effect in a simple supply chain governed by a wholesale price contract. To do so, a non-cooperative differential game model is formulated to study the pricing and capacity investment decisions in a supply chain, which consists of a supplier and a manufacturer. In such a game, there are different decision rules—open-loop, closed-loop, feedback—that are available to the supply chain participants, depending on the observability of the current state of the supply chain. While closed-loop and feedback equilibrium strategies involve the observability of other chain member’s production capacity, open-loop equilibrium strategies do not have such requirement. We examine how the supplier and the manufacturer determine, with the different decision rules, their production capacities and pricing policies to maximize their profits over an infinite planning horizon, and determine how the observability of other supply chain’s members’ production capacity affects the magnitude of the double marginalization effect. Our study suggests that the observability of other chain member’s current production capacity entails a lower production efficiency that results in a greater double marginalization effect. This allows us to conclude that observability of other chain member’s current production capacity is associated with a greater double marginalization effect.     

Cooperation by indirect revelation through strategic behavior

The paper deals with a one-shot prisoners' dilemma when the players have an option to go to court but cannot verify their testimonies. To solve the problem a second stage is added to a game. At the first stage the players are involved in the prisoners' dilemma and at the second stage they play another game in which their actions are verifiable. In such a setup the information about the actions chosen at the prisoners' dilemma stage can be revealed through strategic behavior of the players during second stage. A mechanism for such revelation in the extended game is described. It provides an existence of a unique sequential equilibrium, which may be obtained by an iterative elimination of dominated strategies and has a number of desirable properties.     

Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem

It may seem odd that Abel, a protagonist of Cauchy's new rigor, spoke of “exceptions” when he criticized Cauchy's theorem on the continuity of sums of continuous functions. However, when interpreted contextually, exceptions appear as both valid and viable entities in the early 19th century. First, Abel's use of the term “exception” and the role of the exception in his binomial paper is documented and analyzed. Second, it is suggested how Abel may have acquainted himself with the exception and his use of it in a process denoted critical revision is discussed. Finally, an interpretation of Abel's exception is given that identifies it as a representative example of a more general transition in the understanding of mathematical objects that took place during the period. With this interpretation, exceptions find their place in a fundamental transition during the early 19th century from a formal approach to analysis toward a more conceptual one.     

Strategic response to pollution taxes in supply chain networks: Dynamic,spatial, and organizational dimensions

This paper presents a model of the strategic behavior of firms operating in a spatial supply chain network. The manufacturing and retailing firms engage in an oligopolistic, noncooperative game by sharing customer demand such that a firm’s decisions impact the product prices, which in turn result in changes in all other firms’ decisions. Each firm’s payoff is to maximize its own profit and we show that, in response to such changes in prices and to exogenous environmental taxes, the manufacturing firms may strategically alter a variety of choices such as ’make-buy’ decisions with respect to intermediate inputs, spatial distribution of production, product shipment patterns and inventory management, environmental tax payment vs recycling decisions, and timing of all such choices to sustainably manage the profit and the environmental regulations. An important implication is that effects of a tax depends on the oligopolistic game structure. With respect to methods, we show that this dynamic game can be represented as a set of differential variational inequalities (DVIs) that motivate a computationally efficient nonlinear complementarity (NCP) approach that enables the full exploitation of above-mentioned salient features. We also provide a numerical example that confirms the utility of our proposed framework and shows substantial strategic reaction can be expected to a tax on pollution stocks.     

Discrete hamstrung squad car games

Isaacs introduced a game of perfect information played on the integer grid in which a pursuer with certain restrictions on his movements seeks to capture a slower moving evader. He showed that in a special case it was possible for the evader to avoid capture, and went on to pose the research question as to whether the same situation arose when the pursuer's “speed” was increased. We show that the answer to his question is “No”.     

Teaching Mathematics with a New Curriculum: Changes to Classroom Organization and Interactions

This report describes a high school mathematics teacher's decisions about classroom organization and interactions during his first two years using a new curriculum intended to support teachers' development of student-centered, contributive classroom discourse. In year one, the teacher conducted class and interacted with students primarily in small groups. In year two, he conducted more whole-class instruction. In both years, teacher-student interactions contained univocal and contributive discourse, but in year two the teacher sustained contributive discourse with students for longer periods. The teacher facilitated the most significant changes to classroom discourse in the instructional format with which he had the greatest experience (whole-class instruction). Over the period of this study, two key factors appeared to affect the teacher's decisions about classroom organization and interactions: his perception of students' expectations about mathematics classroom roles and activity, and his own discomfort associated with using a new curriculum. These areas are important candidates for future research about teachers' use of innovative mathematics curricula.     

Designing coalition-based fair and stable pricing mechanisms under private information on consumers’ reservation prices

We model the relation between an aggregator and consumers joining a coalition to reduce the risk resulting from the unpredictability of their base load demand, as a Stackelberg game formulated as a mathematical bilevel program with private information on the consumers’ reservation prices. At the upper-level of the Stackelberg game, the aggregator optimizes his daily price profile so as to reach a net targeted profit which is the maximum value guaranteeing that no consumer will leave the coalition - to contract with a conventional retailer considered here as a fixed alternative - while meeting fairness criterion imposed by the cost-sharing mechanism. At the lower-level, the consumers are asked to provide in day ahead an estimate of their base load hourly demand profile and to schedule their shiftable loads depending on the price signal sent by the aggregator. We provide algorithms that determine the unique price profile and consumer shiftable load schedules as functions of the reservation price estimates. The Stackelberg game between the aggregator and the consumers being repeated for a period of time, the aggregator has the possibility to update his estimates of the reservation prices relying on a feedback function which depends on the percentage of activated loads. A randomized algorithm for consumers’ reservation price learning based on regret minimization is provided. For four cost-sharing mechanisms such as uniform allocation, stand-alone cost, Shapley value, separable and non-separable costs, we determine the closed form of the aggregator’s optimal net targeted profit guaranteeing the stability of the coalition. We also determine conditions guaranteeing the core non-emptiness and prove that for a profit-maximizing aggregator, the stand-alone cost is always preferable to the Shapley value, which coincides with the uniform allocation. Furthermore, the optimal size of the coalition - in terms of the aggregator’s profit - can be determined analytically when the Shapley value is implemented as cost-sharing mechanism. The results are illustrated on a case study where we show that there exists an optimal net targeted profit below which the consumers energy bill is lower when joining the aggregator than with the conventional retailer. Coalition dynamics is also analyzed numerically depending on the consumer inertia in their energy supplier choice process, for each cost-sharing mechanism.     

The simultaneous truel

The simultaneous truel is a three-person game which is a generalization of the simple duel. The players' positions are fixed and their firing is simultaneous. Each player's only decision is which of his opponents will be his target. The (simultaneous) firing continues until there is at most one survivor or until all survivors have fired a specified number of times. Each player is assumed to be concerned only with his own survival; he is indifferent to the fate of his opponents. These games (parametrized by the maximum possible number of shots by each player) are examined for equilibrium points. It is found that, in many cases, the truel has a unique equilibrium point at which the player who is the poorest marksman has the greatest chance of survival.     

COMPARING PROACTIVE AND REACTIVE MANAGEMENT: MANAGING A TRANSBOUNDARY FISH STOCK UNDER CHANGING ENVIRONMENT

Environmental change in general, and climate change in particular, can lead to changes in distribution of fish stocks. When such changes involve transboundary fish stocks, the countries sharing the stock need to reconsider their harvesting policies. We investigate the effects of changing stock distribution on the optimal fishing policies in a two players’ noncooperative game. We compare reactive management, under which the manager ignores future distributional shifts (knowingly or unknowingly), with proactive management where the manager considers such shifts in his decisions. A dynamic programming model is developed to identify closed‐loop Nash strategies. We show that the role of two players is symmetric under reactive management but asymmetric under proactive management where managers anticipate future changes in stock ownership. The player losing the stock tends to harvest more aggressively compared to the player gaining the stock who acts more conservatively. Strategic interactions show tendency for complementary actions that can change abruptly during the ownership transition. The differences between management regimes vary from quantitative to qualitative; differences are minimal for stocks with little or no schooling, whereas highly schooling stocks may avoid collapse only under proactive management.     

Ascending bid auctions with behaviorally consistent bidders

Decision makers whose preferences do not satisfy the independence axiom of expected utility theory, when faced with sequential decisions will act in a dynamically inconsistent manner. In order to avoid this inconsistency and maintain nonexpected utility, we suggest the idea of behavioral consistency. We implement this notion by regarding the same decision maker at different decision nodes as different agents, and then taking the Bayesian — Nash equilibrium of this game. This idea is applied to a finite ascending bid auction game. We show the condition for the existence of an equilibrium of this game, and we also characterize the equilibrium in those cases when it exists. In particular, when the utility functionals are both quasi-concave and quasi-convex, then there is an equilibrium in dominant strategies where each bidder continues to bid if and only if the prevailing price is smaller than his value. In the case of quasi-concavity it is shown that, in equilibrium, each bidder has a value such that he continues with positive probability up to it, and withdraws after that.This research was supported by the NSF under Grant No. SES87-08360. We would like to thank Professor Irving H. LaValle for his helpful suggestions.